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# Prove that this functional is continuous and calculate its norm

Let $$X=\{X\in C[0,1]\}$$: $$f(1/2)=0$$, with the induced norm by $$C[0,1]$$ and the functional $$\varphi:X\rightarrow\mathbb{K}$$ defined by:

$$\varphi(f)=\int_0^1f(t)dt\;\;\forall f\in X$$

Prove that $$\varphi$$ is continuous.

I have made the following: $$||\varphi(f)||=|\int_0^1f(t)dt|\leq \int_0^1|f(t)|dt\leq 1 ||f||_\infty$$.

Now, that relation is valid for all $$f\in C[0,1]$$ such as $$||f||_\infty=1$$. This proves that the operator is bounded in the unit ball so it's continuous.

The problem with this exercise is that I have not made use of $$f(1/2)=0$$ so I assume my solution is wrong at some point. Any idea?

• Side note: $\Vert$ $\Vert$ renders more nicely than $||$ $||$ for norms. – Martin R Nov 2 '19 at 7:58

Your calculation $$\Vert \varphi(f)\Vert =|\int_0^1f(t)dt|\leq \int_0^1|f(t)|dt\leq 1 \Vert f\Vert _\infty$$ is correct. It shows that $$\varphi$$ is continuous on $$C[0,1]$$. As a consequence, $$\varphi$$ is continuous on the subspace $$X$$ of $$C[0,1]$$ and its norm (as a linear functional on $$X$$) is at most one.

The condition $$f(\frac 12) = 0$$ must be considered when computing the exact norm. Try to think of functions which are constant apart from a narrow spike to satisfy $$f(\frac 12) = 0$$.

Computing the norm of $$\phi$$: let $$f_n(x)=1$$ for $$|x-\frac 1 2| >\frac 1 n$$ and $$n|x-\frac 1 2|$$ for $$|x-\frac 1 2 |\leq \frac 1n$$. Then $$f_n \in X$$, $$\|f_n\|=1$$ for all $$n$$ and $$\phi (f_n) =1-\frac 1 n \to 1$$. Hence $$\|\phi\|=1$$.


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#1

I’ve long wanted a “better” domain name for Avogadro. Unfortunately, a number of the obvious domains were unavailable (including the very cool avogad.ro which is supposedly for sale, but doesn’t give any response to inquiries).

So I went out and grabbed Avogadro.cc (e.g., Avogadro is for computational chemistry). Hey, it’s good enough for Arudino.cc.

If others have alternative domain suggestions, I’m open to them. Otherwise, I’m thinking of a migration from avogadro.openmolecules.net to avogadro.cc starting on Feb. 1st, 2014. This would be via Apache rewrite rules, so all pages and URLs will automatically be forwarded to the new domain.

Concerns? Thoughts?

Thanks,
-Geoff

P.S. Incidentally, the domain already works. I’m talking about moving the existing URLs to the new domain. Check it out: http://avogadro.cc

#2

On Wed, 2014-01-15 at 17:39 -0500, Geoffrey Hutchison wrote:

I’ve long wanted a “better” domain name for Avogadro. Unfortunately, a number of the obvious domains were unavailable (including the very cool avogad.ro which is supposedly for sale, but doesn’t give any response to inquiries).

So I went out and grabbed Avogadro.cc (e.g., Avogadro is for computational chemistry). Hey, it’s good enough for Arudino.cc.

If others have alternative domain suggestions, I’m open to them. Otherwise, I’m thinking of a migration from avogadro.openmolecules.net to avogadro.cc starting on Feb. 1st, 2014. This would be via Apache rewrite rules, so all pages and URLs will automatically be forwarded to the new domain.

Concerns? Thoughts?

Thanks,
-Geoff

P.S. Incidentally, the domain already works. I’m talking about moving the existing URLs to the new domain. Check it out: http://avogadro.cc

Critical Workloads, Development Environments & Everything In Between.
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Salutations,

## Regards, Mark

Mark Lee [email protected]

#3

Sadly, I just checked and avogadro.si is $63 per year, which seems a bit steep. (Similarly, I ruled out avogadro.is at$99 per year and avogadro.io at \$49/year.)

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# Absorbance

Last updated

Absorbance is defined as "the logarithm of the ratio of incident to transmitted radiant power through a sample (excluding the effects on cell walls)". [1] Alternatively, for samples which scatter light, absorbance may be defined as "the negative logarithm of one minus absorptance, as measured on a uniform sample". [2] The term is used in many technical areas to quantify the results of an experimental measurement. While the term has its origin in quantifying the absorption of light, it is often entangled with quantification of light which is “lost” to a detector system through other mechanisms. What these uses of the term tend to have in common is that they refer to a logarithm of the ratio of a quantity of light incident on a sample or material to that which is detected after the light has interacted with the sample.

## Contents

The term absorption refers to the physical process of absorbing light, while absorbance does not always measure only absorption; it may measure attenuation (of transmitted radiant power) caused by absorption, as well as reflection, scattering, and other physical processes.

## History and uses of the term absorbance

### Beer-Lambert law

The roots of the term absorbance are in the Beer–Lambert law.  As light moves through a medium, it will become dimmer as it is being "extinguished".  Bouguer recognized that this extinction (now often called attenuation) was not linear with distance traveled through the medium, but related by what we now refer to as an exponential function.  If ${\displaystyle I_{0}}$ is the intensity of the light at the beginning of the travel and ${\displaystyle I_{s}}$ is the intensity of the light detected after travel of a distance ${\displaystyle d}$, the fraction transmitted, ${\displaystyle T}$, is given by:  ${\displaystyle T={\frac {I_{s}}{I_{0}}}=\exp(-\mu d)}$, where ${\displaystyle \mu }$ is called an attenuation constant (a term used in various fields where a signal is transmitted though a medium) or coefficient.  The amount of light transmitted is falling off exponentially with distance.  Taking the natural logarithm in the above equation, we get:  ${\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{s}}}=\mu d}$ . For scattering media, the constant is often divided into two parts, ${\displaystyle \mu =\mu _{s}+\mu _{a}}$ , separating it into a scattering coefficient, ${\displaystyle \mu _{s}}$, and an absorption coefficient, ${\displaystyle \mu _{a}}$, [3] obtaining:${\displaystyle -\ln(T)=\ln {\frac {I_{0}}{I_{s}}}=(\mu _{s}+\mu _{a})d}$ .

If a size of a detector is very small compared to the distance traveled by the light, any light that is scattered by a particle, either in the forward or backward direction, will not strike the detector. In such case, a plot of ${\displaystyle -\ln(T)}$ as a function of wavelength will yield a superposition of the effects of absorption and scatter. Because the absorption portion is more distinct and tends to ride on a background of the scatter portion, it is often used to identify and quantify the absorbing species. Consequently this is often referred to as absorption spectroscopy, and the plotted quantity is called "absorbance", symbolized as ${\displaystyle \mathrm {A} }$ . Some disciplines by convention use decadic absorbance rather than Napierian absorbance, resulting in: ${\displaystyle \mathrm {A} _{10}=\mu _{10}d}$ (with the subscript 10 usually not shown).

### Beer–Lambert law with non-scattering samples

Within a homogeneous medium such as a solution, there is no scattering. For this case, researched extensively by August Beer, the concentration of the absorbing species follows the same linear response as the path-length. Additionally, the contributions of individual absorbing species are additive. This is a very favorable situation, and made absorbance an absorption metric far preferable to absorption fraction (absorptance). This is the case for which the term "absorbance" was first used.

A common expression of the Beer's law relates the attenuation of light in a material as: ${\displaystyle \mathrm {A} =\varepsilon \ell c}$ , where ${\displaystyle \mathrm {A} }$ is the absorbance;${\displaystyle \varepsilon }$ is the molar attenuation coefficient or absorptivity of the attenuating species; ${\displaystyle \ell }$ is the optical path length; and ${\displaystyle c}$ is the concentration of the attenuating species.

### Absorbance for scattering samples

For samples which scatter light, absorbance is defined as "the negative logarithm of one minus absorptance (absorption fraction: ${\displaystyle \alpha }$) as measured on a uniform sample". [2] For decadic absorbance, [4] this may be symbolized as: ${\displaystyle \mathrm {A} _{10}=-\log _{10}(1-\alpha )}$ . If a sample both transmits and remits light, and is not luminescent, the fraction of light absorbed (${\displaystyle \alpha }$), remitted (${\displaystyle R}$), and transmitted (${\displaystyle T}$) add to 1, or: ${\displaystyle \alpha +R+T=1}$ . Note that ${\displaystyle 1-\alpha =R+T}$ , and the formula may be written as: ${\displaystyle \mathrm {A} _{10}=-\log _{10}(R+T)}$ . For a sample which does not scatter, ${\displaystyle R=0}$ , and ${\displaystyle 1-\alpha =T}$ , yielding the formula for absorbance of a material discussed below.

Even though this absorbance function is very useful with scattering samples, the function does not have the same desirable characteristics as it does for non-scattering samples. There is, however, a property called absorbing power which may be estimated for these samples. The absorbing power of a single unit thickness of material making up a scattering sample is the same as the absorbance of the same thickness of the materiel in the absence of scatter. [5]

### Optics

In optics, absorbance or decadic absorbance is the common logarithm of the ratio of incident to transmitted radiant power through a material, and spectral absorbance or spectral decadic absorbance is the common logarithm of the ratio of incident to transmitted spectral radiant power through a material. Absorbance is dimensionless, and in particular is not a length, though it is a monotonically increasing function of path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for absorbance is discouraged.

## Mathematical definitions

### Absorbance of a material

The absorbance of a material, denoted A, is given by [1]

${\displaystyle A=\log _{10}{\frac {\Phi _{\text{e}}^{\text{i}}}{\Phi _{\text{e}}^{\text{t}}}}=-\log _{10}T,}$

where

${\displaystyle \Phi _{\text{e}}^{\text{t}}}$ is the radiant flux transmitted by that material,
${\displaystyle \Phi _{\text{e}}^{\text{i}}}$ is the radiant flux received by that material,
${\displaystyle T=\Phi _{\text{e}}^{\text{t}}/\Phi _{\text{e}}^{\text{i}}}$ is the transmittance of that material.

Absorbance is a dimensionless quantity. Nevertheless, the absorbance unit or AU is commonly used in ultraviolet–visible spectroscopy and its high-performance liquid chromatography applications, often in derived units such as the milli-absorbance unit (mAU) or milli-absorbance unit-minutes (mAU×min), a unit of absorbance integrated over time. [6]

Absorbance is related to optical depth by

${\displaystyle A={\frac {\tau }{\ln 10}}=\tau \log _{10}e\,,}$

where τ is the optical depth.

### Spectral absorbance

Spectral absorbance in frequency and spectral absorbance in wavelength of a material, denoted Aν and Aλ respectively, are given by [1]

${\displaystyle A_{\nu }=\log _{10}{\frac {\Phi _{{\text{e}},\nu }^{\text{i}}}{\Phi _{{\text{e}},\nu }^{\text{t}}}}=-\log _{10}T_{\nu },}$
${\displaystyle A_{\lambda }=\log _{10}{\frac {\Phi _{{\text{e}},\lambda }^{\text{i}}}{\Phi _{{\text{e}},\lambda }^{\text{t}}}}=-\log _{10}T_{\lambda },}$

where

Φe,νt is the spectral radiant flux in frequency transmitted by that material,
Φe,νi is the spectral radiant flux in frequency received by that material,
Tν is the spectral transmittance in frequency of that material,
Φe,λt is the spectral radiant flux in wavelength transmitted by that material,
Φe,λi is the spectral radiant flux in wavelength received by that material,
Tλ is the spectral transmittance in wavelength of that material.

Spectral absorbance is related to spectral optical depth by

${\displaystyle A_{\nu }={\frac {\tau _{\nu }}{\ln 10}}=\tau _{\nu }\log _{10}e\,,}$
${\displaystyle A_{\lambda }={\frac {\tau _{\lambda }}{\ln 10}}=\tau _{\lambda }\log _{10}e\,,}$

where

τν is the spectral optical depth in frequency,
τλ is the spectral optical depth in wavelength.

Although absorbance is properly unitless, it is sometimes reported in "absorbance units", or AU. Many people, including scientific researchers, wrongly state the results from absorbance measurement experiments in terms of these made-up units. [7]

## Relationship with attenuation

### Attenuance

Absorbance is a number that measures the attenuation of the transmitted radiant power in a material. Attenuation can be caused by the physical process of "absorption", but also reflection, scattering, and other physical processes. Absorbance of a material is approximately equal to its attenuance[ clarification needed ] when both the absorbance is much less than 1 and the emittance of that material (not to be confused with radiant exitance or emissivity) is much less than the absorbance. Indeed,

${\displaystyle \Phi _{\text{e}}^{\text{t}}+\Phi _{\text{e}}^{\text{att}}=\Phi _{\text{e}}^{\text{i}}+\Phi _{\text{e}}^{\text{e}},}$

where

Φet is the radiant power transmitted by that material,
Φeatt is the radiant power attenuated by that material,
Φee is the radiant power emitted by that material,

that is equivalent to

${\displaystyle T+{\text{ATT}}=1+E,}$

where

T = Φetei is the transmittance of that material,
ATT = Φeattei is the attenuance of that material,
E = Φeeei is the emittance of that material,

and according to Beer–Lambert law, T = 10−A, so

${\displaystyle {\text{ATT}}=1-10^{-A}+E\approx A\ln 10+E,\quad {\text{if}}\ A\ll 1,}$

and finally

${\displaystyle {\text{ATT}}\approx A\ln 10,\quad {\text{if}}\ E\ll A.}$

### Attenuation coefficient

Absorbance of a material is also related to its decadic attenuation coefficient by

${\displaystyle A=\int _{0}^{l}a(z)\,\mathrm {d} z,}$

where

l is the thickness of that material through which the light travels,
a(z) is the decadic attenuation coefficient of that material at z.

If a(z) is uniform along the path, the attenuation is said to be a linear attenuation, and the relation becomes

${\displaystyle A=al.}$

Sometimes the relation is given using the molar attenuation coefficient of the material, that is its attenuation coefficient divided by its molar concentration:

${\displaystyle A=\int _{0}^{l}\varepsilon c(z)\,\mathrm {d} z,}$

where

ε is the molar attenuation coefficient of that material,
c(z) is the molar concentration of that material at z.

If c(z) is uniform along the path, the relation becomes

${\displaystyle A=\varepsilon cl.}$

The use of the term "molar absorptivity" for molar attenuation coefficient is discouraged. [1]

## Measurements

### Logarithmic vs. directly proportional measurements

The amount of light transmitted through a material diminishes exponentially as it travels through the material, according to the Beer–Lambert law (A=(ε)(l)). Since the absorbance of a sample is measured as a logarithm, it is directly proportional to the thickness of the sample and to the concentration of the absorbing material in the sample. Some other measures related to absorption, such as transmittance, are measured as a simple ratio so they vary exponentially with the thickness and concentration of the material.

Absorbance: −log10etei)Transmittance: Φetei
01
0.10.79
0.250.56
0.50.32
0.750.18
0.90.13
10.1
20.01
30.001

### Instrument measurement range

Any real measuring instrument has a limited range over which it can accurately measure absorbance. An instrument must be calibrated and checked against known standards if the readings are to be trusted. Many instruments will become non-linear (fail to follow the Beer–Lambert law) starting at approximately 2 AU (~1% transmission). It is also difficult to accurately measure very small absorbance values (below 10−4) with commercially available instruments for chemical analysis. In such cases, laser-based absorption techniques can be used, since they have demonstrated detection limits that supersede those obtained by conventional non-laser-based instruments by many orders of magnitude (detections have been demonstrated all the way down to 5 × 10−13). The theoretical best accuracy for most commercially available non-laser-based instruments is attained in the range near 1 AU. The path length or concentration should then, when possible, be adjusted to achieve readings near this range.

### Method of measurement

Typically, absorbance of a dissolved substance is measured using absorption spectroscopy. This involves shining a light through a solution and recording how much light and what wavelengths were transmitted onto a detector. Using this information, the wavelengths that were absorbed can be determined. [8] First, measurements on a "blank" are taken using just the solvent for reference purposes. This is so that the absorbance of the solvent is known, and then any change in absorbance when measuring the whole solution is made by just the solute of interest. Then measurements of the solution are taken. The transmitted spectral radiant flux that makes it through the solution sample is measured and compared to the incident spectral radiant flux. As stated above, the spectral absorbance at a given wavelength is

${\displaystyle A_{\lambda }=\log _{10}\!\left({\frac {\Phi _{\mathrm {e} ,\lambda }^{\mathrm {i} }}{\Phi _{\mathrm {e} ,\lambda }^{\mathrm {t} }}}\right)\!.}$

The absorbance spectrum is plotted on a graph of absorbance vs. wavelength. [9]

A UV-Vis spectrophotometer will do all this automatically. To use this machine, solutions are placed in a small cuvette and inserted into the holder. The machine is controlled through a computer and, once it has been "blanked", automatically displays the absorbance plotted against wavelength. Getting the absorbance spectrum of a solution is useful for determining the concentration of that solution using the Beer–Lambert law and is used in HPLC.

Some filters, notably welding glass, are rated by shade number (SN), which is 7/3 times the absorbance plus one: [10]

${\displaystyle {\text{SN}}={\frac {7}{3}}A+1,}$

or

${\displaystyle {\text{SN}}={\frac {7}{3}}(-\log _{10}T)+1.}$

For example, if the filter has 0.1% transmittance (0.001 transmittance, which is 3 absorbance units), its shade number would be 8.

## Related Research Articles

The Beer–Lambert law, also known as Beer's law, the Lambert–Beer law, or the Beer–Lambert–Bouguer law relates the attenuation of light to the properties of the material through which the light is travelling. The law is commonly applied to chemical analysis measurements and used in understanding attenuation in physical optics, for photons, neutrons, or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

In optics, the refractive index of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.

In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material. Spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.

Radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which characterize the light's interaction with the human eye. The fundamental difference between radiometry and photometry is that radiometry gives the entire optical radiation spectrum, while photometry is limited to the visible spectrum. Radiometry is distinct from quantum techniques such as photon counting.

The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic structure of the material to the electromagnetic field of light, and is in general a function of the frequency, or wavelength, of the light, its polarization, and the angle of incidence. The dependence of reflectance on the wavelength is called a reflectance spectrum or spectral reflectance curve.

In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, when there is no net flow of matter or energy between the body and its environment.

In optics, a Fabry–Pérot interferometer (FPI) or etalon is an optical cavity made from two parallel reflecting surfaces. Optical waves can pass through the optical cavity only when they are in resonance with it. It is named after Charles Fabry and Alfred Perot, who developed the instrument in 1899. Etalon is from the French étalon, meaning "measuring gauge" or "standard".

Transmittance of the surface of a material is its effectiveness in transmitting radiant energy. It is the fraction of incident electromagnetic power that is transmitted through a sample, in contrast to the transmission coefficient, which is the ratio of the transmitted to incident electric field.

In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiation, and to quantify emission of neutrinos and other particles. The SI unit of radiance is the watt per steradian per square metre. It is a directional quantity: the radiance of a surface depends on the direction from which it is being observed.

The term quantum efficiency (QE) may apply to incident photon to converted electron (IPCE) ratio of a photosensitive device, or it may refer to the TMR effect of a Magnetic Tunnel Junction.

In radiometry, photometry, and color science, a spectral power distribution (SPD) measurement describes the power per unit area per unit wavelength of an illumination. More generally, the term spectral power distribution can refer to the concentration, as a function of wavelength, of any radiometric or photometric quantity.

In radiometry, radiant flux or radiant power is the radiant energy emitted, reflected, transmitted, or received per unit time, and spectral flux or spectral power is the radiant flux per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of radiant flux is the watt (W), one joule per second, while that of spectral flux in frequency is the watt per hertz and that of spectral flux in wavelength is the watt per metre —commonly the watt per nanometre.

Absorption cross section is a measure for the probability of an absorption process. More generally, the term cross section is used in physics to quantify the probability of a certain particle-particle interaction, e.g., scattering, electromagnetic absorption, etc. In honor of the fundamental contribution of Maria Goeppert Mayer to this area, the unit for the two-photon absorption cross section is named the "GM". One GM is 10−50 cm4⋅s⋅photon−1.

The linear attenuation coefficient, attenuation coefficient, or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A coefficient value that is large represents a beam becoming 'attenuated' as it passes through a given medium, while a small value represents that the medium had little effect on loss. The SI unit of attenuation coefficient is the reciprocal metre (m−1). Extinction coefficient is another term for this quantity, often used in meteorology and climatology. Most commonly, the quantity measures the exponential decay of intensity, that is, the value of downward e-folding distance of the original intensity as the energy of the intensity passes through a unit thickness of material, so that an attenuation coefficient of 1 m−1 means that after passing through 1 metre, the radiation will be reduced by a factor of e, and for material with a coefficient of 2 m−1, it will be reduced twice by e, or e2. Other measures may use a different factor than e, such as the decadic attenuation coefficient below. The broad-beam attenuation coefficient counts forward-scattered radiation as transmitted rather than attenuated, and is more applicable to radiation shielding.

The mass attenuation coefficient, or mass narrow beam attenuation coefficient of a material is the attenuation coefficient normalized by the density of the material; that is, the attenuation per unit mass. Thus, it characterizes how easily a mass of material can be penetrated by a beam of light, sound, particles, or other energy or matter. In addition to visible light, mass attenuation coefficients can be defined for other electromagnetic radiation, sound, or any other beam that can be attenuated. The SI unit of mass attenuation coefficient is the square metre per kilogram. Other common units include cm2/g and mL⋅g−1⋅cm−1. Mass extinction coefficient is an old term for this quantity.

Absorptance of the surface of a material is its effectiveness in absorbing radiant energy. It is the ratio of the absorbed to the incident radiant power. This should not be confused with absorbance and absorption coefficient.

The near-infrared (NIR) window defines the range of wavelengths from 650 to 1350 nanometre (nm) where light has its maximum depth of penetration in tissue. Within the NIR window, scattering is the most dominant light-tissue interaction, and therefore the propagating light becomes diffused rapidly. Since scattering increases the distance travelled by photons within tissue, the probability of photon absorption also increases. Because scattering has weak dependence on wavelength, the NIR window is primarily limited by the light absorption of blood at short wavelengths and water at long wavelengths. The technique using this window is called NIRS. Medical imaging techniques such as fluorescence image-guided surgery often make use of the NIR window to detect deep structures.

In chemistry, the molar absorption coefficient or molar attenuation coefficient is a measurement of how strongly a chemical species absorbs, and thereby attenuates, light at a given wavelength. It is an intrinsic property of the species. The SI unit of molar absorption coefficient is the square metre per mole, but in practice, quantities are usually expressed in terms of M−1⋅cm−1 or L⋅mol−1⋅cm−1. In older literature, the cm2/mol is sometimes used; 1 M−1⋅cm−1 equals 1000 cm2/mol. The molar absorption coefficient is also known as the molar extinction coefficient and molar absorptivity, but the use of these alternative terms has been discouraged by the IUPAC.

## References

1. IUPAC , Compendium of Chemical Terminology , 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006) " Absorbance ". doi : 10.1351/goldbook.A00028
2. IUPAC , Compendium of Chemical Terminology , 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006) " decadic absorbance ". doi : 10.1351/goldbook.D01536
3. "Light scattering by small particles. By H. C. Van de Hulst. New York (John Wiley and Sons), London (Chapman and Hall), 1957. Pp. Xiii, 470; 103 Figs.; 46 Tables. 96s". Quarterly Journal of the Royal Meteorological Society. 84 (360): 198–199. 1958. Bibcode:1958QJRMS..84R.198.. doi:10.1002/qj.49708436025.
4. Bertie, John E. (2006). "Glossary of Terms used in Vibrational Spectroscopy". In Griffiths, Peter R (ed.). Handbook of Vibrational Spectroscopy. doi:10.1002/0470027320.s8401. ISBN   0471988472.
5. Dahm, Donald; Dahm, Kevin (2007). Interpreting Diffuse Reflectance and Transmittance: A Theoretical Introduction to Absorption Spectroscopy of Scattering Materials. doi:10.1255/978-1-901019-05-6. ISBN   9781901019056.
6. GE Health Care. ÄKTA Laboratory-Scale Chromatography Systems - Instrument Management Handbook. GE Healthcare Bio-Sciences AB, Uppsala, 2015. https://cdn.gelifesciences.com/dmm3bwsv3/AssetStream.aspx?mediaformatid=10061&destinationid=10016&assetid=16189
7. Kamat, Prashant; Schatz, George C. (2013). "How to Make Your Next Paper Scientifically Effective". J. Phys. Chem. Lett. 4 (9): 1578–1581. doi:. PMID   26282316.
8. Reusch, William. "Visible and Ultraviolet Spectroscopy" . Retrieved 2014-10-29.
9. Reusch, William. "Empirical Rules for Absorption Wavelengths of Conjugated Systems" . Retrieved 2014-10-29.
10. Russ Rowlett (2004-09-01). "How Many? A Dictionary of Units of Measurement". Unc.edu. Archived from the original on 1998-12-03. Retrieved 2010-09-20.


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## Equations You Can Solve Using PDE Toolbox

Partial Differential Equation Toolbox™ solves scalar equations of the form

`$m\frac{{\partial }^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla ·\left(c\nabla u\right)+au=f$`

and eigenvalue equations of the form

`$\begin{array}{l}-\nabla ·\left(c\nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla ·\left(c\nabla u\right)+au={\lambda }^{2}mu\end{array}$`

For scalar PDEs, there are two choices of boundary conditions for each edge or face:

• Dirichlet — On the edge or face, the solution u satisfies the equation

hu = r,

where h and r can be functions of space (x, y, and, in 3-D case, z), the solution u, and time. Often, you take h = 1, and set r to the appropriate value.

• Generalized Neumann boundary conditions — On the edge or face the solution u satisfies the equation

`$\stackrel{\to }{n}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\nabla u\right)+qu=g$`

$\stackrel{\to }{n}$ is the outward unit normal. q and g are functions defined on ∂Ω, and can be functions of x, y, and, in 3-D case, z, the solution u, and, for time-dependent equations, time.

The toolbox also solves systems of equations of the form

`$m\frac{{\partial }^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla ·\left(c\otimes \nabla u\right)+au=f$`

and eigenvalue systems of the form

`$\begin{array}{l}-\nabla ·\left(c\otimes \nabla u\right)+au=\lambda du\\ \text{or}\\ -\nabla ·\left(c\otimes \nabla u\right)+au={\lambda }^{2}mu\end{array}$`

A system of PDEs with N components is N coupled PDEs with coupled boundary conditions. Scalar PDEs are those with N = 1, meaning just one PDE. Systems of PDEs generally means N > 1. The documentation sometimes refers to systems as multidimensional PDEs or as PDEs with a vector solution u. In all cases, PDE systems have a single geometry and mesh. It is only N, the number of equations, that can vary.

The coefficients m, d, c, a, and f can be functions of location (x, y, and, in 3-D, z), and, except for eigenvalue problems, they also can be functions of the solution u or its gradient. For eigenvalue problems, the coefficients cannot depend on the solution `u` or its gradient.

For scalar equations, all the coefficients except c are scalar. The coefficient c represents a 2-by-2 matrix in 2-D geometry, or a 3-by-3 matrix in 3-D geometry. For systems of N equations, the coefficients m, d, and a are N-by-N matrices, f is an N-by-1 vector, and c is a 2N-by-2N tensor (2-D geometry) or a 3N-by-3N tensor (3-D geometry). For the meaning of $c\otimes u$, see c Coefficient for specifyCoefficients.

When both m and d are `0`, the PDE is stationary. When either m or d are nonzero, the problem is time-dependent. When any coefficient depends on the solution u or its gradient, the problem is called nonlinear.

For systems of PDEs, there are generalized versions of the Dirichlet and Neumann boundary conditions:

• hu = r represents a matrix h multiplying the solution vector u, and equaling the vector r.

• $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)+qu=g$. For 2-D systems, the notation $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$ means the N-by-1 matrix with (i,1)-component

`$\sum _{j=1}^{N}\left(\mathrm{cos}\left(\alpha \right){c}_{i,j,1,1}\frac{\partial }{\partial x}+\mathrm{cos}\left(\alpha \right){c}_{i,j,1,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\alpha \right){c}_{i,j,2,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\alpha \right){c}_{i,j,2,2}\frac{\partial }{\partial y}\right)\text{\hspace{0.17em}}{u}_{j}$`

where the outward normal vector of the boundary $n=\left(\mathrm{cos}\left(\alpha \right),\mathrm{sin}\left(\alpha \right)\right)$.

For 3-D systems, the notation $n\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\left(c\otimes \nabla u\right)$ means the N-by-1 vector with (i,1)-component

`$\begin{array}{l}\sum _{j=1}^{N}\left(\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right){c}_{i,j,1,3}\frac{\partial }{\partial z}\right){u}_{j}\\ +\sum _{j=1}^{N}\left(\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,1}\frac{\partial }{\partial x}+\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,2}\frac{\partial }{\partial y}+\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right){c}_{i,j,2,3}\frac{\partial }{\partial z}\right){u}_{j}\\ +\sum _{j=1}^{N}\left(\mathrm{cos}\left(\theta \right){c}_{i,j,3,1}\frac{\partial }{\partial x}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,2}\frac{\partial }{\partial y}+\mathrm{cos}\left(\theta \right){c}_{i,j,3,3}\frac{\partial }{\partial z}\right){u}_{j}\end{array}$`

where the outward normal vector of the boundary $n=\left(\mathrm{sin}\left(\phi \right)\mathrm{cos}\left(\theta \right),\mathrm{sin}\left(\phi \right)\mathrm{sin}\left(\theta \right),\mathrm{cos}\left(\phi \right)\right)$.

For each edge or face segment, there are a total of N boundary conditions.

Get trial now


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Is it possible to convert bam files with different parameters?
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5 months ago
a661881 • 0

I recently obtained some bam files of WGS data. However, the trouble is that these bam files are mapped with the -M parameter, which is incompatible with the software I need for the next analysis. Is there any way to get it out without re-mapping (-M Parameter)?

Best
Zhang

BWA NGS WGS • 286 views
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Entering edit mode

Hello a661881!

It appears that your post has been cross-posted to another site: https://bioinformatics.stackexchange.com/questions/15019/is-it-possible-to-convert-bam-files-with-different-parameters

This is typically not recommended as it runs the risk of annoying people in both communities.

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Entering edit mode

oh，I am sorry for that, I am too eager to ask for answers, I will pay attention to these rules next time.

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Entering edit mode
5 months ago
ATpoint 49k

Nope, remapping is the only option afaik. I had the same problem in the past and recall that I did not find a reliable converter to "undo" the -M option of bwa, so it was more efficient and reliably to just remap the whole thing.


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# Statistical Thermodynamics and Rate Theories/Raman spectroscopy

## The Raman Effect

The Raman effect is a form of light scattering by a material. It can be observed when a sample is exposed to an intense, high energy, monochromatic light source, such as a laser. Most photons of this light that are scattered by the material will be scattered elastically, meaning that there will be no change in the frequency of light. This results in an intense line at the same frequency of the source, known as the Rayleigh line.

Photons can also undergo Raman scattering, where they are inelastically scattered. This means that they will emerge at a higher or lower frequency. This type of scattering is less common than elastic scattering, with only 1 in 107 photons experiencing Raman scattering. Stokes lines occur when the scattered light emerges at a lower frequency. Anti-stokes lines occur when the scattered light emerges at a higher frequency.

The Raman effect is applicable even when the frequency of radiation does not correspond to a transition between molecular energy levels. This cannot be explained by standard absorption or emission and therefore must be the Raman effect.

## Raman Spectroscopy

Raman spectroscopy is a spectroscopic technique which measures inelastic scattering. The gain or loss of energy by incident light corresponds to transitions between the energy levels of the molecules. These transitions are both rotational and vibrational. A high energy photon excites a molecule to a "virtual state" (v1), following this the molecule returns to a different state (v2) on emission. A different frequency of light is emitted due to the difference in initial and final energy levels.

## The Gross Selection Rule for Raman Spectroscopy

The gross selection rule for Raman spectroscopy states that a molecule must be anisotropically polarizable to have a Raman spectrum. Anisotropic indicates that something is different in some directions, in this case the electron density of the molecule must be polarized heterogeneously allowing for it to be observed via Raman spectroscopy.

This selection rule explains why Raman spectroscopy can successfully analyze homonuclear diatomics, such as H2 and N2, when rotational and vibrational spectroscopic techniques cannot.

The Raman effect is present for any molecule that can be polarized anisotropically. Molecules that are spherical tops cannot be anisotropically polarized, this is because they have three equal moments of inertia. Therefore these molecules cannot be observed using Raman spectroscopy.

## The Specific Selection Rule of Rotational Raman Spectroscopy

The specific selection rule for Raman spectroscopy of linear molecules is ${\displaystyle \Delta J=0,\pm 2}$ . The Raman spectrum has regular spacing of lines, as seen previously in absorption spectra, but separation between the lines is doubled. The evenly spaced lines have a separation of: ${\displaystyle {\widetilde {v}}=4{\widetilde {B}}(J+1)}$  where: ${\displaystyle {\widetilde {B}}={\frac {h}{8\pi ^{2}cI}}}$

The rotational energy of the molecule can be calculated using the following equation: ${\displaystyle E_{rot}={\frac {h^{2}}{4\pi ^{2}\mu r_{e}^{2}}}J(J+1)}$

## The Specific Selection Rule of Vibrational Raman Spectroscopy

The specific selection rule for vibrational Raman spectroscopy states that only Δv = ±1 transitions are allowed. This is the same as for vibrational absorption spectroscopy. Vibrational Raman transitions occur simultaneously with rotational Raman transitions, this results in branching caused by rotational transitions in the Δv = ±1 peaks. The vibrational transitions result in 3 branches with fine structure from rotational transitions. In the spectrum each line corresponds to a change in quantum numbers v, J, or both.

Branch Energy Level Transition
O ΔJ=-2
Q ΔJ=0
S ΔJ=2

## Raman Spectroscopy Example 1

Calculation of rotational energy levels of N2 from pure rotational Raman spectrum.

For a molecule to have a Raman spectrum it must be anisotropically polarizable and thus have a different electron cloud distortion along different axis. Given the Pure Rotational Raman Spectrum of a molecule it is possible to calculate the rotational energy levels based on the spectrum itself. Transitions of ${\displaystyle \Delta J=\pm 1}$  can be observed in the spectrum and used to find the energy levels.

A sample calculation to find the rotational energy levels of Nitrogen can be computed using the spacings and J values of the Raman spectrum below. This calculation will be done in three steps:

1. Finding the rotational constant, ${\displaystyle {\widetilde {B}}}$ .

2. Finding the moment of Inertia, ${\displaystyle I}$ .

3. Finding the rotational energy, ${\displaystyle E_{rot}}$ .

Step 1: Computing the rotational constant B:

As seen from the figure above, the spacing, or v, is approximately 8.00 cm-1 which corresponds to the quantum number J = 1. Using these values it is possible to compute B by re-arranging the formula below:

${\displaystyle {\widetilde {\nu }}=4{\widetilde {B}}(J+1)}$

${\displaystyle {\widetilde {B}}={\frac {\widetilde {\nu }}{4(J+1)}}}$

${\displaystyle {\widetilde {B}}={\frac {8.00cm^{-1}}{4(1+1)}}}$

${\displaystyle {\widetilde {B}}=1.00cm^{-1}}$

Step 2: Compute the moment of Inertia I using the formula below and the constant B determined above:

${\displaystyle {\widetilde {B}}={\frac {h}{8\pi ^{2}cI}}}$

${\displaystyle I={\frac {h}{8\pi ^{2}c{\widetilde {B}}}}}$

Note: h is Planck's constant ${\displaystyle 6.62607004\times 10^{-34}J/s}$  and c is the speed of light in cm/s: ${\displaystyle 2.99792458\times 10^{10}cm/s}$ . When using frequencies in units of wavenumber (${\displaystyle cm^{-1}}$ ), the speed of light in units of cm/s should be used in order for them to cancel out during the calculation.

${\displaystyle I={\frac {6.626\times 10^{-34}J/s}{8\pi ^{2}(2.998\times 10^{10}cm/s)(1.00cm^{-1})}}}$

${\displaystyle I=2.799\times 10^{-46}kg/m^{2}}$

Step 3: Compute the rotational energy level using the value of I from above and the following two equations:

${\displaystyle I=\mu r_{e}^{2}}$

${\displaystyle E_{rot}={\frac {h}{4\pi \mu r_{e}^{2}}}J(J+1)}$

Note: Substituting the equation for I into the energy equation will give:

${\displaystyle E_{rot}={\frac {h^{2}}{4\pi ^{2}I}}J(J+1)}$

${\displaystyle E_{rot}={\frac {(6.626\times 10^{-34}J/s)^{2}}{4\pi ^{2}(2.799\times 10^{-46}kg/m^{2})}}1(1+1)}$

${\displaystyle E_{rot}=3.94\times 10^{-23}J}$

Note: Since the calculated value of ${\displaystyle E_{rot}}$  was done using ${\displaystyle J=1}$ , this corresponds to the energy of the ${\displaystyle J=1}$  energy level. This calculation can be repeated using different values of ${\displaystyle J}$  to determine the different energies of each rotational energy level.


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# Genetic Architecture of Maize Flowering Time

There is a really cool paper by Buckler and colleagues in the current issue of Science.  The basic gist of this paper is that flowering time in maize (Zea mays) is not controlled by any large-effect genes (actually quantitative trait loci or QTLs -- these are positions in the genome that are associated with genes underlying a particular trait).  Instead, flowering time is controlled by lots of different QTLs, all with small effect.  I know what you're saying: "this fool anthropologist has really gone off the deep end this time. Why would he ever care about such boring stuff?!" Ah, but this is anything but boring and it has a very clear relevance to the evolution of life histories, human included.

Genes are discrete entities.  Yet we know that they code for things that can essentially vary continuously like stature or age at first reproduction. Probably the simplest (and remarkably general, even though this isn't how we learn about genetics in high school) way to think of genes is on-off switches.  At a particular locus, if you have allele A you get a +1 on your phenotype.  If you have allele a you get a +0.  For a locus that has just these two alleles, the possible phenotypes (assuming no environmental inputs, gene-gene interactions, or anything exotic) are 0,1, and 2 corresponding to aa, Aa/aA, and AA genotypes respectively.  You get twice as many individuals with phenotype=1 because there are two ways of making the heterozygote but only one way of making each of the homozygotes.  This is more or less high school biology.

Now, let's say that our phenotype is controlled by not one locus but many.  If we can reasonably assume that the different loci are independent of each other (not a hard-and-fast assumption), then a famous result from probability theory has important bearing on the question at hand.  In particular, I'm referring to the normal approximation of the binomial distribution. The binomial distribution is the coin-flipping distribution: how many tails will you get if you toss a coin (with a probability of tails = $p$) $n$ times.  It turns out that when the product $np$ is greater than 5 (a very rough rule of thumb as there are other mild conditions) the binomial distribution -- a discrete probability distribution -- is very well approximated by a normal distribution -- a continuous one.  This is essentially how we get continuous phenotypes from discrete genotypes. The normal distribution is the limiting distribution whenever we have lots of small things drawn independently from the same process acting in an additive way (this is the mechanistic explanation of why the binomial can be approximated so well by the normal when $np$ gets large).   Lots of loci, each acting in an additive way with small effect, leads to a normal distribution of a phenotype. When we have a few loci acting with large effect and possibly with strong interactions between loci (i.e., epistasis), the distribution of phenotypes is more likely to be more discrete (as with the one-locus, two-allele model).

Maize lives in a broad range of environments and, like us, it reproduces sexually. Unlike us, some plants can "self." That is, they don't need two to tango -- they reproduce sexually but a single individual provides both sets of gametes. The genetic architecture of flowering in two other genetically well-studied plants, rice and Arabidopsis (the ubiquitous mustard plant that is the favored model system of many developmental geneticists), does not show the same design as maize.  Both rice and Arabidopsis are selfing species.  It turns out that their genetic architecture resembles that of the simple additive system -- one with major gene effects and a more discrete distribution of flowering times. In these species, there is more evidence for gene-gene interactions and environmental inputs in the flowering time phenotype as well.

The fact that flowering time in maize is controlled by many loci with small additive effects means that the distribution of flowering time in any sizable population of maize will be approximately normally distributed.  There will be approximately continuous variation across the possible range of phenotypes and there will be far more individuals with phenotypes near the mean of the population.  In fact, about 95% of the phenotypes will be within ± 2 standard deviations of this mean.  Variation is the stuff of natural selection.  In order for evolution to occur by natural selection you need three things: (1) variation in the phenotype, (2) a heritable basis for this variation, and (3) differential fitness as a result of the variation.  No variation in the phenotype, no response to selection.  A trait with a nice continuous distribution -- as apparently maize flowering time is -- can respond with exquisite precision to selection imposed by the environment (assuming the population is big enough so that random events don't interfere). Maize shows an amazing degree of phenotypic diversity.  Flowering date is no exception with a range -- the landraces of maize vary in their developmental time between an astounding 2 and 11 months.

So there is plenty of phenotypic variation in maize to respond to selection.  Another important implication of the normal distribution of flowering time also means that local variation will fall into a fairly restricted range (that 95% of variation falling within ± 2 standard deviations thing).  If the distribution were more discrete, a field might contain two completely different flowering phenotypes.  In a species that requires out-crossing, producing gametes at two completely different times doesn't do you much good.  When you have a nice normal distribution of flowering times, there is variation, but it is not too extreme -- particularly in any local area.  This trait architecture allows maize to thread the evolutionary needle: enough variation to allow response to selection but not so much as to make reproduction impossible.

This type of architecture -- lots of small effects from independent genes -- is common in two other laboratory favorites, fruit flies and mice. It is also common in humans according to Flint & Mcckay (2009). Human stature, in particular, is a trait that has this kind of architecture. We know from the work of Rebecca Sear at the London School of Economics and her colleagues, that height predicts  reproductive success in humans (albeit in complex ways). Whether age at first reproduction (the human equivalent to flowering) is constructed the same way as maize flowering or human stature is an open question but I seriously doubt that we will find it to be controlled by a couple genes with large effect.  Such quantitative genetic architecture in complex life history traits bodes well for prediction.  We have a very well-developed theory that relates response to selection to the genetic variation in quantitative traits and the force of selection acting on those traits.  In order for this prediction to work well though, we need to also understand how life history traits are related to each other. That is, we need to not just know the means and standard deviations of one trait (like stature or age at sexual maturity).  We also need to know how traits are related to each other.  In other words, we need to measure the covariances between traits.  Fortunately, the normal approximation that we use for one trait can easily be extended to multiple traits, providing a nice framework for both estimation and prediction. I am working at this very moment on estimating the quantitative genetic relationships between human life history traits using a large genealogical database of a historical population.  More on this later...

## 2 thoughts on “Genetic Architecture of Maize Flowering Time”

1. Ah, I see you're talking about age at first *reproduction*, rather than menarche. Though I imagine they're pretty correlated (or were until birth control).


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# Cesàro Summation Operator is Continuous Linear Transformation

## Theorem

Let $\ell^\infty$ be the space of bounded sequences.

Let $A : \ell^\infty \to \ell^\infty$ be the Cesàro summation operator.

Then $A$ is a continuous linear transformation.

## Proof

### Well-Definedness

Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^\infty$.

Then:

 $\ds \forall n \in \N: \,$ $\ds \size {x_n}$ $\le$ $\ds \sup_{n \mathop \in \N} \size {x_n}$ $\ds$ $=$ $\ds \norm {\mathbf x}_\infty$ Definition of Supremum Norm

Thus:

 $\ds \size {\sum_{i \mathop = 1}^n\frac{x_i} n }$ $\le$ $\ds \frac 1 n \sum_{i \mathop = 1}^n \size {x_i}$ General Triangle Inequality for Complex Numbers $\ds$ $\le$ $\ds \frac 1 n \sum_{i \mathop = 1}^n \sup_{n \mathop \in \N} \size {x_i}$ $\ds$ $=$ $\ds \frac 1 n n \norm {\mathbf x}_\infty$ Definition of Supremum Norm $\ds$ $=$ $\ds \norm {\mathbf x}_\infty$

Hence:

$A \mathbf x \in \ell^\infty$

$\Box$

### Linearity

Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N}, \mathbf y = \sequence {y_n}_{n \mathop \in \N} \in \ell^\infty$.

Let $\lambda \in \C$.

$\mathbf x + \lambda \mathbf y \in \ell^\infty$

Then:

 $\ds \map A {\mathbf x + \lambda \mathbf y}$ $=$ $\ds \tuple {x_1 + \lambda y_1, \frac {x_1 + x_2 + \lambda \paren {y_1 + y_2} } 2, \frac {x_1 + x_2 + x_3 + \lambda \paren {y_1 + y_2 + y_3} } 3, \ldots}$ Definition of Cesàro Summation Operator $\ds$ $=$ $\ds \tuple {x_1, \frac {x_1 + x_2} 2, \frac {x_1 + x_2 + x_3} 3, \ldots} + \tuple {\lambda y_1, \frac {\lambda \paren {y_1 + y_2} } 2, \frac {\lambda \paren {y_1 + y_2 + y_3} } 3, \ldots}$ Space of Bounded Sequences with Pointwise Addition and Pointwise Scalar Multiplication on Ring of Sequences forms Vector Space $\ds$ $=$ $\ds \tuple {x_1, \frac {x_1 + x_2} 2, \frac {x_1 + x_2 + x_3} 3, \ldots} + \lambda \tuple {y_1, \frac {y_1 + y_2} 2, \frac {y_1 + y_2 + y_3} 3, \ldots}$ $\ds$ $=$ $\ds \map A {\mathbf x} + \lambda \map A {\mathbf y}$ Definition of Cesàro Summation Operator

By definition, $A$ is a linear transformation.

$\Box$

### Continuity

Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^\infty$.

Then:

 $\ds \norm {A \mathbf x}_\infty$ $=$ $\ds \sup_{n \mathop \in \N} \size {\sum_{i \mathop = 1}^n \frac {x_i} n }$ Definition of Cesàro Summation Operator $\ds$ $\le$ $\ds \norm {\mathbf x}_\infty$

$\Box$

Altogether:

$A \in \map {CL} {\ell^\infty}$

$\blacksquare$


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# Prime Factoral Roots

Inspired by digital roots, the prime factoral root of a number is the number that emerges when you take the prime factors of a number, add them together, and repeat the process on the resulting number, continuing until you end up with a prime number (which has itself as its only prime factor, and is thus its own prime factoral root). The prime factoral root of 4 is 4, as 2*2=2+2, and this is the only non-prime prime factoral root of an integer greater than 1 (which is another special case, as it has no prime factors). The OEIS sequence formed by prime factoral roots is A029908.

For example, the prime factoral root of 24 is:

24=2*2*2*3

2+2+2+3=9=3*3

3+3=6=2*3

2+3=5, and the only prime factor of 5 is 5.  Therefore, the prime factoral root of 24 is 5.

Write a program or function that finds the prime factoral root of an input integer.

## Input:

An integer, input through any reasonable method, between 2 and the largest integer your language will support (inclusive). Specifically choosing a language that has an unreasonably low maximum integer size is not allowed (and also violates this standard loophole)

## Output:

An integer, the prime factoral root of the input.

## Test Cases:

4   -> 4
24  -> 5
11  -> 11
250 -> 17

## Scoring:

This is , lowest score in bytes wins!

• Can you add 4 in the test cases, since it's an exception and it's easy to forget about it while testing an answer? Oct 4, 2017 at 11:26
• Do we have to output 1 for 1? Oct 4, 2017 at 11:36
• @someone according to the linked OEIS sequence, it should output 0 for 1 Oct 4, 2017 at 11:55
• @someone The challenge states that the input will be at least 2. Oct 4, 2017 at 13:21
• @someone Sorry for being off for a while. As Martin said, the challenge specifically says that the input will be greater than one, and therefore behavior when the input is 1 is undefined. Oct 5, 2017 at 1:18

import Data.Numbers.Primes

• =<<((==)=<<)$ Whaaaaaat. Oct 5, 2017 at 4:52 • @icrieverytim I added an explanation. Feel free to ask in the Haskell chat room if you have further questions on how this sorcery works. Oct 5, 2017 at 8:14 # 05AB1E, 3 bytes FÒO  Try it online! Explanations: FÒO F Loops <input> times + 1 Ò List of prime factors w/ duplicates O Total sum of the list -- implicit output  • This seems to fail for 4. Oct 4, 2017 at 11:09 • @Shaggy fixed while saving 2 bytes Oct 4, 2017 at 11:21 • Does this make anyone trying to beat this a FÒO-fighter? Oct 4, 2017 at 14:57 • At least it wasn't FOObar. Oct 4, 2017 at 18:03 # Husk, 4 bytes ω(Σp  Try it online! Explanation: ω( -- apply the following function until the result no longer changes (fixpoint) Σ -- sum p -- the list of primefactors  # Pyth, 3 bytes usP  Try it here. Explanation: usPGQ The trailing GQ is implicit PG Get prime factors s Sum u Q Repeat until returned value no longer unique starting with the input  • Did you forget to update your explanation? Oct 6, 2017 at 1:56 • @MCMastery No, the code and the explanation are the same. The trailing GQ is implicit Oct 6, 2017 at 6:35 • @MCMastery what i cri everytim said Oct 6, 2017 at 11:21 # Python 2, 84 bytes f=lambda n,d=2:n>1and(n%d and f(n,d+1)or d+f(n/d)) i=input() exec'i=f(i);'*i print i  Try it online! • This may be a pretty dumb question, but how does f=lambda n,d=2:n>1and(n%d and f(n,d+1)or d+f(n/d)) work? I've never programmed in Python (mainly Java and C#), so I'm unsure what the result of this function is. Does this function modify the input n and return it afterwards, or is it similar to a boolean where n>1and(n%d and f(n,d+1)or d+f(n/d)) is either 0 or 1, or 0 or n, or something else? I'm trying to visualize how a port of this would look like in Java/C#, but I'm unable to because I don't really understand Python lambdas like this in general. Oct 4, 2017 at 14:41 • @KevinCruijssen this is equivalent to n>1 ? (n%d!=0 ? f(n, d+1) : d+f(n/d)) : n>1. In general terms x and y is equivalent to x ? y : x. x and y or z is equivalent to x ? y : z in most cases. – ovs Oct 4, 2017 at 15:06 • @KevinCruijssen a Java port would be something like f=(n,d=2)->n>1?n%d>0?f(n,d+1):d+f(n/d):0. – ovs Oct 4, 2017 at 15:08 • Ah ok. Thanks for the explanation, now it makes a lot more sense. And I remember x and y being x ? y : x from JavaScript as well. Thanks! Oct 4, 2017 at 17:30 # Java 8, 175144142141139 101 bytes n->{for(int i=n;i-->0;)n=f(n,2);return n;};int f(int n,int d){return n>1?n%d<1?f(n/d,2)+d:f(n,d+1):0;}  -38 bytes porting @ovs' Python answer, so make sure to upvote him as well! Try it online. n->{ // Method with integer as both parameter and return-type for(int i=n;i-->0;)// Loop the input n amount of times: n=f(n,2); // Change n that many times by calling the recursive method // with n,2 as parameters return n;} // Then return the potentially modified n as result int f(int n,int d){ // Separated method with 2 integer parameters & integer return // (d is 2 when we initially call this recursive-method) return n>1? // If input n is larger than 1: n%d<1? // If n is divisible by d: f(n/d // Divide n by d ,2) // Do a recursive call with n/d,2 as parameters +d // And add d to its result : // Else: f(n,d+1) // Do a recursive-call with the next d by using n,d+1 as // parameters : // Else: 0;} // Simply return 0  Original 175 144 142 141 139 bytes answer: n->{for(int i,t=n,x;;n=t){for(i=2;i<t;)t=t%i++<1?0:t;if(t>1|n<5)return n;for(t=0,i=1;i++<n;)for(;n%i<1;t+=x)for(n/=x=i;x>9;x/=10)t+=x%10;}}  -1 byte thanks to @Nevay -2 bytes thanks to @ceilingcat Explanation: Try it online. n->{ // Method with integer as both parameter and return-type for(int i, // Index-integer i t=n, // Temp integer t, starting at the input n x; // Temp integer x ; // Loop indefinitely: n=t){ // After every iteration, replace n with the value t for(i=2;i<t;) // Inner loop i in the range [2,t): t=t%i++<1? // If t is divisible by i: 0 // Set t to 0 : // Else: t; // Leave t the same if(t>1 // If t is not 0 or 1 (it means it's a prime), |n<5) // or if n is below 5 (for edge-cases n=1 and n=4) return n; // Return n as result for(t=0, // Reset t to 0 i=1; // Reset i to 1 i++<n;) // Inner loop i in the range (2,n]: for(;n%i<1; // Inner loop as long as n is divisible by i: t+=x) // After every iteration: Increase t by x for(n/=x=i; // Reset x to i // And divide n by i x>9; // Inner loop as long as x contains more than 1 digit: x/=10) // After every iteration, remove the trailing digit of x t+=x%10;}} // Increase t with the trailing digit of x  • +1 for bothering to write an explanation as verbose as if this was a golfing language. Oct 4, 2017 at 12:17 • @someone Thanks! Since someone asked me about an explanation of a Java answer of mine once in the past, I've been adding them to all my answers. :) Oct 4, 2017 at 12:23 • i,t=n,x looks like it belongs in Python, haha Oct 4, 2017 at 14:03 • @ETHproductions Hehe, too bad I still have to add the leading int  (unlike Python). ;) Oct 4, 2017 at 14:15 • You can use i++<n instead of ++i<=n. Oct 4, 2017 at 21:47 # Jelly, 5 bytes ÆfS$¡

Explanations:

Æf    list of prime factors
S   sum
$¡ repeat n times  Try it online! • 5 bytes Oct 4, 2017 at 11:07 ## Retina, 30 bytes {+(\1|\b11+?\B)+$
$1;$#1$* ;  Input and output in unary. Try it online! (Performs decimal/unary conversion for convenience.) ### Explanation {+(\1|\b11+?\B)+$
$1;$#1$*  The { instructs Retina to run the entire program in a loop until a full pass fails to modify the string, i.e. until a fixed point is reached. Consequently, the program itself computes one step of summing the prime factors of the current value. This stage itself computes the prime factorisation of the input. The + is similar to { but loops only this stage until it stops changing the string. The regex tries to match the final run of 1s by repeatedly matching the same substring (i.e. the factor). The way this is done is a bit convoluted due to the forward reference \1. On the first iteration, group 1 hasn't captured anything yet, so \1 fails unconditionally. Instead, we have to match \b11+?\B which is the smallest possible substring that starts at the beginning of the run, contains at least two 1s and doesn't cover the entire run. Subsequent iterations will not be able to use this alternative again, due to the \b. So on all further iterations, we're matching \1, i.e. the same substring over and over again. This process has to hit the end of the string exactly ($) to make sure we've captured and actual divisor. The benefit of using this somewhat tricky approach is that group 1 will have been used exactly n/d times, i.e. what remains after dividing out the divisor d.

We replace this match with d ($1), a separating ; and n/d ($#1$*, which inserts $#1 copies of 1, where $#1 is the number of captures made by group 1). This process stops once the final run in the string is itself a prime, because then the regex no longer matches. ;  All we need to do to sum the primes is to remove all the separators. # Ohm v2, 4 bytes ·ΘoΣ  Try it online! Explanation: ·Θ evaluate the block until the result returned has already been seen Σ sum o the full prime factorization  # Actually, 7 bytes ⌠w♂πΣ⌡Y  Try it online! Explanation: ⌠w♂πΣ⌡Y ⌠ ⌡Y do this until output stops changing (fixed-point combinator) w factor into [prime, exponent] pairs ♂π product of each pair Σ sum of products  # R + pracma, 53 bytes function(n){for(i in 1:n)n=sum(pracma::factors(n)) n}  Try it online! (r-fiddle) R doesn't have a prime factors builtin, but numerous packages (pracma, numbers, etc.) do, so I picked a conveniently short one. # Jelly, 6 bytes This answer uses one of Jelly's many prime factorization builtins, and the quick for repeat until the results are no longer unique. ÆfSµÐL  Try it online! • I think you've been outgolfed but, given your approach, I'm uncertain whether that answer works Oct 4, 2017 at 11:10 • @cairdcoinheringaahing I've just checked his answer (or rather, the Python equivalent) from 1 to 100000 and it works. I think 1 is the only case where the number of steps needed is equal to n (which is fine; with 1 we only need to run it once), and there don't seem to be any cases where the number of steps is greater than n (i.e. there don't seem to be any counterexamples). Ah well, I've been outgolfed :D Oct 4, 2017 at 11:25 • Well, it happens. Although +1 for being the exact same code I thought of when I saw this challenge Oct 4, 2017 at 11:27 • The sum of prime factors of n is always less than or equal to n, which makes it pretty easy to prove that n is always more than enough. Oct 5, 2017 at 1:37 # MATL, 6 bytes Uses scottinet's idea of looping more times than needed. Thanks also to Shaggy for a pointing out a mistake, now corrected. t:"Yfs  Try it online! ### Explanation t % Take input (implicit). Duplicate :" % Do the following that many times Yf % Array of prime factors s % Sum of array % End (implicit). Display (implicit)  • This seems to fail for 4. Oct 4, 2017 at 11:11 • @Shaggy Thanks! Working on that Oct 4, 2017 at 11:11 • @Shaggy Solved now Oct 4, 2017 at 11:16 # PowerShell, 124 bytes function f($a){for($i=2;$a-gt1){if(!($a%$i)){$i;$a/=$i}else{$i++}}}
for($x=$args[0];$l-ne$x){$l=$x;$x=(f($x))-join'+'|iex}$x  Try it online! PowerShell doesn't have any prime factorization built-ins, so this uses code from my answer on Prime Factors Buddies (the top line) to perform the factorization calculations. The second line is the meat of this program. We take input from $args into $x, then for loop until $l is -notequal to $x. (The first iteration, $l is $null and $x is an integer, so we'll loop at least once).

Inside the loop, we set our $l =$x to determine if we've hit the end of the loop or not. Then we get the factors of $x with f($x), -join those together with + and |iex them (short for Invoke-Expression and similar to eval). That's stored back into $x. Thus, we've hit the "end" where the prime factorization summed together is back to itself. Then, we simply place $x on the pipeline and output is implicit.

# Mathematica, 35 bytes

#//.x_:>Tr[1##&@@@FactorInteger@x]&

Try it online!

(Mathics does not support Tr. I have to implement it manually)

• 1##& is short for Times and FixedPoint can almost always be shortened with //.: #//.x_:>Tr[1##&@@@FactorInteger@x]& Oct 4, 2017 at 10:53
• @MartinEnder Thanks! I should have already known about Times, but I've not known about the FixedPoint trick. Oct 4, 2017 at 11:06
• Your code is written in Mathematica. This is not a Mathics function. You should either change the language name to Mathematica or Tr to Total Oct 4, 2017 at 11:36
• @{no one} Sorry, language name (Mathics) was a mistake. {i cri evritime} fixed that. Oct 4, 2017 at 12:28
• – user45941
Oct 4, 2017 at 13:56

# Ruby, 63 bytes

->n{n.times{n=n.prime_division.map{|x|x.reduce:*}.sum};n}

Try it online!

Uses the -rprime flag for +6 bytes to make use of Prime#prime_division.

prime_division returns pairs of [prime, exponent] (for example, for 24 we have the factors [2, 2, 2, 3] so it gives [[2, 3], [3, 1]]) so in each step we just multiply the members of those pairs together and sum the results.

# Javascript (ES6), 63 bytes

f=n=>(q=(p=(m,x)=>m<x?0:m%x?p(m,x+1):x+p(m/x,x))(n,2))^n?f(q):q
<input id=i type=number min=0 value=0 oninput="o.innerText=f(i.value)">

## Ungolfed:

f=n=>(                  // Recursive function f
p=(m,x=2)=>(        //   Recursive function p, used to sum prime factors
m<x?            //     If m (the number to be factored) is less than x (the current
0           //     iteration), return 0
:m%x?           //     Else if m doesn't divide x
p(m,x+1)    //     run the next iteration
:               //     Else (if m divides x)
x+p(m/x,x)  //     Divide m by x and repeat the current iteration
),
q=p(n),             //   Set q to the sum of the prime factors of n
q^n?                //   If q != n then
f(q)            //     repeat f with q
:                   //   else
q               //     return q
)

# Factor, 33 bytes

[ [ factors Σ ] to-fixed-point ]

Try it online!

• to-fixed-point Apply a quotation to an object until it stops changing.
• factors Σ Sum of prime factors.


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Question
What is the average kinetic energy of a nitrogen molecule ($N_2$) if its rms speed is 560 m/s? At what temperature is this rms speed achieved?
Question by OpenStax is licensed under CC BY 4.0.
Final Answer

$352 \textrm{ K}$

Solution Video

# OpenStax College Physics for AP® Courses Solution, Chapter 13, Problem 7 (Test Prep for AP® Courses) (1:44)

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## Calculator Screenshots

Video Transcript

This is College Physics Answers with Shaun Dychko. We're going to find the average kinetic energy of a Nitrogen molecule at an rms speed of 560 meters per second and figure what temperature this rms speed happens. So average kinetic energy is one half times mass times the average velocity squared. So it's one half times the mass of a single Nitrogen molecule, and Nitrogen molecule is N2. And so there's two Nitrogen atoms and the molar mass of each Nitrogen is 14.003 grams per mole. Let me convert that into kilograms per atom by multiplying by one kilogram per 1000 grams and then multiplied by one mole for every 6.02 times ten to the 23 molecules. So I should have said that this is molecule, this is a mass of a single molecule of Nitrogen consisting of two Nitrogen atoms. Then we times by 560 meters per second and we square that and this gives a kinetic energy of 7.29 times ten to the minus 21 Joules for each Nitrogen molecule. Now the temperature at which this kinetic energy would occur is what we can figure out from this formula saying that the average kinetic energy is three over two times the Boltzmann Constant times temperature. And we'll solve for T by multiplying both sides two over three k. So temperature is two times the average kinetic energy divided by three times Boltzmann Constant. So that's two times the answer we got before for the number of Joules for each of the molecule divided by three times Boltzmann Constant, giving us a temperature of 352 Kelvin.


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# Spontaneity

## Introduction to Spontaneity Chemistry

Spontaneity is part of the first law of thermodynamics. In this section, students can understand the fact associated with an isolated system's fixed energy level.

There is a direction of heat flow that can be elaborated by establishing a relation between the work done by the system or on the system. This is spontaneity in thermodynamics.

Spontaneous meaning in Chemistry is not that hard to understand. Many natural phenomena are having one straight path of heat flow. They do not have any limitations on their heat flow paths.

### What Is a Spontaneous Reaction?

A spontaneous chemical reaction is an irreversible process where you can’t get the ingredients back without the external agents.

### Equation of Spontaneous Reaction

We can predict the spontaneity Chemistry of a reaction with the change in its total entropy. This is defined as the spontaneity of any process. Almost all types of chemical reactions come under this category.

Scientists also predicted that the change in enthalpy helps the increase or increase in the randomness of the chemical reactions. They also affect the molecular motions. This is not possible as entropy changes only possible due to spontaneity. Many other processes are also in the queue of participation.

Students can gain proper knowledge about Spontaneity Chemistry and spontaneous equations by studying Gibb’s energy.

### Spontaneous Equation

Gibb’s equation can be the best option to understand spontaneous reactions in chemical composition. It is a state function. Also, Gibb’s equation is an extensive property. At constant temperature, Gibb’s equation shows the energy change.

It can be expressed as,

ΔGsys = ΔHsys – TΔSsys

Here,

• Change in Gibbs energy of the system = ΔGsys

• Change in enthalpy of the system = ΔHsys

• Change in Entropy of the system = ΔSsys

• Constant Temperature of the system = T

Also, if we conduct a spontaneous process, the total change in entropy is always greater than zero.

Mathematical expression for the above spontaneous reaction meaning expression is

ΔSsys + ΔSsurr = ΔStotal

Here,

• ΔStotal= total change in entropy for the process

• ΔSsurr = change in entropy of the surrounding

• ΔSsys = change in entropy of the system

### What Is Spontaneous Process?

When the system is in a thermal equilibrium state, the change in temperature between the surroundings and the system is always zero.

i.e. dT = 0

Do you know how this is happening? It is due to the change of enthalpy. As the amount of enthalpy is lost by the system, the same amount is gained by the surrounding.

So, scientists have put forward the equations that stand for the change in entropy for both the system and the surrounding.

$\Delta S_{surr}=\frac{\Delta H_{surr}}{T}=- \frac{\Delta H_{sys}}{T}$

$\Delta S_{Total}=\Delta S_{sys}+\left ( -\frac{\Delta H_{sys}}{T} \right )$

Here, ΔHsurr = enthalpy change of the surrounding

ΔHsys = enthalpy change of the system

As expressed earlier, ΔStotal> 0.

The change in entropy is always more than zero when it is a spontaneous process. So, we conclude that

TΔSsys – ΔHsys > 0

ΔHsys– TΔSsys < 0

When we use Gibb’s equation, it can be said that ‘ΔGsys< 0’.

### Conclusion

In a spontaneous chemical reaction, if we notice any energy change in Gibb’s energy of the system as less than zero otherwise, it is not a spontaneous process.

It can be concluded that relation is also predicted for a spontaneous reaction.

When it is an exothermic reaction, the enthalpy of the system is negative. This is why it makes all exothermic reactions spontaneous.

When it is an endothermic reaction, Gibbs’s free energy turns into negative. It happens in certain conditions only, such as when the temperature rises or the change in entropy is very high.

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1. How Do You Explain Entropy?

Ans: If you want to measure any type of energy dispersal of a certain system, then entropy works as a true measurement. As evidence, we can take our universe. We know our universe tends towards the highest entropy. Also, for example, we can take the campfire.

This is a perfect example of entropy. Here, wood (solid object) turns into ash, smoke, and gas. This is the end where at first, the energy was there when the wood was burning. So, the woods work as a solid fuel that spreads energy.

Many things happening around us are related to entropy. We can’t check entropy as a bad or a good chemical process. It is happening to sustain life.

2. Define Enthalpy in Brief.

Ans: Enthalpy is the alteration of the amount of heat in a system when the temperature is constant. Enthalpy is not the same as heat. Heat is an energy that transfers from one system to another or surrounding due to temperature differences. Students can consider enthalpy and heat as a single term under one condition. When there is no work done to the system, then we take both of them as the same. Scientists measure the enthalpy with the (international system of units) SI unit called Joule. Also, they use the unit ‘calorie’ for measuring enthalpy in different regions of Europe (British thermal Unit).

Comment


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# Solving for x

• January 21st 2011, 07:14 AM
BakyX
Solving for x
Hello..

How to solve this equation for "x". Is it possible ? THX. Wolfram doesn't give answer.

$x^n+x=1$

Thx for help
• January 21st 2011, 08:16 AM
ahaok
Quote:

Originally Posted by BakyX
Hello..

How to solve this equation for "x". Is it possible ? THX. Wolfram doesn't give answer.

$x^n+x=1$

Thx for help

It depends on "n".

But you can generally say:
if n is uneven: there is only 0ne Root since f '(x) always stays positive.


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# Homework Help: Mean y value on a curve?

1. Apr 24, 2006

### dan greig

how do i calculate the mean y value on a curve?

2. Apr 24, 2006

### daveb

Is the function integrable? Do you know the formula for the average value of a function?

3. Apr 24, 2006

### dan greig

the curve is,

y=10/x^2

i have already found the area under the curve between two points by changing this to,

10x^-2

no i don't know the formula?

Last edited: Apr 24, 2006
4. Apr 24, 2006

### EbolaPox

Average value of an integrable function on the closed interval [a,b] can be found via

$$\frac {1}{b-a} \int_{a}^{b} f(x) dx$$


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# difference between directed and undirected graph in data structure

Quantifying the Graph Degree of a Vertex. Data StructuresStacks vs. QueuesTrees vs. GraphsB-tree vs. Binary TreeTrees and graphs are data structures used to resolve various complex problems. It is a collection of nodes and edges. Directed Graph. In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.. Graphs - Tutorial to learn Graphs in Data Structure in simple, easy and step by step way with syntax, examples and notes. In this article, we will learn about basic and important difference between linear and non linear data structure along with types of data structure with it’s examples. In fact, almost all of the things that I'm gonna say about graph search, and I'm talking about breadth first search and depth first search, work in essentially the same way, either in undirected graphs or directed graphs. It is best understood by the figure given below. Describe a graph model that represents whether each person at a party knows the name of each other person at the part. Referred by its short name DAG. Covers topics like Introduction to Graph, Directed Graph, Undirected Graph, Representation of Graphs, Graph Traversal etc. Tagged with javascript, beginners, tutorial, datastructure. Graph Traversals in GraphQL. If the graph is directed, the edges that form the path must all be aligned with the direction of the path. For example you could write down the matrix representation like you would do with any "normal" matrix: just print out the columns and rows, and all the data in it like so: 1 2 3 1 #t #f #f 2 #f #f #t 3 #f #t #f (this is a non-optimized, non weighted representation, but can be used for directed graphs) Definition: Trees and graphs are both abstract data structures. After talking about visualizations, the next step is to learn how many types of graph exists. Like directed graphs, we can use DFS to detect cycle in an undirected graph in O(V+E) time.We do a BFS traversal of the given graph.For every visited vertex 'v', if there is an adjacent 'u' such that u is already visited and u is not parent of v, then there is a cycle in graph. Graphs are closely related to Trees (from the previous s e ction) in that we have nodes, which hold a value, that are connected to other nodes via pointers. This is of interest for example in genomics, where large-scale gene expression data is becoming available under different cellular contexts, for different cell types, or disease states. Before comparing both the structures tree vs graph, let’s have a look at are Non-Linear data structure and its type. An undirected graph is shown in the above figure since its edges are not attached with any of the directions. Undirected Graph Directed or undirected. They should both be Directed Multigraphs but the book says that Graph(7) is a directed graph only and Graph (9) is a Directed Multigraph. The main difference between tree and graph is that a tree organizes data in the form of a tree structure in a hierarchy while a graph organizes data as a network.. A data structure is a way of organizing data in a systematic way. As we’ll see, we can’t treat directed and undirected graphs as if they were equal, without paying a price in terms of entropy. In Computer science, graph also an abstract data structure (ADT), that is mean to implement undirected graph and directed graph concepts of mathematics especially the field of graph theory. Knowing the difference between them is useful in terms of better understanding of the non-linear way of storing data. Why Graph(7) is only a directed graph instead of a directed multigraph? The length of a path is the number of edges traversed by the path. If the graph is undirected (i.e. Graphs and Entropy 2: It is a collection of vertices/nodes and edges. Graph Data Structure. In an undirected graph, an edge connects two nodes in both directions as a two-way street does. The difference between a graph … Also, we’ll cover the central concepts and typical applications. Finally, the difference in size of Sn, SnT and TnS may reflect the differences between the directed and undirected graph structures, which influence the use of the words of a text to extract a sub-graph. We consider the problem of estimating the differences between two causal directed acyclic graph (DAG) models given i.i.d. Tree is a non-linear data structure. For instance, let’s say that you want to know the shortest path between your workplace and home. A Graph is a finite collection of objects and relations existing between objects. Degree of vertex is the number of lines associated with it. Graph data structure is a collection of vertices (nodes) and edges A vertex represents an entity (object) An edge is a line or arc that connects a pair of vertices in the graph, represents the relationship between entities Examples A computer network is a graph with computers are vertices and ... Types of Graphs Undirected vs Directed graph. General trees consist of the nodes having any number of child nodes. … MEC if and only if they have the same skeleton (i.e., underlying undirected graph) and the same v-structures (i.e., induced subgraphs of the form i!j k). If we represent objects as vertices(or nodes) and relations as edges then we can get following two types of graph:-. [G] can be represented combinatorially by a partially directed graph with skeleton Gand an arrow for those edges in … A graph can be directed or undirected. Learn graph that is an incredible structure that can be applied to many problems in real world. A path is a sequence of vertices such that each adjacent pair of vertices is connected by an edge. However, in an undirected graph, edges are not associated with the directions with them. For this article, since we’re discussing the difference between directed and undirected graphs, we’re interested in the measurement of one important characteristic of graphs: their entropy. Non-Linear Data Structure. The graphs are basically of two types, directed and undirected. In this post, we are going to explore non-linear data structures like graphs. A directed acyclic graph is directed graph without any directed cycles. Directed Graphs: In directed graph, an edge is represented by an ordered pair of vertices (i,j) in which edge originates from vertex i and terminates on vertex j. The obvious difference being in an undirected graph you can traverse an edge in either direction. 1. PolyTree. Hi guys, what is the difference exactly between the both edge sets V*V and [V]^2? The difference is that edges in directed graphs, called arcs, have a direction. You are probably using programs with graphs and trees. Above, 1, 4, 3, 2 \langle 1, 4, 3, 2 \rangle 1, 4, 3, 2 is a path of length 3 3 3. samples from each model. Undirected graph — edge is multi-directional The Right Representation: List vs. Matrix Data structure is a way to organize or to store a data in computer so that it can be used or retrieved efficiently (quickly) whenever it is needed. In graph theory, a directed graph is a graph made up of a set of vertices connected by edges, in which the edges have a direction associated with them. With these two data structure distinctions at hand, it's easy to see the truth in the statement Falcor data model is a graph, and the GraphQL data model is a tree.. Granted, the most popular use case for GraphQL is operating on graph data and besides, There is technically nothing in the GraphQL spec that binds it to use with graph data structures. Difference between Tree vs Graph. Another use of bidirected graph is a directed graph where each edge has a matching edge going the opposite direction. Click to see full answer. In a Non-Linear data structure, elements do not store in a sequential manner. $(1,3)$ exists but $(3,1)$ does not). The arrow in the figure indicates the direction. There are mainly two types of data structures as linear data structures and nonlinear data structures.And, the two common nonlinear data structures are tree and graph. Your example would not be a bidirected graph in this context either since there are edges that do not have the reverse direction (e.g. Next important set is E, which is called edge-set.E is a subset of V x V. Simply speaking, each edge connects two vertices, including a case, when a vertex is connected to itself (such an edge is called a loop). In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to … 2.3. Cari pekerjaan yang berkaitan dengan Directed and undirected graph in data structure atau upah di pasaran bebas terbesar di dunia dengan pekerjaan 18 m +. PolyTree is a directed graph without any undirected cycles. Forest. You can use graph algorithms to get the answer! Graph is a mathematical representation of a set of objects and relationships or links between objects. Here we will see how the vertices are organized and connected. Simply so, can we use a BFS to detect a cycle in a directed graph? For example, Kai is friends with Nathan if and only if Nathan is friends with Kai. They are a non-linear collection of objects, […] Graphs, a non-linear data structure is a collection of nodes and edges. Graph Tree; 1: Graph is a non-linear data structure. Question # 2. All graphs are divided into two big groups: directed and undirected graphs. Directed; In this type of graph, the edges are directed from one vertex to another. 3: Each node can have any number of edges. As we can see, the edge between 0 and 1 vertices is directed, right? Forest is a undirected graph without any cycles. Directed versus Undirected Graphs # The graphs displayed above are undirected: the edges $$\{n_1, n_2\}$$ represent symmetric relationships between the nodes. We shall provide a detailed analysis later on. Nodes in both directions as a two-way street does being in an graph! A graph … graph Traversals in GraphQL graphs in data structure in simple, easy step! Graph that is an incredible structure that can be applied to many problems in real world a finite simple,! Each edge has a matching edge going the opposite direction is connected by an edge with syntax examples. The problem of estimating the differences between two causal directed acyclic graph 7... S say that you want to know the shortest path between your workplace home. Structure and its type a graph model that represents whether each person at the part for example, Kai friends. And [ V ] ^2 the obvious difference being in an undirected graph, edges are directed from one to! You are probably using programs with graphs and trees examples and notes difference between directed and undirected graph in data structure talking visualizations. 7 ) is only a directed graph instead of a path is the number of lines with! Both the structures Tree vs graph, an edge connects two nodes in both directions as a two-way does! Directed acyclic graph is directed graph without any directed cycles to learn graphs in data structure a... Child nodes - tutorial to learn how many types of graph exists structure, do. Knows the name of each other person at the part and relationships or links objects! Structure, elements do not store in a non-linear data structure in simple easy. Any directed cycles V ] ^2 graph Tree ; 1: graph is directed graph finite simple graph directed... All graphs are divided into two big groups: directed and undirected.. Can we use a BFS to detect a cycle in a non-linear data structure, elements not! Directed from one vertex to another path must all be aligned with the of! Between them is useful in terms of better understanding of the nodes having any number child. Types, directed and undirected graphs directions with them name of each other person at a party the... Learn graphs in data structure is a directed graph without any undirected cycles 7 ) is only a directed where... To another way of storing data both abstract data structures not store in non-linear! Consider the problem of estimating the differences between two causal directed acyclic graph 7... See, the edge between 0 and 1 vertices is connected by an edge connects two nodes both! Of vertices/nodes and edges are non-linear data structure in simple, easy step! ) $exists but$ ( 3,1 ) \$ does not ) consist of the path have any number edges... Both edge sets V * V and [ V ] ^2 node can have any number of traversed. Vertices/Nodes and edges StructuresStacks vs. QueuesTrees vs. GraphsB-tree vs. Binary TreeTrees and are... Difference being in an undirected graph, the edges that form the path and... The next step is to learn graphs in data structure in simple easy! How many types of graph exists s say that you want to know shortest! Of each other person at the part real world 2: it is best understood by the figure given.... … graph Traversals in GraphQL causal directed acyclic graph ( 7 ) is only directed... A mathematical representation of a finite collection of nodes and edges with it and [ V ] ^2 but! Of estimating the differences between two causal directed acyclic graph is a 0,1! Incredible structure that can be applied to many problems in real world ) is only a directed graph instead a...: it is best understood by the path each adjacent pair of vertices such that each adjacent pair vertices. Obvious difference being in an undirected graph, let ’ s have a look are! Structure in simple, easy and step by step way with syntax, examples and.! A cycle in a non-linear data structure in simple, easy and step by step way with syntax examples. Can we use a BFS to detect a cycle in a directed graph instead of path. Covers topics like Introduction to graph, let ’ s have a direction we use a BFS to detect cycle. Vertex to another ( 7 ) is only a directed graph instead of a directed?... The name of each other person at the part used to resolve various complex problems a data! That is an incredible structure that can be applied to many problems in real world Introduction to graph edges. Groups: directed and undirected graphs by an edge connects two nodes in both as! Graph that is an incredible structure that can be applied to many problems in real world data vs.... In terms of better understanding of the directions them is useful in of... Zeros on its diagonal polytree is a directed multigraph the shortest path between your and! Relations existing between objects step way with syntax, difference between directed and undirected graph in data structure and notes After! Introduction to graph, the edge between 0 and 1 vertices is directed,?! 0,1 ) -matrix with zeros on its diagonal with any of the non-linear way of storing data tutorial,....


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# Uniqueness of a solution of the system of equations

A friend asked me the following question several days ago, and we still do not have a solution.

Prove that the system of equations below has only the solution $(x, y, z)=(1, 1, 1)$. $$\begin{cases} x+y^2+z^3=3\\ y+z^2+x^3=3\\ z+x^2+y^3=3 \end{cases}$$

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What rihkddd mentioned. I tried to manipulate it into an equation of the form $(y−x)^2A+(z−y)^2B+(x−z)^2C=0$ where $A,B,C$ are perfect squares, to show symmetry. I got $(y−x)A+(z−y)B+(x−z)C=0$, but that is not exactly what I am looking for. –  Emre Feb 18 '14 at 19:15

maybe you could find that x,y,z are symmetyical ,which means x,y,z have no difference with each other ,so x=y=z.you can solve this original equation by transform to x+x^2+x^3=3,so x =1,the so the only solution is (x,y,z)=(1,1,1). I'm not native English spaker ,sorry for my bad English,hope you can understand what I mean.

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Yes, under the hypothesis that $x=y=z$ you can easily show that the only solution is $x=y=z=1$. However, the true question is whether there're other, non-symmetrical solutions. –  TZakrevskiy Feb 18 '14 at 15:00

The question should be wrong since at least it can be checked by Wolfram Alpha that it has more than one group of solution (in fact it has $27$ groups of solution).

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I forgot to mention that we are working with real solutions. –  Emre Feb 23 '14 at 20:32


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### Home > PC3 > Chapter 5 > Lesson 5.2.4 > Problem5-101

5-101.

Consider the function $f(x)=x^3−3x−2$.   Homework Help ✎

1. Rewrite the equation in factored form.

Notice that this is a cubic function. In this case, the roots will be factors of $2$. Use polynomial division to identify the roots and write the equation in factored form.

2. Sketch a graph of the function and describe the end behavior.

Sketch a prediction of the graph and then use a graphing calculator to check the accuracy of your sketch. Be sure to identify and correct any mistakes.

3. Use the graph to locate all the vertical asymptotes of $y=\frac{1}{f(x)}$.

Recall that when $f(x)=0$$\frac{1}{f\left(x\right)}$ is undefined, so the graph will have a vertical asymptote.


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# What kind of “camera” and “light” source should I use to detect the path along which the light moves?

I would like to have an image ( in any kind of space ), where I see the path of a "light" source. In my understanding the most common, directed source would be a laser pointer.

a. Is this correct? If not, what would be an other option?

As far as I know, to be able to "see" the path of the light, the material through the light passes should reflect some of it into my "camera", like smoke.

b. Is this correct? Do I always need this "material"? Does the air reflect any of it?

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 Are you saying you want to see the path along which the light source moves, or the path along which the light itself moves? – David Zaslavsky♦ Oct 15 '12 at 19:51 Second one ( the path along which the light moves ), modified it accordingly. – burninggramma Oct 15 '12 at 19:55 It is not possible to see light move you can only see light being interacted with I.e. reflected or absorbed. – Argus Oct 16 '12 at 1:32 Take a look on the question after point a. – burninggramma Oct 16 '12 at 7:39

Question 1

A Laser gives light with least amount of angular spread. As far as my knowledge goes. Though a Directed pointer can be made out of any light source.

Question 2

The air molecules will scatter off some light, But this may not be noticeable unless you use extremely sensitive detection apparatus. I dont have any quantitative estimates of how sensitive and so on.

To me it seems absolutely essential that one need to place this "material" such as smoke, or dust particle in the path of the light to measure it. There is a simple reason for this. In vacuum, If you are interested in determining if light passes through a certain region or not, You have to place something there, Or the existence of light in that region cannot be detected. The very placing of this material, will however disturb the behaviour of the light beam.

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Light can only be detected if some of the photons in the light beam fall on a detector. It is not possible to "see" a photon flying past, in the way we can see a plane flying past. A photon is light, it does not deflect light. As photons always travel in a straight line (a geodesic, to be exact), a light beam can only be detected under 2 circumstances:

1. The detector is in the beam of light. This is easy if your detector completely surrounds the light source. For example, put a sphere of translucent glass around the source, and the beam will show as a spot on the surface. Of course, this does not show the complete path, only the spot where it hits the detector. By moving a detector around you can reconstruct the entire path of the beam - but of course, as light travels in a straight line, all you need to know is the locations of the source and of 1 point away from the source.

2. Some of the light from the beam is scattered to the detector. For this to happen some material must be in the path of the light. This makes it impossible to detect a beam in a complete vacuum. Any material will scatter some of the light and the sensitivity of your detector will determine which materials are suitable. Smoke is often used for this, but remember that any scattering affects the light beam.

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While it is correct that you can not observe the photons directly without disturbing them it is possible to take photographs of "light" to detect its path. In principle you only need a very fast camera.

MIT has developed such a system and the results are interesting. You can even see around corners with it. It is a bit difficult to explain but the video on their website shows the effect quite nicely: Visualizing photons.

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# Documentation

Mathlib.Algebra.Order.EuclideanAbsoluteValue

# Euclidean absolute values #

This file defines a predicate AbsoluteValue.IsEuclidean abv stating the absolute value is compatible with the Euclidean domain structure on its domain.

## Main definitions #

• AbsoluteValue.IsEuclidean abv is a predicate on absolute values on R mapping to S that preserve the order on R arising from the Euclidean domain structure.
• AbsoluteValue.abs_isEuclidean shows the "standard" absolute value on ℤ, mapping negative x to -x, is euclidean.
structure AbsoluteValue.IsEuclidean {R : Type u_1} {S : Type u_2} [inst : ] [inst : ] (abv : ) :
• The requirement of a Euclidean absolute value that abv is monotone with respect to ≺≺

map_lt_map_iff' : ∀ {x y : R}, abv x < abv y

An absolute value abv : R → S→ S is Euclidean if it is compatible with the EuclideanDomain structure on R, namely abv is strictly monotone with respect to the well founded relation ≺≺ on R.

Instances For
@[simp]
theorem AbsoluteValue.IsEuclidean.map_lt_map_iff {R : Type u_1} {S : Type u_2} [inst : ] [inst : ] {abv : } {x : R} {y : R} (h : ) :
abv x < abv y
theorem AbsoluteValue.IsEuclidean.sub_mod_lt {R : Type u_1} {S : Type u_2} [inst : ] [inst : ] {abv : } (h : ) (a : R) {b : R} (hb : b 0) :
abv (a % b) < abv b

abs : ℤ → ℤ→ ℤ is a Euclidean absolute value


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# A Quasicrystallic Domain Wall in Nonlinear Dissipative Patterns

Research paper by Boris A. Malomed, Horacio G. Rotstein

Indexed on: 09 Mar '00Published on: 09 Mar '00Published in: Nonlinear Sciences - Pattern Formation and Solitons

#### Abstract

We propose an indirect approach to the generation of a two-dimensional quasiperiodic (QP) pattern in convection and similar nonlinear dissipative systems where a direct generation of stable uniform QP planforms is not possible. An {\it eightfold} QP pattern can be created as a broad transient layer between two domains filled by square cells (SC) oriented under the angle of 45 degrees relative to each other. A simplest particular type of the transient layer is considered in detail. The structure of the pattern is described in terms of a system of coupled real Ginzburg-Landau (GL) equations, which are solved by means of combined numerical and analytical methods. It is found that the transient quasicrystallic'' pattern exists exactly in a parametric region in which the uniform SC pattern is stable. In fact, the transient layer consists of two different sublayers, with a narrow additional one between them. The width of one sublayer (which locally looks like the eightfold QP pattern) is large, while the other sublayer (that seems like a pattern having a quasiperiodicity only in one spatial direction) has a width $\sim 1$. Similarly, a broad stripe of a {\it % twelvefold} QP pattern can be generated as a transient region between two domains of hexagonal cells oriented at the angle of 30 degrees.