text:
Gamma rays and neutrinos from a cosmic ray source in the Galactic Center region

# Gamma rays and neutrinos from a cosmic ray source in the Galactic Center region

A. D. Supanitsky Instituto de Astronomía y Física del Espacio (IAFE), CONICET-UBA, CC 67, Suc. 28, C1428ZAA Buenos Aires, Argentina.
July 6, 2019
###### Abstract

The center of the our Galaxy is a region where very energetic phenomena take place. In particular powerful cosmic ray sources can be located in that region. The cosmic rays accelerated in these sources may interact with ambient protons and/or low energy photons producing gamma rays and neutrinos. The observation of these two types of secondary particles can be very useful for the identification of the cosmic ray sources and for the understanding of the physical processes occurring during acceleration.

Motivated by the excess in the neutrino spectrum recently reported by the IceCube Collaboration, we study in detail the shape of the gamma ray and neutrino spectra originated from the interaction of cosmic ray protons with ambient protons for sources located in the Galactic Center region. We consider different models for proton acceleration and study the impact on the gamma ray and neutrino spectra. We also discuss the possibility to constrain and even identify a particular neutrino source by using the information given by the gamma ray spectrum taking advantage of the modification of the spectral shape, caused by the interaction of the gamma rays with the photons of the radiation field present in the interstellar medium, which strongly depends on the source distance.

## I Introduction

Gamma rays and neutrinos can be generated in cosmic ray sources due to the interaction of the cosmic rays with ambient photons and/or protons. Powerful cosmic ray accelerators can be located in the Galactic Center region; therefore, the observation of gamma rays and neutrinos coming from the Galactic Center is of great importance for the identification and understanding of the Galactic cosmic ray sources. It is worth noting that, in contrast with charged cosmic rays, these two types of secondary particles are not deviated by the magnetic fields present in the interstellar medium and then they point back to its source.

The IceCube Collaboration has recently reported the observation of 28 neutrino events in the energy range from 30 TeV to 1.2 PeV IceCubeA:13 (); IceCubeB:13 (), while only events were expected considering a conventional atmospheric neutrino background. This corresponds to a excess. Despite the large angular uncertainty of these events (), we can observe that 5 of the 28 events come from the Galactic Center region. However, there is a degeneracy about the origin of these neutrinos because they can also originate outside our Galaxy. It has been pointed out that the gamma ray flux associated with the neutrino flux can be useful to constrain or even identify the candidate neutrino sources Gupta:12 (); Anchordoqui:13 (); Murase:13 (); Ahlers:13 (). In particular, in Refs. Neronov:13 (); Razzaque:13 () it is found that the the gamma ray data taken by Fermi LAT at lower energies are consistent with a Galactic origin of these five events. However, the upper limits on the diffuse gamma ray flux obtained by different experiments at higher energies disfavor this hypothesis Ahlers:13 ().

On the other hand, the region in the cosmic ray energy spectrum where the transition from the Galactic to extragalactic cosmic rays takes place is still unknown (see Ref. Aloisio:12 () for a review). Two possible regions for this transition are the second knee, a steepening of the spectrum which is given at an energy of EeV, and the ankle, a hardening of the spectrum placed at an energy of EeV. In Ref. Anchordoqui:13 () it has been pointed out that the identification of Galactic neutrino sources and the observation of their spectra can contribute to finding the region where the transition from Galactic to extragalactic cosmic rays takes place. This is due to the fact that the end of the neutrino energy spectrum is correlated with the maximum energy at which the cosmic rays are accelerated in the source. Following Ref. Anchordoqui:13 (), the observation of the end of the neutrino spectrum, of a given Galactic source, at PeV favors the scenario in which the transition takes place in the second knee region, whereas the observation of PeV favors the scenario in which the transition is given at the ankle region.

The most probable mechanism for the production of gamma rays and neutrinos in Galactic cosmic ray sources is the interaction of cosmic rays with ambient protons Gupta:12 (). In this work we study in detail the shape of the gamma ray and neutrino spectra that originate from the proton-proton interaction in Galactic cosmic ray sources placed in the Galactic Center region, which is motivated by the recent IceCube results. We consider different shapes of the proton spectrum based on acceleration models. We also consider the values for the maximum energy of protons corresponding to the Galactic to extragalactic transition in the second knee region and in the ankle region. The propagation of the gamma rays and neutrinos from the Galactic Center to the Solar System is included in these calculations. While neutrinos suffer flavor oscillation during propagation, gamma rays can interact with the photons of the Galactic interstellar radiation field (ISRF) and the cosmic microwave background (CMB) which is the most important one Moskalenko:06 (). In particular, we find that the ratio between the neutrino and gamma ray spectra depends on the cosmic ray proton spectrum, which can be used to study the conditions under which the cosmic ray protons are accelerated in the source. Finally, we study the possibility to discriminate between the Galactic and extragalactic origins of a neutrino spectrum observed in the direction of the Galactic Center by observing the associated gamma ray flux, which in principle will be possible with the planned high energy gamma ray observatories.

## Ii Gamma rays and neutrinos injected by a cosmic ray source

Gamma rays and neutrinos can be produced as a result of the interaction of the cosmic rays with ambient protons present in the sources. Gamma rays are mainly produced by the decay of neutral pions; mesons also contribute to the gamma ray flux. The generation of neutrinos is dominated by the decay of charged pions. Positive (negative) charged pions mainly decay into an antimuon and a muon (antimuon) neutrino,

 π+ → μ++νμ, π− → μ−+¯νμ.

The subsequent decay of the muons and antimuons produces more neutrinos,

 μ+ → e++νe+¯νμ, μ− → e−+¯νe+νμ.

It is believed that the Galactic cosmic rays are accelerated principally at supernova remnants (see e. g. Ref. Hillas:06 ()) by means of the first order Fermi mechanism. Also, the experimental evidence indicates that the injected composition of the Galactic sources is dominated by protons Longair (). The spectrum of protons that escape from the acceleration region accelerated via the first order Fermi mechanism, without taking account of energy losses, can be written as

 Φp(Ep) = A E−γp (1+(EpEmax)2δ) (1) ×exp(−γ−12δ(EpEmax)2δ),

where is a normalization constant, is the spectral index, refers to the maximum energy which indicates the end of the spectrum (note that protons can be accelerated to higher energies than ), and comes from the energy dependence of the diffusion coefficient in the acceleration region, (see Appendix A for details).

Cosmic rays can suffer energy losses during the acceleration process due to the presence of strong magnetic fields and ambient low energy photons Protheroe:98 (); Protheroe:04 (). The spectrum near its end can have different shapes, including pileups and sudden cutoffs, depending on the type of energy loss dominating in the acceleration region. Then, in the subsequent analyses a proton spectrum with a sudden cutoff in is also considered,

 Φp(Ep)=A⎧⎪⎨⎪⎩E−γpEp≤Emax0Ep>Emax. (2)

Figure 1 shows the proton spectra for and for (Kolmogorov spectrum), (Kraichnan spectrum), and (completely disordered field) Protheroe:04 (). The spectrum corresponding to the sudden cutoff is also shown. The maximum energy considered is TeV, which corresponds to the transition between the Galactic and extragalactic cosmic rays in the region of the second knee. As expected, the steepening of the spectra is more pronounced for increasing values of .

Given the proton spectrum the spectra of gamma rays and neutrinos generated by the interaction with the ambient protons are calculated following Ref. Kelner:06 (). In that work the proton-proton interaction was simulated by using the hadronic interaction model Sibyll 2.1 Sibyll (). The gamma ray spectrum is calculated from

 Φ0γ(Eγ)=C∫∞EγdEpEp σinel(Ep) Φp(Ep) Fγ(EγEp,Ep), (3)

where is a normalization constant and is the proton-proton inelastic cross section. and are taken from Ref. Kelner:06 (). The neutrino and antineutrino spectra are obtained in a similar way, and the corresponding functions are also taken from Ref. Kelner:06 ().

Figure 2 shows the gamma ray and neutrino111Hereafter, neutrino spectrum will refer to the sum of the neutrino and antineutrino spectra of a given flavor. spectra at the source for , TeV, and . Note that the results obtained in this work are in very good agreement with the ones obtained in Ref. Kappes:07 ().

From Fig. 2 it can be seen that the gamma ray spectrum is larger than the one corresponding to the electron neutrino and muon neutrino. Also, the end of the neutrino spectra takes place at a smaller energy than the one corresponding to gamma rays.

## Iii Gamma rays and neutrinos at Earth

### iii.1 Precise calculation

Gamma rays and neutrinos produced at a given source propagate through the interstellar medium to reach the Earth. The electron and muon neutrinos oscillate during propagation; therefore, the flavor ratio observed at the Earth is different from the one in the source. It is believed that the neutrino oscillations originate from the fact that the flavor states, with , are not the mass eigenstates, with . The flavor and the mass basis are related by Giunti:07 ()

 |να⟩=3∑j=1U∗αj|νj⟩, (4)

where is a unitary matrix which can be parametrized by the mixing angles and and a violating phase . The transition probability can be written as Giunti:07 ()

 Pνα→νβ=3∑j=1|Uαj|2 |Uβj|2. (5)

In the so-called tribimaximal mixing approximation Harrison:02 () the parameters of the mixing matrix are taken as , , , and . By using this approximation it can be seen that a flavor ratio in the source goes to after propagation. In this work a more precise estimation of the mixing angles is used in which is different from zero PDG:12 (): , , and . It is also assumed that . By using this new estimation of the parameters the flavor ratio goes to , which is very close to the tribimaximal mixing approximation. Therefore, the propagated spectra are calculated from

 ⎛⎜⎝ΦνeΦνμΦντ⎞⎟⎠=P⎛⎜ ⎜⎝Φ0νeΦ0νμΦ0ντ⎞⎟ ⎟⎠ (6)

where the elements of the matrix are given by and are the neutrino spectra at the source. Note that in this case .

Gamma rays interact with photons of different backgrounds when they propagate from the source to the Earth; the most important ones are the CMB and the ISRF Moskalenko:06 (). The most important process at the energies considered is the electron-positron pair production. Figure 3 shows the mean free path of photons, , as a function of energy for the CMB. The calculation has been done following Ref. Protheroe:86 (). From the figure it can be seen that for energies of order of 1 PeV kpc, which is very close to the distance from the Solar System to the Galactic Center ( kpc). Therefore, at these energies the attenuation in the CMB becomes quite important.

The attenuation factor is given by , where is the distance from the source to the Earth and

 τ(Eγ,D)=∫D0dℓλγ(Eγ,ℓ). (7)

Thus, the gamma ray flux at the Earth is given by

 Φγ(Eγ)=T(Eγ,D) Φ0γ(Eγ)4πD2. (8)

Figure 4 shows the attenuation factor for the CMB and ISRF (taken from Ref. Moskalenko:06 ()) as a function of the energy for two different positions of the source: One is placed at the Galactic Center and the other at 4 Mpc from the Solar System in the direction of the Galactic Center. The attenuation factor for the ISRF is taken from Ref. Moskalenko:06 (); the one corresponding to the extragalactic source is obtained by integrating from a distance of 28.5 kpc in the direction of the Galactic Center (see Eq. (7)). It can be seen that the attenuation caused by the interaction with the CMB photons is more important than that corresponding to the ISRF.

The top panel of Fig. 5 shows the gamma ray and neutrino spectra at the Earth for a Galactic cosmic ray source, like the one considered in Sect. II (, TeV, and ), located in the Galactic Center. In this figure the effects of the attenuation in the gamma ray spectra are evident. Also, the neutrino spectra for the different flavors become similar after propagation. The bottom panel of Fig. 5 shows the ratio of the neutrino spectra to the gamma ray spectrum. It can be seen that it is not constant in a large energy range due to the attenuation suffered by the gamma rays and the different ends of the neutrino and gamma ray spectra. In particular, the first maximum of corresponds to the attenuation of the gamma rays in the ISRF, whereas the second and larger one corresponds to the attenuation in the CMB.

The shape of the end of the gamma ray and neutrino spectra depends on the shape of the proton spectrum. Figure 6 shows the gamma ray and the total neutrino spectra (summed over all flavors) at the Earth as a function of energy for the different values of considered and for the case in which the proton spectrum presents a sudden cutoff. The maximum energy of the proton spectrum is TeV. As expected, when the proton spectrum ends faster, the gamma ray and the neutrino spectra also end faster.

Figure 7 shows the ratio between the total neutrino spectrum and the gamma ray spectrum as a function of energy, corresponding to the spectra shown in Fig. 6. It can be seen that presents two maxima; as mentioned above, the first one corresponds to the attenuation due to the interaction of the gamma rays with the photons of the ISRF and the second one to the CMB. Also, the shape of depends on the type of proton spectrum considered, which can be useful, from the observational point of view, to study the end of the proton spectrum at the source, which is intimately related to the diffusion coefficient and the energy losses suffered by protons in the acceleration region.

The gamma ray and neutrino spectra are modified when the maximum energy of the accelerated protons increases. Figure 8 shows the ratio between the total neutrino spectrum and the gamma ray spectrum as a function of energy for the maximum energy of the proton spectrum TeV, which corresponds to the case in which the transition between the Galactic and extragalactic cosmic rays is given in the ankle region. It can be seen that in this case is larger than one corresponding to the case considered above. This is due to the fact that, in this case, the region in which the attenuation on the CMB is more important ( TeV), the gamma ray and neutrino spectra approximately follow the shape of the proton spectrum; i.e. the end of the spectra is given at higher energies. Also in this case, the ratio can be useful to study the end of the proton spectrum at the source, which is related to the physical conditions under which the protons are accelerated.

It can be seen that by increasing the spectral index , the steepening of the gamma ray and neutrino spectra takes place at lower energies, which makes decrease.

### iii.2 Approximate calculation

It is quite common to calculate the gamma ray spectrum from the neutrino spectrum (and the other way around), making use of several approximations (see, for instance, Ref. Ahlers:13 ()). In this section the accuracy of this approximation is studied by means of the numerical calculations presented above. In proton-proton collisions , , and are created with the same multiplicity. As a result of the decay of charged pions and the subsequent muon or antimuon decay four particles are created; three of those are neutrinos, and then . In the case of neutral pion decay two photons are created; then . Therefore, , and the differential flux of gamma rays can be obtained from Ahlers:13 ()

 Φγ(Eγ) ≅ 13∑αΦνα+¯να(Eγ2) dEνdEγ (9) ≅ 16∑αΦνα+¯να(Eγ2),

where . Here the effects of the propagation of gamma rays in the interstellar medium are not taken into account in order to better compare the spectral shape of the numerical calculation and the approximation.

Figure 9 shows the gamma ray spectrum as a function of the energy calculated as in Sec. III.1 (solid line) and from the neutrino spectra, also calculated as in Sec. III.1, by using Eq. (9) (dashed line) for , TeV, and . The approximation underestimates the gamma ray spectrum at lower energies and overestimates the spectrum at higher energies. Note that the larger differences between the approximated calculation and the numerical one are given at high energies, in the region at the end of the spectrum. In particular, the approximated spectrum is less steep in that energy region, which makes the difference between the approximated spectrum and the numerical one to increase with energy.

In order to quantify these differences let us introduce the following parameter,

 ε(Φγ)(E)=1−Φappγ(E)Φγ(E), (10)

where is the approximated spectrum and is the one obtained numerically. Figure 10 shows the relative error as a function of energy for the four cases of the proton spectrum considered with and TeV. At lower energies the differences between the approximated spectra and the numerical one are smaller than %. The larger differences take place in the region at the end of the gamma ray spectra because, as mentioned above, the approximated spectra is less steep than the numerical one. Note that in all cases considered, the behavior of the relative error is qualitatively similar; in the region corresponding to the end of the spectra, the absolute value of the relative error increases with energy, reaching values of the order of very fast. Note also that the approximation underestimates the spectrum at lower energies and overestimates it in the region corresponding to the end of the spectrum, i.e.  changes its sign at a given energy depending on the proton spectrum considered.

## Iv Galactic versus extragalactic origin of the observed neutrino spectrum

It is not possible to elucidate the origin of neutrinos coming from the Galactic Center region with just the observation of their energy spectrum and arrival direction distribution. There is a degeneracy; these neutrinos can originate in other regions of the Galaxy or even outside the Galaxy. This degeneracy can be broken by observing the companion gamma ray flux because the attenuation of the gamma rays on the CMB and ISRF depends on the distance of the source.

The top panel of Fig. 11 shows the integral spectra at the Earth of neutrinos (all flavors) produced in proton-proton interactions for the two values of the maximum energy of the proton spectrum considered, for , and . The bottom panel of Fig. 11 shows the corresponding gamma ray flux for these two cases and also for the case in which the gamma rays and neutrinos originate in an extragalactic source. For the extragalactic case it is assumed that the source is located at 4 Mpc from the Solar System; the attenuation in the CMB and ISRF used for the calculation corresponds to the extragalactic case of Fig. 4. Also in this case, we use . The maximum energy for the proton spectrum used in this case is TeV, but a very similar spectrum is obtained for TeV due to the sudden cutoff at TeV produced by the CMB and ISRF attenuation. The normalization of the spectra is chosen such that the gamma ray flux is very well observed by the different experiments. The bottom panel of Fig. 11 also shows the sensitivity of different planned or under-construction experiments: CTA (Cherenkov Telescope Array) cta (), HiSCORE (Hundred*i Square-km Cosmic ORigin Explorer) hiscore (), HAWC (High Altitude Water Cherenkov) hawc (), and LHAASO (Large High Altitude Air Shower Observatory) lhaaso (). Note that only CTA and HiSCORE are planned for the Southern Hemisphere222The CTA project consists in the construction of two observatories, one in each hemisphere, but the southern observatory will be constructed first. which is ideal for the observation of the Galactic Center.

From Fig. 11 it can be seen that, thanks to the attenuation of the gamma rays in the photon fields, it is possible to at least decide about the Galactic or extragalactic origin of the neutrino flux observed at the Earth. Moreover, in the case of a Galactic neutrino source not necessarily placed in the Galactic Center region, the measurement of the corresponding gamma ray flux can give an estimation of the distance of the source, provided that the gamma ray flux reaches energies larger than a few PeV.

It is worth noting that the test proposed here requires clear evidence of the existence of a neutrino source and a good measurement of its spectrum. Although the present IceCube data show some hints of the existence of one or more neutrino sources in the Galactic Center region, the present statistics is still poor for this type of test. This situation can change in the near future when more data are collected.

## V Conclusions

In this work we have studied in detail the shape of the gamma ray and neutrino energy spectra generated by the interaction of cosmic ray protons with ambient protons. We have considered two possible maximum energies for the proton spectrum: one consistent with the Galactic to extragalactic transition of the cosmic rays given in the second knee region and the other in the ankle region. We have studied the propagation of gamma rays and neutrinos from the source to the Earth, paying special attention to the case in which the source is located in the Galactic Center region. We have shown that the shape of the gamma ray and neutrino spectra at the Earth reflects the properties of the spectrum of protons accelerated in the sources; then the observation of these two types of secondaries can be very useful for the study of the cosmic ray sources.

We have studied the accuracy of a common approximation to infer the gamma ray spectrum from the neutrino spectra. We have found that at low energies the approximate calculation underestimates the gamma ray spectrum in less than % (for the four different types of proton spectra considered in this work) and that at higher energies the approximate calculation overestimates the gamma ray spectrum. The differences between the approximate spectrum and the one obtained numerically are larger in the region corresponding to the end of the spectrum and increase with energy. This is due to the fact that the approximate spectrum is less steep than the one obtained numerically.

We have also discussed the existing degeneracy about the origin of neutrinos observed in the direction of Galactic Center region. We have shown that such degeneracy can be broken by observing the companion gamma ray spectrum (above a few PeV), which strongly depends on the distance to the source due to the interaction of the gamma rays with the low energy photons of the CMB and ISRF.

###### Acknowledgements.
A. D. S. is a member of the Carrera del Investigador Científico of CONICET, Argentina. This work is supported by CONICET PIP 114-201101-00360 and ANPCyT PICT-2011-2223, Argentina.

## Appendix A Energy spectrum of cosmic ray protons

It is believed that cosmic rays are accelerated by the diffusive shock acceleration mechanism. Following Refs. Protheroe:98 (); Drury:99 (); Protheroe:04 () the differential energy spectrum of the accelerated particles () fulfills the equation,

 ∂n∂t+∂∂E(E racc n)=Q−resc n, (11)

where is the acceleration rate, is the escape rate, and is the source term. The solution of Eq. (11) for the stationary case () and for is given by,

 n(E)=˙N0E racc(E) exp(−∫EE0dE′resc(E′)E′ racc(E′)). (12)

The spectrum of the accelerated particles that escape from the acceleration zone is given by .

If the diffusion coefficients downstream, , and upstream, , have the same power-law dependence with the energy of the accelerated particles, and , the acceleration rate takes the form Protheroe:98 (),

 racc(E)=a E−δ. (13)

The escape rate can also be written as Protheroe:98 (); Protheroe:04 ()

 resc(E)=b E−δ+c Eδ, (14)

where the first term takes into account the particles that escape downstream and the second term is related to the escape of particles due to the finite size of the acceleration region Protheroe:04 ().

From Eqs. (12), (13), and (14) the energy spectrum of the escaping particles can be written as,

 nesc(E) = ˙N0E0(γ−1)(1+(EEmax)2δ)exp[−γ−12δ (15) ×((EEmax)2δ−(E0Emax)2δ)],

where and . Therefore, Eq. (1) is obtained from Eq. (15) by using

 A=˙N0E0(γ−1)exp[γ−12δ(E0Emax)2δ]. (16)

Note that possible energy losses suffered by the particles during the acceleration process are not taken into account in the present calculation.

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# How to tell which function affect behavior of a function the most?

The problem that got me thinking. $\left(\dfrac{\sin x}{x}\right)^2$

The overall pattern of this function is a parabola but it has been pulled down by $1/x$ and it wobbles because of $\sin x$.

So, in general is there any mathematical method/way to determine which function is the most dominate?

so for any $\left(f\left(g\right)\right)^{h}$ how could I tell which function $f,g,h$ will affect the shape of the graph most?

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This seems to really depend on $f,g,h$ to give anything general –  Belgi Mar 6 '13 at 1:15

Your problem is an ill-posed one.

Note that there are many properties of graphs that you can call "shape" (for example regularity, it is clearly something different than the large scale geometry of the graph). Even if you choose one of the shape properties you have to specify how you measure its changes.

Last but not least: if you pick a function $\psi$ it is very unlikely that it has a unique representation $\psi=(f\circ g)^h$...

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I knew my question was not high quality, sorry for that. So it really just depends upon which fucntions are involed then. –  yiyi Mar 6 '13 at 1:33
I am not saying that your question is not high quality :) I'm just saying that such a general method does not exist. Imho all you can do is to analyze each case of the representation $(f\circ g)^h$ separately. –  Godot Mar 6 '13 at 2:04

format: plain


text:
# Checkbox with numerical label inside

I'm trying to create a form with labeled checkboxes.

Like this:

I have a nice pdf form written out using eforms, but I am unable to overlay anything on top of a checkbox or textfield. I was almost able to create the desired effect with the following code:

\documentclass{article}

\usepackage{eforms}

\begin{document}
\begin{Form}

\fbox{
\CheckBox[
name=course1,
borderwidth=0.0,
bordercolor={1 1 1},
width=12px,
height=14px]
{\mbox{}}}
\null \hspace{2px}
\hspace{-33px}
\raisebox{-1px}{\framebox(22,16){\raisebox{-13px}{\hspace*{-16px}{\tiny 1a}}}}

\end{Form}
\end{document}

Which generates this:

I know this is a horrendous hack-job. And the first fbox has a border that I don't want. So I am sending this off to the internet gods in hopes of some mercy.

• Could you please provide a complete, compilable MWE (minimal working example)? A MWE begins at \documentclass and ends at \end{document} – Skillmon Apr 3 '17 at 18:31
• @Skillmon, thanks for the request. I have updated the code to be an MWE. – jdods Apr 3 '17 at 18:53

There are a number of problems with your code. First, 1px is an incredibly small unit. Second, \CheckBox and the Form environment are defined in hyperref, which is loaded by eforms.

The only way to move the letters closer to the checkmark is to reduce the width of the \CheckBox. 6pt is about as narrow as you can go.

\documentclass{standalone}

\usepackage{hyperref}

\begin{document}
\begin{Form}

\fboxsep=1pt
\fbox{\hspace{-\fboxsep}\raisebox{\dimexpr \fboxrule-\fboxsep}[0pt][0pt]{\tiny 1a}%
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text:
# Measuring cold/cryogenic objects without touching them?

A common way to measure temperature on the scale of household objects and cooking is to use an infrared thermometer. The basic idea is to look at the thermal emission of infrared light from an object which is related to the temperature (like blackbody radiation).

I am interested in ways to use non-contact thermometers at low temperatures (below 50 Kelvin). Blackbody radiation quickly becomes very weak in total energy as you cool down, so I don't think that would work. My question is, are there non-contact thermometers that work at cryogenic temperatures (<50 Kelvin)?

For example, I know Raman spectroscopy can be used to measure temperature by looking at frequency shifts and the ratio between energy absorption and loss events (which scales as $$e^{E/k_B T}$$). I can't find any examples of this technique for low-temperature thermometry though.

As a note, I am not necessarily looking for a cheap or convenient way to do this measurement. I am more interested to know if such devices exist and how they work physically.

• Is there a good reason to suspect that the device will be significantly out of thermal equilibrium with its immediate surroundings? – catalogue_number Jun 22 at 14:22
• @catalogue_number, good question: yes. It is quite technical and unrelated to the question, so I won't dwell on that though. i suspect that in the end a diode or thermocouple will have to be used practically speaking, but the question still stands about non-contact thermometry at low temperature – KF Gauss Jun 22 at 20:24
• It's a lot of pain to build a combined cryo-laser system. Better ask plainly your question (what is the specific set-up, what are the possible configurations), not to get lost in good answers to your implicit question, that may not be relevant to your actual case. Depending on the actual situation, you can place calibrated resistors in adjecency to the object (what is the object? is it a lithographic sample?), with substrate with decent heat conductivity – Alexander Jun 24 at 12:05
• @Alexander, I decided to divorce the fundamental question with the application in mind. I know in reality the best route in practice is to integrate a thermocouple and that will eventually happen in our case. But I am interested to know if a "cryo-laser system" (whatever that means to you) actually exists and the physical principles involved. I've edited the question to emphasize that – KF Gauss Jun 24 at 12:29

Raman spectroscopy is the answer. One may exploit the asymmetric temperature dependence of Stokes and anti-Stokes phonon scattering intensities to infer temperature: $$\frac{I_\text{Stokes}}{I_\text{Anti-Stokes}} \propto \exp \left(\frac{hc\bar{\nu}}{kT}\right)$$ Optical methods for measuring low temperatures have been published in this open access paper.


text:
## CryptoDB

### André Chailloux

#### Publications

Year
Venue
Title
2020
PKC
The GPV construction [ GPV08 ] presents a generic construction of signature schemes in the Hash and Sign paradigm and is used in some lattice based signatures. This construction requires a family $mathcal {F}$ of trapdoor preimage sampleable functions (TPSF). In this work we extend this notion to the weaker Average TPSF (ATPSF) and show that the GPV construction also holds for ATPSF in the Random Oracle Model (ROM). We also introduce the problem of finding a Claw with a random function (Claw(RF)) and present a tight security reduction to the Claw(RF) problem. Our reduction is also optimal meaning that an algorithm that solves the Claw(RF) problem breaks the scheme. We extend these results to the quantum setting and prove this same tight and optimal reduction in the QROM. Finally, we apply these results to code-based signatures, notably the Wave signature scheme and prove security for it in the ROM and the QROM, improving and extending the original analysis of [ DST19a ].
2017
EUROCRYPT
2017
ASIACRYPT
2008
TCC
2007
EPRINT
We study the role of help in Non-Interactive Zero-Knowledge protocols and its relation to the standard interactive model. In the classical case, we show that help and interaction are equivalent, answering an open question of Ben-Or and Gutfreund (\cite{BG03}). This implies a new complete problem for the class SZK, the Image Intersection Density. For this problem, we also prove a polarization lemma which is stronger than the previously known one. In the quantum setting, we define the notion of quantum help and show in a more direct way that help and interaction are again equivalent. Moreover, we define quantum Non-Interactive Zero-Knowledge with classical help and prove that it is equal to the class of languages that have classical honest-Verifier Zero Knowledge protocols secure against quantum Verifiers (\cite{Wat06, HKSZ07}). Last, we provide new complete problems for all these quantum classes. Similar results were independently discovered by Dragos Florin Ciocan and Salil Vadhan.
2007
EPRINT
We show that interactive and noninteractive zero-knowledge are equivalent in the `help model' of Ben-Or and Gutfreund ({\em J. Cryptology}, 2003). In this model, the shared reference string is generated by a probabilistic polynomial-time dealer who is given access to the statement to be proven. Our results do not rely on any unproven complexity assumptions and hold for statistical zero knowledge, for computational zero knowledge restricted to AM, and for quantum zero knowledge when the help is a pure quantum state.


text:
Korean Journal of Food Preservation
The Korean Society of Food Preservation
Article

# Measurement of brown eggs freshness using non-destructive UV/VIS spectroscopy and viscosity of egg white

Sanghee Kim1,*
1Division of Mechanical and Electronics Engineering, Hansung University, Seoul 02876, Korea
*Corresponding author. Sanghee Kim, Tel: +82-2-760-4326, E-mail: s-kim@hansung.ac.kr

Citation: Kim S. Measurement of brown eggs freshness using non-destructive UV/VIS spectroscopy and viscosity of egg white. Korean J Food Preserv, 29(3), 353-366 (2022)

Copyright © The Korean Society of Food Preservation. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received: Dec 24, 2021; Revised: Mar 24, 2022; Accepted: Mar 25, 2022

Published Online: Jun 30, 2022

## Abstract

The present research aimed to determine the freshness of brown eggs using non-destructive ultraviolet-visible (UV/VIS) transmission spectroscopy (λ = 350-1020 nm) and viscosity measurement during storage for 7 days. As per human visual perception, the eggshell color varied from light yellow-brown to dark red-brown, distributed near the edge of the spectral locus of orange to a red region in the CIE 1931 chromaticity diagram. Quantitatively characterized luminance (CIE Y) values of the eggshell brightness were 7.77 (darkest) and 49.1 cd/cm2 (brightest). The L* value (brownness) of the eggshell increased while the a* value (redness) decreased. The correlation between luminance and transmittance values was evaluated using a linear regression model which showed a correlation coefficient of 0.92. The transmittance spectra collected daily showed a gradual increase from 36.0% (day 1) to 38.8% (day 3). The slope of this increment became steeper after day 3 and reached 50.8% (day 7). The freshness of the eggs calculated based on the transmittance value of 100% (day 1) showed an inverse relation with the storage time and was reduced by 59% (day 7). Egg white exhibited non-Newtonian shear thinning behavior and showed a reduction in viscosity by 13.7% reduction on day 7 as compared to that on day 1. Our results clearly demonstrated that the freshness of the eggs declined with an increase in storage time. This analytical method of measuring egg freshness has potential applications in designing an automatic egg selection machine, to identify stale eggs from the fresh ones on-site in small and medium egg farms by simply measuring the transmittance of eggshells of varying darkness.

Keywords: freshness of brown eggs; CIE 1931; light transmittance; egg white viscosity

## 1. Introduction

Eggs are an excellent protein sources and are consumed daily worldwide. Several factors, such as egg size, egg shape, eggshell color, and eggshell quality influence consumers’ acceptability and preference. In the egg industry, the internal qualities of eggs such as egg white cleanliness and viscosity, size of the air cell, yolk shape, and yolk strength are considered important parameters, which indicate egg freshness (Mehdizadeh et al., 2014). The freshness of eggs is influenced by several factors, such as temperature, humidity, storage time, and age of the hen. Among these factors, the storage time and temperature play important roles in maintain the freshness of eggs, since the internal quality of eggs starts deteriorating immediately after the eggs are laid (Severa et al., 2010; Singh et al., 2014, Yimenu et al., 2017). Currently, many studies have focused on evaluating the freshness of eggs via non-destructive and non-invasive techniques because the traditional destructive Haugh unit method requires a large number of eggs. The higher the HU value, the better is the egg freshness because higher quality eggs always have thicker egg whites.

The brown color of eggshells is due to the deposition of protoporphyrin, formed due to the degradation of hemoglobin in blood (Yang et al., 2009). This brown pigment is included in the white calcium carbonate shell typically during the last phase of egg shell formation. The intensity of this brown color varies with diet, age and strain of the hens (Alasahan et al., 2016; Ingram et al., 2008; Seo et al., 2010). The brown layer is still calcium based and the eggshell color irreversibly changes upon washing (Leleu et al., 2011).

The difference in darkness intensity is characterized by International Commision on Illumination (CIE) L* (blackness, 0 or whiteness, 100), a* (+: redness or −: greenness), and b* (+: yellowness, −: blueness) values (Baylan et al., 2017; Kim et al., 2015; Luvkanov et al., 2015). Moreover, the brownness of eggshells is quantitatively determined as the brightness value, which is measured as luminance (CIE Y, cd/cm2) (Withouck et al., 2013). CIE is a color matching system which quantitatively links wavelength distribution in the visible spectrum with the colors perceived by human vision. This color matching process provides a mathematical generalization of the parameters recorded as points across the visible spectrum. Detection using an instrument perceives the reflected light at a specific wavelength as numerical values. The color information from the spectral data can be mapped as a flat and tongue-shaped CIE chromaticity diagram, which is an internationally accepted method of color identification. The CIE system characterizes colors by the luminance parameter Y and two color coordinates x and y, which specify the position in the chromaticity diagram.

The non-destructive visible-infrared transmittance spectroscopy is used to estimate the freshness of eggs and the resulting spectral data shows an increased transmittance value as the freshness of eggs decreases with aging (Giunchi et al., 2008; Kemps et al., 2006; Yao et al., 2014). The brightness of the eggshells is an important parameter in optical detection because darker the eggshells, lesser is the amount of light transmitted. Previous research has shown that the transmission of incident light reduces as the intensity of the brown color, i.e., brownness (darkness) increases. In contrast, the brighter eggshells allow a higher degree of light transmission at wavelength λ=625 nm (Lorentec etal., 2019; Shafey et al., 2004). Thus, it is assumed that most of the incident light was absorbed by the darker eggshells, thus reducing transmission of the incident light.

Furthermore, the egg white viscosity (thinning) also changes with age, leading to deterioration in the internal quality of eggs. To note, a longer storage time increases the pH inside the egg and decreases the egg white viscosity. Moreover, aeration through the eggshell also increases the pH level and causes thinning of the gel-like thick albumen inside the egg by disrupting the linkages between ovomucin and lysozyme (Chen et al., 2019; Mehdizadeh et al., 2014). The changes in transmittance and viscosity value of the egg white are often used to determine egg freshness and are known to be directly related to the shelf life of eggs.

The non-destructive spectroscopic method is a promising technique for investigating eggshell quality and freshness. This method provides rapid and accurate results, and thus, is suitable for industrial applications (Loffredi et al., 2021). However, the use of eggshell color variability for real-time detection of the freshness of eggs under different storage conditions is still a concern (Akowuah et al., 2020; Dai et al., 2020; Hansoongnoen et al., 2012).

In the present study, the freshness of eggs was investigated via non-destructive optical detection using ultraviolet-visible (UV/VIS) transmission spectroscopy with eggs collected daily from the farm. The color parameters of diverse brown eggshells were summarized using the CIE 1931 color system, and the correlation between luminance (brightness, CIE Y) and transmittance value was obtained and analyzed using a linear regression model, to minimize the interference of eggshell brightness in the transmittance of the incident light. Lastly, the viscosity of egg white was also measured to determine the freshness of brown eggs during storage and verify the decreased transmittance value. Therefore, the aim of this study was the real-time detection of the freshness of brown eggs having various shell brightness, to separate old eggs from the fresh ones in a farm. This study provides a novel, rapid, and simple method for the non-destructive and on-site testing of egg freshness.

## 2. Materials and methods

2.1. Materials

Fresh brown eggs were obtained from grass-fed Hy-Line hens (<50 week-old) raised in an organic farm (Rfood Agricultural Corporation, Chungbuk, Korea). The farm management provided an additional dietary supplement to the hens, composed of 80% corn, and 20% complex of protein, amino acids and vitamins. After collecting the eggs, they were washed thoroughly using the farm’s automatic washer (Nine Systems, Chungnam, Korea) equipped with a spray nozzle and brushes and dried. Following this, the cracked and abnormally shaped eggs were removed using an automatic crack detector. A total of 1,000 eggs were used to determine the degree of darkness, among which 54 eggs were chosen by random sampling and used to determine the relationship between eggshell brightness (CIE Y) and transmittance value of incident light (T%). For freshness measurement, 70 eggs were collected on the day they were laid and divided into 7 groups (10 eggs/group). The eggs were then stored at 20°C with 55% RH (relative humidity) for 7 days.

2.2. UV-VIS spectroscopy and viscosity measurement

Eggshell color information was collected using UV/VIS spectrophotometer (SE2020-050-VNIR, OTO Photonics, Hsinchu, Taiwan) in the wavelength range of 350-1,020 nm with a resolution of 0.38 nm. The light source was a halogen lamp equipped with an infrared filter (150 Watt, GSS5-1000-F, Fiber Optic, Cheonan, Korea) and connected to an illuminator (FOK-150W, Fiber Optic), which provided continuous light supply during the measurement. The spectrum of light transmitted through the egg white was obtained using this instrument daily for 7 days, to determine the freshness of eggs. The transmittance value was measured daily from eggs in each group (1 group/day) and once an egg was used for analysis, it was discarded. The measurement was repeated during the 7-day cycles. The software SpectraSmart® provided by the manufacturer was used to control the color information and spectral acquisition as shown in Fig. 1. The experiment for freshness measurement was conducted twice with 140 eggs, to obtain statistically significant data. The transmittance value on Day 7 was set as the criteria to identify the stale eggs. To confirm the accuracy of determination, tests were conducted with a total 120 eggs mixed with the already known 15 stale eggs. At the end of the test, all the 15 stale eggs were successfully by measurement with 0% error. To further validate the results, the eggs were cracked and egg white thinning was evaluated. After the spectral assessment, the eggs were carefully broken and the egg white component was separated by removing the yolk and chalaza, for daily viscosity measurement. The egg white viscosity measurement was performed at 25±0.5 °C using the Brookfield viscometer (DV2T, Brookfield Engineering, Middleboro, USA) with spindle SC4-18 at 60 rpm at a shear rate of 79.2/s. The measurements were conducted for 30 s at intervals of 5 s. In addition, shear rate-dependent changes in viscosity were obtained for different shear rates from 10 to 100/s.

Fig. 1. Experimental setup used for the measurement of eggshell color and egg white transmission spectra.
2.3. CIE 1931 color parameters

The eggshell CIE 1931 color parameters were obtained via UV/VIS spectrophotometer in transmittance mode with a standard light source and viewing angles. The two-chromaticity coordinates (x, y) and luminance CIE Y values were obtained from 54 eggs under illuminant A at a specific viewing angle of 2°. The tristimulus values of XYZ were calculated using x, y coordinates and the CIE Y values from the following equation (Westland, 2003):

(1)

Diversity in the eggshell darkness was measured, and the color components of the brown eggshells (L*, a*, b*) were calculated to correct certain experimental errors due to diverse eggshell surface darkness. The brightness of each brown eggshell was calculated in terms of CIE L*, a*, b* coordinates from equation (2) (Westland, 2003):

$\begin{array}{c}{L}^{*}=116{\left(\frac{Y}{{Y}_{n}}\right)}^{1/3}-16\\ {a}^{*}=500{\left(\frac{X}{{X}_{n}}\right)}^{1/3}-{\left(\frac{Y}{{Y}_{n}}\right)}^{1/3}\\ {b}^{*}=200{\left(\frac{Y}{{Y}_{n}}\right)}^{1/3}-{\left(\frac{z}{{z}_{n}}\right)}^{1/3}\end{array}$
(2)

where Xn, Yn, Zn are the tristimulus values, viz. 109.85, 100.00, and 35.58, respectively, of a specific white color stimulus of the illuminant A. In addition, the color difference value (ΔE) was calculated from equation (3) (Eleroğlu et al., 2016):

$\Delta E={\left({L}^{2}+{a}^{2}+{b}^{2}\right)}^{1/2}$
(3)

The obtained spectral color parameters of CIE XYZ under illuminant A were converted to the CIE standard illumination D65 through a mathematical transformation using the Bradford matrix as shown below (Pascale, 2003), to map the coordinates in the CIE 1931 chromaticity diagram:

${\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]}_{D65}=\left[\begin{array}{rrr}\hfill 0.8477& \hfill -0.1179& \hfill 0.3948\\ \hfill -0.1367& \hfill 1.1042& \hfill 0.1292\\ \hfill 0.0799& \hfill -0.1349& \hfill 3.1924\end{array}\right]\cdot {\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]}_{A}$
(4)

The seven stimulus colors (yellow, orange, red, aqua, pink, purple, and blue) were presented in the CIE 1931 chromaticity diagram to compare the physiologically perceived brown eggshell color via human vision with quantitative description of each color. The CIE xy coordinates of the stimulus color were adapted from McCamy et al. (1976). The CIE xyY values (illuminant C) were converted to those of illuminant D65 by mathematical conversion using Bradford matrix to compare the parameters reported by Xiao et al. (2011) and showed close approximation.

${\left[\begin{array}{l}X\hfill \\ Y\hfill \\ Z\hfill \end{array}\right]}_{D65}=\left[\begin{array}{rrr}\hfill 0.9904& \hfill -0.0072& \hfill -0.0116\\ \hfill -0.0124& \hfill 1.0156& \hfill -0.0029\\ \hfill -0.0036& \hfill 0.0068& \hfill 0.9182\end{array}\right]\cdot {\left[\begin{array}{l}X\hfill \\ Y\hfill \\ Z\hfill \end{array}\right]}_{A}$
(5)
2.4. Statistical analysis

The CIE color data were collected from 54 eggs chosen by random sampling. The mean and the corresponding dispersion of brownness of eggshells (standard deviation) were evaluated. The measurement precision was determined as coefficient of variation (CV) using Microsoft Excel. The data showing a correlation between transmittance and viscosity values as of storage period represented mean± standard error and were analyzed on Microsoft Excel using one-way analysis variance (ANOVA) with a 95% confidence level to estimate the significance of the results.

## 3. Results and discussion

3.1. Eggshell color characterization

The freshness of brown eggs during storage was evaluated by the non-destructive UV/VIS spectroscopic technique. In optical detection, the eggshell brightness was considered as an important parameter because darker eggshells allowed lesser transmission of the incident light than the brighter ones did. Thus, the brightness of the eggshell color was investigated prior to spectroscopic measurements. The brown eggs collected in the current study showed diverse eggshell brightness and the color ranged from light yellow-brown (brightest, bottom) to dark red-brown (darkest, top) in human visual perception (Fig. 2(A), inset). The CIE 1931 (x, y) coordinates of the brightest (0.51, 0.42) and the darkest (0.64, 0.35) eggshell color were obtained through spectral assessment. These coordinates were equivalent to the stimuli color positions of orange (0.54, 0.40) and red (0.55, 0.33), respectively, as previously reported by Xiao et al. (2011). The (x, y) coordinates were then mapped in a 2D CIE 1931 chromaticity diagram and a cluster was built near the border of the spectral locus. This clustered region was assigned to orange and red spectral regions which were characterized by distinctive stimuli colors in the CIE 1931 chromaticity diagram (Fig. 2(A)). The darkness of the eggshells was measured and obtained as quantitative luminance (CIE Y, cd/cm2), a photometric value, signifying the luminous flux of perceived power of light per unit projected area (Withouck et al., 2013). The luminance values of the 54 eggs ranged from 7.77 cd/cm2 (dark) to 49.01 cd/cm2 (light) (Table 1). A higher luminance value indicated a brighter eggshell, and the CIE coordinate (x, y) tended to shift towards the orange stimulus region in the chromaticity diagram (Fig. 2(B)).

Fig. 2. Color coordinate mapping of eggshell color and quantification of brightness. (A) CIE 1931 chromaticity coordinates of shell color from 54 brown eggs with visual stimuli: yellow (Y), red (R), orange (O), pink (P), purple (P), aqua (A) and blue (B). Inset: image of the brightest (bottom) and darkest (top) brown eggs with color scale. (B) Distribution of measured luminance values (CIE Y) of eggshells from 54 brown eggs according to CIE 1931 chromaticity coordinates.
Table 1. Measured and calculated values of CIE color parameters from brown eggs
Parameters Minimum Maximum Mean SD (standard deviation) CV (coefficient of variation)
Y*1) (cd/cm2) 7.77 49.01 25.01 9.21 0.37
L*2) 33.49 75.46 55.87 9.41 0.177
a*3) 19.47 49.30 31.74 7.93 0.249
b*4) 46.39 97.95 66.19 10.21 0.162
E*5) 71.10 130.51 92.86 12.07 0.131

Y*, luminance value (0=dark, 100=light).

L*, egg’s lightness (0=dark, 100=white).

a*, egg’s color tone as a function of a red-green scale (<0: green, >0: red).

b*, egg’s color tone as a function of a blue-yellow scale (<0: blue, >0: yellow).

E*, color difference value (lower number is dark).

In addition, the different degrees of brownness of the eggshells were further measured quantitatively as changes in the L*, a*, b* values. The characterized color information of the eggshell in the form of CIE L*, a*, b* and color difference (ΔE) are summarized in Table 1. As mentioned earlier, the eggshell color was light yellow-brown to dark red-brown and these colors were quantitatively characterized using the CIE color system. Brown is mixture of three primary colors (red, yellow, and blue) and the range of brownness is affected by the ratio of these three components. These three primary colors were expressed as CIE a* and b* values and their quantitative relation generated different degrees of color. In addition, the degree of brightness was represented as L* value from the darkest (0) to the brightest (100). The gradient of brownness of the eggshells was inversely associated with the L* and a* value. The ratio between the red and green component was displayed a* value and a shift towards a higher a* value was represented as deep red.

The spectral measurements and analysis results showed that the eggshell lightness, L* value was 33.49 (minimum, dark) and 75.46 (maximum, light). The range of L* value indicated that brown eggshells exhibited medium brightness. The decrease in the L* value for the dark brown eggshells has been directly related to increase in the a* value, which represented redness. The light yellow-brown color is the combined brighter and greenish property, represented as large L* and small a* value.

In contrast, dark red-brown color indicates dark and reddish, resulting in a small L* and large a* value.

Experimental results clearly demonstrated that an increase in the L* value correlated with a decrease in eggshell redness, i.e., the a* value (Fig. 3(A)). The largest a* value (49.30) was observed when the L* value was the lowest (33.49). The smallest a* value (19.47) was linked with the highest L* value (75.46). The minimum and maximum eggshell color tone (b* values) were 46.39 and 97.95, respectively. The eggshell color showed diverse brownness ranging from light yellow-brown (brightest) to dark red-brown (darkest) under human visual perception and spectrophotometric quantitation displayed as changes in the L*, a*, b* values. Results displayed relatively higher a* and b* values than those reported previously (Eleroğlu et al., 2016; Li et al., 2016; Seo et al., 2010). These values possibly resulted from the different strains of egg-laying hens, and the corresponding dietary condition (Ingram et al., 2008; Seo et al., 2010). Seo et al. (2010) showed that lightness (L*) decreased from 52.2 to 49.6, while the redness (a*) increased from 14.7 to 15.7 significantly when Fe-soy protein (Fe-SP) supplementary treatment (Fe-SP 100 and Fe-SP 100+MgO) was provided to the Hy-Line brown egg-laying hens. In the present study, the measured L* and a* values for the grass-fed, egg-laying hens were higher than those reported by Seo et al. (2010). Thus, different dietary conditions induced varied eggshell colors although the eggs were collected from the same strain of Hy-Line hens.

Fig. 3. Characteristics of eggshell color components and brightness. (A) Scatter plot representation of CIE L* and a* values showing the changes in brightness of brown eggshell color tones as a function of red-green component. [L*: 0=black, 100=white, a*: lower numbers=more green (less red)]. (B) Scatter plot representing the relation between CIE L* and the calculated ΔE values of the brown eggshells (L*: 0=black, 100=white, ΔE: lower number=dark). Scale bar of different colors represents the range of ΔE values.

The color difference (ΔE) values were calculated based on the obtained CIE L*, a*, and b* values. The ΔE value utilizes a generally known linear formula to calculate the distance between two color points and a lower ΔE value indicates a darker color. In Fig. 3(B), the ΔE values showed a linear increased with the L* values, representing brightness. The darkest eggshell showed the smallest ΔE (71.10) and L* (33.49) values and increased gradually until it reached the maximum value of ΔE (130.51) and L* (75.46). In the plot, most of the ΔE values were scattered between 80 and 110, signifying intermediate to a lighter level of brownness of the eggshells because the darker eggshells should exhibit lower ΔE values (Eleroğlu et al., 2016).

3.2. Spectral analysis of brown egg freshness

The internal quality of the brown eggs in terms of altered freshness was measured spectrophotometrically and the maximum absorbance (λmax) was observed at 622 nm (Fig. 4(A)). The optical measurement of the transmittance value of egg white was aimed to determine the actual amount of light passing through ignoring the effect of different eggshell darkness. As stated above, the intensity of brownness (brightness) of the eggshells was diverse in range. The correlation between the transmittance and luminance values (CIE Y) was determined prior to the transmittance values of the eggs was quantified because the brighter eggshells allowed more transmission of the incident light. The relationship between transmittance and luminance (CIE Y) showed a correlation coefficient of 0.92 in linear regression (Fig. 4(B)). The actual quantity of the incident light transmitted through the eggs was obtained from the following equation (6):

${Y}_{T}\text{=1}\text{.3957}x\text{+9}\text{.7912}$
(6)
Fig. 4. Spectra of brown eggs and eggshell brightness effect on light transmittance value. (A) Transmittance spectrum of brown eggshells with different brightness. The main peak position was measured at λmax = 622 nm. (B) Scatter plot represents the correlation between transmittance values of the brown eggs and luminance values (CIE Y) of their shell color.

where, YT = transmittance of incident light passing through an egg and x = luminance (CIE Y) of the eggshell. The variation in the transmittance values with respect to storage time was further measured to investigate the freshness of the eggs up to 7 days from laying. The obtained spectrum of the transmittance values from the brown eggs (λmax = 622 nm) were corrected using the linear regression model (Equation 6) to adjust the brightness of each eggshell. The longer the storage time, the greater the transmittance value, due to decreased viscosity of the liquid-like egg white protein inside the egg. The transmittance increased gradually from 36.0% (day 1) to 38.8% (day 3), followed by a much steeper slope after day 3, and finally reached 50.8% on Day 7 (Fig. 5(A)). The increased transmittance value is related to a loose egg white protein network allowing more incident light than that with a dense structure. Additionally, the egg white freshness, calculated based on the transmittance value of day 1 as 100%, showed a negative relation with the storage time. Results showed that the freshness of the eggs decreased gradually to 59% on day 7 (Fig. 5(B)). The relation between the freshness value of the eggs (YF) and the storage time (x) can be represented as follows (correlation coefficient of 0.8433):

${Y}_{F}=-23.86Ln\left(x\right)+98.554$
(7)
Fig. 5. Relation between spectral analysis results and freshness of eggs. (A) Plot showing inverse relation between transmittance and viscosity with increase in storage period (day). The increase in transmittance and decrease in viscosity during storage period was significant (p<0.05). (B) Decreased freshness of brown eggs with increase in the storage time (day). Error bars represent mean values±standard error.

where, YF = freshness of egg followed by normalized transmittance value (%), x = storage time (day). The transmittance values from different storage days were normalized using the day 1 value as the standard. The difference in the transmittance value with respect to day 1 was directly proportional to the storage time. Therefore, the increased quantity of incident light passing through the egg white reflected that the eggs became aged and thus their freshness declined. Spectrophotometric measurement of the internal quality of the brown eggs in terms of altered freshness showed the main peak position at 622 nm (λmax=622 nm), as reported previously (Lorentec et al., 2019). The actual amount of incident light passing through the eggshells was measured by establishing the linear regression model between the luminance (CIE Y) and transmittance values. After correcting for the brownness of the eggshells, the actual light transmitted through the egg white was measured and showed a significant increase for a longer storage period (Fig. 5(A)). This could be attribute to increased passage of light through a loose and less dense egg white protein network. It is known that decreased CO2 concentration during storage induces a shift in egg white pH towards alkalinity (maximum - 9.5) (Yao et al., 2014). This increase in pH results in the gel-like thick egg white albumin losing its viscosity (egg white thinning) caused by partial disruption of the egg white protein network which allows more light to pass through.

3.3. Viscosity measurement

The effect of storage time on the viscosity of egg white is shown in Fig. 6(A). At a constant shear rate, the viscosity of the egg white decreased with increasing storage time. The measured viscosity of egg white was 4.95 centipoise (cp) on day 1 and gradually decreased to 4.69 cp on day 5. The viscosity value decreased more rapidly after day 5 and reached 4.28 cp on Day 7. Egg white is an aqueous solution with a gel-like structure and exhibits a non-Newtonian, pseudoplastic flow behavior which decreases the viscosity with increasing shear rate and time. The egg white viscosity was determined as a function of shear rate (10 to 100/s) and time (up to 30 s). Results clearly demonstrated that the shear-thinning behavior of egg white led to a gradual decrease in viscosity with increasing shear rate (Chen et al., 2019). Initially, the viscosity values for short (20 h) and long storage time (80 h) were 16.4 cp and 15.2 cp, respectively, and the difference in viscosity was maintained up to a shear rate of 80/s (Fig. 6(A)). The two plots showed gaps which indicated different viscosity values during gradual decrease until they merged after a shear rate of 80/s. Such shear-thinning behavior can originate from the breakdown of linkages between protein molecules. Further shearing would stabilize the internal protein structure by completing rearrangement and forming a loose structure. Egg white samples from different storage periods exhibited a time-dependent non-Newtonian thixotropic behavior showing a decrease in viscosity with respect to shearing time at a fixed centrifugation speed (Fig. 6(B)). Viscosity showed a time-dependent decrease, signifying thixotropic behavior (Atilgan and Unluturk, 2008; Singh et al., 2014).The rapid decrease in viscosity at the beginning of shearing however became constant after 15 s. This can be explained as the shear-induced rearrangement of the internal structure towards the direction of shear causing a decrease in flow resistance. The storage time is closely related to changes in the egg white viscosity. Our results also confirmed a significant reduction in viscosity with increased storage time at a constant shear rate, as reported earlier (Severa et al., 2010; Singh et al., 2014). The decrease in egg white viscosity with an increase in storage time reflects the disruption of linkages between protein molecules, which unwind and become relatively flexible. As eggs age, water gradually evaporates through the eggshell and loose inter-molecular linkages rupture, thereby causing a decrease in flow resistance by the progressive liquefaction and thinning of the egg white component (Cancer and Yüceer, 2015). In some portions, the loose inter-molecular linkages collapse and rearrange towards the shear direction (Chen et al., 2019; Spada et al., 2012). As a consequence, the loose and partially disrupted protein network in aged egg white facilitates the transmittance of incident light through the eggs. A prior study also reported a negative linear correlation between the freshness of eggs and the storage time (Liu et al., 2007). Our results prove the inverse relation between viscosity and transmittance value with increase in storage time and provide evidence that the freshness of brown eggs decreases within the storage time up to 7 days.

Fig. 6. Shear thinning behavior of egg white. (A) Non-Newtonian behavior of egg white under different shear rates. The decrease in viscosity value was apparent as shear rate increased. (B) Shear time dependent decrease in viscosity value demonstrate the thixotropy of egg white. Error bars represent means±standard error (n=3).

## 4. Conclusions

In the current study, non-destructive transmittance detection, using the direct radiation of incident light through the short axis of the eggshell surface, and the viscosity value of egg white were used to evaluate the change in egg freshness during 7 days. Our results prove the inverse relation between viscosity and transmittance value with increase in storage time and provide evidence that the freshness of brown eggs decreases within the storage time up to 7 days. This novel optical-based technique to measure egg freshness has potential applications in designing an automatic egg selection machine, which can separate stale eggs from the fresh ones on-site in small and medium egg farms by simply measuring the transmittance of eggshells of varying darkness.

## Acknowledgements

The author is thankful to Dr. Lord and Mr. Mario for preparing the experimental setup and manuscript. This work was financially supported by Hansung University.

## Conflict of interests

The authors declare no potential conflicts of interest.

## Author contributions

Conceptualization; Data creation; Formal analysis; Methodology; Validation; Writing: Kim S.

## Ethics approval

This article does not require IRB/IACUC approval because there are no human and animal participants.

## ORCID

ORCID

Sanghee Kim (First & Corresponding author) https://orcid.org/0000-0001-6208-8931

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Seo YM, Shin KS, Rhee AR, Chi YS, Han J, Paik IK. Effects of dietary Fe-soy proteinate and MgO on egg production and quality of eggshell in laying hens. Asian-Australas J Anim Sci, 23, 1043-1048 (2010)

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Severa L, Nedomovas, Buchar J. Influence of storing time and temperature on the viscosity of an egg yolk. J Food Eng, 96, 266-269 (2010)

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Shafey TM, Ghannam MM, Al-Batshan HA, Al-Ayed MS. Effect of pigment intensity and region of eggshell on the spectral transmission of light that passes the eggshell of chickens. Int J Poult Sci, 3, 228-233 (2004)

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Singh J, Sharma HK, Premi M, Kumari K. Effect of storage conditions of egg on rheological properties of liquid whole egg. J Food Sci Technol, 51,543-550 (2014)

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Spada FP, Gutierrez EMR, de Souza MC, Brazaca SGC, Leme DEA, Fischer FS, Coelho AAD, Savino VJM. Viscosity of egg white from hens of different strains fed with commercial and natural additives. Food Sci Technol, 32, 47-51 (2012)

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, ,


text:
Parameters of comparison of the first sort (through a gap)
 The set equality Will be true if x =

In number theory, which deals with the study of integer values, there is another problem that we will try to solve.

$(A*x)mod(B)=C$

if we know A, B, C then for what value of x this equality will be true?

As an example

$(58*x)mod(47)=87$

The solution of such problems is inextricably linked with the Euler function  . Although of course there is an alternative method of solution (according to Euclid), but we will not consider it.

How to solve such equations. Recall that, according to the Fermat-Euler formula, there is the following dependence. If a and m are coprime numbers (i.e. not having common divisors), then

$a^{\phi(m)}mod(m)=1$

Given that the Euler function of a prime m is m-1, we get the famous formula for any prime

$a^{m-1}mod(m)=1$

where, as already mentioned, a must be coprime with m.

Euler’s method for solving similar comparisons in formulas looks like this

$a^{\phi(m)}mod(m)=1 =>a*a^{\phi(m)-1}mod(m)=1$

$a(a^{\phi(m)-1}b)mod(m)=b$

Then, solving the equation $(a*x)mod(m)=b$

find out what is x

$x=ba^{\phi(m)-1}$

Let's try to solve our first example.

$(58*x)mod(47)=87$

$x=87*58^{\phi(m)-1}$

the Euler function for 47 is 46

and the final formula is equal $x=87*58^{46-1}$

If you count "looser" you get a huge number, but we need to find out only $xmod(m)=>xmod(47)$

To solve such a problem, we will use the material The number rest in degree on the module and find out that our solution

equally $x=87*30mod(47)=25$

check by substituting the resulting value

$\frac{(58*25-87)}{(47)}$

completely divided, which means our decision is right.

As you can see, we can also solve a similar problem along the way, which is called the inverse value modulo the residue class and which is expressed by the formula

$(A*x)mod(B)=1$

Good luck!

Copyright © 2020 AbakBot-online calculators. All Right Reserved. Author by Dmitry Varlamov


text:
# Maths Formulas for Class 10 PDF Chapterwise

Here you will find all the maths formulas for class 10. You can also download these formulas of class 10 in a PDF file and take the printout for fast practice. There are fifteen chapters in NCERT class 10 Maths book. We have provided all the formulas for these chapters, including Real Numbers, Polynomials, Linear Equations, Arithmetic Progressions, Quadratic Equations, Trigonometry, Area & Volume, and other important formulas.

Maths formulas for class 10 are the basic formulas that you will need to prepare for the mathematics exam and improve your mathematical skills for higher classes. These basic maths formulas are crucial for every student. We suggest you note down all the important maths formulas for class 10 or take the printout to save your time while studying.

Page Contents

## Real Numbers

Real number maths formulas include important theorems, algorithms, and Euclid’s division lemma to learn more about real numbers.

### Euclid’s Division Lemma

Theorem: For given positive integers a and b, there exist unique integers q and r satisfying,

a = bq + r, 0 ≤ r < b.

• Algorithm: Series of well-defined steps which gives a procedure for solving a type of problem
• Lemma: Proven statement used for proving another statement

### HCF using Euclid’s Division Algorithm

Finding HCF of two number c and d where c > d

• Step 1: Apply division lemma to get the whole numbers (q and r) such that, c=dq+r, 0≤r
• Step 2: If r=0, d is the HCF of c and d. If r ≠0, apply division lemma to d and r.
• Step 3: Continue the process till the remainder is zero. The divisor at the final stage will be HFC.

### Theorem of Arithmetic

Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

• The prime factorisation of a natural number is unique, except for the order of its factors.
• For any two positive number p and q, HCF (p,q) x LCM (p,q) = p x q

### Rational Number Theorems

• If p is a prime number and a is a positive integer, then if p divides a2, the p also divides a.
• √2 is irrational
• If x is a rational number whose decimal expansion terminates, then x can be expressed in the form of p/q, where p and q are coprime, and the prime factorisation of q is in the form 2n5m, when n,m are non-negative integers.

LCM (p,q,r) = p.q.r HCF(p,q,r) / HCF(p,q) . HCF(q,r) . HCF(p,r)

HCF (p,q,r) = p.q.r LCM(p,q,r) / LCM(p,q) . LCM(q,r) . LCM(p,r)

## Polynomials

### Degree of polynomials

Highest power of x in a polynomial p(x) is the degree of polynomial.

• Linear polynomials: Polynomials of degree 1, i.e., 2x-3
• Quadric Polynomials: Polynomials of degree 2, i.e., 3x2-4x+7
• Cubic Polynomials: Polynomials of degree 3, i.e., 4x3+3x2+2x+9

If k is zero of polynomial p(x) = 2x-3, then p(k) = 2x-3 = 0.

Zero of the lineal polynomial ax+b is -b/a = -(constant term)/coefficient of x

• A quadratic polynomial in x with real coefficients is of the form ax2+ bx + c, where a, b, c are real numbers with a ≠ 0.
• The zeroes of a polynomial p(x) are precisely the x-coordinates of the points, where the graph of y = p(x) intersects the x-axis.
• A quadratic polynomial can have at most 2 zeroes and a cubic polynomial can have at most 3 zeroes.

If α and β are the zeroes of the quadratic polynomial ax2 + bx + c, then,

If α, β, γ are the zeroes of the cubic polynomial ax3+ bx2+ cx + d, then

### Division Algorithm:

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that,

p(x) = g(x) × q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).

## Pair of Linear Equations in two Variables

Here you will find maths formulas for class 10 for chapter 3. Download all the formulas of class 10 in PDF file or read online.

### Linear Equations:

An equation between variables that produce a straight line when plotted on a graph is called linear equation.

### Pair of linear equations:

Two linear equations in the same two variables are called a pair of linear equations in two variables.

### Graphical Method of solving pair of linear equations

It is represented by two lines on the graph.

• Consistent pair of equations: If the lines intersect at a point, then that point gives the unique solution of the two equations.
• Dependent (consistent) Pair: If the lines coincide, then there are infinitely many solutions – each point on the line being a solution.
• Inconsistent Pair: If the lines are parallel, then the pair of equations has no solution.

### Algebraic Method

• Substitution Method:
• Find the value of one variable, say y in terms of the other variable, i.e., x from either equation, whichever is convenient.
• Substitute this value of y in the other equation, and reduce it to an equation in one variable, i.e., in terms of x, which can be solved.
• If this statement is true, you can conclude that the pair of linear equations has infinitely many solutions. If the statement is false, then the pair of linear equations is inconsistent.
• Substitute the value of x (or y) obtained in Step 2 in the equation used in Step 1 to obtain the value of the other variable.
• Elimination Method
• Multiply both the equations by a number(non-zero) to make the coefficient of one variable equal. Let us say is y.
• Subtract the equation to eliminate variable y, and find the value of x
• Substitute the value of x in one of the equations to get the value of y.
• Cross-Multiplication Method
• Multiply Equation (1) by b2 and Equation (2) by b1, to get new equations.
• b2a1x + b2b1y + b2c1 = 0                                   eq. 3
• b1a2x + b1b2y + b1c2= 0                                    eq. 4
• Subtract eq.4 from eq. 3 to get
• (b2a1– b1a2) x + (b2b1– b1b2)y + (b2c1– b1c2) = 0
• Find the value of x from there and put it in eq. 1 or eq. 2 to find value of y.

## Quadratic Equations Formulas for Class 10

Solve quadratic equation problems by using Quadratic Equations Maths Formulas for class 10. Learn about finding solutions and roots of the equations.

Quadratic equation in the form of variable x is of the form ax2+bx+c=0, where a,b,c are real numbers and a≠0.

### Solution of Quadratic Equation by Factorisation

• For any quadratic equation, split the middle term and factorize the equation into two linear factors
• Equate each factor to zero

e.g., 2x2-5x+3=0 can be factorize to linear terms as 2x2-2x-3x+3=0 or 2x(x-1)-3(x-1)=0

Thus, (2x-3)(x-1)=0 or x=1,3/2. Hence, 1 and ½ are roots of the equation.

• a real number α is called a root of the quadratic equation ax2+bx+c =0, a ≠ 0 if aα2+ bα + c =0.

For an equation ax2+bx+c=0,

• Two distinct real roots: if b2– 4ac > 0,
• Two equal roots: if b2– 4ac=0
• No real roots: if b2– 4ac <0

## Algebraic Maths Formulas For Class 10

Algebraic maths formulas for class 10 are helpful to solve quadratic equations and cubical equations easily. You have already learned about some of these formulas in previous classes.

## Arithmetic Progression

list of numbers in which each term is obtained by adding a fixed number to the preceding term except the first term.

e.g., a1, a2, a3, . . ., aor 2,4,6,8,….., ∞

• Term: Every number in the list is a term
• Common Difference: The fixed number which is added to the terms to make the list. It can either be positive, negative or zero.
• Common Difference (d) = an – an-1

#### General Form of AP

a, a+d, a+2d, a+3d,…..

where d is the common difference.

• Finite AP: AP having limited number of terms
• Infinite AP: AP having unlimited number of terms

#### nth Term of AP

an = a+(n-1)d, where a = first term, d = common difference, n = number of term

e.g., finding 10th terms will be, a10 = a+(10-1)d

#### Sum of n Terms of AP

S = n/2 [2a+(n-1)d] or     S = n/2 (a+an) or, S = n/2(a+l), where l is the last term.

Sum of first n Positive Integers, Sn = n(n+1)/2

## Coordinate Geometry

Distance between two points P(x1,y1) and Q(x2,y2) is,

$$\sqrt{\left( x_{2} – x_{1} \right)^{2} + \left( y_{2} – y_{1} \right)^{2}}$$

Distance of P(x,y) from Origin O(0,0) is,

$$\sqrt{x^{2} + y^{2}}$$

The coordinates of the point P(x, y) which divides the line segment joining the points A(x1,y1) and B(x2, y2) internally in the ratio m1 : m2 are

and x = $$\frac{m_{1}x_{2} + m_{2}x_{1}}{m_{1} + m_{2}}$$ , y = $$\frac{m_{1}y_{2} + m_{2}y_{1}}{m_{1} + m_{2}}$$

The mid-point of the line segment joining the points P(x1, y1) and Q(x2, y2) is

The area of the triangle formed by the points (x1, y1), (x2, y2) and (x3, y3) is the numerical value of the expression

## Trigonometry Formulas for Class 10

This section includes all maths formulas for class 10 trigonometry. Studying trigonometry in class 10 is really necessary as it will get more complex in upcoming classes (class 11 and class 12). Make sure you practice well with all trigonometry formulas for class 10.

For a right angled triangle ABC, which is right angled at B. and have ∠A = ∠θ

Sin θ = $$\frac{\text{Perpendicular}}{\text{Hypotenus}}$$ = $$\frac{\text{Side opposite to angle }\theta\ }{\text{Hypotenus}}$$ = $$\frac{P\ }{H}$$
Cos θ = $$\frac{\text{Base}}{\text{Hypotenus}}$$ = $$\frac{\text{Side }\text{adjacent}\text{ to angle }\theta\ }{\text{Hypotenus}}$$ = $$\frac{B\ }{H}$$
Cot θ = $$\frac{\text{Base}}{\text{Perpendicular}}\$$=$$\ \frac{\text{Side adjacent to angle}\ \theta}{\text{Side opposite to angle}\ \theta}$$ = $$\frac{B\ }{P}$$ = $$\frac{1\ }{\tan\ \theta}$$ = $$\frac{\cos\ \theta\ }{\sin\ \theta}$$
Tan θ = $$\frac{\text{Perpendicular}}{\text{Base}}\$$=$$\ \frac{\text{Side }\text{Opposite}\text{ to angle }\theta}{\text{Side }\text{Adjacent}\text{ to angle }\theta}$$ = $$\frac{P\ }{B}$$ = $$\frac{1\ }{\cot\ \theta}$$ = $$\frac{\text{sing}\ \theta\ }{\cos\ \theta}$$
Sec θ = $$\frac{\text{Hypotenus}}{\text{Base}}$$ = $$\frac{\text{Hypotenus}\ }{S\text{ide }\text{adjacent}\text{ to angle }\theta}$$ = $$\frac{H\ }{B}$$ = $$\frac{1\ }{\text{co}s\ \theta}$$
Cosec θ = $$\frac{\text{Hypotenus}}{\text{Perpendicular}}$$ = $$\frac{\text{Hypotenus }}{\text{Side opposite to angle }\theta}$$ = $$\frac{\text{H }}{P}$$ = $$\frac{1\ }{\sin\ \theta}$$

### Other Trigonometry Formulas and Identities

• Line of Sight: the line drawn from the eye of an observer to the point in the object viewed by the observer.
• Angle of Elevation: angle formed by the line of sight with the horizontal when it is above the horizontal level. (e.g., when we raise our head to look any object)
• Angle of Depression: the angle formed by the line of sight with the horizontal when it is below the horizontal level. (e.g., when we look down at any object)

## Circles Formulas of Class 10

• Area of a circle = πr2
• Circumference (perimeter) of the circle = πD or 2πr
• Radius   = D/2, where D is the diameter
• Area of a sector of a circle with radius r and angle θ = (θ/360) x πr2
• Length of an arc formed by the sector of angle θ = (θ/360) x 2πr
• Area of a segment of a circle = Area of the corresponding sector – Area of the corresponding triangle.

π = 22/7 or 3.14159265

• The tangent to a circle is perpendicular to the radius through the point of contact.
• The lengths of the two tangents from an external point to a circle are equal.

## Surface Areas and Volumes formulas

You have already studied about surface areas and volumes in previous classes. These problems get a bit complex here in class 10. Download all maths formulas for class 10 for this chapter. It would be best if you learn these formulas as soon as possible so that you can solve the problems easily in no-time.

### Cylinder

r = radius of circular base, h = height of cylinder

### Cone Formula

r = radius of circular base, l = slant height

### Sphere Formulas

r = radius, Semi-sphere = half of sphere

### Formulas for Cuboid

l = length, b = breadth (width), h = height

### Cube Formulas

Length = breadth = height = a

## Statistics formulas for Class 10

### Formulas to find mean of grouped data

#### Direct Method of finding mean of grouped data

$\overline{x} = \ \frac{f_{1}x_{1} + f_{2}x_{2} + \ldots\ + f_{n}x_{n}}{f_{1} + f_{2} + \ldots\ + f_{n}}$

Or we can also write it as,

$$\overline{x} = \ \frac{\sum_{i = 1}^{n}{f_{i}x_{i}}}{\sum_{i = 1}^{n}f_{i}}$$ or $$\overline{x} = \ \frac{\Sigma f_{i}x_{i}}{\Sigma f_{i}}$$ , where i varies from 1 to n

#### Assumed mean method

$\overline{x} = \ a + \ \frac{\Sigma f_{i}d_{i}}{\Sigma f_{i}}$

#### Step-deviation method

$\overline{x} = \ a + \left( \frac{\Sigma f_{i}u_{i}}{\Sigma f_{i}} \right)\text{x h}$

### Mode of grouped data

Mode = $$l + \ \frac{f_{1} – f_{0}}{2f_{1} + f_{0} – f_{2}}$$

### Median of Grouped data

Median = $$l + \ \left( \frac{\frac{n}{2} – cf}{f} \right)\text{x h}$$

## Probability Formulas for Class 10

You are studying probability since class 7. Here you will find more probability maths formulas for class 10. Make sure to practice these questions well.

### Experimental Probability

P(E) = $$\frac{\text{Number of trials in which the event happened}}{\text{Total Number of trials}}$$

### Theoretical Probability

P(E) = $$\frac{\text{Number of outcomes favourable to E}}{\text{Number of all possible outcomes of the experiment}}$$

Where, E is an event.

For an event E,

P ($$\overline{E}$$) = 1 — P(E), where $$\overline{E}$$ denotes “not E” or the complement of event E.

Important Points:

• The sum of the probabilities of all the elementary events of an experiment is 1
• The probability of any impossible event is 0
• Probability of an event that is sure to occur is 1. Such event is called a sure event or certain event.
• A playing die has six outcomes (1,2,3,4,5,6)

### Playing Cards – total 52 in sets of four

Those were all of the maths formulas for class 10. You can download the maths formulas PDF by clicking on the download link shared below. We will soon upload more study materials including notes and solutions. Till then practice the maths problems and memorize the formulas quickly.

## Artificial Intelligence Learning at Vector Tutorials

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Summary

The physics of a tree blowing in the wind is very difficult to model because the system is highly nonlinear. I made a numerical model of the system by expressing it as a fractal.

Max wind speed: 0.50
Wavelength (time steps per cycle): 350
Rotational k: .05

Fractal Tree

Use the sliders to change the model's parameters and look for different types of resonance. You'll find that the tree quickly stops acting like a tree but does show several interesting types of chaotic and periodic motion.

I made an ipython notebook to analyze the motion of this tree more rigorously for my applied math course. We also made a poster.

Each circle represents a base unit of a fractal. The lines connecting these circles have restoring forces both in the angular and lengthwise directions. For each time step, the fractal uses a recursive tick() function to update physics at every node and a recursive draw() function to re-render the fractal. The wind's force is given by a force object.