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# Quantum surface-response of metals revealed by acoustic graphene plasmons

### Idea

We think about a GDM heterostructure (see Fig. 1) composed of a graphene sheet with a floor conductivity σ ≡ σ(q,ω) separated from a metallic substrate by a skinny dielectric slab of thickness t and relative permittivity ϵ2 ≡ ϵ2(ω); lastly, the system is roofed by a superstrate of relative permittivity ϵ1 ≡ ϵ1(ω). Whereas the metallic substrate could, in precept, be represented by a nonlocal and spatially non-uniform (close to the interface) dielectric perform, right here we summary its contributions into two elements: a bulk, native contribution through ({epsilon }_{textual content{m}}equiv {epsilon }_{textual content{m}}(omega )={epsilon }_{infty }(omega )-{omega }_{textual content{p}}^{2}/({omega }^{2}+textual content{i}omega {gamma }_{textual content{m}})), and a floor, quantum contribution included via the d-parameters. These parameters are quantum-mechanical surface-response capabilities, outlined by the primary moments of the microscopic induced cost (d) and of the conventional by-product of the tangential present (d); see Fig. 1 (Supplementary Be aware 1 offers a concise introduction). They permit the leading-order corrections to classicality to be conveniently integrated through a floor dipole density ( d) and a floor present density ( d)9,15,16, and may be obtained both by first-principles computation20,21, semiclassical fashions, or experiments15.

The electromagnetic excitations of any system may be obtained by analyzing the poles of the (composite) system’s scattering coefficients. For the AGPs of a GDM construction, the related coefficient is the p-polarized reflection (or transmission) coefficient, whose poles are given by (1 – {r}_{p}^{2|{mathrm{g}}| 1} {r}_{p}^{2|{mathrm{m}}} {textual content{e}}^{{textual content{i}}2{okay}_{z,2}t}=0) (ref. 22). Right here, ({r}_{p}^{2| textual content{g}| 1}) and ({r}_{p}^{2| textual content{m}}) denote the p-polarized reflection coefficients for the dielectric–graphene–dielectric and the dielectric–metallic interface (detailed in Supplementary Be aware 2), respectively. Every coefficient yields a material-specific contribution to the general quantum response: ({r}_{p}^{2|textual content{g}| 1}) incorporates graphene’s through σ(q,ω) and ({r}_{p}^{2| textual content{m}}) incorporates the metallic’s through the d-parameters (see Supplementary Be aware 2). The advanced exponential [with ({k}_{z,2}equiv {({epsilon }_{2}{k}_{0}^{2}-{q}^{2})}^{1/2}), where q denotes the in-plane wavevector] incorporates the results of a number of reflections inside the slab. Thus, utilizing the above-noted reflection coefficients (outlined explicitly in Supplementary Be aware 2), we acquire a quantum-corrected AGP dispersion equation:

$$left[frac{{epsilon }_{1}}{{kappa }_{1}}+frac{{epsilon }_{2}}{{kappa }_{2}}+frac{,text{i},sigma }{omega {epsilon }_{0}}right]left[{epsilon }_{text{m}}{kappa }_{2}+{epsilon }_{2}{kappa }_{text{m}}-left(right.{epsilon }_{text{m}}-{epsilon }_{2}left)right.left(right.{q}^{2}{d}_{perp }-{kappa }_{2}{kappa }_{text{m}}{d}_{parallel }left)right.right] =left[frac{{epsilon }_{1}}{{kappa }_{1}}-frac{{epsilon }_{2}}{{kappa }_{2}}+frac{,text{i},sigma }{omega {epsilon }_{0}}right]left[{epsilon }_{text{m}}{kappa }_{2}-{epsilon }_{2}{kappa }_{text{m}}+left(right.{epsilon }_{text{m}}-{epsilon }_{2}left)right.left(right.{q}^{2}{d}_{perp }+{kappa }_{2}{kappa }_{text{m}}{d}_{parallel }left)right.right]{textual content{e}}^{-2{kappa }_{2}t},$$

(1)

for in-plane AGP wavevector q and out-of-plane confinement components ({kappa }_{j}equiv ( {q}^{2}-{epsilon }_{j}{okay}_{0}^{2} )^{1/2}) for j {1, 2, m}.

Since AGPs are exceptionally subwavelength (with confinement components as much as nearly 300)8,10,11, the nonretarded restrict (whereby κj → q) constitutes a superb approximation. On this regime, and for encapsulated graphene, i.e., the place ϵd ≡ ϵ1 = ϵ2, Eq. (1) simplifies to

$$left[1+frac{2{epsilon }_{text{d}}}{q}frac{omega {epsilon }_{0}}{text{i},sigma }right]left[frac{{epsilon }_{text{m}}+{epsilon }_{text{d}}}{{epsilon }_{text{m}}-{epsilon }_{text{d}}}-qleft(right.{d}_{perp }-{d}_{parallel }left)right.right]=left[right.1+qleft(right.{d}_{perp }+{d}_{parallel }left)right.left]proper.{textual content{e}}^{-2qt}.$$

(2)

For simplicity and concreteness, we are going to think about a easy jellium therapy of the metallic such that d vanishes on account of cost neutrality21,23, leaving solely d nonzero. Subsequent, we exploit the truth that AGPs usually span frequencies throughout the terahertz (THz) and mid-infrared (mid-IR) spectral ranges, i.e., properly beneath the plasma frequency ωp of the metallic. On this low-frequency regime, ωωp, the frequency dependence of d (and d) has the common, asymptotic dependence

$${d}_{perp }(omega )simeq zeta +,textual content{i}frac{omega }{{omega }_{textual content{p}}}xi ,qquad (textual content{for},,omega ll {omega }_{textual content{p}}),$$

(3)

as proven by Persson et al.24,25 by exploiting Kramers–Kronig relations. Right here, ζ is the so-called static image-plane place, i.e., the centroid of induced cost underneath a static, exterior discipline26, and ξ defines a phase-space coefficient for low-frequency electron–gap pair creation, whose price is qωξ21: each are ground-state portions. Within the jellium approximation of the interacting electron liquid, the constants ζ ≡ ζ(rs) and ξ ≡ ξ(rs) rely solely on the provider density ne, right here parameterized by the Wigner–Seitz radius ({r}_{s}{a}_{textual content{B}}equiv {(3{n}_{textual content{e}}/4pi )}^{1/3}) (Bohr radius, aB). Within the following, we exploit the straightforward asymptotic relation in Eq. (3) to calculate the dispersion of AGPs with metallic (along with graphene’s) quantum response included.

### Quantum corrections in AGPs on account of metallic quantum surface-response

The spectrum of AGPs calculated classically and with quantum corrections is proven in Fig. 2. Three fashions are thought of: one, a totally classical, local-response approximation therapy of each the graphene and the metallic; and two others, during which graphene’s response is handled by the nonlocal RPA4,9,17,18,19 whereas the metallic’s response is handled both classically or with quantum surface-response included (through the d-parameter). As famous beforehand, we undertake a jellium approximation for the d-parameter. Determine 2a reveals that—for a set wavevector—the AGP’s resonance blueshifts upon inclusion of graphene’s quantum response, adopted by a redshift because of the quantum surface-response of the metallic (since ({rm{Re}} {,}{d}_{perp } > ,0) for jellium metals; digital spill-out)13,15,16,21,27,28. This redshifting because of the metallic’s quantum surface-response is reverse to that predicted by the semiclassical hydrodynamic mannequin (HDM) the place the result’s at all times a blueshift14 (comparable to ({rm{Re}}{,}{d}_{perp }^{textual content{HDM}} < ,0); digital “spill-in”) because of the neglect of spill-out results29. The imaginary a part of the AGP’s wavevector (that characterizes the mode’s propagation size) is proven in Fig. 2b: the online impact of the inclusion of d is a small, albeit constant, improve of this imaginary element. However this, the modification of ({rm{Im}}, q) just isn’t impartial of the shift in ({rm{Re}}{,}q); because of this, a rise in ({rm{Im}}{,}q) doesn’t essentially suggest the presence of a major quantum decay channel [e.g., an increase of ({rm{Im}}{,}q) can simply result from increased classical loss (i.e., arising from local response alone) at the newly shifted ({rm{Re}}, q) position]. Due to this, we examine the standard issue (Qequiv {rm{Re}}{,}q/{rm{Im}}{,}q) (or “inverse damping ratio”30,31) as an alternative32 (Fig. 2c), which gives a complementary perspective that emphasizes the efficient (or normalized) propagation size somewhat than absolutely the size. The incorporation of quantum mechanical results, first in graphene alone, after which in each graphene and metallic, reduces the AGP’s high quality issue. Nonetheless, the affect of metal-related quantum losses within the latter is negligible, as evidenced by the practically overlapping black and pink curves in Fig. 2c.

To raised perceive these observations, we deal with the AGP’s q-shift because of the metallic’s quantum surface-response as a perturbation: writing q = q0 + q1, we discover that the quantum correction from the metallic is q1q0d/(2t), for a jellium adjoining to hoover within the ({omega }^{2}/{omega }_{textual content{p}}^{2}ll {q}_{0}tll 1) restrict (Supplementary Be aware 3). This straightforward outcome, along with Eq. (3), gives a near-quantitative account of the AGP dispersion shifts on account of metallic quantum surface-response: for ωωp, (i) ({rm{Re}}, {d}_{perp }) tends to a finite worth, ζ, which will increase (decreases) ({rm{Re}}, q) for ζ > 0 (ζ < 0); and (ii) ({rm{Im}}{,}{d}_{perp }) is (mathop{propto}omega) and subsequently asymptotically vanishing as ω/ωp → 0 and so solely negligibly will increase ({rm{Im}}, q). Furthermore, the previous perturbative evaluation warrants ({rm{Re}}{,}{q}_{1}/{rm{Re}}{,}{q}_{0} approx {rm{Im}}{,}{q}_{1}/{rm{Im}}{,}{q}_{0}) (Supplementary Be aware 3), which elucidates the rationale why the AGP’s high quality issue stays basically unaffected by the inclusion of metallic quantum surface-response. Notably, these outcomes clarify current experimental observations that discovered considerable spectral shifts however negligible extra broadening on account of metallic quantum response10,11.

Subsequent, by contemplating the separation between graphene and the metallic interface as a renormalizable parameter, we discover a complementary and instructive perspective on the affect of metallic quantum surface-response. Particularly, inside the spectral vary of curiosity for AGPs (i.e., ωωp), we discover that the “naked” graphene–metallic separation t is successfully renormalized because of the metallic’s quantum surface-response from t to (tilde{t}equiv t-s), the place sdζ (see Supplementary Be aware 4), comparable to a bodily image the place the metallic’s interface lies on the centroid of its induced density (i.e., ({rm{Re}}{,}{d}_{perp })) somewhat than at its “classical” jellium edge. With this method, the type of the dispersion equation is unchanged however references the renormalized separation (tilde{t}) as an alternative of its naked counterpart t, i.e.:

$$1+frac{2{epsilon }_{textual content{d}}}{q}frac{omega {epsilon }_{0}}{textual content{i}sigma }=frac{{epsilon }_{textual content{m}}-{epsilon }_{textual content{d}}}{{epsilon }_{textual content{m}}+{epsilon }_{textual content{d}}} {textual content{e}}^{-2qtilde{t}},$$

(4)

This angle, for example, has substantial implications for the evaluation and understanding of plasmon rulers33,34,35 at nanometric scales.

Moreover, our findings moreover counsel an fascinating experimental alternative: as all different experimental parameters may be well-characterized by impartial means (together with the nonlocal conductivity of graphene), high-precision measurements of the AGP’s dispersion can allow the characterization of the low-frequency metallic quantum response—a regime that has in any other case been inaccessible in standard metal-only plasmonics. The underlying concept is illustrated in Fig. 3; relying on the signal of the static asymptote ζ ≡ d(0), the AGP’s dispersion shifts towards bigger q (smaller ω; redshift) for ζ > 0 and towards smaller q (bigger ω; blueshift) for ζ < 0. As famous above, the q-shift is ~q0ζ/(2t). Crucially, regardless of the ångström-scale of ζ, this shift may be sizable: the inverse scaling with the spacer thickness t successfully amplifies the attainable shifts in q, reaching as much as a number of μm−1 for few-nanometer t. We stress that these regimes are properly inside present state-of-the-art experimental capabilities8,10,11, suggesting a brand new path towards the systematic exploration of the static quantum response of metals.

### Probing the quantum surface-response of metals with AGPs

The important thing parameter that regulates the affect of quantum floor corrections stemming from the metallic is the graphene–metallic separation, t (analogously to the observations of nonclassical results in standard plasmons at slender metallic gaps13,36,37); see Fig. 4. For the experimentally consultant parameters indicated in Fig. 4, these come into impact for t 5 nm, rising quickly upon reducing the graphene–metallic separation additional. Mainly, ignoring the nonlocal response of the metallic results in a constant overestimation (underestimation) of AGP’s wavevector (group velocity) for d < 0, and vice versa for d > 0 (Fig. 4a); this conduct is per the efficient renormalization of the graphene–metallic separation talked about earlier (Fig. 4b). Lastly, we analyze the interaction of each t and EF and their joint affect on the magnitude of the quantum corrections from the metallic (we take d = −4 Å, which is affordable for the Au substrate utilized in current AGP experiments7,8,11); in Fig. 4c we present the relative wavevector quantum shift (excited at λ0 = 11.28 μm32). Within the few-nanometer regime, the quantum corrections to the AGP wavevector method 5%, growing additional as t decreases—for example, within the excessive, one-atom-thick restrict (t ≈ 0.7 nm11, which additionally roughly coincides with fringe of the validity of the d-parameter framework, i.e., t 1 nm15) the AGP’s wavevector can change by as a lot as 10% for average graphene doping. The pronounced Fermi stage dependence exhibited in Fig. 4c additionally suggests a complementary method for measuring the metallic’s quantum surface-response even when an experimental parameter is unknown (though, as beforehand famous, all related experimental parameters can in reality be characterised utilizing presently obtainable strategies8,10,11,15): such an unknown variable may be fitted at low EF utilizing the “classical” idea (i.e., with d = d = 0), because the affect of metallic quantum response is negligible in that regime. A parameter-free evaluation of the metallic’s quantum surface-response can then be carried out subsequently by growing EF (and with it, the metal-induced quantum shift). We emphasize that this may be completed in the identical system by doping graphene utilizing normal electrostatic gating8,10,11.


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## The p tag

This tag defines a paragraph of text.

Some text

It’s a block element.

Inside it, we can add any inline element we like, like span or a.

We cannot nest p elements one into another.

By default browsers style a paragraph with a margin on top and at the bottom. 16px in Chrome, but the exact value might vary between browsers.

This causes two consecutive paragraphs to be spaced, replicating what we think of a “paragraph” in printed text.

## The span tag

This is an inline tag that can be used to create a section in a paragraph that can be targeted using CSS:

A part of the text and here another part

## The br tag

This tag represents a line break. It’s an inline element, and does not need a closing tag.

We use it to create a new line inside a p tag, without creating a new paragraph.

And compared to creating a new paragraph, it does not add additional spacing.

Some text
A new line

HTML provides us 6 heading tags. From most important to least important, we have h1, h2, h3, h4, h5, h6.

Typically a page will have one h1 element, which is the page title. Then you might have one or more h2 elements depending on the page content.

Headings, especially the heading organization, are also essential for SEO, and search engines use them in various ways.

The browser by default will render the h1 tag bigger, and will make the elements size smaller as the number near h increases:

All headings are block elements. They cannot contain other elements, just text.

## The strong tag

This tag is used to mark the text inside it as strong. This is pretty important, it’s not a visual hint, but a semantic hint. Depending on the medium used, its interpretation will vary.

Browsers by default make the text in this tag bold.

## The em tag

This tag is used to mark the text inside it as emphasized. Like with strong, it’s not a visual hint but a semantic hint.

Browsers by default make the text in this italic.

## Quotes

The blockquote HTML tag is useful to insert citations in the text.

Browsers by default apply a margin to the blockquote element. Chrome applies a 40px left and right margin, and a 10px top and bottom margin.

The q HTML tag is used for inline quotes.

## Horizontal line

Not really based on text, but the hr tag is often used inside a page. It means horizontal rule, and it adds an horizontal line in the page.

Useful to separate sections in the page.

## Code blocks

The code tag is especially useful to show code, because browsers give it a monospaced font.

That’s typically the only thing that browsers do. This is the CSS applied by Chrome:

code {
font-family: monospace;
}

This tag is typically wrapped in a pre tag, because the code element ignores whitespace and line breaks. Like the p tag.

Chrome gives pre this default styling:

pre {
display: block;
font-family: monospace;
white-space: pre;
margin: 1em 0px;
}

which prevents white space collapsing and makes it a block element.

## Lists

We have 3 types of lists:

• unordered lists
• ordered lists
• definition lists

Unordered lists are created using the ul tag. Each item in the list is created with the li tag:

First

Second

Ordered lists are similar, just made with the ol tag:

First

Second

The difference between the two is that ordered lists have a number before each item:

Definition lists are a bit different. You have a term, and its definition:

Flavio

The name

Copes

The surname

This is how browsers typically render them:

I must say you rarely see them in the wild, for sure not much as ul and ol, but sometimes they might be useful.

## Other text tags

There is a number of tags with presentational purposes:

• the mark tag
• the ins tag
• the del tag
• the sup tag
• the sub tag
• the small tag
• the i tag
• the b tag

This is an example of the visual rendering of them which is applied by default by browsers:

You might wonder, how is b different than strong? And how i is different than em?

The difference lies in the semantic meaning. While b and i are a direct hint at the browser to make a piece of text bold or italic, strong and em give the text a special meaning, and it’s up to the browser to give the styling. Which happens to be exactly the same as b and i, by default. Although you can change that using CSS.

There are a number of other, less used tags related to text. I just mentioned the ones that I see used the most.

format: plain


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# 500Cm To Meter

500Cm To Meter. 1 millimeters to meter = 0.001 meter. 500 centimeters x 0.01 = 5 meters. 1 m 2 = 1 cm * 1 cm / 10 000. Type in your own numbers in the form to convert the units! To convert 4500 cm to m multiply the length in centimeters by 0.01. Quick conversion chart of cm to meter 1 cm to meter = 0.01 meter 10 cm to meter = 0.1 meter 50 cm to meter = 0.5 meter 100 cm to. 20 centimetres = 0.2 meters:

Given that 500cm to convert into meter solution as cm and m both are the unit of measurement we know that. If we want to calculate how many meters are 500 centimeters we have to multiply 500 by 1 and divide the product by 100. Next, let's look at an example showing the work and calculations that are involved in converting from centimeters to meters (cm to m). A centimeter, or centimetre, is a unit of length equal to one hundredth of a meter. 1 m 2 = 1 cm * 1 cm / 10 000. 500 centimeters x 0.01 = 5 meters. 1 centimeter persegi = 0.0001 meter persegi:

## How far is 50 centimeters in meters?

500 centimeter = 5 meter. 2500 centimetres = 25 meters: 1 m 2 = 1 cm * 1 cm / 10 000. Looking at the bottom numbers (denominators) we need to be able to. $$\mathrm{{1}\;m = {100}\;cm}$$ ${1}$ centimeter sama dengan ${0.01}$ meter. 500 cm is equal to how many meters? Centimeter to meter conversion example task:. The 4500 cm in m formula is [m] = 4500 * 0.01.

### Konversi Dari Cm Ke M:

To convert centimeters to meters, multiply the centimeter value by 0.01 or divide by 100. How to convert from centimeters to meters. 1 centimeter (cm) is equal to 0.01 meter (m). If we want to calculate how many meters are 500 centimeters we have to multiply 500 by 1 and divide the product by 100.

## Also, Explore Many Other Unit Converters Or Learn.

We can also form a simple. A centimeter, or centimetre, is a unit of length equal to one hundredth of a meter. Centimeter to meter conversion example task:. Next, let's look at an example showing the work and calculations that are involved in converting from centimeters to meters (cm to m).

## Conclusion of 500Cm To Meter.

The distance d in meters (m) is equal to the distance d in centimeters. 1 m 2 = 1 cm * 1 cm / 10 000. Formula to convert 500 cm to m is 500 / 100.. Also, explore many other unit converters or learn. Centimeter to meter conversion example task:.

Source


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# F-BF,IF The Canoe Trip, Variation 2

Alignments to Content Standards: F-IF.B.4 F-IF.B.5 F-BF.A.1.a

Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle.

1. Mike guesses that the current is flowing at a speed of 50 feet per minute. Assuming this is correct, how long will it take for Mike to reach his destination?

2. Mike does not really know the speed of the current. Make a table showing the time it will take him to reach his destination for different speeds:

Speed of Current
(feet per minute)
Mike’s Speed
(feet per minute)
Time for Mike to travel 30,000 feet (minutes)
0
50
100
140
149
$s$
3. The time $T$ taken by the trip, in minutes, as a function of the speed of the current is $s$ feet/minute. Write an equation expressing $T$ in terms of $s$. Explain why $s = 150$ does not make sense for this function, both in terms of the canoe trip and in terms of the equation.

4. Sketch a graph of the equation in part (c). Explain why it makes sense that the graph has a vertical asymptote at $s = 150$.

## IM Commentary

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function. The canoe context focuses attention on the variables as numbers, rather than as abstract symbols. Variation 1 of this task uses function notation and expects students to derive the formula for the function directly, without the aid of a table.

The task also provides an opportunity to discuss mathematical models, their interpretation, and their limits. For example, teachers could ask if it makes sense for $s$ to be negative. This might correspond to a flow of water moving in the same direction as Mike, and indeed the equation in the solution gives the correct answer in that case.

More fanciful, and requiring a longer discussion, is the question of whether it makes sense to consider values of $s$ larger than 150. If $s=300$, for example, a naive application of the formula predicts that Mike will arrive at his destination in $-200$ minutes! It is reasonable to say that negative times do not make sense and to exclude values of $s$ greater than $150$. However, value $-200$ could also be interpreted as referring to an event that takes place 200 minutes before the trip starts. If Mike had been at his destination 200 minutes ago, then a river which was flowing at 300 feet per minute against his direction of travel would push him precisely the 30,000 feet from his destination that the problem began with.

## Solution

1. The current is working against Mike’s paddling, so he’ll move 150 - 50 = 100 feet per minute. Thus, it will take him 30000/100 = 300 minutes = 5 hours of paddling to reach his destination.

2. Speed of Current
(feet per minute)
Mike’s Speed
(feet per minute)
Time for Mike to travel 30,000 feet (minutes)
0150200
50100300
10050600
140103,000
149130,000
$s$$150-s$$\frac{30,000}{150-s}$
3. From the last row of the table we see that $$T = \frac{30,000}{150-s}.$$ If the speed of the current is 150 feet per minute, then Mike isn’t strong enough to paddle against it. Paddling his hardest, he would just sit still so he would never reach his destination; that is, there is no “time required for Mike to travel 30,000 feet” because he can’t travel. If you put $s = 150$ into the equation you will find yourself trying to divide by zero, which also tells you that the answer is undefined.

4. The graph shows the values of $T$ getting larger and larger as $s$ gets close to 150. The graph can never touch the vertical line at $s = 150$ because that would imply that there is a “time it takes Mike to travel 30000 ft” when the current is 150, but there isn’t. When $s$ is close to 150 the speed of the current is close to Mike’s paddling speed. That means that Mike is traveling very slowly. Thus, when we calculate the time it takes him to travel 30,000 ft we are dividing by a very small number, which gives a very large answer. The closer $s$ is to 150, the closer Mike’s actual speed is to zero, and the larger the time it takes.


text:
5,780 views

. You are given a set of n points on the number line. They are given in arbitrary order. The task is to find the points that are closest to each other.

To solve the problem you decide to take one point and compute its distance to all other points and repeat this process for all points. During the process you track the closest pair.

What is the running time of this algorithm ?

1. O(nlgn)
2. O(n^2)
3. Runtime is independent of n.
4. O(log n)

The answer is B $O(n^2)$.

The reasoning behind this is that you essentially calculate the distance between all the pairs of points. Assuming there are n numbers, you calculate the distance (based on this algorithm) for $n \dfrac{(n-1)}{2}$ pairs, hence $O(n^2)$.

Since all numbers are on the number line, this can be done in a more efficient way, i.e. sort the points in the line and the traverse the list of points to find the closest.

1 vote


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# skrf.network.Network.z0¶

Network.z0

Characteristic impedance[s] of the network ports.

This property stores the characteristic impedance of each port of the network. Because it is possible that each port has a different characteristic impedance each varying with frequency, z0 is stored internally as a fxn array.

However because z0 is frequently simple (like 50ohm), it can be set with just number as well.

Returns: z0 – characteristic impedance for network numpy.ndarray of shape fxn


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# Uncertainty Principle, Energy and Position

1. Mar 30, 2010

### MrJohnnyPrice

1. The problem statement, all variables and given/known data

Electrons in atoms are known to have energies in the range of a few eV. Show that the uncertainty principle allows electrons of this energy to be confined in a region the size of an atom (0.1 nm)

2. Relevant equations

$$\Delta$$x$$\Delta$$p $$\geq$$ h bar / 2

$$\Delta$$E$$\Delta$$t $$\geq$$ h bar / 2

3. The attempt at a solution

h bar = 6.58 x 10^-16 eV * s
$$\Delta$$x = .1 nm

I know that if there was an uncertainty relationship between Energy and displacement I could set h bar and $$\Delta$$x to the values given and show that this is possible, but I can't figure out how to find an energy position uncertainty relation.

2. Mar 31, 2010

### Feldoh

1/2mv^2 and mv look an awful lot a like...


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Calculating Discount C++

The following question is from the book Problem Solving With C++. This is given as an exercise in the chapter "Functions for all subtasks":

Write a function that calculates a discount applicable on the price of an item. Your function should have three arguments: the price as a reference to a double, the discount as a double value, and a bool to indicate if the discount is calculated as a percentage or a fixed amount. You should calculate the discount and modify the price of the item accordingly. Your program should check that the discount is not negative, and that the price of the item does not drop to zero after applying discount. Use assert statement for this.

Here's my solution:

#include<iostream>
#include<cassert>

double calcDiscount(double& price, double discount, bool type);

int main()
{
char type;
double price, discount, disAmt;
std::cout<<"Enter price of the item = ";
std::cin>>price;
disAmt = calcDiscount(price, discount, type);
price = price - disAmt;
assert(price > 0);
std::cout<<"Price after discount = "<<price<<"\n";
}

double calcDiscount(double& price, double discount, bool type)
{
std::cout<<"Enter 1 to calculate discout as percentage.\n"
<<"OR\n"
<<"Enter 0 to calculate discount as fixed amount.\n";
std::cin>>type;
if(type)
{
std::cout<<"Enter discount percentage = ";
std::cin>>discount;
discount = (price * (discount/100.0));
}
else
{
std::cout<<"Enter discount amount = ";
std::cin>>discount;
}
assert(discount >= 0);
return(discount);
}

Although it gives the correct output. But still what improvements could be made?

• It does not give the correct output, as the logic for the discount amount is completely missing. And how would you know whether it computes correctly since there is no "output"? Feb 1 at 15:30

You should put main at the bottom. It's not useful to declare your functions in a CPP file, since C++ supports overloading. If you defined the actual function with a different signature that would not be a compile-time error. But you would have a linker error -- the pre-declaring of the function did not help you, and only caused more confusion.

So, put functions in the order they need to be. Declarations in the CPP files are only for mutual recursion.

So much of what's poor with the solution is due to the specification! I understand that a beginner can't swallow everything at once, and it's trying to teach functions and decomposition in this lesson.

Your function should have three arguments: the price as a reference to a double, ...

Never never never use double for money.
I was shocked at how many pros in business did not learn this. So don't flirt with such an insidious habit even in toy code.

And making it a reference is suspicious.

...the discount as a double value, ...

That seems like it might be OK for a rate, but it runs into some of the same issues. E.g. you can't represent 10% exactly.

...and a bool to indicate if the discount is calculated as a percentage or a fixed amount.

You're using the same type to represent an amount or a discount. Are they the same type of thing? Even if they can be represented by the same underlying primitive type, they are not the same kind of thing. You're writing code using primitives in the declaration, not domain-specific types. You are not passing a double but an Amount, logically. A book that teaches problem solving should introduce that idea early, too.

As for using a bool to indicate two modes of operation for the function: Well, that does happen. A bool can be OK here, but only just. It's really poor design to have two different functions under one wrapper, without having a majority of the code in common.

BTW, I happen to have exactly that in my own code! But, it's not passing a amount-or-rate as a single parameter. The amount, or the rate, is found in a database. In my case, I use an enum to indicate Rate or Discount (or whatever I called them). And the function in question called the underlying calculation as one line, and the rest of the function was doing work that didn't matter which it was, so it's sensible to combine them into one function like that.

You should calculate the discount and modify the price of the item accordingly.

Don't use "in-out" parameters! This is poor design. It should be returning the new value. You are returning double too even though this spec does not say anything about a return value.

Your program should check that the discount is not negative, and that the price of the item does not drop to zero after applying discount. Use assert statement for this.

Checking is good. The assert statement, not so much. It only has an effect in a debug build and is ignored in a release build! It's difficult to unit test that your code performs that check, as it simply aborts the program. An exception would be better. There is an exception type designed for exactly this.

Are the requirements clear?

You write (price * (discount/100.0)); interpreting the discount parameter as 30.0 meaning 30%. But it's a double, and it's normal to use a floating-point value to mean a value between 0.0 and 1.0. That is, 30% would naturally be 0.3.

Jarod42 already pointed out that you are asking for the type parameter inside the function. You are not passing it to the function! In main you declare it as char (not bool) but don't use it.

You are missing the point of having a function. You should collect the needed information and hand it all to the function. The function should look only at its parameters and not worry about where the information came from.

Then you prompt for discount! You are completely missing the point of using a function that takes discount as a parameter!

Worse, you are not computing the discount amount in the non-percent case. You are neither computing the amount nor checking for a negative value as directed. Your code gives the wrong answer and fails.

You should not post on Code Review until code is complete and working correctly. You don't even know... this shows why you need testing. One big advantage of using functions (correctly) is you can call them from test code with built-in values which are checked automatically, not prompting you for anything at all. This lets the program auto-test the logic without you having to repeatedly type in values.

You write: return(discount); But return is not a function. Write return discount; without redundant parens around the whole thing. This does make a difference in some cases as of C++11.

Don't Give Up

This is the way to learn these things. The books don't tell you everything, and trying things for yourself is the only way to figure it out.

Sorry as expected function interface is bad (having bool parameter instead of 2 functions, mixing in/out parameter and return value, function doing several stuff with naming not reflecting that, ...).

All I/O should not be in calcDiscount:

double calcDiscount(double& price, double discount, bool as_percentage)
{
assert(discount >= 0);
if (as_percentage) {
discount = price * discount / 100.0;
}
price -= discount;
assert(price > 0);
return discount;
}

Then in main, you might do the I/O

int main()
{
double price;
std::cout << "Enter price of the item = ";
char type;
std::cin >> price;
std::cout << "Enter 1 to calculate discount as percentage.\n"
<< "OR\n"
<< "Enter 0 to calculate discount as fixed amount.\n";
std::cin >> type;
double effective_discount;
if (type == '1') {
std::cout << "Enter discount percentage = ";
double discountPercent;
std::cin >> discountPercent;
effective_discount = calcDiscount(price, discountPercent, true);
} else {
std::cout << "Enter discount amount = ";
double discount;
std::cin >> discount;
effective_discount = calcDiscount(price, discount, false);
}
std::cout << "discount = " << effective_discount << "\n";
std::cout << "Price after discount = " << price << "\n";
}


text:
# Math Help - What do I do to simplify this polynomial?

1. ## What do I do to simplify this polynomial?

4x^4-4x^2+1.

How do I simplify this one?
What really throws me is the whole +1 thing.

2. Originally Posted by quikwerk
4x^4-4x^2+1.

How do I simplify this one?
What really throws me is the whole +1 thing.
Hi quikwerk,

$4x^4-4x^2+1$

You might see if better if we

Let $u=x^2$

$4u^2-4u+1=(2u-1)(2u-1) \Longrightarrow (2x^2-1)(2x^2-1)=(2x-1)^2$

3. ## How do I simplify these polynomial fractions?

thanks!
That really helps! I think I see how that question works now!


text:
# Ubuntu – Thunar Icons Missing

i3-wmiconsthunar

I installed thunar yesterday but on starting up this morning, I found that all of the folder icons had vanished. I'm using i3wm (but the icons were visible in i3 as of yesterday). Any idea why they might have disappeared or how I could retrieve them?

Thanks.

Try installing gnome-icon-theme-full.


text:
### Home > A2C > Chapter 8 > Lesson 8.1.1 > Problem8-9

8-9.

Switch x and f(x) and solve for f(x).

$f^{-1}(x)=\frac{x^{3}+1}{4}$

See part (a).


text:
# Diofant equation

In the set of integers (Z) solve the equation:

$212x+316y=0$

Write result with integer parameter
$t\in Z$
(parameter t = ...-2,-1,0,1,2,3... if equation has infinitely many solutions)

Correct result:

x =  -79 t
y =  53 t

#### Solution:

We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you!

Tips to related online calculators
Do you want to calculate greatest common divisor two or more numbers?
Do you solve Diofant problems and looking for a calculator of Diofant integer equations?

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text:
# Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\circ B=\frac{1}{2}(AB+BA)$. According to the above article in wikipedia P. Jordan introduced this notion in 1933 to formalize the notion of an algebra of observables in quantum mechanics.

Question 1. Did this operation become really useful in quantum mechanics in any non-trivial way?

Question 2. I would be happy to see some explanations why the notion of Jordan algebra is useful/ natural. Are there applications of it to other parts of mathematics?

Here are very few interesting facts I was able to find so far.

1) There is a really beautiful classification of the finite dimensional formally real Jordan algebras due to Jordan, von Neumann & Wigner (1934).

2) The above wikipedia paper mentions a classification of some special class of infinite dimensional Jordan algebras (Zelmanov, 1979).

3) I have recently heard about the Koecher-Vinberg classification of symmetric cones: such cones are precisely cones of squares in Euclidean Jordan algebras.

Thus the only way I heard Jordan algebras are related to other parts of mathematics is via the classification lists of various subclasses of them.

• The book "A taste of Jordan algebras" by Kevin McCrimmon contains a very interesting historical introduction. In particular, he points out that the physicists were rather disappointed with the fact that there is no infinite series of exceptional Jordan algebras, and apparently this made the concept less useful for them than they had hoped for. Feb 4, 2016 at 12:50
• Just for general culture: Jordan here is the German mathematical physicist Pascual Jordan (and not the earlier and French mathematician Camille Jordan). en.wikipedia.org/wiki/Pascual_Jordan
– YCor
Feb 4, 2016 at 13:01
• added to YCor : consequently, one must pronouce Yordan', while Camille Jordan is pronounced Geordã. Feb 4, 2016 at 14:12

They turn up quite often in the study of (exceptional) linear algebraic groups. The most famous instance of this is the fact that algebraic groups of type $F_4$ are precisely the automorphism groups of Albert algebras, i.e. $27$-dimensional exceptional Jordan algebras. One possible reference for this fact is the book "Octonions, Jordan Algebras and Exceptional Groups" by Springer and Veldkamp.

I have used Jordan algebras (and also some generalization known as structurable algebras) quite often in my own research, in particular in connection to Moufang sets (these are groups "of rank one" in some sense). For instance, in my paper with Richard Weiss, "Moufang sets and Jordan division algebras", Math. Ann. 335 (2006), no. 2, 415–433, we show (among other things) that every Jordan division algebra $J$ gives rise to a Moufang sets, i.e. there exists a group that you could sensibly denote as $\mathrm{PSL}_2(J)$. (When $J$ is an Albert division algebra, for instance, this group is a linear algebraic group of absolute type $E_7$ of $k$-rank one.)

Another, completely different instance where Jordan algebras played a crucial role, was in Zel'manov's solution to the restricted Burnside problem. See, for instance, here.

I would like to elaborate on the link to "associative" problems (such as the Zelmanov's solution of the restricted Burnside problems mentioned by Tom De Medts) - mainly because by saying that Jordan algebras are nonassociative algebras satisfying a strange list of axioms, you are forgetting that some reverse engineering takes place here.

First of all, if $A$ is an associative algebra, considering the new operation $x\circ y=x\circ y+y\circ x$ is not a completely unnatural thing from the algebraic point of view. Moreover, some important classes, like Hermitian matrices, form a subalgebra with respect to $\circ$, but not with respect to the matrix product!

Then, it is natural to try and axiomatise this operation. For instance, we all know that the operation $xy-yx$ defines Lie algebras, so the Jacobi identity holds. Once you play around with elements a little bit, you discover that the operation $\circ$ satisfies the identity $((x\circ x)\circ y)\circ x = (x\circ x)\circ (x\circ y)$, the Jordan identity you see in the literature. (And that's how that identity emerged.)

What however is very different from the case of Lie algebras is that in fact $\circ$ satisfies many other identities that do not follow from the Jordan identity by algebraic manipulations, for example an identity of degree $8$ discovered by Glennie in early 1960s (https://en.wikipedia.org/wiki/Glennie%27s_identity). But it turns out that working within the larger class of algebras satisfying the Jordan identity only is actually not bad, and a lot of things can be accomplished in that more general context.

Jordan algebras were originally introduced by Pascual Jordan in a hope to generalize the orthodox formulation of quantum mechanics, but this program was not successful as far as the generalization of quantum mechanics is concerned. In this respect, Jordan algebras are neither essential nor terribly useful in understanding quantum mechanics.

On the other hand the importance of Jordan algebras in mathematical physics is undisputed and related to the fact that they are tightly interconnected with another nonassociative structures: Lie algebras. As remarked by Kevin McCrimmon, "if you open up a Lie algebra and look inside, 9 times out of 10 there is a Jordan algebra (of pair) which makes it work”.

In addition to McCrimmon's book "A taste of Jordan algebras", mentioned in comments, the following works might be useful in understanding the role of Jordan algebras in mathematical physics and mathematics:

http://projecteuclid.org/euclid.bams/1183540925 (Jordan algebras and their applications, by Kevin Mccrimmon).

http://arxiv.org/abs/1106.4415 (Jordan structures in mathematics and physics, by Radu Iordanescu).

http://arxiv.org/abs/0809.4685 (Black Holes, Qubits and Octonions, by L. Borsten, D. Dahanayake, M.J. Duff, H. Ebrahim and W. Rubens).

http://www.worldscientific.com/worldscibooks/10.1142/3282 (On the Role of Division, Jordan and Related Algebras in Particle Physics, by F. Gursey and C.-H. Tze).

The biography of Pascual Jordan can be found here: http://arxiv.org/abs/hep-th/0303241 (Pascual Jordan, his contributions to quantum mechanics and his legacy in contemporary local quantum physics, by Bert Schroer). Jordan was a very important figure in developing quantum mechanics and quantum field theory but his reputation was greatly damaged by his relations to the Nazi regime, although, according to Schroer, "he never received benefits for his pro-NS convictions and the sympathy remained one-sided. Unlike the mathematician Teichmueller, whose rabid anti-semitism led to the emptying of the Gottingen mathematics department, Jordan inflicted the damage mainly on himself".

Jordan algebras and more generally Jordan pairs and triple systems have applications in the theory of Riemannian symmetric spaces. Some relevant links:

https://en.wikipedia.org/wiki/Hermitian_symmetric_space#Jordan_algebras

https://en.wikipedia.org/wiki/Triple_system#Jordan_triple_systems

• Tucek: this is item 3) in sva question. Namely, the application to the classification of irreducible Hermitian symmetric spaces of noncompact type. Historically such classification goes back to Elie Cartan by using Lie theory. Was M. Koecher who realized that by using Jordan algebras such spaces can be constructed as a bounded symmetric domains starting by a Jordan triple system: mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0279.0284.ocr.pdf Feb 4, 2016 at 12:26


text:
# Multi-heterodyne two dimensional coherent spectroscopy using frequency combs

## Abstract

Optical multi-dimensional coherent spectroscopy is a powerful technique for studying the structure, properties and ultrafast dynamics of atoms, molecules, semiconductor materials and complex systems. Current implementations of multi-dimensional coherent spectroscopy have long acquisition times and/or limited spectral resolution. In addition, most of the techniques utilize complex geometries or phase cycling schemes to isolate non-linear signals. We demonstrate a novel approach of using frequency combs to perform rapid, high resolution and background free multi-dimensional coherent spectroscopy of semiconductor materials. Our approach is inspired by dual-comb spectroscopy, which has been proven to be a versatile tool for obtaining one dimensional absorption spectra with high resolution in a short acquisition time. We demonstrate the method using a GaAs multi-quantum well sample.

## Introduction

Rapid and precise measurements of light-matter interactions are crucial in understanding the complex properties of materials1,2,3,4,5,6,7,8. Since the development of frequency combs (typically mode-locked lasers), the technique known as dual comb spectroscopy (DCS)9,10 has attracted substantial attention as a revolutionary approach to optical spectroscopy. DCS is analogous to Fourier Transform Infrared (FTIR) spectroscopy but does not require moving optical elements, which limit the acquisition speed. DCS is a powerful technique that combines high spectral resolution, high sensitivity, broad spectral coverage and fast acquisition speeds. Because of these qualities, DCS is being developed for use in many fields, from atomic and molecular spectroscopy, to precision metrology, to light detection and ranging (LIDAR) and even atmospheric monitoring11,12,13,14,15. In DCS, one frequency comb interrogates the sample, and the linear response is sampled in time with a LO comb with a slightly different repetition rate. The resulting interferogram is captured by a single photodetector. In the frequency domain, this arrangement produces a Radio Frequency (RF) comb spectrum that results from these two optical combs beating against each other on a single photodetector. With the development of micro-resonator combs16 DCS is becoming transportable as well. However, DCS suffers from the limitations inherent in one dimensional techniques arising from inhomogeneous broadening and the inability to discern if resonances are coupled.

Optical multi-dimensional coherent spectroscopy (MDCS) has been proven to be a very powerful tool for addressing these limitations. The concept of MDCS originated in Nuclear Magnetic Resonance17 where it is widely used today. Optical MDCS is an advanced non-linear technique that uses a sequence of laser pulses (typically three) incident to the sample and detects the Four-Wave-Mixing (FWM) signal generated by the sample as a function of the time delay(s) between the incident pulses. A multi-dimensional spectrum is then constructed by Fourier transforming the signal with respect to the emission time and the delays between the pulses. Optical MDCS has many advantages including 1) unfolding congested one dimensional spectra and isolating different spectral contributions, 2) identifying couplings and interactions between excited states, 3) separating homogeneous and inhomogeneous linewidths, and 4) providing important insights about many-body physics. However, high resolution MDCS techniques (implemented using long delay stages to probe narrow resonances such as atoms, molecules, Nitrogen vacancy centers, self-assembled quantum dots etc.) have long acquisition times (tens of minutes to hours), whereas rapid MDCS techniques provide much lower spectral resolution18,19,20,21. Spectral resolution limitation is either due to time delays (between pulses) achievable using acousto-optic pulse shapers, spatial spectral interferometry or due to the spectrometer resolution. In addition, the experimental apparatus is bulky and uses complex geometries and phase cycling schemes to suppress background signals.

Recently we have performed a marriage between MDCS and DCS, which we call M-DSC2, that allows us to perform multi-dimensional coherent spectroscopy that is simultaneously rapid, background free and high resolution22. The method was applied to atomic samples and here, we extend its applications for semiconductor materials. DCS technique has previously been applied to coherent anti-stokes Raman spectroscopy and time resolved pump-probe spectroscopy23,24 to study molecules and semiconductor materials. M-DCS2 extends its applications for multi-dimensional coherent spectroscopy.

## Method

The concept of comb based MDCS is schematically shown in Fig. 1 (please the supplementary material for details). Two combs with different offset frequencies, produced from a single source comb using an Acousto-Optical Modulator (AOM), generate a FWM signal in a collinear geometry. Inset (a) shows the generation of a FWM signal by a pair of pulses (delayed pulse interacts twice (supplementary material) in the photon echo sequence25. The emitted FWM comb is then spectrally separated in the RF domain after heterodyne detection with the Local Oscillator (LO) comb26 that has a slightly different repetition rate (frep_1 = 93.544290 MHz and frep_LO = 93.544290 MHz + 220 Hz locked to a direct digital synthesizer). The repetition rate difference allows the LO delay to be scanned rapidly over a long range (Fig. 1 (inset b)) and is a distinction from AOM based approaches that only use a single laser27. The delay between the excitation pulses is then varied using a retroreflector mounted on a mechanical translation stage to generate the second dimension for the two dimensional spectrum. The phase fluctuations due to fluctuations in offset frequency, optical path length and/or repetition frequency are measured and subtracted in real time26.

For this proof of concept experiment, we used a sample containing 10 layers of 10 nm thick GaAs quantum wells (QW) separated by the same thickness Al0.3Ga0.7As barriers cooled down to 7 K. The optical bandwidth of the beams are filtered such that they excite both the Heavy Hole (HH) and the Light Hole (LH) excitonic resonances (level diagram shown in Fig. 1). The optical bandwidth that can unambiguously be mapped to the RF domain strongly depends on how tightly the lasers are phase locked. Our phase locking loops and cancelation scheme26 allowed us to obtain a comb resolution in a limited optical range. For this proof of concept experiment we applied real time phase correction in the optical region covering the HH resonance and hence an optical bandpass filter (3 nm full width at half maximum) covering only the HH resonance was inserted before the photodetector.

## Results and Discussion

Figure 2(a) shows the spectrum of the FWM signal at zero delay remapped to the optical domain. This spectrum corresponds to the Fourier transform of the emitted FWM signal with respect to the emission time (Fig. 1 inset (a)). A portion of the spectrum on an expanded scale is shown in Fig. 2(b), the comb structure is clear. This spectrum was obtained in 100 ms and the width of a single comb tooth is a few MHz (~10 neV). In traditional time-domain spectroscopy, similar resolution in the same acquisition time would require a >30 m long delay stage moving with nearly the speed of sound. Currently available commercial FTIR spectrometers have the maximum path length difference up to 10 m and can provide the resolution of 30 MHz.

To generate a two dimensional spectrum, we scanned the delay between excitation pulses with 16.7 fs steps and recorded the FWM RF signal at each delay. Figure 2(c) shows the evolution of the real and imaginary parts of a single comb line’s (the middle comb tooth on Fig. 2(b)) electric field as a function of the excitation delay. It displays a beat pattern that corresponds to the interference of two FWM signals emitted at the HH excitonic resonance frequency that have different absorption (evolution) frequencies, specifically the HH and LH excitonic resonances. The electric field of the signal is

$$\begin{array}{c}{{\boldsymbol{E}}}_{{\boldsymbol{sig}}} \sim {{\boldsymbol{E}}}_{1}^{\ast }({{\boldsymbol{\omega }}}_{{\boldsymbol{lh}}}){{\boldsymbol{E}}}_{2}({{\boldsymbol{\omega }}}_{{\boldsymbol{hh}}}){{\boldsymbol{E}}}_{2}({{\boldsymbol{\omega }}}_{{\boldsymbol{lh}}}){{\boldsymbol{e}}}^{-{\boldsymbol{i}}{{\boldsymbol{\omega }}}_{{\boldsymbol{lh}}}{\boldsymbol{\tau }}}+{{\boldsymbol{E}}}_{1}^{\ast }({{\boldsymbol{\omega }}}_{{\boldsymbol{lh}}}){{\boldsymbol{E}}}_{2}({{\boldsymbol{\omega }}}_{{\boldsymbol{lh}}}){{\boldsymbol{E}}}_{2}({{\boldsymbol{\omega }}}_{{\boldsymbol{hh}}}){{\boldsymbol{e}}}^{-{\boldsymbol{i}}{{\boldsymbol{\omega }}}_{{\boldsymbol{lh}}}{\boldsymbol{\tau }}}+\\ 2{{\boldsymbol{E}}}_{1}^{\ast }({{\boldsymbol{\omega }}}_{{\boldsymbol{hh}}}){{\boldsymbol{E}}}_{2}({{\boldsymbol{\omega }}}_{{\boldsymbol{hh}}}){{\boldsymbol{E}}}_{2}({{\boldsymbol{\omega }}}_{{\boldsymbol{hh}}}){{\boldsymbol{e}}}^{-{\boldsymbol{i}}{{\boldsymbol{\omega }}}_{{\boldsymbol{hh}}}{\boldsymbol{\tau }}}\end{array}$$
(1)

where $${{\boldsymbol{\omega }}}_{{\boldsymbol{hh}}}$$ and $${{\boldsymbol{\omega }}}_{{\boldsymbol{lh}}}$$ correspond to HH and LH frequencies and $${\boldsymbol{\tau }}$$ is the evolution time (Supplementary text). The Fourier transform of the beat signal with respect to the evolution time is shown in Fig. 2(d). The peaks correspond to HH and LH absorption energies. The negative sign on the absorption energy axis is due to the negative phase evolution during the evolution period (Fig. 1 inset (a)).

Figure 3 shows the two dimensional coherent spectrum obtained by Fourier transforming all comb lines in the FWM spectrum with respect to the evolution time. On the bottom plane, the same data is shown using a contour plot. On the side plane, the measured linear absorption spectrum (blue) and excitation laser spectrum (red) are shown. The two-dimensional spectrum covers a rectangular region in frequency space because of the restricted the bandwidth over which we get comb resolution, improving to the repetition rate lock or implementing adaptive sampling28 will increase the bandwidth.

The two-dimensional spectrum shows two peaks. The peak at −1549 meV absorption photon energy and 1549 meV emission photon energy corresponds to absorption and emission at the HH exciton energy whereas the coupling between the HH and LH exciton resonances results in the peak at −1557 meV absorption photon energy and 1549 emission photon energy. Similar coupling information can be obtained if the emission of the FWM signal is detected at LH energy (1557 meV). In either case, if HH and LH resonances represented 2 uncoupled resonances, then the coupling peak would be absent. The coupling information, in addition to the rapid acquisition and high resolution associated with frequency combs, can potentially be used as an efficient way to separate different species in a sample containing a mixture, which cannot be achieved using only one dimensional high resolution energy spectra.

In addition to the high resolution, our method provides a route to very short acquisition times, which is critical in many applications. In this experiment, the full two dimensional spectrum can be obtained under 3 minutes, which can be improved by optimizing the stage performance. A more dramatic speed-up could be realized by using three frequency combs with different repetition rates to improve the resolution and acquisition speed in the evolution time as well. Three combs would allow the same spectrum with comb resolution in both energy axes to be obtained in a few seconds (Supplementary text).

## References

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Li, H., Bristow, A. D., Siemens, M. E., Moody, G. & Cundiff, S. T. Unraveling quantum pathways using optical 3D Fourier-transform spectroscopy. Nature Communications 4, 1390, https://doi.org/10.1038/ncomms2405, http://www.nature.com/articles/ncomms2405#supplementary-information (2013).

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Hamm, P. & Zanni, M. Concepts and methods of 2D infrared spectroscopy. (Cambridge University Press, 2011).

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Thämer, M., De Marco, L., Ramasesha, K., Mandal, A. & Tokmakoff, A. Ultrafast 2D IR spectroscopy of the excess proton in liquid water. Science 350, 78–82, https://doi.org/10.1126/science.aab3908 (2015).

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Cundiff, S. T. Coherent spectroscopy of semiconductors. Opt Express 16, 4639–4664, https://doi.org/10.1364/Oe.16.004639 (2008).

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Turner, D. B. & Nelson, K. A. Coherent measurements of high-order electronic correlations in quantum wells. Nature 466, 1089–1092, http://www.nature.com/nature/journal/v466/n7310/abs/nature09286.html#supplementary-information (2010).

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Wong, C. Y. & Scholes, G. D. Biexcitonic Fine Structure of CdSe Nanocrystals Probed by Polarization-Dependent Two-Dimensional Photon Echo Spectroscopy. The Journal of Physical Chemistry A 115, 3797–3806, https://doi.org/10.1021/jp1079197 (2011).

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Schlau-Cohen, G. S., Dawlaty, J. M. & Fleming, G. R. Ultrafast Multidimensional Spectroscopy: Principles and Applications to Photosynthetic Systems. IEEE Journal of Selected Topics in Quantum Electronics 18, 283–295, https://doi.org/10.1109/JSTQE.2011.2112640 (2012).

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Zhuang, W., Hayashi, T. & Mukamel, S. Coherent Multidimensional Vibrational Spectroscopy of Biomolecules; Concepts, Simulations and Challenges. Angewandte Chemie (International ed. in English) 48, 3750–3781, https://doi.org/10.1002/anie.200802644 (2009).

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Baumann, E. et al. Spectroscopy of the methane ν3 band with an accurate midinfrared coherent dual-comb spectrometer. Physical Review A 84, 062513 (2011).

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Boudreau, S., Levasseur, S., Perilla, C., Roy, S. & Genest, J. Chemical detection with hyperspectral lidar using dual frequency combs. Opt. Express 21, 7411–7418, https://doi.org/10.1364/OE.21.007411 (2013).

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## Acknowledgements

The project was funded by the Intelligence Advanced Research Projects Activities (IARPA). Contract 2016-16041300005. We thank Eric Martin for helpful discussions.

## Author information

Authors

### Contributions

Authors contributed equally for designing and building the experimental setup, acquiring and analyzing the results and writing the manuscript.

### Corresponding author

Correspondence to Steven T. Cundiff.

## Ethics declarations

### Competing Interests

Bachana Lomsadze and Steven T. Cundiff are inventors on provisional patent application 62/394,771 submitted by the University of Michigan that covers “Frequency comb based multi-dimensional coherent spectroscopy”.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Lomsadze, B., Cundiff, S.T. Multi-heterodyne two dimensional coherent spectroscopy using frequency combs. Sci Rep 7, 14018 (2017). https://doi.org/10.1038/s41598-017-14537-z

• Accepted:

• Published:

• ### Elastic tape behavior of a bi-directional Kerr-lens mode-locked dual-comb ring laser

•  & Richard S. Arnold

Optics Letters (2020)

• ### Dual frequency-comb spectroscopy of chromophores in condensed phases

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• , Tai Hyun Yoon
•  & Minhaeng Cho

Chemical Physics (2019)

• ### Frequency comb-based multidimensional coherent spectroscopy

• T. Cundiff Steven
• , G. Cerullo
• , J. Ogilvie
• , F. Kärtner
• , M. Khalil
•  & R. Li

EPJ Web of Conferences (2019)

• ### Theory of three-pulse photon echo spectroscopy with dual frequency combs

• Jonggu Jeon
• , JunWoo Kim
• , Tai Hyun Yoon
•  & Minhaeng Cho

Journal of the Optical Society of America B (2019)

• ### Single-scan acquisition of multiple multidimensional spectra

• Travis M. Autry
• , Galan Moody
• , James Fraser
• , Corey McDonald
• , R. P. Mirin
•  & Kevin Silverman

Optica (2019)


text:
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Syntax

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Description

Defines a new typing environment and the corresponding \start and \stop commands. The first argument is the name of the typing environment, the second argument are the options to apply for this environment.

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Example

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# Question about Similarity vs Dissimilarity Matrix

Right now, I'm working on a coming up with a similarity vs dissimilarity matrix for a set of data points for a clustering algorithm. My question is, if I want to use one of the many clustering algorithms given in $$R$$, such as the K-Medoids algorithm, does it require a similarity or dissimilarity matrix as its parameter?

What's the difference between the two?

If I use the Gower Distance from the Daisy function in R, does it output a similarity or dissimilarity matrix?

Also lets assume that I have $$n$$ features and they are all categorical (this is just an example) I a custom distance function where when comparing two data points $$G$$ and $$H$$, I use the formula $$\sum_i^nX_i$$ where $$X_i = 1$$ if feature $$i$$ of $$G$$, $$G_i$$ and feature i of $$H$$, $$H_i$$ are equal to each other. So, $$X_i=1$$if and only if $$G_i=H_i$$ for feature $$i$$ for all of the $$n$$ categorical features. Will this result in getting a similarity or dissimilarity matrix?

Also, as mentioned above, if I want to use one of the many clustering algorithms given in $$R$$, such as the K-Medoids algorithm, does it require a similarity or dissimilarity matrix as its parameter?

In general does the similarity or dissimilarity matrix get used for these

A similarity is larger if the objects are more similar.

A dissimilarity is larger if the objects are less similar.

This sounds trivial, but if you get the sign wrong, you suddenly search for the worst rather than the best solution...

It's easy to see that a distance is always a dissimilarity.

K-medoids could be implemented for similarities, but I am not aware of any implementation that does not expect the data to be a dissimilarity. It may be fine to simply pass -similarity to many implementations. Because all they care for is to minimize a sum of dissimilarities, which can trivially be shown then to be equivalent to maximizing the sum of similarities.

In many machine learning packages dissimilarity, which is a distance matrix, is a parameter for clustering (sometimes semi-supervised models).

However the real parameter is type of the distance. You need to tune distance type parameter like k in kmeans. (You need to optimize the distance type according to your business objective).

Check https://en.wikipedia.org/wiki/Distance for distance types. Additionally in some cases, correlation is used for similarity.


text:
# Brandeis Graduate Student Seminar (GSS) – Fall 2015

### Fall 2015

The GSS meets Fridays at 2pm in GS300. All math graduate students are welcome. Free pizza is provided, and beer is often available too.

• Aug 28 – Introduction and Panel Discussion
• Featured panelists: Aru Ray, Yan Zhuang, Devin Murray & Shaunak Deo
• Moderator: Jordan Tirrell
• We will introduce the Graduate Student Seminar and organize speakers and our weekly teas. This will be followed by a Q&A panel for graduate students. Our goal is to encourage discussion and disseminate graduate school wisdom (topics may include courses, requirements, advisors, thesis writing, job applications, socializing, sanity etc). Beer will be free.
• Sept 4 – Yan ZhuangThrough the Probabilistic Lens
• Abstract: In this expository talk, we present probabilistic proofs of three non-probabilistic results: Erdős’s theorem on sum-free sets (number theory), Sperner’s theorem (extremal combinatorics), and the Weierstrass approximation theorem (analysis). No background knowledge is assumed beyond the very basics of probability theory.
• Sept 11 – Tue Ly: Geometric and probabilistic ideas in Diophantine approximation.
• Abstract: In this talk, I will briefly introduce some classical results of Diophantine approximation on the real numbers, and discuss about some tools in geometry and probability to prove their higher dimensional analogues.
• Sept 18 – Duncan Levear: Linear Programming and Applications
• Abstract: Many problems in Operations Research can be framed as linear programs: optimizing a linear function subject to linear constraints. By proving some results about these problems, chiefly the Fundamental Theorem of Linear Programming, we will discover shortcuts to finding an optimal solution. This will shed light on ubiquitous problems from our daily lives: such as scheduling, dieting/mixing, and cooking Thanksgiving dinner.
• Sept 25 – McKee Krumpak: Regular polytopes and the reflection groups associated to them
• Abstract: Approximately covered topics will be the enumeration of the regular polytopes and some geometric intuition for their symmetry groups. Possibly something about the associated Coxeter diagram of a reflection group, depending on how things go.
• Oct 2 – Job Rachowicz: Applications of Category Theory
• Abstract: In this talk we’ll address the concern that category theory is just `abstract nonsense.’ We’ll (very) briefly remind ourselves of the basics and some well-known categories. Then we’ll talk about some applications of category theory in and out of mathematics.
• Oct 9 – Frankie Petronella: Why you should care about PDE’s
• Abstract: In this talk I will be discussing PDE’s. I will provide a short introduction into the subject area. I will discuss some of the methods used to prove the existence and uniqueness of solutions; including one that I have used. I will provide an example that illustrates why existence of solutions is not guaranteed. Finally, I will discuss some of the more fun applications of PDE’s.
• Oct 16 – Devin Murray: Using non-standard analysis to study groups and metric spaces
• Abstract: There are two normal formal foundations for analysis, the standard one uses the delta-epsilon definition of limits and goes forward from there. The non-standard approach (which is perhaps more in line with Newtons original idea) is to enrich the real numbers with ‘infinitesimals’ and to define everything with these infinitesimals instead of limits. From my exposure to non-standard analysis in the past, I got the impression that it was mostly for overly-enthusiastic undergraduates, model theorists, and quacks. I will attempt to correct this misconception and show that non-standard analysis contains a powerful set of tools which aid in discussing and solving many delicate and interesting problems in groups theory and metric geometry.
• Oct 30 – Josh Eike: Counter-Intuitive Games and Their Strategies
• Abstract: Mathematicians, economists, and others like to come up with clever games that challenge or expose people’s intuitions. The Monty Hall Problem and the Prisoner’s Dilemma are classics, but there are many more. We will see how the Brandeis math department fares against such challenges. There will be tricks and there will be treats. Keep your wits about you and you might leave with a prize!
• Nov 6 – Tarakaram Gollamudi: 4 Riemann – Roch’s I know
• Abstract: It has long been understood that there are deep connections between algebraic number theory and algebraic geometry. In this talk, I introduce Arakelov theory which expose some of these deep connections. In particular, I will talk about how we can extend definition of “genus” to Number Fields, and I will introduce Reimann-Roch for Number Fields, which is an analogue of RR for Curves. If time permits I will also introduce Alexander Grothendieck RR for Number Fields. I will try to make the talk self contained. But little exposure to RR for algebraic curves, Minksowski Theory and extensions of valuations would be helpful. (Thanks to Carl Wang for his help over the Summer).
• Nov 13 – Ying Zhou: A formula for the permutation of maximal green sequences in $A_n$ straight orientation and beyond.
• Abstract: In this talk I will define quivers, quiver mutations, maximal green sequences, the associated permutation of a maximal green sequence and then give a formula for the permutation of maximal green sequences in $A_n$ straight orientation. This is a joint work with Professor Kiyoshi Igusa. Recent research such as the work in one-sink orientations of $A_n$ and the technique of quiver cutting will also be discussed.
• Nov 20 – Special Guest Jessica Sorrell: Lattice-based Cryptography from the Learning with Errors Problem
• Abstract: The most ubiquitous cryptographic algorithms today rely on the hardness of factoring or the discrete log problem, but these can be efficiently solved by a quantum computer implementing Shor’s algorithm. The search for a secure and efficient cryptographic successor has motivated renewed interest in the study of hard problems on lattices. We will talk about one such problem, the learning with errors problem, and show how one may use the hardness of this problem and its ring variant to construct public key encryption systems.
• Dec 4 – Jordan Awan: Graph Coloring
• Abstract: We will introduce the problem of coloring the faces (or vertices) of a planar undirected graph. After developing some machinery, including Euler’s formula, we will prove the 5-color theorem.
• Dec 4 – Angelica Deibel: The generic free basis property in Right-Angled Coxeter Groups
• Abstract: I will prove that every Right-Angled Coxeter Group either is virtually abelian or satisfies the generic free basis property.


text:
# Semi-circle

Geometry Level 3

In a right triangle, a semi-circle is inscribed so that its diameter lies on the hypotenuse and its center divides the hypotenuse in two segments of lengths 15 and 20.the length of the arc of the semicircle between the points at which the legs touch the semicircle is $$K \pi$$. Then find the numerical value of $$K$$.

×