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# p-adic poly-Bernoulli numbers

We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}_p$ and Taekyun Kim's method.

But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $\mathbb{Z}_p$?

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## 1 Answer

I think what you are looking for is:

$$B_n^{k}(x)=\frac{Li_k (1-e^{-t})}{t}\int_{\mathbb{Z}_p}(x+y)^ndy$$

for $n\geq0$, and $Li_k$ the polylogarithmic function.

See for example, "Poly-Bernoulli Polynomials and Their Applications", by Kim, Jang and Seo.

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Are those $t$'s supposed to be $n$'s? – Hurkyl Sep 30 at 13:18
@Hurkyl Mmmm, no, I don't think so. The paper is avaible online, so I've included the link. – Myshkin Sep 30 at 13:30


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You are currently browsing the tag archive for the ‘pair of pants’ tag.

Jeremy Kahn kindly sent me a more detailed overview of his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission, this is reproduced below in its entirety.

Editorial note: I have latexified Jeremy’s email; hence “dhat-mu” becomes $\hat{d}\mu$, “boundary-hat” becomes $\hat{d}$, and “boundary-tilde” becomes $\tilde{d}$. I also linkified the link to Caroline Series’ paper.

Hi Danny,

I was busy with the conference on Thursday and Friday, and taking a break on Saturday, and now I’ve finally had a chance to read your blog, and reply to your message. I decided (especially as Jesse had requested it) to write out a complete outline of the theorem. I’m sending a copy of this message to you, Jesse Johnson, Ian Agol, and Francois Labourie: you are all welcome to reproduce it, as long as it is reproduced in its entirety, and states clearly that this is joint work with Vladimir Markovic. Of course, time and energy permitting, I’ll be happy to answer any questions.

Here is an outline of the argument, working backwards to make it clearer:

1. We want to construct a surface made out of skew pants, each of which has complex half-length close to $R$, and which are joined together so that the complex twist-bends are within $o(1/R)$ of $1$. Using a paper of Caroline
Series (published in the Pacific J. of Mathematics) we show that these surfaces are quasi-isometrically embedded in the universal cover of the three-manifold.

2. Consider the following two conditions on two Borel measures $\mu$ and $\nu$ on a metric space $X$ with the same (finite) total measure:

A. For every Borel subset $A$ of $X$, $\mu(A)$ is less than or equal to the $\nu$-measure of an $\epsilon$ neighborhood of $A$.

B. There is a measure space $(Y, \eta)$ and functions $f: Y \to X$ and $g: Y \to X$ such that $\mu$ and $\nu$ are the push-forwards by $f$ and $g$ respectively of the measure $\eta$, and the distance in $X$ between $f(y)$ and $g(y)$ is less than $\epsilon$ for almost every $y \in Y$.

It is easy to show that B implies A (also that A is symmetric in $\mu$ and $\nu$!). In the case where $\mu$ and $\nu$ are discrete and integral measures (the measure of every point is a non-negative integer), we can show that A implies B (and $Y$ will be a finite set with the counting measure) using Hall’s marriage theorem. In fact, the statement that A implies B for discrete and integral measures is easily shown to be equivalent to Hall’s marriage theorem. I don’t know if A implies B in general because I don’t know how to replace the inductive algorithm for Hall’s marriage theorem with a method that works for a relation between two general measure spaces.

We call $\mu$ and $\nu$ $\epsilon$-equivalent if they satisfy condition A, and note that the condition is additively transitive: if $\mu$ is $\epsilon$-equivalent to $\nu$, and $\nu$ is $\delta$-equivalent to $\rho$, then $\mu$ and $\rho$ are $(\epsilon+\delta)$-equivalent.

3. Suppose that $\gamma$ is one boundary component of a pair of skew pants $P$. We can form the common orthogonals in $P$ from $\gamma$ to each of other other two cuffs. For each common orthogonal, at the point where it meets $\gamma$, we can find a unit normal vector to $\gamma$ that points along this common orthogonal. The two resulting normal vectors are related by a translation along the half-length of $\gamma$ (the suitable square root of the loxodromic element for $\gamma$), so we will call them a pair of opposite unit normal vectors (or pounv for short) and they live in the live in the bundle of pounv’s which is conformally equivalent to the complex plane mod the lattice generated by the half-length of $\gamma$ and $2\pi i$. We give the bundle of pounv’s the Euclidean metric inherited from the complex plane, and also the Lebesgue measure.

4. Given a measure on pants we can produce a measure on the union pounv bundles of the boundary geodesics as follows: if the measure is a unit atom on one pair of skew pants, the resulting measure on pounv bundles is a unit atom on the pounv bundle of each the cuffs, at the pounv described in step 3. We extend to a general measure by linearity. This produces a linear operator we will call the $\hat{d}$ operator.

If we are given a positive integral formal sum of pants (or a multi-set of pants) we can think of it as an integral measure on the space of pants.

5. On the pounv bundle for each closed geodesic we can apply a translation of $1 + i \pi$; we will call this translation $\tau$. We can think of $\tau$ as a map from the union of the pounv bundles to itself.

6. Let $\mu$ be an integral measure on pants with cuff half-lengths close to $R$. We can apply the $\hat{d}$ operator described in step 4 to obtain a measure on the union of pounv bundles of all the boundary geodesics; we will call the measure $\hat{d}\mu$. If $\hat{d}\mu$ and the translation of $\hat{d}\mu$ by $\tau$ are $\epsilon/R$ equivalent, then we can take two oriented pants for each pair of pants in our multi-set (taking each of the two possible orientations) and then fit all of these oriented pants into an oriented surface of the type described in step 1. We use Hall’s marriage theorem as described in step 2, and a very small amount of combinatorics.

If the measure $\hat{d}\mu$, restricted to a given pounv bundle, is $\epsilon/R$ equivalent to a rescaling of Lebesgue measure on that torus, then $\hat{d}\mu$ and $\tau$ of $\hat{d}\mu$ are $2\epsilon/R$-equivalent, which is what we wanted.

******************

This is as far as I got in the first talk at Utah, so it would be best to stop and take a breath for a moment. We haven’t really done anything, but we’ve reformulated the problem: the type of surface we want has been well-defined, and the problem of finding this surface has been reformulated as finding a measure on pairs of pants that satisfies a given criterion.

*****************

7. A two-frame for $M$ will comprise a tangent vector and a normal vector both at the same point, unit length and orthogonal. Given a two-frame we can rotate the tangent vector 120 degrees around the normal vector, using the right-hand rule; the orbit of this action is an ordered triple of two-frames, which will call a tripod. We can also rotate 120 degrees in the opposite direction, and obtain an anti-tripod.

8. A connected pair of two-frames is a pair of two frames along with a geodesic segment connecting them. Given $\epsilon$ and $r$, with $r$ large in terms of $\epsilon$, we can find a weighting function on connected two-frames such that the following properties hold whenever the weight is non-zero:

A. The length of the connecting segment is within $\epsilon$ of $r$.

B. If the normal vector of one two-frame is parallel translated along the connecting segment, then it forms an angle of less then $\epsilon$ with the normal vector of the other two-frame.

C. The angle between the the tangent vector of the two frame and (the tangent vector to) the connecting geodesic segment is exponentially small in $r$.

Moreover,

D. Given a pair of two-frames, the sum of the weights of the connecting geodesic segments is exponentially close (in $r$) to 1.

E. The weighting is geometrically natural, in that it depends only the length of the connecting segment, the angle between the parallel translated normal vectors, and the angles between the connecting segment and the tangent vectors.

We will describe the (relatively simple) weighting function in the end; we will use the exponential mixing of geodesic flow to obtain property D.

9. Given a tripod and an anti-tripod, we can form three pairs of two-frames by pairing the frames in order, and then we can measures (or weightings) on the connected pairs of two-frames, and then form the product measure (or weighting) by multiplying the weights of the three connections. This gives us a weighting on “connected pairs of tripods” (really a tripod and an anti-tripod) that is supported on connections that satisfy properties A, B, and C.

10. We call a perfect connection between two two-frames a geodesic segment that has a length of $r$, and angle of zero between the segment and the tangent vectors, and translates one normal vector to the other. If a tripod and an anti-tripod were connected by three perfect connection, then they would be a 1-dimensional retract of a flat pair of pants with three cuffs of equal length $R$, where $R$ is approximately $r + \log \cos \pi/6$ when $r$ is large. If the tripod and anti-tripod are connected by arcs that satisfy properties A and B, then the connected pair of tripods is still a retract of a skew pair of pants, whose cuffs have half-length within $\epsilon$ (or $10\epsilon$) of $R$. Thus there is a map from good connected pairs of tripods to good pairs of pants, which we will denote by $\pi$.

11. We can let $\tilde{\mu}$ be the measure on connected pairs of tripods, given by integrating the weighting of steps 8 and 9 with respect to the Liouville measure on pairs of tripods (or pairs of two-frames). We then push this measure forward by $\pi$ to obtain a measure $\mu$ on pairs of pants; after finding a rational approximation and clearing denominators, it will be the $\mu$ that was asked for in step 6. We will show that $\hat{d}\mu$ (taking the original irrational $\mu$) is $\epsilon/R$-equivalent to a rescaling of Lebesgue measure on each pounv bundle and thereby complete the proof.

12. A partially connected pair of tripods $T$ is a pair of tripods where we have connected two out of the three pairs of two-frames. To a partially connected pair of tripods we can assign a single closed geodesic $\gamma$ that is homotopic to the concatenation (at both ends) of the two connecting segments. If we connect the third pair of two-frames and apply $\pi$ we obtain a pair of pants $P$, and we can then find a pair of opposite unit normal vectors for gamma pointing to the two cuffs of $P$ (as described in step 3). We will describe a method for predicting the pounv for $\gamma$ and $P$ knowing only the partially connected tripod $T$: First, lift $T$ to the solid torus cover of $M$ determined by $\gamma$, and then follow geodesic segments from the tangent vectors of the two unconnected two frames of (the lift of) $T$ to the ideal boundary of this $\gamma$-cover. We can connect these two points in the boundary by two geodesics, each of which goes about half-way around this solid torus cover. We can then find the common orthogonals from each of these geodesics to (the lift of) $\gamma$, and then obtain two normal vectors to $\gamma$ pointing along these common orthogonals; it is easy to verify that these are half-way along $\gamma$ from each other (in the complex sense) and hence form a pounv. Property C of the connections between two-frames (and hence tripods) implies that this predicted pounv will be exponentially close (in $r$) to the actually pounv of any pair of pants $P$.

To summarize: given a good connected pair of tripods, we get a good pair of pants $P$, and taking one cuff gamma of $P$, we get a pounv for $\gamma$ as described in step 3. But we only need two out of the three connecting segments to get $\gamma$, and using the third pair of two frames, without even knowing the third connecting segment, we can predict the pounv for $\gamma$ and $P$ to very high accuracy.

13. We can then define the $\tilde{d}$ operator from measures on partially connected pairs of tripods to measures on the pounv bundles for the associated geodesics; this operator is just the linear extension of the operation in step 12. Given a connected pair of tripods, we can get three partially connected pairs of tripods in the obvious way; we can thereby extend $\tilde{d}$ to map measures on connected pairs of tripods to measures on the bundles of pounv’s; because the predicted pounv described in step 12 is exponentially close to the actual pounv described in step 3, the two measures $\tilde{d} \tilde{\mu}$ and $\hat{d}\mu$ are $\exp(-\alpha r)$-equivalent, by the B => A of step 2.

14. For each closed geodesic $\gamma$, we can lift all the partially connected tripods that give $\gamma$ to the $\gamma$ cover of $M$ described in step 12. There is a natural torus action on the normal bundle of $\gamma$, and this extends to an action on all of the solid torus cover associated to $\gamma$. Moreover, it acts on the (lifts of) partially connected tripods, and it does not change the weightings of the two established connecting segments, because of property E of the weighting function.

This is the crucial point: the effective weighting on a partially connected pair of tripods is not just the product of the weights of the two established connections, but that product times the sum of the weights of all possible third connections. By property D of the weighting function, this sum, while not constant, is exponentially close to being constant, so the effective weighting is exponentially close to being invariant under the torus action. Because the predicted pounv for a partially connected pair of tripods is equivariant for the torus action, the measure $\tilde{d} \tilde{\mu}$ is exponentially close to a torus invariant measure on the pounv bundle (which is necessary a rescaling of Lebesgue measure), in the sense that the Radon-Nikodym derivative is exponentially close to 1. It is then an easy lemma that the two measures are exponentially close in the sense of step 2. And then we’re finished: $\hat{d}\mu$ is exponentially close to $\tilde{d} \tilde{\mu}$, which is exponentially close to a rescaling of Lebesgue measure, which is what we wanted (with
overkill) in step 6.

15. It remains only to define the weighting function described in step 8, which is surprisingly simple: We take some left-invariant metric on $\text{PSL}_2(\bf{C})$, and hence on the two-frame bundle for $M$ and its universal cover. Given a connected pair of two-frames in $M$, we lift to the universal cover, to obtain two two-frames $v$ and $w$. We then flow $v$ and $w$ forward by the frame flow for time $r/4$ to obtain $v'$ and $w'$. We let $V$ be the $\epsilon$ neighborhood of $v'$, and $W$ be the $\epsilon$ neighborhood of $w'$, with the tangent vector of $w'$ replaced by its negation. Then the weighting of the connection is the volume of the intersection of $W$ with the image of $V$ under the frame flow for time $r/2$.

Properties A, B, and C are not difficult to verify. Property D follows immediately from exponential mixing: If we have $v$ and $w$ downstairs without any connection, and similarly define $v'$, $w'$, $V$ and $W$, then the sum of the weights of the possible connections will just be the volume of the intersection of the downstairs $W$ with the frame flow of $V$. By exponential mixing, this converges at the rate $\exp(-\alpha r)$ to the square of the volume of an $\epsilon$ neighborhood, divided by the volume of $M$.

We can normalize the weights by dividing by this constant.

Jeremy

One obvious comment to make is that the argument is remarkably short, and does not depend on any very delicate or complicated analytic estimates (maybe the argument that the glued up surfaces are quasi-geodesic is the most delicate part). It is fair to say that it defies the conventional wisdom in that respect — I was personally very surprised that the general method could be made to work, especially in light of the failure of Bowen’s program. Kudos to Jeremy and Vlad for their boldness and ingenuity.

Another comment to make is that the matching argument is surprisingly robust and general, and I expect it to have many broader applications. One thing I was confused about in my last post seems to be resolved by Jeremy’s sketch above — if I understand it correctly, one first (almost) pairs continuous measures, and only then approximates them by discrete integral measures (with a little bit of combinatorics at the end). And one really does need exponential mixing rather than just mixing.

Incidentally, apropos the matching argument, there are some interesting and well-known variations where things go haywire. For example, papers by Burago-Kleiner and (Curt) McMullen show that there are examples of separated nets in Euclidean space which are not bilipschitz to a lattice (though, interestingly, Curt shows that they are Holder equivalent). No such examples exist in hyperbolic space, because of — nonamenability and Hall’s marriage theorem! Roughly, when trying to match up points in two nets in hyperbolic space, one doesn’t need to look very far because the number of options grows exponentially. This is one reason why Kahn-Markovic need to control the matchings of their measures carefully, because it must be done on a very small scale (where the exponential growth does not kick in).

I thought I would also mention that in case my previous comments lead one to believe otherwise, exponential mixing of the geodesic flow on a hyperbolic manifold is somewhat delicate. Exponential mixing under a flow $g_t$ on a space $X$ preserving a probability measure $\mu$ means that for all (sufficiently nice) functions $f$ and $h$ on $X$, the correlations $\rho(h,f,t):= \int_X h(x)f(g_tx) d\mu - \int_X h(x) d\mu \int_X f(x) d\mu$ are bounded in absolute value by an expression of the form $C_1e^{-tC_2}$ for suitable constants $C_1,C_2$ (which might depend on the analytic quality of $f$ and $h$). For example, one takes $X$ to be the unit tangent bundle of a hyperbolic manifold, and $g_t$ the geodesic flow (i.e. the flow which pushes vectors along the geodesics they are tangent to, at constant speed). Exponential mixing should be contrasted with the much slower mixing of the horocycle flow on a hyperbolic surface, for which the correlation is bounded by an expression like $C_1(\log t)^{C_2}t^{-1}$. The geodesic flow on a hyperbolic manifold is an example of what is called an Anosov flow; i.e. the tangent bundle $TM$ splits equivariantly under the flow into three subbundles $E^0, E^s, E^u$ where $E^0$ is $1$-dimensional and tangent to the flow, $E^s$ is contracted uniformly exponentially by the flow, and $E^u$ is expanded uniformly exponentially by the flow. The best one knows for (certain) Anosov flows (by Chernov) is that the flow is stretched exponentially mixing, i.e. with an estimate of the form $C_1e^{-\sqrt{t}C_2}$. One knows exponential mixing for the geodesic flow on variable negative curvature surfaces by Dolgopyat, and on certain locally symmetric spaces, using representation theory. See Pollicott’s lecture notes here for more details. I don’t know if exponential mixing for geodesic flows is known on manifolds of variable negative curvature in high dimensions. Also I’d appreciate it if any reader who knows some ergodic theory can confirm/deny/clarify this paragraph . . .

(Update 8/12): Jeremy tells me that he and Vladimir only need “sufficiently high degree polynomial” mixing, so perhaps there is a decent chance the methods can be extended to variable negative curvature.

(Update 10/29): The paper is now available from the arXiv.

I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that every complete hyperbolic $3$-manifold $M$ contains a closed $\pi_1$-injective surface. Equivalently, $\pi_1(M)$ contains a closed surface subgroup. Apparently, Jeremy made the announcement at an FRG conference in Utah. This answers a long-standing question in $3$-manifold topology, which is a variation on some problems originally posed by Waldhausen. If one further knew that hyperbolic $3$-manifold groups were LERF, one would be able to deduce that all hyperbolic $3$-manifolds are virtually Haken, and (by a recent theorem of Agol), virtually fibered. Dani Wise (and others) have programs to show that hyperbolic $3$-manifold groups are LERF; if successful, this would therefore resolve some of the most important outstanding problems in $3$-manifold topology (in fact, I would say: the most important outstanding problems, by a substantial margin).

In fact, the argument appears to work for hyperbolic manifolds of every dimension $\ge 3$, and possibly more generally still. Details on the argument of Markovic-Kahn are scarce (Vlad informs me that they expect to have a preprint in a few weeks) but the sketch of the argument presented by Kahn is compelling. Roughly speaking, the argument (as summarized by Ian Agol in a comment at Jesse’s blog) takes the following form:

1. Given $M$, for a sufficiently big constant $R$, one can find “many” immersed, almost totally-geodesic pairs of pants (i.e. thrice-punctured spheres) with geodesic boundary components (i.e. “cuffs”) of length very close to $2R$. In fact, one can further insist that the complex length of the boundary geodesic is very close to $2R$ (i.e. holonomy transport around this geodesic does not rotate the normal bundle very much).
2. Conversely, given any geodesic of complex length very close to $2R$, one can find many such pairs of pants that it bounds, and moreover one can find them so that the normal to the geodesic pointing in to the surface is prescribed.
3. If one takes a sufficiently big collection of such geodesic pairs of pants, one has enough of them in oppositely-aligned pairs along each boundary component, that they can be matched up (by some version of Hall’s marriage theorem), and furthermore, matched up with a definite prescribed “twist” along the boundary components
4. One checks that the resulting (closed) surface is sufficiently close to totally geodesic that the ambient negative curvature certifies it is $\pi_1$-injective

Many aspects of this argument have a lot in common with some previous attempts on the surface subgroup conjecture, including one recent approach by Bowen (note: Bowen’s approach is known to have some fatal difficulties; the “twist” in 3. above specifically addresses some of them). All of these points deserve some comments.

First, where do the pairs of pants come from? If $P$ is a totally geodesic pair of pants with boundary components of length close to $2R$, the pants $P$ retract onto a geodesic spine, i.e. an immersed totally geodesic theta graph, whose edges all have length close to $2R$, and which meet at angles very close to $120$ degrees. One can cut this spine up into two pieces, which are obtained by exponentiating the edges of an infinitesimal (almost)-planar tripod for length $R$.

Given a tripod $T$ in some plane in the tangent space at some point of $M$, one can exponentiate the edges for length $R$ to construct such a half-spine; if $T$ and $T'$ are a pair of tripods for which the exponentiated endpoints nearly match up, with almost opposite tangent vectors, then the resulting half-spines can be glued up to make a spine, and thickened to make a pair of pants. One key idea is to use the exponential mixing property of the geodesic flow on a hyperbolic manifold, e.g. as proved by Pollicott. Given some tolerance $\epsilon$, once $R$ is sufficiently large, the mixing result shows that the set of such pairs of tripods for which such a matching occurs have a definite density in the space of all pairs (and in fact, are more and more equidistributed in this space, in probability). In fact, one may even insist that two of the pairs of prongs join up to make some specific closed geodesic of length almost $2R$, and vary the pair of third prongs a very small amount so that they glue up. This takes care of the first two points; this seems quite uncontroversial (exponential mixing comes in, I suspect, to know that one doesn’t need to wiggle the pair of third prongs much, having paired the first two pairs).

The matching (i.e. the gluing up of opposite pant cuffs) apparently is done by some variant of Hall’s marriage theorem. One needs to know (I think) that for any finite set of cuffs to be glued, the set of other cuffs that they could potentially be glued to is at least as big in cardinality. This probably needs some thought, but it is plausibly true: given a cuff, it can be glued to any cuff which is almost oppositely aligned to it, and since there is some tolerance in the angle of gluing — this is where dimension at least $3$ is necessary — and moreover, since oriented cuffs are almost equidistributed, one can always find “more” cuffs that are opposite, up to a bit of tolerance, to any given subset of cuffs (of course, more details are necessary here). There is an extra wrinkle to the argument, which is that the gluing must be done with a “twist” of a definite amount, so that cuffs are not glued up in such a way that the perpendicular geodesic arcs joining pairs of cuffs match up.

(Update 8/8: I think there must necessarily be more details to the matching argument, as very loosely described above. There are at least two additional issues that must be dealt with in order to perform a matching: a parity issue (since each pants has an odd number of cuffs) and a homology issue (if the argument relativizes, so that one fixes some collection of cuffs in advance and glues up everything else, one concludes a posteriori that the union of the unglued cuffs is homologically inessential). Probably the parity issue (and more subtle divisibility issues) can be solved by gluing with real-valued weights, then approximating a real solution by a rational solution, and multiplying through to clear denominators. Maybe the homology issue does not arise, if in fact the argument doesn’t relativize.) Both these issues suggest that one does not specify in advance a collection of pants to be glued up, but rather wants to glue up a definite number of pants from some subset.)

This issue of a twist is important for the 4th point, which is perhaps the most delicate. In order to know that the resulting surface is $\pi_1$-injective, one must use geometry. A closed (immersed) surface in a hyperbolic manifold which is (locally) very close to being totally geodesic is $\pi_1$-injective. One way to see this is to observe that a geodesic loop in the surface is almost geodesic in the manifold; the ambient negative curvature means that the geodesic can be shrunk (by the negative of the gradient of length in the space of loops) to become geodesic in the ambient manifold; if it is close to being geodesic at the start, it very quickly becomes totally geodesic, without getting much shorter. Any closed geodesic in a hyperbolic manifold is essential.

If one builds a surface by gluing up almost totally geodesic pieces in such a way that there is almost no angle along the gluing, the resulting surface is almost geodesic, and therefore injective. However, one must be very careful to control the geometry of the pieces that are glued, and this is hard to do if the injectivity radius is very small. A geodesic pair of pants has area $2\pi$ no matter how long its boundary components are. So if the boundary components have length $2R$, then at the points where they are thinnest, they are only $e^{-R}$ across. If cuffs are glued where the pants are thinnest, even if the gluing angle is very small, the surfaces themselves might twist through a big angle in a very short time. So one needs to make sure that the thinnest part of one pants are glued up to a thicker part of the next, which is glued to a thicker part of the next . . . and so on. This is the point of introducing the twist before gluing: the twists accumulate, and before one has glued $R$ pieces together, one has entered the thick part of some pants, where the injectivity radius is bounded below by some universal constant.

Anyway, this seems like a really spectacular development, with an excellent chance of working out. Some of the ingredients — e.g. the exponential mixing of the geodesic flow — work just as well in variable negative curvature. In fact, some version of it should work for arbitrary hyperbolic groups (using Mineyev’s flow space). Without knowing more details of the argument, one can’t say how delicate the last part of the argument is, and how far it generalizes (but readers are invited to speculate . . .)

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# Derive fluorescent scattering cross section as an approximation when $\omega \sim \omega_0$

The scattering cross section for an bounded electron with resonant frequency $\nu_0 =\frac{\omega_0}{2 \pi}$ under the effect of an EM with frequency $\nu=\frac{\omega}{2\pi}$ is $$\sigma_d=\bigg( \frac{8}{3} \pi r_0^2 \bigg) \frac{\omega^4}{(\omega ^2 -\omega_0^2)^2+(\Gamma \omega)^2 }$$

Where $\Gamma$ is the dissipation factor and $r_0$ is electron classical radius.

On textbook, when fluorescent cross section is introduced, it is said that the previous expression is approximated to the following when $\omega \sim \omega_0$

$$\bigg( \frac{8}{3} \pi r_0^2 \bigg) \frac{\omega^4}{(\omega ^2 -\omega_0^2)^2+(\Gamma \omega)^2 }\to_{\omega \sim \omega_0} \bigg( \frac{3}{2} \pi \frac{\lambda_0^2}{ 4 \pi^2} \bigg) \frac{\gamma^2}{(\omega -\omega_0)^2+(\Gamma /2)^2 }$$

Where $\lambda_0 =\frac{c}{\nu_0}$ and $\gamma$ represents the part of $\Gamma$ related to dissipation due to irradiation. (In general $\Gamma= \gamma +\gamma'$ where, $\gamma'$ is dissipation due to other effects, such as the presence of atoms near the electron).

How to prove the previous approximation mathematically?

• Since you are adding $(\omega^2-\omega_0^2)^2$ and $(\Gamma\omega)^2$, then $[\Gamma] = [\gamma] = [\omega]$. Then, $[\bigg( \frac{3}{2 \cdot 4 \pi^2} \pi \lambda_0^2 \bigg) \frac{\gamma}{(\omega -\omega_0)^2+(\Gamma /2)^2 }] = \frac{[\sigma_d]}{[\omega]}$. Your formula can't be right. – Spirine Aug 19 '17 at 15:29
• @Spirine Thanks a lot, I apologize it is $\gamma^2$, I fixed it – Sørën Aug 19 '17 at 16:43

Maybe this helps.

$$\frac{\omega ^4}{(\omega ^2 - \omega_0^2)^2+(Γ\omega)^2} = \frac{\omega ^4}{(\omega - \omega_0)^2(\omega + \omega_0)^2+(Γ\omega)^2}$$

Since $\omega \approx\omega_0$, we can consider that $(\omega + \omega_0)^2\approx(\omega + \omega)^2 \approx (\omega_0 + \omega_0)^2\approx4\omega^2$ because good ol' physics and totally valid approximations. So, the relation becomes

$$\frac{\omega ^4}{4\omega^2(\omega - \omega_0)^2+(Γ\omega)^2} = \frac{1}{4} \frac{\omega ^2}{(\omega - \omega_0)^2+(Γ/2)^2}$$

This can also be considered as $$\frac{1}{4} \frac{\omega_0 ^2}{(\omega - \omega_0)^2+(Γ/2)^2}$$

Based on the same approximation made above.

Now you can see that the main fraction is similar to what you wanted. All that's left is to use the relations for the other parameters and it should get you to the result (as long as there are no typos there). I did not do it myself because it's not clear to me what $\gamma$ is.

In his answer, Victor Palea gave you the mathematical development that is needed for your derivation. Now, what you need is a answer from physics, that is what is $\gamma$ in your case?

From what I read, I understand non relativistic classical model of dipolar radiation with radiation dampening at the rate $\Gamma$. Something that classically correspond to an equation looking like the following:

$$\stackrel{..}{x} - \, \Gamma \stackrel{.}{x} + \, \omega_0^2 x = C$$

Short answer: $\gamma$ is the classical damping factor due to the radiation of moving charges (see Larmor's formula or Abraham–Lorentz force). For an electron it reads:

$\gamma = \dfrac{2}{3}\omega_0^2 \dfrac{r_0}{c}$ (check homogeneity)

So $\left(\dfrac{8}{3} \pi r_0^2 \right) \omega_0^2 = \left(\dfrac{8}{3} \pi\right) \dfrac{9}{4} \dfrac{\gamma^2 c^2}{\omega_0^2} = \left( 6 \pi\right)\dfrac{c^2}{4\pi^2\nu_0^2} \, \gamma^2 = 6 \pi \dfrac{\lambda_0^2}{4\pi^2} \, \gamma^2$

And you have a factor 1/4 to add from the Taylor expansion done by Victor Palea.

Finally, notice that I am not neglecting other effects represented through $\gamma'$, so we still have $\Gamma = \gamma + \gamma'$.


text:
Examining the slide

Imagine that the we are examining bacteria on a slide under a microscope. The slide is represented as a two-dimensional grid of cells, each of which is either empty or filled. If the cell is empty we associate it with $$0$$ and if the cell is filled we associate it with $$1$$. Any group of cells which are connected horizontally,vertically or diagonally is called a blob. We want to write a program to estimate the severity of infection.

For example, there are $$2$$ blobs in the example above, the largest of which constitutes of $$5$$ cells. What is the size of the largest blob in in this square matrix?

×


text:
# Wildcard to get files by exact extension

While diagnosing a larger batch script that needs to loop files with *.log extension I've found a funny behaviour. In a sample directory with files like this:

bar.log
foo.log
foo.log.ignore
foo.log.log-1676521099
not-related

... my little test script:

@echo off
setlocal enabledelayedexpansion
set DEF_LOG="C:\test\*.log"
for %%i in (%DEF_LOG%) do (
echo %%i
)

... prints this:

C:\test\bar.log
C:\test\foo.log
C:\test\foo.log.log-1676521099

Digging deeper, I've found that's how Windows wildcards have been designed:

C:\>dir "C:\test\*.log" /b
bar.log
foo.log
foo.log.log-1676521099

My question is: how can I list all files that end exactly with .log?

-
However, it is interesting that the file named test.log.txt do not pass C:\test*.log filter... –  mihai_mandis Dec 24 '13 at 13:29

The source of the behavior is short file names. Unless disabled, file names that exceed the old 8.3 format will be assigned a short name that does match the 8.3 format. If the extension exceeds 3 characters, then the short name will consist of the first 3 characters of the long name extension. Execute the dir /x "C:\test\*.log" to see the short names.

All commands will test both the long and the short name when looking for matches.

A common way to get your desired result is to use DIR /B piped to FINDSTR. Note that you should avoid using delayed expansion when expanding a FOR variable because that will corrupt values that contain !. The ! character may appear in a file name.

@echo off
pushd "C:\test"
for /f "eol=: delims=" %%F in ('dir *.log^|findstr /lie ".log"') do echo %%~fF
popd

-
Thanks for the bonus tip about enabledelayedexpansion—I've just verified that it mangles file names that contain exclamation marks. (Since I need that for another feature, I suspect my script needs a full rewrite.) –  Álvaro G. Vicario Dec 30 '13 at 13:27

Use a check inside loop to get only files with desired extension

@echo off
setlocal enabledelayedexpansion

rem set desired extension for additional filter
set extn=.log
set DEF_LOG="C:\temp\*.log"

for %%i in (%DEF_LOG%) do (
rem if file extension is equal to our ext
if "%%~xi"=="%extn%" echo %%i
)

-

The Easyiest and faster way to do that is to test the modifier %%~x with your extension.

@echo off&cls
for /f "delims=" %%a in ('dir /b') do if /i "%%~xa"==".log" echo %%a
pause

-


text:
# Covariance of relativistic free particle Lagrangian under Lorentz transformation

How does the Lagrangian of a free particle: $$L = - \sqrt{1- \Big(\frac{dx}{dt}\Big)^{2}}$$ transform under the co-ordinate transformation $(q,\tau)$? $$q = \cosh(\psi)x + \sinh(\psi)t,$$ $$\tau = \sinh(\psi)x + \cosh(\psi)t.$$

I've seen this worked out in the case where the boost is expressed in terms of $\gamma$ and relative velocity but not when there are hyperbolic trig functions.

My attempt:

Writing x and t interms of $q$ and $\tau$ and then trying to write $\frac{dx}{dt}$ in terms of $\frac{dq}{d\tau}$.

However I am not able to do that! Am I missing something?

• Using the chain rule you can show that $\frac{dx}{dt}$ and $\frac{dq}{d\tau}$ are both equal to $c(\tanh{\psi}+\tanh^{-1}{\psi})$, thus your Lagrangian density is indeed invariant. – G K Sep 6 '18 at 14:09
• @FredericThomas He is talking about the Lagrangian, while you are reffering to the Action. – G K Sep 6 '18 at 14:21
• @GK Thanks ! But surely you do mean $\tanh \psi + \coth \psi$ i.e. the second term is $(\tanh \psi )^{-1}$ instead of $\tanh^{-1} \psi$, right ? – Astronomer Sep 6 '18 at 16:15
• Yes thats right! I am sorry for the confusion. – G K Sep 6 '18 at 16:20

I think while doing the calculations, I have found the answer. @GK gave an answer but I think there is a subtle mistake in that approach because both q and $\tau$ are coupled transformations of x and t. Thus the total derivative is not equal to the partial one. So this is my solution.

$q$ and $\tau$ both can be expressed a function of $(x,t)$. From the definition of a total derivative $$\frac{dq}{dt} = \Big(\frac{\partial q}{\partial x}\Big)\frac{dx}{dt} + \frac{\partial q}{\partial t}$$

$$\Rightarrow \frac{dq}{dt} = \cosh(\psi) \frac{dx}{dt} + \sinh(\psi)$$

Similarly we have $$\frac{d\tau}{dt} = \Big(\frac{\partial \tau}{\partial x}\Big)\frac{dx}{dt} + \frac{\partial \tau}{\partial t}$$

$$\Rightarrow \frac{d\tau}{dt} = \sinh(\psi) \frac{dx}{dt} + \cosh(\psi)$$

From (1) and (2) we have $$\frac{dq}{d\tau} = \dfrac{\cosh(\psi) \frac{dx}{dt} + \sinh(\psi)}{\sinh(\psi) \frac{dx}{dt} + \cosh(\psi)}$$

$$\Rightarrow 1- \Big(\frac{dq}{d\tau}\Big)^{2}= 1- \Bigg(\dfrac{\cosh(\psi) \frac{dx}{dt} + \sinh(\psi)}{\sinh(\psi) \frac{dx}{dt} + \cosh(\psi)}\Bigg)^{2}$$

Going through the algebra and simplifying using (2) we get

$$\Rightarrow 1- \Big(\frac{dq}{d\tau}\Big)^{2}= \dfrac{(\sinh^{2}(\psi) - \cosh^{2}(\psi))(\frac{dx}{dt})^{2}+(\cosh^{2}(\psi) - \sinh^{2}(\psi))}{(\frac{d\tau}{dt})^{2}}$$

Using the well know trig identity $\cosh^{2}(\psi) - \sinh^{2}(\psi) = 1$ we have,

$$\Rightarrow 1- \Big(\frac{dq}{d\tau}\Big)^{2}= \dfrac{1-(\frac{dx}{dt})^{2}}{(\frac{d\tau}{dt})^{2}}$$

We know the Lagrangian $L^{\prime} = - \sqrt{1- \Big(\frac{dq}{d\tau}\Big)^{2}}$

Thus $$L^{\prime} = (\frac{dt}{d\tau})L$$

$$\Rightarrow L^{\prime}(\dot{q}) d\tau = L(\dot{x}) dt$$

$$\Rightarrow S^{\prime} [L^{\prime}] = S[L]$$

Even though the Lagrangian doesn't stay the same, the action does !


text:
# M. Faierman

Search this author in Google Scholar

Articles: 1

### Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$

M. Faierman

Methods Funct. Anal. Topology 22 (2016), no. 4, 330-345

We consider a spectral problem over $\mathbb{R}^n$ for a Douglis-Nirenberg system of differential operators under limited smoothness assumptions and under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ in the complex plane with vertex at the origin. We pose the problem in an $L_p$ Sobolev-Bessel potential space setting, $1 < p < \infty$, and denote by $A_p$ the operator induced in this setting by the spectral problem. We then derive results pertaining to the Fredholm theory for $A_p$ for values of the spectral parameter $\lambda$ lying in $\mathcal{L}$ as well as results pertaining to the invariance of the Fredholm domain of $A_p$ with $p$.


text: Tetration FAQ Discussion - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Tetration FAQ Discussion (/showthread.php?tid=5) Pages: 1 2 3 RE: Intro and Formula sheet - andydude - 11/15/2007 I just realized that the nth tetrate makes an incredibly unified terminology system if the other suffix used with tetration was used for iteration as well! I've been using the term "orbit" for the third way of viewing iteration but this is classically viewed as an indexed set, not a function per se. So this could be the new terminology: iteration (binary), nth iterate $f^2(x)$, iterational $f^x(0)$ (formerly orbit). tetration (binary), nth tetrate ${}^{3}x$ (formerly hyperpower, tower), tetrational ${}^{x}e$. Andrew Robbins RE: Intro and Formula sheet - jaydfox - 11/15/2007 Yes, I like the iterate/iterational and tetrate/tetrational terminology. I've been using "iterator" where you've been using orbit, though I like all three terms if I had to adopt one of them and stick with it. And "multiple" at least rhymes with iterational, tetrational, and exponential (in the sense that the suffixes "-al" and "-le" sound the same, at least for a Californian like myself), making for a loosely consistent pattern. One place it breaks down with other precedents is power/exponential. Calling x^3 the third exponentiate doesn't quite sound right (even if making an effort to pronounce it differently from the verb "to exponentiate"). But here, there is an ingrained terminology that may be causing a bias to what "sounds right". Edit: Already the term "exponentiate" is starting to sound okay to me, used as a noun. Funny how the mind adapts. Literature - Gottfried - 11/23/2007 Just came across an article of Helmuth Kneser, which G.A.Edgar referenced in his reply: Kneser Reelle analytische Lösungen der Gleichung ... und verwandter Funktionalgleichungen. Autor: Kneser, Hellmuth Journal für reine und angewandte Mathematik Zeitschrift / Zeitschriften-Band 187 / Zeitschriften-Artikel It covers many of our discussions here, but is in german lang. Gottfried RE: Intro and Formula sheet - andydude - 11/30/2007 @Henryk What do you mean by "iteration formula"? Andrew Robbins RE: Literature - andydude - 12/17/2007 The more I think about it, the more I think that the FAQ and the formula sheet should be combined. Here is a combined outline for the possible merge of the two: Preamble Introduction to Hyper-operators What is the Ackermann function? Hyper-operator notation Hyper-operator terminology Hyper-operator identities Introduction to Tetration Tetration as the hyper-4-operator Tetration notation and terminology Sequence of n-th tetrates Standard extensions of tetration Extending tetration to zero base Extending tetration to complex bases Extending tetration to negative heights Extending tetration to infinite heights Introduction to Iteration Dynamical systems translation equation Abel's functional equation Schroeder's functional equation Boettcher's functional equation Julia's functional equation Tetration vs. Iterated Exp Exponentiation and factorials Table of exponentiation Extension of exponentiation Extension of factorials Extension of super-factorials Topological conjugacy Lambert-W-like conjugates (srt2(x), W(x), x^^oo) Exponential-like conjugates (b^x = (h^(1/h))^x, h^x - 1) Topological conjugacy of the super-logarithm Other extensions of Tetration Ingolf Dahl's solution Ioannis Galidakis' solution Daniel Geisler's solution Gottfried Helms' solution Robert Munafo's solution Cliff Nelson's solution Peter Walker's solution Andrew Robbins' solution Jay D. Fox's solution Other extensions of Hyper-operators Bromer's mixed hyper-operators Munafo's lower hyper-operators Trappmann's binary-tree hyper-operators Trappmann's associative/commutative hyper-operators Trappmann's balanced hyper-operators @Henryk If you could provide the TeX source for your FAQ, I will combine it with the formula sheet... Andrew Robbins RE: Tetration FAQ - andydude - 01/13/2008 It has become large. I think our intent is to make this a short introduction, not a full book on tetration, but at least the version I'm posting today has grown in length from the version Henryk started this thread with, but Henryk's content still needs to be merged in with this content. I have included Henryk's introduction, and in the $\LaTeX$ file I have indicated where Henryk's sections would fit in with this outline. In the version that I am posting, I have included many of my own results, results which are easy to verify, and results that do not need a great deal of mathematical background to verify. This is a list of my own work I have included: Derivation of Munafo class using super-logarithms. Extending hyper-N-operations to real number N. Extending hyper-N-logarithms to real number N. Extending Galidakis' Puiseux series to iterated exponentials. (trivial) My Recurrence equation for Taylor series of tetration. My Recurrence equation for Taylor series of iterated exponentials. Topological conjugacy of exp., dec. exp., and scaled exp. (Exponential-like conjugates). Topological conjugacy of prod. exp., self-power, and self-root (Lambert-like conjugates). Of all of these, the theorem I was most reluctant to include is the Taylor series of tetration and iterated exponentials (pages 21-22). This recurrence equation took me several months of research to find, and as such, I want a little credit for it, but thats all. I have other results that could be much more valuable for publication, so I decided to include this recurrence equation after all. Much of the content, like reviews of other people's work, I had actually written some time before, and was planning on writing a book about Tetration. Although this never happened, I thought these sections were appropriate for a FAQ, so I have included some of these sections verbatim. Since the topological conjugacy section involves commutative diagrams, I have included the package (diagrams.sty) in the folder I uploaded. Also included are the images used within the FAQ. To everyone: please look over the material, see what you can improve, and what is missing, wrong, inappropriate, too advanced, misplaced, out of order, and so on. We may have to split this into two articles again, one for the FAQ and one for the collection of theorems, but I like the idea of having a single article to describe our progress. Andrew Robbins PS. If you don't want the sources, just download the PDF. RE: Tetration FAQ - Ivars - 01/13/2008 Hi Andy I know I have to keep head cool, but the first fabulous graph on Your summary article is just the most general phase diagram possible in reals, and I just know it is true. It explains all existing phase diagrams and hints on some new. Next task is just to adapt parameters to fit experimental data-any. And the physics will pop out themselves from mathematical models You use. Now imagine what it would look like when imaginary values are included and what would that mean. It can be done as slowly as possible, of course Ivars RE: Tetration FAQ - GFR - 01/13/2008 I appreciate the evolution of the FAQ draft. May I submit to you all, and particularly to Henryk and Andrew, the attached notes that I prepared few months ago for my discussions with KAR ? They contain a bidimensional display trying to define the domains of existence of the "towers" and "superroots" and trying to show that they are disjoint. It needs to be carefully finalized. I shall appreciate your kind comments on that. Thank you. GFR RE: Tetration FAQ - andydude - 01/16/2008 Moderator Note: I split off the discussion about bifurcation, because this is mathematical discussion, not discussion about the FAQ. RE: Tetration FAQ - bo198214 - 01/19/2008 andydude Wrote:It has become large. Hey Andrew, you put really a lot of work in. Thanks for your contribution. It seems however that we have to discuss what to put in the FAQ to keep it in a reasonable size. Basically a FAQ shell prevent people starting to deal with this topic from reinventing the wheel, by clearly and elementary explaining the basics and give known answers to frequently asked questions or frequently occuring problems. This should be our directive for designing the FAQ. We should also be careful to not explain things based on our advanced background but instead being very explicit, elementary (and easy to read) and give a lot of examples. As such I would seperate a (formula, theorem) reference from the FAQ. I would ban the more exotic notations into a history chapter. Also note that the box notation is not commonly used but rather our invention. The box tex-notation is also very difficult to use without predefined commands (for example between the tex-tags in this forum). I think we need a simple TeX-representation like a sub- or superscripted operator symbol (together with srt, slg, hrt, hlg for the inverse functions). Also we need for everything an ASCII equivalent which people can use when posting (for the non-tex-experienced people). I append my own tex-source.


text:
# Statistics - One Proportion Z Test

The test statistic is a z-score (z) defined by the following equation. ${z = \frac{(p - P)}{\sigma}}$ where P is the hypothesized value of population proportion in the null hypothesis, p is the sample proportion, and ${\sigma}$ is the standard deviation of the sampling distribution.

Test Statistics is defined and given by the following function:

## Formula

${ z = \frac {\hat p -p_o}{\sqrt{\frac{p_o(1-p_o)}{n}}} }$

Where −

• ${z}$ = Test statistics

• ${n}$ = Sample size

• ${p_o}$ = Null hypothesized value

• ${\hat p}$ = Observed proportion

### Example

Problem Statement:

A survey claims that 9 out of 10 doctors recommend aspirin for their patients with headaches. To test this claim, a random sample of 100 doctors is obtained. Of these 100 doctors, 82 indicate that they recommend aspirin. Is this claim accurate? Use alpha = 0.05.

Solution:

Define Null and Alternative Hypotheses

${ H_0;p = .90 \\[7pt] H_0;p \ne .90 }$

Here Alpha = 0.05. Using an alpha of 0.05 with a two-tailed test, we would expect our distribution to look something like this:

Here we have 0.025 in each tail. Looking up 1 - 0.025 in our z-table, we find a critical value of 1.96. Thus, our decision rule for this two-tailed test is: If Z is less than -1.96, or greater than 1.96, reject the null hypothesis.Calculate Test Statistic:

${ z = \frac {\hat p -p_o}{\sqrt{\frac{p_o(1-p_o)}{n}}} \\[7pt] \hat p = .82 \\[7pt] p_o = .90 \\[7pt] n = 100 \\[7pt] z_o = \frac {.82 - .90}{\sqrt{\frac{ .90 (1- .90)}{100}}} \\[7pt] \ = \frac{-.08}{0.03} \\[7pt] \ = -2.667 }$

As z = -2.667 Thus as result we should reject the null hypothesis and as conclusion, The claim that 9 out of 10 doctors recommend aspirin for their patients is not accurate, z = -2.667, p < 0.05.


text:
## WeBWorK Problems

### Recognizing equivalent implicit differentiation answers

by Doug Torrance -
Number of replies: 3
Suppose we want to compute the second derivative of y when y is defined implicitly by:

x2 + y2 = 1

Using implicit differentiation, the answer is:

y'' = (-x2 - y2) / y3

A clever student might recognize that the numerator is just the opposite of the left-hand side of our original implicit equation, and enter:

y'' = -1 / y3

Is there a way for WeBWorK to recognize both solutions as correct?

From what I understand, if we use MathObjects and define a Formula for our answer, the answers will be checked by picking some random test points for x and y. But in this case, we'd want to make sure that those test points satisfy the original implicit equation. For example, if we pick x = 1/2 as one of the test points, then we would want to pair it with y = sqrt 3 / 2 or y = - sqrt 3 / 2.
In reply to Doug Torrance

### Re: Recognizing equivalent implicit differentiation answers

by Danny Glin -
For this case I use parserOneOf.pl, which allows you to give a list of correct answers, of which the student can submit any one.

You will need to add parserOneOf.pl to the loadMacros section, then the usage would be something like:

$correct1 = Compute("(-x^2-y^2)/y^3");$correct2 = Compute("-1/y^3");
ANS(OneOf($correct1,$correct2)->cmp());

One could try to write a more sophisticated answer checker that tried to substitute the original formula where possible, but in the example you provide and the ones I'm using I find it very unlikely that the student would find a correct answer other than the two forms mentioned.
In reply to Doug Torrance

### Re: Recognizing equivalent implicit differentiation answers

by Alex Jordan -
Here is an example of what I have done in the past for this issue.

Say the problem asks to find dy/dx for x^4+y^2+y=7.

Consider this code:

Context() -> variables -> add(y=>'Real');
$dydx = Formula("-4x^3/(2y+1)"); @y = (random(-2,2,0.2), random(-2,2,0.2), random(-2,2,0.2), random(-2,2,0.2), random(-2,2,0.2)); @x = map{(7-($_)-($_)**2)**(1/4)*(-1)**(random(-1,1,2))} (@y); @xy = map{[$x[$_],$y[$_]]} (0..4);$dydx -> {test_points} = ~~@xy;

Explanation:
1. Add y to the context (this assumes x is there to start).
2. Make the canonical correct answer formula. This will display as "the" correct answer.
3. Make some random y-values coming from an interval where 7 - y - y^2 is positive. This part means you need to consider the specifics of the original curve.
4. Find x-values that pair with those y-values on the curve. This was doable with this curve because x can essentially be solved for. Note it is randomly choosing positive or negative fourth roots. In other settings, it may make more sense to randomize x-values and solve for y-values. In yet other settings, this approach just isn't quite doable.
5. Make an array where each element is an array reference to an x,y-pair that is on the curve.
6. Set the test points for the answer to be this array.

So during a check, instead of choosing a bunch of random x and y-values to plug in to -4x^3/(2y+1), it is forced to only choose (x,y) pairs that are on the curve. Therefore if a student uses an expression that has made use of the defining relationship between x and y, it won't matter. With this example, it is OK to answer with:
-4x^3/(2y+1)

or with:
-4x^4/[x(2y+1)]

or with:
-4(7 - y - y^2)/[x(2y+1)]

etc.

In reply to Alex Jordan

### Re: Recognizing equivalent implicit differentiation answers

by Doug Torrance -
Thank you for the reply! There was one problem in particular in the OPL which was giving my students trouble because of this issue.

I've modified your example and submitted a pull request so hopefully future students don't run into the same issue!


text:
Solving Percent Equations – Practice ProblemsHaving fun while studying, practice your skills by solving these exercises!

For now, Practice Problems are only available on tablets and desktop computers. Please log in on one of these devices.

Do you need help? Watch the Video Lesson for this Practice Problem.

To calculate percent equations, it’s helpful to use what you know of ratios and proportions. To solve for unknown values when calculating percent of a number equations, use the proportion: Part/whole = percent/100. Notice the second ratio is the fraction that represents a percent.

What is the part and whole? For example, if you eat 3 slices of pizza, the part you eat is equal to 3 slices and the whole pie is equal to 4 slices. To solve for any unknown value in the proportion, you can use cross product.

To use cross product, also known as cross multiplication, imagine a cross written across the proportion a/b = c/d, when there are two equal ratios making a proportion, the products of the cross values are equal, so for this example, ad = bc. Using cross product to set up an equation then use opposite operations to isolate the variable. For some equations, not all the information will be provided in percent form, so before you can set up an equation using cross product, you may have to convert fractions and decimals to percent.

Just to refresh your memory: To convert a decimal to a percent, multiply by 100 then add the percent sign. To convert a fraction to a percent, create an equivalent fraction with 100 in the denominator. To learn more about percent equations, take a look at what Richard the Mange is making in his lab.

Convert between formats to solve percent equations. CCSS.MATH.CONTENT.HSA.SSE.B.3

Exercises in this Practice Problem
 Describe how you can determine the percent of honey. Calculate how much is $17.5~\%$ of $200~ml$. Define the whole, the part, and the percentage. Help Richard convert the percentage into $ml$. Explain the meaning of percent. Determine the percentage.


text:
## Fregean concepts

January 25, 2008

I’m currently reading up on Frege’s theory of concepts for one of my courses. There are several things that really puzzle me about the theory though.

The first confusing thing is to do with relations. Frege says that relations are doubly unsaturated. You can fill a diadic relation with an object, and get a monadic relation. For example if I have the relation $x < y$ I may fill it out with four to get $x < 4$, the monadic concept of being less than four, or alternatively I may put four in the other slot and get $4 < x$ the monadic concept of being greater than four. And it clearly made a difference, for the former but not the later is true of 3. But what if we consider $x = y$? What about $4 = x$ and $x = 4$ – are they the same or different? If we think analogously to the last case, we might think they were different. Relations are doubly unsaturated, they have two “object shaped holes”*, in the first case the first hole was filled, in the second the second hole was filled – they denote different concepts because in each case a different hole is covered. But this cannot be the case, since for Frege, concepts are extensional. If they’re true of the same things, they are the same concept.

There is the puzzling section 70 in the Grundlagen where he suggests that relations might actually be monadic concepts taking composite objects as arguments. I assume the composite objects would have to have as much structure as ordered pairs if they were to do the job. But I don’t think that really helps us, for if $x < y$ was really a monadic concept of pairs, the notion of filling in only one of the argument places doesn’t make too much sense. The most natural way to do that would be to compose $<$ with $\langle 4, x \rangle$, the function that pairs an object with 4. But this means treating < as a second level concept.

That brings me onto the second thing I find confusing. In “Function and Concept” Frege tries to compositionally derive the truth values of various formulae from the Begriffsschrift from the concepts/functions each expression denotes. This works fine for the simple formulae he tries like $\forall x x=x$. But what about: $\forall x(f(x) = 4)$? It should be derived compositionally from the following pieces:

1. The second level concept $\forall$ true of a first level concept F(x) iff F gives true for each object.
2. The first level function $f(x)$ which for each object gives the number four.
3. $4$ the number four.
4. $x = y$, the first level diadic relation-concept that gives the true when supplemented with x and y, where x and y are identical.

However, I cannot see any plausible way to piece these concepts together – they just don’t fit. Crucially $f(x) = 4$ is not type correct, since = is a first level relation, it cannot take a first level function for its left argument, only an object. But even if you try combining (1) – (4) in other ways (i.e. not in the obviously compositional fashion), you don’t seem to get the right results.

Quite generally, there are all kinds of typing problems with the way Frege uses concepts. For example, in $\forall x \exists y (x < y)$, $\exists y$ is supposed to take a monadic concept as argument like $\forall x$ does, but instead it gets a diadic relation-concept.

I guess I don’t really know the literature, or Frege’s writings well enough to know whether he addressed this, or if there is a straightforward way to get around the type mismatches. Does anyone have any ideas? Any pointers to the literature would be very welcome :D.

*This is my metaphor, not Frege’s. You should probably that argument with a pinch of salt.


text:
# neat-interpolation: A quasiquoter for neat and simple multiline text interpolation

[ library, mit, quasiqoutes, quasiquotes, string ] [ Propose Tags ]

NeatInterpolation provides a quasiquoter for producing Text data with a simple interpolation of input values. It removes the excessive indentation from the input text and accurately manages the indentation of all lines of interpolated variables.

## Modules

• NeatInterpolation


text:
# Integral estimate for Hankle´s Contour

I have to proof the following estimate

$$\vert \int\limits_{H_k}z^{s-1}(e^{z}-1)^{-1}dz\vert \leq k^{\sigma}$$

Where $$H_k$$ ist the Hankel Contour with radius $$\rho_k = (2k+1) \pi$$

From another estimate, which I already have proofen I got

$$\vert \frac{z^{s-1}}{e^z-1}\vert \leq \rho^{\sigma-2}$$ for every radius $$> 0$$. where $$\sigma = Re(s)$$

For the searched estimate I tried

$$\vert \int\limits_{H_k}z^{s-1}(e^{z}-1)^{-1}dz\vert \leq \int\limits_{H_k}\vert z^{s-1}(e^{z}-1)^{-1}\vert dz \leq \int\limits_{H_k} \rho_k^{\sigma-2}dz$$

but that don´t lead me further. Anyone has a clue for me hwo I can go on?


text:
## cosmomc: n_t error when adding new parameter unrelated to n_t/inflation

Use of Healpix, camb, CLASS, cosmomc, compilers, etc.
Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Hi,

I have added a new parameter to CAMB and cosmoMC (Nov2016 version). CAMB runs fine, but when I try to run cosmoMC, it stops with the following error message:

Fast divided into 1 blocks
21 parameters ( 7 slow (0 semi-slow), 14 fast (0 semi-fast) )
Starting Monte-Carlo
Error: inflation_consistency but n\_t not set to zero
MpiStop: 0

I have not altered any inflation-settings, so e.g. in the /batch2/params_CMB_defaults.ini, inflation_consistency is set to T and nt = 0 is also set (although a comment notifies that this will be ignored when inflation_consistency is set to T).
The new parameter has nothing to do with inflation or nt, so I really donâ€™t understand why this error occurs.

I would greatly appreciate any ideas as to why this error occurs.

P.S. When adding the new parameter, I follow the description given in http://cosmocoffee.info/viewtopic.php?t ... c&start=15 . I have made no other changes to cosmoMC and the changes I have made in CAMB are not related to inflation or nt.

Antony Lewis
Posts: 1364
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

### Re: cosmomc: n_t error when adding new parameter unrelated t

I expect the mapping between parameter array indices and physical parameters got mixed up when adding your new parameter, e.g. check your CosmologyParameterizations.f90

Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Hi,

Thank you for the quick reply.
Yes, this could very well be the case, because I am a little confused about the .paramnames file: I keep simply using the params_CMB.paramnames and add my new parameter, a_switch, at the bottom of this file. This makes it parameter number 47 in this file. But before adding a_switch, there were only 16 theory-parameters (line 74 in CosmologyParameterizations.f90). I change this to 17 and have tried adding a_switch in SetForH both as "CMB%a_switch = Params(17)" and "CMB%a_switch = Params(47)" (I have also tried adding a_switch as a real(mcp) in SetForH and then simply setting "a_switch = Params(17)" or "a_switch = Params(47)" because this is what is done in the posting I linked to before). I get the same error regardless of which I choose, but maybe I am missing some important detail of how to work with the .paramnames file?

Antony Lewis
Posts: 1364
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

### Re: cosmomc: n_t error when adding new parameter unrelated t

If you are adding a theory parameter you want to add it in the corresponding place in the paramnames file (line 17).

Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Ok, I have made sure that a_switch is added in line 17 of params_CMB.paramnames and that it is entered as Params(17) in SetForH. The error message is different now, and is as follows:

.
.
.
Computing tensors: F
max_eta_k = 14000.00
transfer kmax = 5.000000
SetTheoryParameterNumbers: parameter numbers do not match
MpiStop 0

I have already changed line 74 in CosmologyParameterizations.f90 to include 17 instead of 16 parameters (i.e. the line now reads "call this%SetTheoryParameterNumbers(17,last_power_index)"), so this should not be the problem. Are there other places where I have to change such numbers to let cosmoMC know that there has been added a parameter?

Erick Almaraz
Posts: 25
Joined: June 10 2012
Affiliation: University of Mexico

### cosmomc: n_t error when adding new parameter unrelated to n_

I don't know if this is mandatory, but in my case I declared the extra parameter in

Code: Select all

Type, extends(TTheoryParams) :: CMBParams
in CosmologyTypes.f90

Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Hi,

I'm pretty sure this is mandatory and I have already done it, but thank you anyway.

Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Hi,

I looked a little more in source/CosmologyTypes.f90 and tried adding "+1" in line 141 so that it now reads:

num_initpower = semi_slow_num+1

The code runs without errors now, but this change was not mentioned in the topic I linked to in my initial post (i.e. in http://cosmocoffee.info/viewtopic.php?t ... c&start=15 ) so I am a little unsure if I am doing the correct thing, or if I have added my new parameter incorrectly somehow.

I would really appreciate any thoughts on whether or not it seems reasonable that I had to change num_initpower.

Antony Lewis
Posts: 1364
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

### Re: cosmomc: n_t error when adding new parameter unrelated t

You shouldn't have to change that. It sounds as though your .paramnames is still inconsistent, make sure the number of lines in params_CMB.paramnames matches the sum of the arguments you are passing to SetTheoryParameterNumbers.

Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Hi,

Thank you for letting me know that the "solution" is NOT a solution!

Originally (before I changed anything in the code), line 74 in sources/CosmologyParameterization.f90 read:

call this%SetTheoryParameteNumbers(16,last_power_index)

I added a_switch as line 17 in paramnames/params_CMB.paramnames and therefore changed the above line to:

call this%SetTheoryParameteNumbers(17,last_power_index)

I.e. I changed "16" to "17" after adding a single parameter. Should I have done it differently?

P.S. paramnames/params_CMB.paramnames is entirely unaltered compared to its original version except that I add a_Switch in line 17 (there seems to be some parameters in the file that aren't used, but I haven't deleted any because I don't want to accidentally delete a line that is in fact used somewhere...)

Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Could it be that I am supposed to change lines 12-13 in sources/CosmologyTypes.f90? Those lines currently read:

integer, parameter :: As_index=1, ns_index=2, nrun_index=3, nrunrun_index =4, amp_ratio_index = 5, &
& nt_index = 6, ntrun_index = 7, Aphiphi_index = 8, last_power_index = Aphiphi_index

Do I need to change these indexes now that I have inserted a_switch in line 17 in paramnames/params_CM.paramnames such that A_s, ns, nrun etc. have all been moved down one line in paramnames/params_CM.paramnames?..."last_power_index" is used as the second input in SetTheoryParameterNumbers in source/CosmologyParameterizations.f90, so if Aphiphi_index and hence last_power_index need to be changed, this could be why SetTheoryParameterNumbers gives the error?

As it is currently, SetTheoryParemeterNumbers gets the inputs "17" and "last_power_index" = "Aphiphi_index" = 8, i.e. the total number is 17+8 = 25 which is the number of lines with theory parameters (parameters without a "*") in paramnames/params_CMB.paramnames.

Sofie Marie Koksbang
Posts: 9
Joined: February 27 2017
Affiliation: Aarhus University

### cosmomc: n_t error when adding new parameter unrelated to n_

Hi Antony and Erick,

I think the problem is fixed now: I removed the paramnames file and copied a new one from a cosmoMC-version that I haven't altered at all. Then I added a_Switch in line 17 of the paramnames-file and now it's running. So, it seems that the problem was originally my misplacement of a_switch in params_CMB.paramnames and then before that got fixed, I must have messed up something in the file or entire directory (I sometimes accidentally "vim" into directories and start typing before I realize my mistake - maybe that has messed up the paramnames-directory...)


text:
Godel’s incompleteness theorem deals with “first order logic”, so it’s worth a post explaining what that is. In particular, we will distinguish it from zeroth order logic as well as from second and higher order logics.

What distinguishes these levels of logic from one another is what they’re allowed to quantify over. The two most common quantifiers are the symbols $$\forall$$ which stands for “for all” and $$\exists$$ which stands for “there exists”. One might use these quantifiers to make a statement like so:

$\forall x ((Prime(x) \land x > 2) \supset Odd(x))$

which is a formal way of stating that all primes greater than 2 are odd. If I translated this formula into a sentence, it would read: “for all $$x$$, if $$x$$ is prime and $$x$$ is greater than 2, then $$x$$ is odd”. I’m assuming I’ve already defined the $$Prime$$ and $$Odd$$ properties elsewhere, but you get the idea.

When we quantify over variables ($$x$$ in this case), we first need to specify the domain over which these variable can range. Maybe the domain of $$x$$ is the natural numbers, maybe it’s the real numbers, or maybe it’s something altogether different like colors. I can quantify over colors. There exists a color $$x$$ that I like more than red. QED.

Let’s say that we’ve specified the domain is the natural numbers, as Godel did.

Zeroth order logic is not allowed to quantify over the domain at all. By the way, propositional logic is another name for zeroth order logic.

First order logic is allowed to quantify over individuals of the domain (e.g. over natural numbers). The statement I made above about primes is a statement of first order logic in that $$x$$ is to be interpreted as a natural number.

Second order logic is allowed to quantify over sets of the domain. Recall that a property of the natural numbers can be thought of as the set of numbers which have that property. So, the ability to quantify over sets gives one the ability to quantify over properties (or relations, more generally). Here is a statement that requires second order logic:

$\exists P. P(5) \land P(7)$

In words, there exists a property $$P$$ such that both $$5$$ and $$7$$ have the property $$P$$. Maybe $$P$$ is the property of being odd, or the property of being prime, or the property of being less than 10, … the list can go on.

Higher order logics: You can extend this pattern to higher order logics. Third order logic can quantify over sets of sets… and so on.


text:
Conference paper Open Access

# Flexible Hybrid Process for Combined Production of Heat, Power and Renewable Feedstock for Refineries

Kurkela, Esa; Tuomi, Sanna; Kurkela, Minna; Hiltunen, Ilkka

### Dublin Core Export

<?xml version='1.0' encoding='utf-8'?>
<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
<dc:creator>Kurkela, Esa</dc:creator>
<dc:creator>Tuomi, Sanna</dc:creator>
<dc:creator>Kurkela, Minna</dc:creator>
<dc:creator>Hiltunen, Ilkka</dc:creator>
<dc:date>2019-10-22</dc:date>
<dc:description>A flexible combined heat, power and fuel production concept, FLEXCHX, is being developed for managing the seasonal mismatch between solar energy supply and demand for heat and power that is highly pronounced particularly in the Northern and Central Europe. This process produces heat, power and an intermediate energy carrier (Fischer-Tropsch hydrocarbon product), which can be refined to transportation fuels using existing oil refining equipment. The FLEXCHX process can be integrated to various combined heat and power production systems, both industrial CHPs and communal district heating units. In the summer season, renewable fuels are produced from biomass and hydrogen; the hydrogen is produced from water via electrolysis that is driven by low-cost excess electricity from the grid. In the dark winter season, the plant is operated only with biomass in order to maximize the production of the much needed heat, electricity and FT hydrocarbons. Most of the invested plant components are in full use throughout the year, only the electrolysis unit is operated seasonally. © 2019 ETA-Florence Renewable Energies.</dc:description>
<dc:identifier>https://zenodo.org/record/3515920</dc:identifier>
<dc:identifier>10.5281/zenodo.3515920</dc:identifier>
<dc:identifier>oai:zenodo.org:3515920</dc:identifier>
<dc:relation>info:eu-repo/grantAgreement/EC/H2020/763919/</dc:relation>
<dc:relation>doi:10.5281/zenodo.3515919</dc:relation>
<dc:relation>url:https://zenodo.org/communities/flexchx</dc:relation>
<dc:rights>info:eu-repo/semantics/openAccess</dc:rights>
<dc:source>Proceedings of the 27th European Biomass Conference and Exhibition: EUBCE 2019 539-543</dc:source>
<dc:subject>Gasification</dc:subject>
<dc:subject>Fischer-Tropsch</dc:subject>
<dc:subject>Reforming</dc:subject>
<dc:subject>Biomass</dc:subject>
<dc:subject>Biofuels</dc:subject>
<dc:title>Flexible Hybrid Process for Combined Production of Heat, Power and Renewable Feedstock for Refineries</dc:title>
<dc:type>info:eu-repo/semantics/conferencePaper</dc:type>
<dc:type>publication-conferencepaper</dc:type>
</oai_dc:dc>

47
45
views


text:
## CryptoDB

### Paper: The COS Stream Ciphers are Extremely Weak

Authors: Steve Babbage URL: http://eprint.iacr.org/2001/078 Search ePrint Search Google A new family of very fast stream ciphers called COS (for "crossing over system") has been proposed by Filiol and Fontaine, and seems to have been adopted for at least one commercial standard. In this note we show that the COS ciphers are very weak indeed ? it requires negligible effort to reconstruct the state of the keystream generator from a very small amount of known keystream.
##### BibTeX
@misc{eprint-2001-11490,
title={The COS Stream Ciphers are Extremely Weak},
booktitle={IACR Eprint archive},
keywords={secret-key cryptography / COS, stream cipher, nonlinear feedback shift register, cryptanalysis},
url={http://eprint.iacr.org/2001/078},
note={ steve.babbage@vodafone.com 11576 received 11 Sep 2001},
author={Steve Babbage},
year=2001
}


text:
# How do you solve 7x^2 = 21x using the quadratic formula?

May 17, 2016

#### Answer:

The solutions are:
$x = 3$

$x = 0$

#### Explanation:

$7 {x}^{2} = 21 x$

$7 {x}^{2} - 21 x = 0$

The equation is of the form color(blue)(ax^2+bx+c=0 where:

$a = 7 , b = - 21 , c = 0$
(the expression has no constant term)

The Discriminant is given by:

$\Delta = {b}^{2} - 4 \cdot a \cdot c$

$= {\left(- 21\right)}^{2} - \left(4 \cdot 7 \cdot 0\right)$

$= 441$

The solutions are found using the formula
$x = \frac{- b \pm \sqrt{\Delta}}{2 \cdot a}$

$x = \frac{- \left(- 21\right) \pm \sqrt{441}}{2 \cdot 7} = \frac{21 \pm 21}{14}$

$x = \frac{21 + 21}{14} = \frac{42}{14} = 3$

$x = \frac{21 - 21}{14} = \frac{0}{14} = 0$


text:
Scott, your snake equation proof looks good, except that you only show that the two profunctors are equal along the diagonal. That is, you only show that for all \$$x\$$,
$(1_X \times \cup_X ) (\cap_X \times 1_X )((1,x),(x,1)) = 1_X(x,x)$
You need to show that for all \$$x,\tilde{x}\$$,
$(1_X \times \cup_X ) (\cap_X \times 1_X )((1,x),(\tilde{x},1)) = 1_X(x,\tilde{x}).$
Replacing \$$x\$$ by \$$\tilde{x}\$$ in the appropriate places in your proof, you get that
\$$(1_X \times \cup_X ) (\cap_X \times 1_X )((1,x),(\tilde{x},1))\$$
if and only if there exist \$$x'', x^*, x'\$$ such that \$$x\le x'' \le x^* \le x' \le \tilde{x}\$$,
and of course this hold if and only if \$$x\le\tilde{x}\$$ or in other words \$$1_X(x,\tilde{x})\$$.

On a stylistic note, I think 0 is better than 1 as the name for the only element in \$$\mathbf{1}\$$, as John used in the last lecture. I've seen others here use a bullet or star symbol, which I think are also better than 1.


text:
# Landau–Lifshitz–Gilbert equation

In physics, the Landau–Lifshitz–Gilbert equation, named for Lev Landau, Evgeny Lifshitz, and T. L. Gilbert, is a name used for a differential equation describing the precessional motion of magnetization M in a solid. It is a modification by Gilbert of the original equation of Landau and Lifshitz.

The various forms of the equation are commonly used in micromagnetics to model the effects of a magnetic field on ferromagnetic materials. In particular it can be used to model the time domain behavior of magnetic elements due to a magnetic field.[1] An additional term was added to the equation to describe the effect of spin polarized current on magnets.[2]

## Landau–Lifshitz equation

The terms of the Landau–Lifshitz–Gilbert equation: precession (red) and damping (blue). The trajectory of the magnetization (dotted spiral) is drawn under the simplifying assumption that the effective field Heff is constant.

In a ferromagnet, the magnetization M can vary internally but at each point its magnitude is equal to the saturation magnetization Ms. The Landau–Lifshitz–Gilbert equation predicts the rotation of the magnetization in response to torques. An earlier, but equivalent, equation (the Landau–Lifshitz equation) was introduced by Landau & Lifshitz (1935):[3][4][5]

${\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma \mathbf {M} \times \mathbf {H_{\mathrm {eff} }} -\lambda \mathbf {M} \times \left(\mathbf {M} \times \mathbf {H_{\mathrm {eff} }} \right)}$

(1)

where γ is the electron gyromagnetic ratio and λ is a phenomenological damping parameter, often replaced by

${\displaystyle \lambda =\alpha {\frac {\gamma }{M_{\mathrm {s} }}},}$

where α is a dimensionless constant called the damping factor. The effective field Heff is a combination of the external magnetic field, the demagnetizing field (magnetic field due to the magnetization), and some quantum mechanical effects. To solve this equation, additional equations for the demagnetizing field must be included.

Using the methods of irreversible statistical mechanics, numerous authors have independently obtained the Landau–Lifshitz equation.[6][7][8]

## Landau–Lifshitz–Gilbert equation

In 1955 Gilbert replaced the damping term in the Landau–Lifshitz (LL) equation by one that depends on the time derivative of the magnetization:

${\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma \left(\mathbf {M} \times \mathbf {H} _{\mathrm {eff} }-\eta \mathbf {M} \times {\frac {d\mathbf {M} }{dt}}\right)}$

(2b)

This is the Landau–Lifshitz–Gilbert (LLG) equation, where η is the damping parameter, which is characteristic of the material. It can be transformed into the Landau–Lifshitz equation:[3]

${\displaystyle {\frac {d\mathbf {M} }{dt}}=-\gamma '\mathbf {M} \times \mathbf {H} _{\mathrm {eff} }-\lambda \mathbf {M} \times (\mathbf {M} \times \mathbf {H} _{\mathrm {eff} })}$

(2a)

where

${\displaystyle \gamma '={\frac {\gamma }{1+\gamma ^{2}\eta ^{2}M_{s}^{2}}}\qquad {\text{and}}\qquad \lambda ={\frac {\gamma ^{2}\eta }{1+\gamma ^{2}\eta ^{2}M_{s}^{2}}}.}$

In this form of the LL equation, the precessional term γ' depends on the damping term. This better represents the behavior of real ferromagnets when the damping is large.[9][10]

## Landau–Lifshitz–Gilbert–Slonczewski equation

In 1996 Slonczewski expanded the model to account for the spin-transfer torque, i.e. the torque induced upon the magnetization by spin-polarized current flowing through the ferromagnet. This is commonly written in terms of the unit moment defined by m = M / MS:

${\displaystyle {\dot {\mathbf {m} }}=-\gamma \mathbf {m} \times \mathbf {H} _{\mathrm {eff} }+\alpha \mathbf {m} \times {\dot {\mathbf {m} }}+\tau _{\parallel }{\frac {\mathbf {m} \times (\mathbf {x} \times \mathbf {m} )}{\left|\mathbf {x} \times \mathbf {m} \right|}}+\tau _{\perp }{\frac {\mathbf {x} \times \mathbf {m} }{\left|\mathbf {x} \times \mathbf {m} \right|}}}$

where ${\displaystyle \alpha }$ is the dimensionless damping parameter, ${\displaystyle \tau _{\perp }}$ and ${\displaystyle \tau _{\parallel }}$ are driving torques, and x is the unit vector along the polarization of the current.[11][12]

## References and footnotes

1. ^ Yang, Bo. "Numerical Studies of Dynamical Micromagnetics". Retrieved 8 August 2011.
2. ^ d’Aquino, Massimiliano (2004). "2.6.1 Landau-Lifshitz-Gilbert equation with Slonczewski spin-transfer torque term". Nonlinear Magnetization Dynamics in Thin-films and Nanoparticles. PhD Thesis, University of Naples Federico II.
3. ^ a b
4. ^ *Brown, Jr., William Fuller (1978) [Originally published in 1963]. Micromagnetics. Robert E. Krieger Publishing Co.
5. ^ *Chikazumi, Sōshin (1997). Physics of Ferromagnetism. Clarendon Press. ISBN 978-0-19-851776-4.
6. ^ Iwata, Takao (1983). "A thermodynamical approach to the irreversible magnetization in single-domain particles". Journal of Magnetism and Magnetic Materials. 31–34: 1013–1014. Bibcode:1983JMMM...31.1013I. doi:10.1016/0304-8853(83)90774-6.
7. ^ Iwata, Takao (1986). "Irreversible magnetization in some ferromagnetic insulators". Journal of Magnetism and Magnetic Materials. 59 (3–4): 215–220. Bibcode:1986JMMM...59..215I. doi:10.1016/0304-8853(86)90415-4.
8. ^ Saslow, W. M. (2009). "Landau–Lifshitz or Gilbert damping? That is the question". Journal of Applied Physics. 105 (7): 07D315. Bibcode:2009JAP...105gD315S. doi:10.1063/1.3077204.
9. ^ For details of Kelly's non-resonant experiment, and of Gilbert's analysis (which led to Gilbert's modifying the damping term), see Gilbert, T. L. and Kelly, J. M. "Anomalous rotational damping in ferromagnetic sheets", Conf. Magnetism and Magnetic Materials, Pittsburgh, PA, June 14–16, 1955 (New York: American Institute of Electrical Engineers, Oct. 1955, pp. 253–263). Text references to Figures 5 and 6 should have been to Tables 1 and 2. Gilbert could not fit Kelly's experiments with fixed usual gyromagnetic ratio γ and a frequency-dependent λ=αγ, but could fit that data for a fixed Gilbert gyromagnetic ratio γG=γ/(1+α2) and a frequency-dependent α. Values of α as large as 9 were required, indicating very broad absorption, and thus a relatively low-quality sample. Modern samples, when analyzed from resonance absorption, give α's on the order of 0.05 or less. J. R. Mayfield, in J. Appl. Phys. Supplement to Vol. 30, 256S-257S (1959), at the top left of p.257S, writes “As was first pointed out by J. C. Slonczewski, the observed torque peak can be interpreted in terms of rotational switching effects (abrupt reorientations of M) which must occur when K/M ≤ H ≤ 2K/M.” Therefore the interpretation given by Gilbert was not universally accepted.
10. ^ J. Mallinson, "On damped gyromagnetic precession," in IEEE Transactions on Magnetics, vol. 23, no. 4, pp. 2003-2004, July 1987, doi: 10.1109/TMAG.1987.1065181.
11. ^ Slonczewski, John C. (1996). "Current-driven excitation of magnetic multilayers". Journal of Magnetism and Magnetic Materials. 159 (1): –1–L7. Bibcode:1996JMMM..159L...1S. doi:10.1016/0304-8853(96)00062-5.
12. ^ Wolf, S. A. (16 November 2001). "Spintronics: A Spin-Based Electronics Vision for the Future". Science. 294 (5546): 1488–1495. Bibcode:2001Sci...294.1488W. doi:10.1126/science.1065389. PMID 11711666. S2CID 14010432. Archived from the original on June 18, 2017.


text:
# [OS X TeX] print the latex source filename

Peter Dyballa Peter_Dyballa at Web.DE
Sat Mar 21 20:36:28 CET 2009

Am 21.03.2009 um 15:40 schrieb Alain Schremmer:

> That's the trouble: I am not getting any error. I am not getting
> anything … other than
>
> This is a text body!

That's correct!

Everything in the preamble does not find a way to materialise in the
body. You only see side-effects (font size, text width, etc.).

Heiko Oberdiek's code has one "executable:" \SourceFile. When it
executes, it executes the \hypersetup (and overwrites variables'
values, as mentioned before). \pdffilemoddate{} returns a date in a
form like that: D:20090304121314+01'00', i.e., D(ate):YYYYMMDDHHMMSS
+TZh'TZmin'. It's what PDF uses internally.

In the lines before the final \makeatother you can see variable names
which should look familiar to you (\year, \month, \day, \time). You
are using them for your time stamps. While these reflect the time of
compilation, \SourceFile overwrites them with the last modification
or save time of the source file. So, when you want to print the time
of the source file's compilation, you should not use Heiko Oberdiek's
code – or comment the "side-effect" code.

The four mentioned variables always exist and are initialised. \time
contains the minutes since midnight – to easily distinguish between
AM and PM, to whom it may concern! You could also use this:

\newcount\hours
\newcount\minutes
\hours\time \divide\hours 60
\edef\timehhmm{\ifnum\hours<10 0\fi\the\hours
:\ifnum\minutes<10 0\fi\the\minutes}
\newcommand\daytime{\edef\today{\today\,~\timehhmm}\today}

When you now put \daytime somewhere in your text body, the definition
of \today is updated, then \today is executed and something like
"<your \today definition> hh:mm" will be printed. You don't need the
calc package – because you can calculate, presumingly, and TeX also.
(Found somewhere, lhelp.sty?)

--
Greetings

Pete

"We need a president who's fluent in at least one language."
– Buck Henry


text:
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# All Kings, Queens and Aces Are Removed from a Pack of 52 Cards. the Remaining Cards Are Well-shuffled and Then a Card is Drawn from It. Find the Probability that the Drawn Card is a Red Card. - Mathematics

Short Note

All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well-shuffled and then a card is drawn from it. Find the probability that the drawn card is a red card.

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#### Solution

There are 4 kings, 4 queens and 4 aces. These are removed.

Thus, remaining number of cards = 52 − 4 − 4 − 4 = 40.

Number of red cards now = 26 − 6 = 20.

∴ P(getting a red card) =("Number of favourable outcomes")/"Number of all possible outcomes"

= 20/40 = 1/2

Thus, the probability that the drawn card is a red card is 1/2.

Is there an error in this question or solution?
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#### APPEARS IN

RS Aggarwal Secondary School Class 10 Maths
Chapter 15 Probability
Exercise 15B | Q 23.2 | Page 698
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text:
# How to color ParametricPlot3D based on the parameter?

I have a set of ODEs I am solving, and then plotting (several) trajectories of in ParametricPlot3D. I want to have a color of each plot reflect the time (i.e. the Parameter) at that point.

I tried what something along the lines of:

ParametricPlot3D[{{Sin[t], Cos[t], t/10}, {Sin[2*t], 3*Cos[t], Tan[t]}}, {t, 0, 20},
ColorFunction -> Function[t, Hue[t]]]

which colors the function in a seemingly arbitrary way.

Cheers!

-
Look in the details section too see what arguments are given to the ColorFunction. Function[{x, y, z, t}, ...] – ssch Nov 8 '13 at 14:25

ParametricPlot3D[{{Sin[t], Cos[t], t/10}, {Sin[2*t], 3*Cos[t], Tan[t]}}, {t, 0, 2 Pi},

Please note that extending the plot till t == 20 as you've done will cause the second function to plot along the same path several times, so occluding the "real" coloring – Dr. belisarius Nov 8 '13 at 16:20


text:
### Home > PC > Chapter 9 > Lesson 9.3.4 > Problem9-144

9-144.
1. Remember our toy car with distance function d(t) = 10t2? Remember to label your answers with the proper units. Homework Help ✎

1. Find the velocity of the car at 1 second, that is, find d ′(1).

2. Find the velocity of the car at 3 seconds, or d ′(3).

3. Find the velocity of the car at s seconds for any fixed time s, or d ′(s).

Find the slope function.

$d'(t)=\lim_{h\rightarrow 0}\frac{10(t+h)^{2}-10t^{2}}{h}$

Substitute t = 1 into the slope function.

Substitute t = 3 into the slope function.

Substitute t = s into the slope function.


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# Tools to bound the singular values of a finite sum of random matrices from below?

Matrix Chernoff bounds (see also this arXiv paper) are usually used to give upper bounds on the largest eigenvalue of a finite sum of random matrices. Sometimes it can also be used to give a lower bound on the smallest eigenvalue of a finite sum of random positive semidefinite (PSD) matrices.

For PSD matrices, the eigenvalues are the same as the singular values. My question is:

Is there any tool, or any other special cases (apart from the PSD case) for which there is a tool, to obtain lower bounds on the singular values of a finite sum of random matrices, or random symmetrical matrices?

• Crossposted on Math.SE. Please do not crosspost the same question, on mutliple StackExchange sites at once. Otherwise, welcome! – Niel de Beaudrap Jan 5 '15 at 13:34
• This question is interesting, but potentially borderline for being on-topic: Chernoff bounds and bounds on singular values are clearly of interest in computer science (as are algorithms to compute them), but analytic upper/lower bounds on strictly mathematical objects would seem more a proper subject of mathematics. Perhaps this would be better answered on Math.SE. – Niel de Beaudrap Jan 5 '15 at 13:36
• or also MathOverflow – vzn Jan 8 '15 at 16:11