How to simplify this rational expression? [closed]

i was wondering how to solve this question

could you please tell me how to solve this question

\begin{align*} \frac xy &= \frac 23\\ a&=0,\overline{xy}\\ b&=0,\overline{yx} \end{align*}

$\Rightarrow$ $\min(a-b)=$?

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a closed projection on a C*-algebra is compact iff it is closed on the multiplier algebra

I'm trying to understand the proof for the equivalence of (i) and (v) in the following picture. I don't quite understand what the highlighted sentence means. I want to know why there is a surjection from $B$ onto $M(A)/A$. Can anyone explain for me, or provide some other proof? Thanks in advance.

The picture can be found in the following paper:

Brown, Lawrence G., Semicontinuity and multipliers of (C^*)-algebras, Can. J. Math. 40, No. 4, 865-988 (1988). ZBL0647.46044. 1: https://i.sstatic.net/v3UGY.png

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What is geometric engineering in quantum field theory?

Could someone help me to understand what geometric engineering in quantum field theory is? I didn't find any introductory articles online. Thank you!

Edit : Here is my background. I am math major. I know representation theory of quantum groups, quiver moduli space and hall algebra and some of algebraic geometry. I know very little physics except some brief mention from articles. That's why I feel difficult to follow the original paper posted on arxiv. What I want to know basically is 1) what they want to do 2)what they did 3)what part of math they used.

Ps: Thank Urs for your suggestions.

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Torelli for divisors in products of projective space

Let $Z_{m,n}$ be a divisor of bidegree $(m,n)$ inside $\mathbb{P}^1 \times \mathbb{P}^2$. Do any Torelli theorems hold for $Z_{m,n}$? I'm particularly interested in the case $(m,n)=(2,4)$.

It would also be interesting to know the answer for divisors of tridegree $(m,n,l)$ in $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$.

One thing that I know of is a Torelli theorem for the divisor of bidegree $(2,2)$ (a.k.a. the Verra threefold) in $\mathbb{P}^2 \times \mathbb{P}^2$ due to Iliev, but this is one higher dimension than I'm interested in.

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Continuity of perimeter of sets

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^1$-boundary. For any $r>0$, define $$\Omega_r=\{x\in\Omega: 0<d(x,\partial\Omega)<r\}.$$ Under what minimal regularity assumptions on the boundary can we prove $$\lim_{r\to 0}\text{Per}(\Omega_r)=\text{Per}(\Omega),$$ where $\text{Per}$ denotes the perimeter? For example, when $\Omega$ has smooth boundary, the above limit is true according to Lemma 3 in https://www.math.cmu.edu/%7Etblass/CNA-PIRE/Modica1987.pdf

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How to extend $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ similarity test to tensors?

Similarity test $\textrm{Tr}(A^k)=\textrm{Tr}(B^k)$ for $k=1,\ldots,d$ allows to conclude $d\times d$ symmetric matrices $A,B$ are similar: there exists orthogonal $O$ such that $A=OBO^T$.

How to extend such similarity test from order-2 matrix to higher order-$r$ symmetric tensors: ensuring that they differ by same orthogonal $O$ applied to all $r$ coordinates?

While the above $\textrm{Tr}(A^k)$ can be represented as length $k$ cycle with summation over edges, we can extend such rotation invariants to general graphs with degree $r$ vertices for order-$r$ tensors - getting necessary condition for tensors. But the real difficulty is getting sufficient condition - e.g. a complete set of invariants from graphs, such that their agreement ensures similarity?

ps. Practical application are e.g. shape descriptors modulo rotation: https://arxiv.org/abs/2601.03326

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Is there a Lipschitz and complete functional calculus for tracial von Neumann algebras?

This is an attempt at understanding why, precisely, the free entropy dimension approach to solving the free group factor problem fails. My understanding of the problem is that not all elements of, say, $L(\mathbb{F}_2)$, are "Lipschitz functions" applied to the standard free generators. However, I've never seen a formalized version of this assertion, let alone a proof of it. So, here is my understanding of what it actually means, in formal terms. To start with, let me define, in very general terms, what a functional calculus is:

Definition: Fix a positive integer $n$. Let $\mathcal{C}$ be a category whose objects are all of the form $(M, x_1, \cdots, x_n)$, where $M$ is a (complex, associative) algebra and $x_1, \cdots, x_n \in M$; and whose morphisms from $(M, x_1, \cdots, x_n)$ to $(N, y_1, \cdots, y_n)$ are all of the form $\phi: M \to N$, algebra homomorphisms s.t. $\phi(x_i) = y_i$ for all $1 \leq i \leq n$. (Note that $\text{Obj}(\mathcal{C})$ needs not contain all such $(M, x_1, \cdots, x_n)$, nor does the sets of morphisms need to contain all such $\phi$. To put it another way, $\mathcal{C}$ is a (not necessarily full) subcategory of the category of $n$-pointed algebras.) A functional calculus over $\mathcal{C}$ is an algebra $A$ with distinguished elements $a_1, \cdots, a_n$ (note that $(A, a_1, \cdots, a_n)$ needs not be in $\text{Obj}(\mathcal{C})$), together with an algebra homomorphism $\pi_{(M, x_1, \cdots, x_n)}: A \to M$ for each $(M, x_1, \cdots, x_n) \in \text{Obj}(\mathcal{C})$, called an evaluation map, s.t.,

1. $\pi_{(M, x_1, \cdots, x_n)}(a_i) = x_i$ for all $1 \leq i \leq n$;
2. $\phi \circ \pi_{(M, x_1, \cdots, x_n)} = \pi_{(N, y_1, \cdots, y_n)}$ for every $\phi \in \text{Mor}_\mathcal{C}((M, x_1, \cdots, x_n), (N, y_1, \cdots, y_n))$.

All the usual functional calculi can be put into this framework, such as polynomial functional calculus, holomorphic functional calculus, continuous functional calculus, (bounded or unbounded) Borel functional calculus, etc. Various universal (Banach / $C^\ast$- / von Neumann) algebras also fall within this framework.

For our specific case, we are concerned with functional calculi whose ranges are within tracial von Neumann algebras. Since we are concerned with the free entropy dimension, we can even restrict to the ones that are separable and Connes-embeddable:

Definition: Let $\mathcal{C}_\tau$ be the category whose objects are all tuples of the form $(M, x_1, \cdots, x_n)$, where $M$ is a separable, Connes-embeddable, tracial von Neumann algebra with a specified faithful, normal, tracial state $\tau_M$, and $x_1, \cdots, x_n$ are self-adjoint contractions in $M$. Morphisms from $(M, x_1, \cdots, x_n)$ to $(N, y_1, \cdots, y_n)$ are all trace-preserving $\ast$-homomorphisms from $M$ to $N$ that send $x_i$ to $y_i$ for all $1 \leq i \leq n$. A functional calculus is called tracial if it is over the category $\mathcal{C}_\tau$, its underlying algebra $A$ is a normed (but not necessarily Banach) $\ast$-algebra, the distinguished elements $a_1, \cdots, a_n$ of $A$ are all self-adjoint contractions, and the evaluation maps are all contractive $\ast$-homomorphisms.

Now, moving on to the actually important definitions. We want the functional calculus to be "Lipschitz". For technical reasons related to the definition of the free entropy dimension, we also want it to be “close to polynomials” in a suitable sense.

Definition: Consider a tracial functional calculus $(A, \pi_{(M, x_1, \cdots, x_n)})_{(M, x_1, \cdots, x_n) \in \text{Obj}(\mathcal{C}_\tau)}$. We say:

1. The functional calculus is weakly continuous if for each $a \in A$, $(M, x_1, \cdots, x_n) \in \text{Obj}(\mathcal{C}_\tau)$, and $\varepsilon > 0$, there exists a (noncommutative) polynomial $P \in \mathbb{C}\langle T_1, \cdots, T_n\rangle$ and an open (under the weak$^\ast$ topology) neighborhood $\mathcal{O}$ of the law of $(x_1, \cdots, x_n)$, s.t. $\sup_{(M, x_1, \cdots, x_n) \in \mathcal{C}_\tau} \|P(x_1, \cdots, x_n)\|_\infty \leq \|a\|_A$, and whenever $(N, y_1, \cdots, y_n) \in \text{Obj}(\mathcal{C}_\tau)$ satisfies the law of $(y_1, \cdots, y_n)$ is in $\mathcal{O}$, we have $\|\pi_{(N, y_1, \cdots, y_n)}(a) - P(y_1, \cdots, y_n)\|_2 < \varepsilon$.
2. An element $a \in A$ is continuous if for each separable, Connes-embeddable, tracial von Neumann algebra $(M, \tau_M)$, the map $(M_{\text{sa}})_1^n \ni (x_1, \cdots, x_n) \mapsto \pi_{(M, x_1, \cdots, x_n)}(a) \in M$ is $L^2$-to-$L^2$ uniformly continuous, uniformly across all choices of $M$, i.e., for every $\varepsilon > 0$, there exists $\delta > 0$ depending only on $a$ and $\varepsilon$, s.t. whenever $(M, \tau_M)$ is a separable, Connes-embeddable, tracial von Neumann algebra and $(x_1, \cdots, x_n), (y_1, \cdots, y_n) \in (M_{\text{sa}})_1^n$ with $\|x_i - y_i\|_2 < \delta$ for all $1 \leq i \leq n$, we have $\|\pi_{(M, x_1, \cdots, x_n)}(a) - \pi_{(M, y_1, \cdots, y_n)}(a)\|_2 < \varepsilon$.
3. The functional calculus is strongly continuous if it is weakly continuous and every $a \in A$ is continuous.
4. An element $a \in A$ is Lipschitz if for each separable, Connes-embeddable, tracial von Neumann algebra $(M, \tau_M)$, the map $(M_{\text{sa}})_1^n \ni (x_1, \cdots, x_n) \mapsto \pi_{(M, x_1, \cdots, x_n)}(a) \in M$ is $L^2$-to-$L^2$ Lipschitz, uniformly across all choices of $M$, i.e., there exists $C > 0$ depending only on $a$, s.t. whenever $(M, \tau_M)$ is a separable, Connes-embeddable, tracial von Neumann algebra and $(x_1, \cdots, x_n), (y_1, \cdots, y_n) \in (M_{\text{sa}})_1^n$, we have $\|\pi_{(M, x_1, \cdots, x_n)}(a) - \pi_{(M, y_1, \cdots, y_n)}(a)\|_2 \leq C\max_{1 \leq i \leq n}\|x_i - y_i\|_2$.
5. The functional calculus is Lipschitz if it is weakly continuous and every $a \in A$ is Lipschitz.
6. The functional calculus is complete if the range of $\pi_{(M, x_1, \cdots, x_n)}$ is $W^\ast(x_1, \cdots, x_n)$ for every $(M, x_1, \cdots, x_n) \in \text{Obj}(\mathcal{C}_\tau)$.

Now I can formally state my question:

Question: Does there exist a Lipschitz and complete tracial functional calculus (for every $n \geq 2$)?

Remarks:

1. An affirmative answer will immediately imply the free entropy dimension is an $W^\ast$-invariant and thus resolves the free group factor problem. So, I'm not expecting an easy yes answer. What I'm really asking is whether there is an easy proof that no Lipschitz and complete tracial functional calculus can exist.
2. The reason I called the first property weakly continuous despite it seemingly not related to continuity is because of the following equivalent characterization of weakly continuous functional calculus: A tracial functional calculus is weakly continuous iff for every $a \in A$, the law of $(\pi_{(M, x_1, \cdots, x_n)}(a), x_1, \cdots, x_n)$ depends only on the law of $(x_1, \cdots, x_n)$ and the mapping from the law of $(x_1, \cdots, x_n)$ to the law of $(\pi_{(M, x_1, \cdots, x_n)}(a), x_1, \cdots, x_n)$ is continuous w.r.t. the weak$^\ast$ topology. I didn't use this version in the definition because I feel it is harder to work with given that the goal is to disprove the existence of some functional calculus, but that's just my own subjective opinion.
3. If we weaken Lipschitz to strongly continuous, then the answer is actually yes: There exists a strongly continuous and complete tracial functional calculus. See Appendix section A.2 of this paper. One can restrict to its Lipschitz elements to obtain a Lipschitz tracial functional calculus, of course. But the issue is then we may no longer have completeness.
4. Since we are working with only Connes-embeddable algebras, for a weakly continuous tracial functional calculus, in the definition of both continuous and Lipschitz elements, it suffices to restrict $M$ to full matrix algebras with their standard normalized traces.

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Postnikov tower for $S^3$

I'm trying to look into the Postnikov tower for the $3$-sphere with the obstruction classes shown explicitly, at least till degree $6$. Can anyone suggest me some reference where it might have been done explicitly? Also, can someone please explain secondary and tertiary obstructions and how to detect them? Any help would be greatly appreciated. Thanks.

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Presentation of the mod-2 cohomology ring of the exceptional Coxeter group H_4

Let $H_4$ denote the exceptional finite Coxeter group of type $H_4$ (order $14400$), realized as the reflection group of the $120$--cell. One can describe $H_4$ as a semidirect product $$ H_4 \cong \mathbb{Z}/2\mathbb{Z} \ltimes (A_5 \ltimes \mathbb{Z}/2\mathbb{Z}). $$ I am interested in an explicit description of the group cohomology ring $$ H^*(H_4;\mathbb{F}_2) $$

as a polynomial ring. Is there a known presentation (generators and relations) of the mod--$2$ cohomology ring $H^*(H_4;\mathbb{F}_2)$?

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Generating functions for tensor L-functions of $\mathrm{GL}_m \times \mathrm{GL}_n$

Let $F$ to be a non-archimedean local field and $\pi_n$ be an unramified principal series of $\mathrm{GL}_n(F)$.

Let $\sigma_n$ be the normalised spherical matrix coefficient of $\pi_n$ such that $\sigma_n(I_n) = 1$, and define $\Delta_n$ as the bi-$\mathrm{GL}_n(\mathfrak o)$-invariant by \begin{equation*} \Delta_n^s(g) := \begin{cases} |\det g|^{s+(n-1)/2} &\text{if $g\in \mathrm{Mat}(\mathfrak o)$}\\ 0 & \text{otherwise}. \end{cases} \end{equation*}

In Godement-Jacquet's Zeta Functions of Simple Algebra Lemma 6.10, we have $$\int_{\mathrm{GL}_n(F)} \sigma_n(g) \Delta_n^s(g) \, dg = L(s, \pi_n)$$ where $L(s, \pi_n)$ is the standard unramified L-function of $\pi_n$. For $m,n\geq2$ does there exists a function $\Delta_{m,n}(g_1,g_2)$ for $(g_1,g_2) \in \mathrm{GL}_m(F) \times \mathrm{GL}_n(F)$ that is bi-$\mathrm{GL}_m(\mathfrak o) \times \mathrm{GL}_n(\mathfrak o)$-invariant such that we have $$\int_{\mathrm{GL}_m(F) \times \mathrm{GL}_n(F)} \sigma_m(g_1) \sigma_n(g_2) \Delta_{m,n}(g_1,g_2) \, dg_1 \, dg_2$$ evaluates to the tensor unramified L-function $L(s, \pi_n \times \pi_m)$ of $\pi_n \times \pi_m$? Is there any reference for such a construction?

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A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$

$\def\S{\mathbb{S}}$ Dear all,

So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is: for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$

Is there a general formula (or a nice combinatorial description) for the number of conjugacy classes produced in this case?

Some thoughts on the problem: for a fixed $n$ one can compute this by hand as follows: Note that if we consider the action of $\S_n \times \S_n$ on $\S_n \times \S_n$ by conjugation, then we have classes of types $(\lambda, \mu)$ for each configuration $\lambda$ and $\mu$ of the Young diagram. Here we restrict ourselves to the diagonal action, so this would split the $(\lambda,\mu)$ pair further. To compute this splitting for a fixed pair $(x,y)$ where $x$ is of type $\lambda$, y is of type $\mu$, let $C(x)$ denotes the centralizer of $x$. For any $g \in \S_n$, write $g = g_1g_x$ for some $g_x \in C(x)$. Then $(x,y) \sim (g_1xg_1^{-1}, g_1(g_xyg_x^{-1})g_1^{-1})$ Then the number of equivalence classes we have for this pair is the number of orbits of the class $\mu$ acted upon by $C(x)$ via conjugation.

But is there a general formula?

Thanks,

Ngoc

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(Non)existence of Schrödinger operators with eigenvalues of infinite multiplicity at the bottom of the essential spectrum

Let $(\Omega,g)$ be a connected Riemannian manifold of finite volume (possibly with boundary). This includes domains in $\mathbb{R}^n$ and compact manifolds without boundary; noncompact and noncomplete manifolds are also allowed. Suppose we have a potential $V\in L^1(\Omega)$ with $V\ge 0$ such that $$ -C \int_{\Omega} \varphi^2 \leq Q[\varphi, \varphi] := \int_{\Omega} |\nabla\varphi|^2 - \int_{\Omega} \varphi^2V \qquad \forall \varphi \in H^1_0(\Omega) \cap L^{\infty}(\Omega) $$ for some constant $C \geq 0$. In other words, $Q$ is an unbounded quadratic form on $L^2(\Omega)$ that is bounded from below, and the domain of $Q$ can be extended to the whole $H^1_0(\Omega) := \overline{C^\infty_0(\Omega)}^{H^1}$. Suppose $Q|_{H^1_0(\Omega)}$ is closable, and let $\Delta - V$ be the Friedrichs extension of $Q$. Here, $\Delta \varphi = - \operatorname{div} \nabla \varphi$ (dependencies on the metric $g$ are omitted).

Question

Does there exist an example of $(\Omega, g)$ and $0\leq V \in L^1(\Omega)$ as above such that the Schrödinger operator $\Delta - V$ has finitely many negative eigenvalues (counting multiplicities) and an infinite-dimensional kernel? In other words, $0 = \inf \sigma_\text{ess}(\Delta - V)$ is an eigenvalue of infinite multiplicity.

Feel free to consider the following modifications of the question if it helps to prove existence/nonexistence:

• $\Omega$ is compact without boundary, so $H^1_0(\Omega) = H^1(\Omega)$;
• $\dim [\ker (\Delta - V) \cap H^{1}_0(\Omega)] = \infty$, that is, we consider $\ker Q$ instead of $\ker (\Delta - V)$;
• no restriction on the sign of $V$, but it still defines a bounded bilinear form on $H^1_0(\Omega)$;
• no restriction on $\operatorname{Vol} \Omega$ (so, $\mathbb{R}^n$ is included).

I know how to construct eigenvalues of infinite multiplicity by considering tensor products of $1$-dimensional operators (a.k.a. separation of variables). However, in this case, the number of negative eigenvalues should always be infinite if zero is an eigenvalue of infinite multiplicity.

Motivation

This question is related to the following optimization problem: $$ \int_{\Omega} V \to \max \quad \text{on}\ \left\{0\leq V \in L^\infty(\Omega)\colon \ \lambda_k(\Delta - V) \geq 0\right\}. $$ The maximizing potentials may lie outside of $L^\infty(\Omega)$ and are related with harmonic maps to spheres. In fact, when $\Omega$ is compact without boundary, one can find such $V$ in the form $V = |\nabla u|^2$ for some harmonic map $u \colon \Omega \to \mathbb{S}^n$ with $n = \operatorname{mult} \lambda_k(\Delta - V)$. I am wondering whether the case $n = \infty$ is possible?

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Definition of Thom isomorphism in topological K-theory

I have a confusion regarding the definition of the thom isomorphism map in topological K-theory.

Suppose $\pi: F \rightarrow X$ is a complex vector bundle. Let $i:X \rightarrow F$ denote the zero section. Then $\pi \circ i : X \rightarrow X$ is the identity map on $X$ and $i \circ \pi : F \rightarrow F$ is homotopic to the identitity map on $F$. Thus it follows that both the induced maps $i^* : K(F) \rightarrow K(X)$ and $\pi^* : K(X) \rightarrow K(F)$ are isomorphisms.

My question is:

Is the above argument correct? If so, why can't we simply define the Thom isomorphism as $\pi^*$?

The reason why I ask this question is as follows. The book Differential Topology and Quantum Field Theory by Charles Nash describes the construction of a sequence of vector bundles over F: \begin{equation} 0 \xrightarrow{\wedge} F \xrightarrow{\wedge} F \wedge F \xrightarrow{\wedge} \dots \xrightarrow{\wedge} F \wedge \dots \wedge F \to 0 \end{equation} This determines an element $\lambda_F \in K(F)$. Thom isomorphism $\phi: K(X) \to K(F)$ is then defined as multiplying the element of $K(X)$ with $\lambda_F$. If I understand correctly this is the map $[E] \mapsto [\pi^*E] * \lambda_F$. Why is multiplication by $\lambda_F$ necessary?

Thanks!

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What is known about the Ramsey number of the sum?

I'm wondering whether anyone can point me to any paper (if they exist) that bounds $R(a+b)$ given $R(a)$ and $R(b)$. If one had, for instance

$$R(a+b)\leq R(a)R(b)$$

Then we could conclude via Fekete's lemma that $\lim_n\sqrt[n]{R(n)}$ exists, this sadly, isn't true, as known bounds already imply $R(3+3)>R(3)R(3)$. There are also weakened versions of Fekete's lemma, but I'm guessing they're either harder conditions to verify or false. Of course there's easy bounds that just come from the growing properties of $R(n)$, like

$$R(a+b)\leq (R(a)R(b))^4$$

but growing properties cannot ever be enough, as there's of course there are functions that also satisfy the best known Ramsey bounds, are increasing, but don't have the desired limit. I'm looking for something that somehow uses the definition of $R(n)$ to show that a graph on $f(R(a),R(b))$ vertices must contain a monochromatic $K_{a+b}$

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A variant of Ramsey numbers

The well known Ramsey number $R(k)$ is the least integer $n$ so that every 2-edge coloring of $K_n$ contains a monochromatic $K_k.$

Another interpretation of the above definition is that every graph on $R(k)$ vertices has a $K_k$ or $\overline{K_k}$ as a (induced) subgraph. There are many generalizations of Ramsey numbers and I am curious to see what happens if we push this in the direction of complete multipartite graphs.

Let $\widetilde{R}(k)$ be the least integer $n$ so that every graph on $n$ vertices must contain an induced complete multipartite graph on $k$ vertices.

Since $K_k$ and $\overline{K}_k$ are complete multipartite graphs we have $\widetilde{R}(k) \leq R(k)$. What I am wondering is whether allowing other complete multipartite graph reduces the order of $\widetilde{R}(k)$ significantly. More precisely

Is it true that $\widetilde{R}(k) = o(R(k))$?

I am still looking at the available literature so if anyone is aware of results in this direction that is also appreciated. In particular is anybody aware of non-obvious bounds for $\widetilde{R}(k)$?

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Reference Request: Dirichlet Style Counting Function

Reference Request: While working on a project the below function started appearing in several places and I was wondering if it is already a named equation and if there are any references or sources that people would recommend for learning more about it. It seems to have some similarities to the Dirichlet Divisor Problem, Von Mangoldt function, and Riemann Prime counting function.

\begin{equation} f(N) = \sum\limits_{k=2}^{N} D(k)\lfloor \frac{N}{k} - 1\rfloor = \frac{\Lambda(k)}{\log(k)} \lfloor \frac{N}{k} - 1\rfloor \end{equation}

Where $D(k)$ is similar to the Von Mangoldt function, $\Lambda(k)$:

\begin{equation} D(n) = \frac{\Lambda(n)}{\log(n)} = \begin{cases} \frac{1}{r}, & \text{if } n = p^r \text{ for some prime } p \text{ and integer } r \ge 1, \\ 0, & \text{otherwise}. \end{cases} \end{equation}

It appears to be counting the number of numbers divisible by prime powers less than $N$ inversely weighted by the powers but I am very curious to learn more about what properties of this function are already in the literature. Thank you!

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Product of central binomial coefficients

I have a question about an equality involving products of central binomial coefficients. If $x_1,...,x_n$ and $y_1,...,y_n$ are positive integers, with $\sum_i x_i = \sum_i y_i$ and $$ \binom{2x_1}{x_1} \cdots \binom{2x_n}{x_n} = \binom{2y_1}{y_1}\cdots \binom{2y_n}{y_n}\,, $$ what are the restrictions on the $x_i$ and $y_i$, and is there any solution other than the trivial one $\{x_1,...,x_n\}=\{y_1,...,y_n\}$?

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Second fundamental form and geodesic curvatures of frame lines

Consider a smooth surface $S$ in $\mathbb{R}^3$, if necessary simply connected, parametrized by $\mathbf{r}(u,v)$. With the second fundamental form written as $\text{II}=L\:\mathrm{d}u^2+2M\:\mathrm{d}u\:\mathrm{d}v+N\:\mathrm{d}v^2$, $L$ is the normal curvature of the $v-$constant curves, $\kappa_n^{(u)}$. We can also relate normal curvatures to geodesic curvatures using the identities

\begin{align} \kappa_n^{(u)}&=\kappa^{(u)}\:\cos\theta^{(u)}\\ \kappa_g^{(u)}&=\kappa^{(u)}\:\sin\theta^{(u)} \end{align}

where $\theta^{(u)}$ is the angle between the surface normal and the normal to the curve, or $\cos\theta^{(u)}=\widehat{\mathbf{N}}_S\cdot \widehat{\partial^2_{uu}}\mathbf{r}$, and $\kappa_g^{(u)}$ is the geodesic curvature and $\kappa^{(u)}$ the space, total or just curve's 'curvature'. The same holds identically for $u-$ constant lines, so we can write

\begin{align} \text{II}=\kappa_g^{(u)}\cot\theta^{(u)}\:\mathrm{d}u^2+2M\:\mathrm{d}u\:\mathrm{d}v+\kappa_g^{(v)}\cot\theta^{(v)}\:\mathrm{d}v^2 \end{align}

Now, the $\kappa_g$ can be calculated in terms of the metric only via the Christoffel symbols, and the $\theta^{(u)},\:\theta^{(v)}$ can be expressed in terms of dot products of the Frenet–Serret frames vectors. The question is, what about $M$ ? I haven't seen a similar way to express this diagonal term in terms of either geodesic curvatures of the coordinates lines or something similar as with $L$ and $N$.

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Finite groups for which the maximum degree of the prime graph is 2

Does there exist a finite non-solvable and non-almost-simple group satisfying the following conditions?

1. The degree of every vertex in its prime graph is at most $2$,
2. If a vertex $p$ in its prime graph has degree $2$, then its Sylow $p$-subgroup has exponent $p$.

(Note: The prime graph of a finite group is the graph whose vertices are the prime numbers dividing the order of the group, with two vertices being linked by an edge if and only if their product divides the order of some element of the group.)

I believe that, if such a group exists, it should be realized as a direct product of copies of the simple group $A_5$.

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Equivalence of gauge-fixed linear measure and KKS symplectic measure in loop space: proof or plausible conjecture?

Can you answer the following simple question: Take the measure ${DC}$ in loop space in $\mathbf R^4$ , and factorize if by diffeomorphisms, using conformal gauge as follows. Consider a minimal surface bounded by $C$, parametrized by a holomorphic map $f_\mu(z)$ of a disk: $$ C_\mu(\theta) = 2 \Re f_\mu(e^{i\theta}). $$ The map is subject to Virasoro constraint $f'^2=0$, which serves as a gauge condition to fix diffeomorphisms. The null vector $f'(z)$ can be parametrized by two twistors $ (\lambda, \mu)$ :
$$ f'_\alpha(z) = \bar \lambda(z) \sigma_\alpha \mu(z). $$ Now, my question is how to go from $DC/\operatorname{Vol}(\operatorname{Diff})$ to the KKS symplectic measure $$ D\lambda D\mu \exp\left(i \int_{d D} (\bar \lambda d \lambda+\bar \mu d \mu )\right)?$$ This conjecture seems reasonable geometrically, but I would like to have a proof, say by honest computation of the Faddeev-Popov Jacobian for this Virasoro as a gauge condition for parametrization of $C$.
In short, my loop is parametrized by these two twistors, and I would like to convert the linear measure ${DC}$ modulo diffeomorphisms to the $\lambda,\mu$ variables.

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