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Proof theory and subsystems of second-order arithmetic: in particular the reverse mathematics of Godel's system $T$

While doing some research on reverse mathematics, I came across the following document under the address, www.andrew.cmu.edu>user>avigad>Talks>survey1: Proof theory and Subsystems of Second-Order ...
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2 views

The graph and sign of $p_n-\operatorname{ali}(n)$, where $p_n$ is the $n$-th prime and $\operatorname{ali}(n)$ the inverse of the logarithmic integral

I'm inspired in [1] to ask the following question. My problem is that I have not an implementation of the inverse of the logarithmic integral $\operatorname{Li}(x)=\int_2^x\frac{dt}{\log t}$, that ...
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0answers
35 views

On the existence of a globally generated vector bundle on a $K3$ surface

This is on some confusion on the proof of lemma $1.6$ of the paper titled Special divisors on curves on a $K3$ surface(For convenience I am attaching the link here: https://link.springer.com/article/...
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18 views

Collapsing an independet set in the icosahedral graph

Let $G$ be the icosahedron graph. If $I \subseteq V(G)$ is an independent set with $|I|>1$, does contracting $I$ necessarily increase the Hadwiger number?
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0answers
27 views

Show that the lub of a set of negative numbers cannot be positive [on hold]

How is this conclusion? If a set $S$ contains only negative numbers then $0$ by definition is an upper bound. Any positive number would be greater than $0$ therefore, the lub of set $S$ can never be ...
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0answers
35 views

Searching for an example of semicontinous function $\dots$

QUESTION. Can you construct a bounded, semi-continuous function that is nowhere continuous? Any reference is appreciated.
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16 views

history background forThe Cauchy–Schlömilch transformation

Do you know a history background of The Cauchy–Schlömilch transformation that I use for my thesis? contect me:hossein264.1375@email.kntu.ac.ir
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0answers
40 views

What is the value of the partition function of CFT on a compact conformal manifold?

Is the value of the partition function of a 2d CFT on a compact conformal manifold well defined? Or is there some kind of "anomaly" that makes it dependent on a bulk or some other kind of further ...
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41 views

Is there a name for this equivalence relation?

Let $M$ be an arbitrary set and let $\mathscr{F}$ be a family of subsets of $M$. Is there a known name for the following equivalence relation or its corresponding partition? $\sim_{M,\mathscr{F}}\,=\...
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52 views

Is there something wrong with this definition of principal bundle?

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (...
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1answer
35 views

Simple System of linear diophantine inequalities

Is there a simple way to find the first integer solution to the following system of equations? $$ \begin{cases} \begin{align} a_1x>y \\ a_2x<y\\ \end{align} \end{cases} $$...
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1answer
100 views

Laplace spectrum of the $2$-Sphere

The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$ acting on functions. The eigenvalues of this ...
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81 views

Are epimorphic endomorphisms of noetherian commutative rings always injective?

This question was asked, but not answered, on Mathematics Stackexchange. [In this post "ring" means "commutative ring with one".] Let $A$ be a noetherian ring, and let $f:A\to A$ be an endomorphism ...
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42 views

In 1d is fractional Sobolev space H^(3/2) a vector space?

I am trying to find some informations on the fractional Sobolev spaces in one dimension. My question is about $H^{3/2}$ and the homogeneous version $\dot{H}^{3/2}$: is the homogeneous version a ...
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0answers
31 views

About p-laplacian and variations

Let $\Omega \subset \mathbb{R^{n}}$ be a domain (open and connected set), for $p\geq 2$, the $p$-laplacian is defined by: $\Delta_p u= div (|\nabla u|^{p-2} \nabla u)$, in non-divergence form the $p$-...
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51 views

Dehn and the Jordan curve theorem

I am looking for a manuscript of Max Dehn entitled "Beweis des Satzes, dass jedes geradlinige geschlossene Polygon ohne Doppelpunkte 'die Ebene in zwei Teile teilt'". According to Heinrich ...
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1answer
97 views

$d^3$ in the Atiyah-Hirzebruch spectral sequence for (twisted) $KO$

Cross posted from here after no responses and a bounty being placed on the question. Let $h^n(-)$ be a generalised cohomology theory. For a space $X$ there is a spectral sequence known as the Atiyah-...
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25 views

Every point in a regular polytope has its own antipodal point or antipodal face

I apologize for using non-common language. When this problem comes to my mind, it seems quite easy but It's not. Maybe It can be rewritten as, There exists a unique facet containing the most far ...
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44 views

Universal property of finitely generated graded algebras

Let $k$ be a field, and let $R=\oplus_{i\geq 0}R_{i}$ be a commutative finitely generated graded $k$-algebra with $R_{0}=k$. Suppose that the submodule $\oplus_{i=0}^{l}R_{i}$ generates $R$ as a $k$-...
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35 views

Finiteness of decompositions of tensor-powers of a MCM module

Let $X$ be an isolated, Gorenstein, surface singularity and $M$ a maximal Cohen-Macaulay module on $X$. Let $S$ be the set of all indecomposable maximal Cohen-Macaulay modules $N$ on $X$ such that ...
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1answer
79 views

Sobolev embedding in the space of continuous functions

Let $I = \mathbb{R}$ and let $W^{1,2}(I,\mathbb{R})$ be the Sobolev space of function from $I$ to $\mathbb{R}$ (one time weakly differentiable and contained in $L^{2}$) and $C^{0}(I,\mathbb{R})$ be ...
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0answers
34 views

The Bivariate Risk of Ruin Problem

Consider a random process $\mathbf{S} = (S_0, S_1, \ldots, S_k) : S_0 = 1$ and $$ S_{i+1} = \sum_{i = 1}^{S_i} \; U_i, $$ with each of the $U_i$ having distribution $$ \begin{align*} U_i &= (...
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0answers
95 views

Is the Lie algebra of a flat group scheme still flat?

Let $\mathcal{O}$ be a discrete valuation ring, and $\mathcal{G}$ is a flat group scheme over $\mathcal{O}$, we may assume the generic fiber is reductive. Then can we define its Lie algebra, and ...
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0answers
34 views

discrete Fourier transform for composition of differential operators on a grid

This question pertains to stability analysis of finite difference methods using the discrete Fourier transform. Suppose I have a convection diffusion equation of the form: (1) $\hspace{.5in}u_t + \...
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0answers
18 views

Smallest nonzero eigenvalue of Laplacian with Neumann boundary conditions of a ball in simply-connected pinched negative curvature space

In the paper, Small eigenvalues of geometrically finite manifolds, http://www.math.uni-bonn.de/people/ursula/eigenvalue.pdf , page 11, in the last paragraph, the author made a claim that "Now the ...
3
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1answer
158 views

Divisibility of certain polynomials

Consider the finite sums $$F_n(q)=\sum_{k=1}^nq^{\binom{k}2}$$ with exponents the triangular numbers $\binom{k}2$. When $n$ is odd, it appears that $F_n(q)$ does not factorize over $\mathbb{Z}[q]$. On ...
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1answer
74 views

Does the inequality $\mu F(\mu)^2\geq \int_{\mu}^{b}F(x)\cdot(1-F(x))\,\mathrm dx$ hold for compactly supported continuous random variables?

This question was originally asked on the Mathematics StackExchange by User smcc Consider a continuous random variable $V$ with cumulative distribution function $F$ and density function $f$. Suppose ...
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25 views

On an estimate of hitting times

I have a question of hitting times of Brownian motions. Let $\mathbb{D} \subset \mathbb{C}$ be the open unit disk centered at the origin. We divide $\mathbb{D}$ into open sets $A$ and $B$, which are ...
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12 views

Fritz-John conditions: Equality-constrained case as special case of inequality constraints

In Chapter 4 of Nonlinear Programming: Theory and Algorithms by Bazarra, Sherali, and Shetty, the following claim is made after Theorem 4.3.2 (Fritz-John necessary conditions): "Note also that these ...
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1answer
42 views

A question about harmonic measure 2

Suppose $W$ is a bounded open subset of $\mathbb{R}^{n}$ and $n\geq2$. Let $V$ be the interior of the closure of $W$ and $E$ a subset of the boundary of $V$. If $\omega(x,W)(E)=0$ ($\omega(x,W)$ is ...
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1answer
147 views

Delooping a fibration sequence with loopspace fiber and finite CW complexes

The following question is somewhat similar to a previous one on MathOverflow, except that my application does not directly involve Eilenberg-MacLane spaces $K(G,n)$, and so I don't see the immediate ...
2
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1answer
49 views

Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic

Let $(W,S)$ be a Coxeter system. Let $q=(q_s)_{s\in S} \in \mathbb{R}^{\text{#}S}$ be a tuple of positive real numbers with $q_s=q_t$ whenever $s$ and $t$ are conjugate to each other. Follwing Davis', ...
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0answers
33 views

Notions of integrability for affine Lie algebras and positive energy representations

Let $\mathfrak{g}$ be a simple (complex) Lie algebra. Given an invariant bilinear form $\kappa : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{C}$, we can form the central extension $\hat{\mathfrak{g}}...
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0answers
147 views

On actions by hyperbolic group

Can I have a virtually cyclic infinite hyperbolic group $G$ such that hyperbolic group acting on $l^{\infty}(G)$ faithfully and if $g\rightarrow \lambda_{g}$ is the left regular representation such ...
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1answer
506 views

How can I improve my mathematical creativity? [on hold]

NOTE: This post has been completely rewritten, but the ideas remain the same. I've been trying to figure out the divide between "good" and "great" mathematicians, and one metric I see repeatedly is "...
5
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0answers
124 views

Hilbert's solution of Waring's problem

What actually is the strategy of Hilbert's method of solution to Waring's problem? I do not read German, so I do not understand what he says. But I was told that it is different from the Hardy ...
5
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1answer
136 views

Yau's conjecture on nodal sets for manifolds with boundary

I've just read a review paper about Yau's conjecture on nodal sets of the eigenfunctions for the Laplace operator on manifolds. Briefly, if $\phi_\lambda$, $\lambda$ are an eigenpair for the Laplace-...
2
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1answer
98 views

Can Mellin transform be applied in this function? What's the result?

$$f(x) = \mathop {\lim }\limits_{T \to \infty } {i}\int_{-1/2-i\,T}^{-1/2+i\,T} \frac{(x-1)^{s}}{2^{s+1}}\,\frac{1}{sin(\pi*s)\,}\,\frac{ds}{s}$$
1
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2answers
42 views

Opial type inequalities

Let $x(t)\in C^1[0,h]$ be such that $x(0)=x(h)=0$ and $x(t)$ in (0,h) ,then the following inequality $ \int^h_0 |x(t)x^{'}(t)|dt \leq \frac{h}{4}\int^h_0(x^{'}(t))^2dt$ my question: I would like ...
3
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0answers
53 views

Boolean functional equations

My current approach to investigating reversible quantum gates requires the solution of Boolean functional equations. For example, $$f(x,y,z) = f(x,y \oplus f(x,y,z), z \oplus f(x, y, z)),$$ where $f\...
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2answers
107 views

Lower bound for solutions of Vinogradov's Mean Value Theorem

I have a doubt about the number of solutions for the system $$x_1^j+\cdots+x_s^j=y_1^j+\cdots+y_s^j,\quad(1\leq j\leq k)$$ with $1\leq x_i,y_i\leq X$. It is a big breakthrough of Bourgain-Demeter-Guth,...
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0answers
14 views

Bando model vehicles lengths

Why does Bando model $x''_j=V(x_{j+1} - x_j) - x'_j$ ignore vehicles lengths? Is it because of trying to make model simple as possible?
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0answers
37 views

Existence of limit computable map

Is there a limit computable function $\Phi$ with the following properties? Let $T\subset \mathbb{N}^{<\mathbb{N}}$ be a tree (coded as its characteristic function) and $f\in\mathbb{N}^\mathbb{N}$ ...
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1answer
49 views

Recognition of a graph as a product of its quotients

Is there an algorithm to determine whether a given simple graph $G$ is a product graph, typically, say a cartesian product graph of two smaller simple graphs $G_1, G_2$, such that the two simple ...
13
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0answers
158 views

Finite version special case Jacobi triple product formula

In this paper, Shanks uses the following formula: $$ \sum_{s=0}^{n-1}q^{s(2n+1)} \times \left( \prod_{k=s+1}^{n} \dfrac{1-q^{2k}}{1-q^{2k-1}}\right) = \sum_{s=1}^{2n} q^{\frac{s(s-1)}{2}}$$ to get a ...
3
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1answer
119 views

outer automorphism classification

I am trying to understand Bestvina's "A Bers-like proof of the existence of train tracks for free group automorphisms". I'm going to ask a probably trivial question ... Here we go: The automorphism $\...
6
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1answer
127 views

Embeddability of all graphs of cardinality $\kappa$ into one graph of cardinality $\kappa$

Does every infinite cardinal $\kappa$ have the following property? There is a simple, undirected graph $G_0=(\kappa, E_0)$ such that every simple, undirected graph $G=(\kappa, E)$ is isomorphic to ...
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0answers
27 views

Which properties, a warped product manifold $M$, can benefit from in having a complex subspace in its tangent space?

I have a Riemannian warped product manifold $M=B \times_f F$ where $M$ is not compatible with an almost-complex structure $J$, but (for example) $B$ is compatible with an almost-complex structure $J$. ...
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0answers
43 views

Quadratic factors of $l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$

Related to sum of three squares and this question. Let $l_1,l_2,l_3 \in \mathbb{Z}[x,y]$ and $n \in \mathbb{Z}$. Assume that $n$ is not a cube and not twice cube. Let $f=l_1(x,y)^3+l_2(x,y)^3+l_3(x,...
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0answers
37 views

Measure/Volume in length space

Is it possible that there is a non-trival doubling measure $u$ on the doubling length space $(M^n,d)$, where $M^n$ is a closed topological n-manifold, satisfies: (1) Let $d_1:=\lambda d$, then the ...

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