# All Questions

107,274
questions

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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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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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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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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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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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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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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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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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**1**answer

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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**1**answer

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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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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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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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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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 ...

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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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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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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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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 ...

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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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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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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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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 "...

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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 ...

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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-...

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**1**answer

98 views

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

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

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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 ...

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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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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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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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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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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 ...

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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 ...

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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 $\...

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**1**answer

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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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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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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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 ...