# All Questions

**0**

votes

**0**answers

9 views

### Pasch axiom and Pythagorian field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorian field condition.

**1**

vote

**0**answers

15 views

### Does the link of a hypersurface singularity determine its analytic type?

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a ...

**-2**

votes

**0**answers

18 views

### Formula for getting a value that doubles the amount of the previous value? [on hold]

I am new to Math overflow. I have a question that I cannot seem to answer whatever formula I try.
I don't know how to explain it so I'll just graph it:
Let 'x' be an increasing number.
x = y
...

**0**

votes

**0**answers

14 views

### Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote D being a set {a,b} of such pairs of parameters that NOT ALL zeroes of corresponding ...

**0**

votes

**0**answers

16 views

### Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$:
\begin{cases}
u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\
u(0)=x_0,
\end{cases}
Suppose $A$ generates an analyitc ...

**7**

votes

**1**answer

115 views

### Is it possible to have a research career while checking the proof of every theorem that you cite?

A colleague raised the above question with me; more precisely he said:
Suppose that a mathematician were resolved not to publish any theorems
unless she had checked the proof of every theorem ...

**0**

votes

**0**answers

7 views

### Smoothness of Value function for SDE with discontinuous coefficients

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function ...

**-5**

votes

**0**answers

20 views

### Discrete Math proof problem, unsure where to start [on hold]

Let { m1, m2, ....., mk } be pairwise relatively prime positive integers. Prove that there cannot be more than one solution to the system of congruence's $$ \langle x ≡ ai (mod mi) \rangle $$ in ...

**0**

votes

**0**answers

37 views

### connectedness of semi algebraic sets

We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, ...

**0**

votes

**0**answers

38 views

### Moving a result from the unconditional to the conditional

I'm generally wary when lifting a result stated unconditionally to a situation where I'm conditioning on a random variable. Consider the following classical result in weak convergence:
Theorem. Let ...

**0**

votes

**0**answers

26 views

### On Schrijver Lower bound

Shrijver lower bound gives number of perfect matchings on a $k$-regular bipartite graphs as $\Big(\frac{(k-1)^{k-1}}{k^{k-2}}\Big)^n$. What is the corresponding lower bound for min-degree $k$ and ...

**2**

votes

**0**answers

33 views

### Algebraic $K_1$ group for a $C^*$-algebra

Let $A$ be a $C^*$-algebra: then one defines topological $K_1$ group as $GL_{\infty}(A^+)/\Big(GL_{\infty}(A^+)\Big)_0$ where $A^+$ denotes $A$ with the unit adjointed (even if $A$ already had a unit: ...

**1**

vote

**0**answers

29 views

### First variation on double integral

Currently I am trying to fully understand the paper of munk1921.
In the derivation of the minimum induced drag theorem it is at one point stated (p.378) that in order to minimize drag the following ...

**-4**

votes

**0**answers

40 views

### Example of infinite field of characteristic prime is not algebraically closed field [on hold]

I know that if $F$ is an algebraically closed field, then $F$ is infinite. The converse is not true, so what is the example of an infinite field of characteristic prime $p>0$ not algebraically ...

**-4**

votes

**0**answers

59 views

### Rational power Napier number [on hold]

Help me with the following question.
Prove that $2^e$ irrational, where $e$ is the Napier number.

**0**

votes

**1**answer

83 views

### Relation between Harmonic vector field and Harmonic 1-form

Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function
$$E(X)=\int_M\|dX\|dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^2dvol_g.$$
...

**-6**

votes

**0**answers

31 views

### Proof about a measure zero set [on hold]

Let $A$ be a Lebesgue measurable subset of $\Bbb R$.
Show that there exists a Borel measurable subset $B$ of $\Bbb R$ such that $A\subseteq B$ and such that $l^*(B\setminus A)=0$. Also, show that ...

**-6**

votes

**0**answers

128 views

### Great Mathematicians Without a PhD [on hold]

While listing to some music, I was wondering which great mathematicians did not have or do not have a PhD. This is a very subjective question, since "great" is not formally defined. But to describe it ...

**16**

votes

**1**answer

374 views

### Who proved that $l^1$ and $L^1[0,1]$ are not isomorphic?

$l^1$ has the Schur property (every weakly convergent sequence is norm convergent) and $L^1[0,1]$ does not, so the two spaces cannot be isomorphic.
Is this folklore, or is it credited to someone? ...

**-9**

votes

**0**answers

107 views

### I have no experience with math research [on hold]

Consider the polynomial expansion for $\frac {\sin x}{x} = p(x)= 1-\frac {x^2}{3!}+\frac {x^4}{5!}–\frac {x^6}{7!} + \cdots$
$p(x) = \prod(x-a_i)$ for $i = 1 →∞$ where the $a_i$ are the roots of ...

**3**

votes

**0**answers

37 views

### Average minimum number of random k-sparse vectors in GF(2) to span the whole space?

What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...

**1**

vote

**0**answers

31 views

### Decomposition of flat homogeneous Kahler manifolds

In a paper I am reading, it is claimed that a flat homogeneous Kahler manifold is a Kahler product $\mathbb C ^k \times T_1 \times \cdots \times T_s $ where $\mathbb C ^k $ is considered with its ...

**1**

vote

**1**answer

85 views

### Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?

Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$.
Consider the condition $$\omega_C\simeq N_{C/S}$$
For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and ...

**-9**

votes

**0**answers

36 views

### Translation services for math texts [on hold]

I'm looking for translation services that has expertise in translating math texts for popular world languages. The problem with common translation services is that such translations are not of very ...

**2**

votes

**0**answers

66 views

### Why do we care about simplicity of the spectrum in Oseledets' theorem?

Oseledets' theorem is a fundamental result in Ergodic theory (see for example here, or Chapter 4 of Lectures on Lyapunov Exponents by Marcelo Viana).
The simplicity of the spectrum has been studied ...

**7**

votes

**1**answer

104 views

### Style guide for referring to past work

Has anyone written or expressed a coherent position on how to refer to mathematical results (theorems, proofs) by past authors? Even if there are no hard and fast rules, I find it helpful to have a ...

**-1**

votes

**1**answer

46 views

### Finding functional equations that a given function satisfies

Suppose we're given a function, for example a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x)=ax+b $ with $a,b \in \mathbb{C} $. I would like to know which functional equations are ...

**0**

votes

**0**answers

12 views

### How can I approximate this in terms of Gauss-Hermite abscissa and weights?

I am having the following expression. This is the PDF of Nakagami-Lognormal Distribution. I want to express in terms of Gauss-Hermite abscissas and weights. How can I do it?
...

**1**

vote

**0**answers

17 views

### Is the equicontinuous weak-star topology locally convex on the dual of an LF-space?

The Banach-Dieudonné theorem states that if $X$ is a metrizable locally convex Hausdorff space then the equicontinuous weak-* topology on $X'$ coincides with the topology of precompact convergence and ...

**5**

votes

**0**answers

55 views

### Smoothing a continuous section in 1-jet bundle

Here is a question I encountered when reading the book "Convex Integration Theory by D.Spring". My question lies in the second paragraph to the proof of theorem 4.2($C^{0}$-dense $h$-principle).
I ...

**-8**

votes

**0**answers

52 views

### What are singular value of $A$? [on hold]

Let $
A = \left( {\begin{array}{*{20}{c}}
{x + (\frac{3}{4} + y)i}&1&1\\
0&{(x - \frac{5}{4}) + iy}&1\\
0&0&{(x + \frac{3}{4}) + iy}
\end{array}} \right)$, and $x,y\in ...

**5**

votes

**0**answers

62 views

### Historical perspectives on CAT(0) spaces

Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of ...

**2**

votes

**0**answers

84 views

### Geometry of Rogers-Ramanujan continued fraction

I'd like to understand the underlying geometry of the Rogers-Ramanujan continued fraction from the point of view of integrable systems (ideally Toda type theories).
The generating function $R(z) = ...

**4**

votes

**2**answers

419 views

### Langlands program vs Shimura-Taniyama-Weil conjecture

Edward Frenkel said that "we can see Langlands program as a generalization of Shimura-Taniyama-Weil conjecture in the case of elliptic curves"
I hope I'm not distorting his phrase, can someone ...

**3**

votes

**1**answer

39 views

### On the eigenvalues' distribution of random unitary

Fix an integer $d$, let $\mathbb{U}_d$ be the $d\times d$ unitary group.
For any $U\in \mathbb{U}_d$, define $\Omega(U)$ be the length of the smallest arc containing all the eigenvalues of $U$ on the ...

**5**

votes

**1**answer

99 views

### Poincare-Hopf theorem for polytopes?

Is there an analogue of Poincare-Hopf theorem for polytopes?
I want to apply it in the following situation.
I have a polytope in $R^n$ and a smooth explicitly given vector field in $R^n$.
I want to ...

**1**

vote

**0**answers

16 views

### Maximum principle of the gradient of harmonic extension under weak regularity assumption

I have a question which is very likely to be trivial, but I'm stuck on it! Suppose $f \in W^{1, 2}(B_2(0))$ and $\|{\nabla f\|}_{L^{\infty}(B_2(0)\setminus B_1(0))} < \infty$. Consider then the ...

**2**

votes

**1**answer

65 views

### embeds in $ L(c_{0},\ell_{1}) $

Let $ c_{0}:=\lbrace x:\mathbb{N}\rightarrow \mathbb{R} :\lim_{j\rightarrow\infty} x_{j}=0 \rbrace$ denote the usual Banach sequence spaces. Given Banach spaces $X,Y$ let $L(X,Y)$ denote the Banach ...

**2**

votes

**1**answer

78 views

### Can there be a numerical system in which logarithms can be expressed in terms of exponentials in closed form?

The invention of complex numbers allowed to express trigonometric functions through hyperbolic ones in closed form.
Is there possible an extension of real/complex numbers in which logarithms and ...

**0**

votes

**0**answers

58 views

### Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why
$d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...

**2**

votes

**2**answers

124 views

### Definable curves in definable sets

Suppose that I have an unbounded subset $X \subset \mathbb{R}^n$, definable in the $o$-minimal structure $\mathbb{R}_{an, exp}$. Is it possible to find an unbounded, analytic and definable curve (i.e. ...

**4**

votes

**1**answer

54 views

### Ergodic, non-atomic measure on the circle which are $\times 2$ and $\times \frac12$ invariant

There any many ergodic, $T$-invariant, non-atomic measures on the space $X = [0,1)$, where $Tx = 2x \pmod 1$ is the doubling map.
My question is: are any such measures also $T^{-1}$-invariant? BYO ...

**0**

votes

**0**answers

54 views

### Generating-bijective groups

We may say that two finitely generated groups $G$ and $H$ are generating-bijective when there exist homomorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$ such that, for each ordered generating ...

**0**

votes

**2**answers

138 views

### How to write $\mathbb{C}[G/U_-]=\oplus_{\lambda} V_{\lambda}$ explicitly?

Let $G=GL_n$ and $U_-$ the set of all lower unipotent triangular matrices. Then by Gauss Decomposition, we have $G = U_-B$, where $B$ is the set of all upper triangular matrices. The group $U_-$ acts ...

**0**

votes

**0**answers

31 views

### Free cocompact action of discrete group gives a covering map [migrated]

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...

**6**

votes

**0**answers

82 views

### Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2.
I was wondering if there exists any reference concerning the ...

**0**

votes

**1**answer

33 views

### Example of a Schur-nontrivial group with no abelian subgroup of the form $H\times H$?

A group $G$ is Schur-nontrivial if the Schur multipler $H^2(G,U(1))$ is not the trivial group.
I am trying to find an example of a Schur-nontrivial group which does not contain a subgroup of the form ...

**-1**

votes

**0**answers

10 views

### Sequences of random variables converging in probability to the same limit a.s [migrated]

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, for ...

**0**

votes

**0**answers

37 views

### Differential categories vs McBride's notion of derivative

Has anyone done an analysis to see if Blute, Cockett, and Seely's differential categories suffice to provide a notion of 1-hole context in the symmetric monoidal setting?

**1**

vote

**0**answers

50 views

### Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process
$$
dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0
$$
where $a\in (-\infty,+\infty), ...