0
votes
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
0answers
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), ...

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