0
votes
0answers
3 views

Pollard's construction of measures from set functions on lattices of sets

Theorem 12 in Appendix A of Pollard's A User's Guide to Measure Theoretic Probability gives conditions under which a set function defined on a family of sets $\mathscr{K}$ which is closed under finite ...
0
votes
0answers
3 views

Analytic continuation of an integral

Let $$f(y)=\frac{y_1^{1/3}y_2^{1/3}}{y_1+y_2+1}$$. Consider the following integral: $$F(s_1,s_2)=\int_{\mathbb{R}_+^2}f(y)^{s_1}f(y^{-1})^{s_2}\frac{dy_1dy_2}{y_1y_2}$$ where ...
1
vote
0answers
5 views

Around Vopěnka: Accessible category with small full discrete subcategories of arbitrary size?

I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C_\lambda$ of $\lambda$-many models, no two of ...
-1
votes
0answers
6 views

A question about holomorphic fiber space when the base space has positive first Chern class

Let $\pi:X\to B$ be a holomorphic fiber space where $X$ and $B$ are Kaehler manifolds and $c_1(B)>0$ then what can we say about fibers and also about moduli space of fibers? We know by adjunction ...
0
votes
0answers
4 views

help with an asymptotic estimate for a certain product

(I apologize in advance if this question is not suitable for Math Overflow, but it came up in a research problem and thought perhaps I could find some help here.) I'm having difficulty finding an ...
0
votes
0answers
12 views

cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian. Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where ...
2
votes
0answers
11 views

Convex hulls of quasi-convex sets in proper CAT(0) spaces

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...
3
votes
1answer
42 views

Simultaneous approximation by rationals with relatively prime numerators

The following seems hard to me (or perhaps just not true), but perhaps I am mistaken. It is known that given irrational numbers $x_1$ and $x_2$, there are infinitely many simultaneous rational ...
1
vote
1answer
51 views

Paper of Denis Simon on quadratic equations in dimensions 4, 5?

In several places I have come across references to a 2005-6 preprint of Denis Simon entitled Quadratic equations in dimensions 4, 5, and more This paper gives fast algorithms to find isotropic ...
4
votes
0answers
25 views

Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...
1
vote
1answer
47 views

Zero-dimensional spaces and clopen separations

Let $X$ be a topological space. (All of the spaces I'm considering are $T_0$, but in general they are not $T_1$. To be even more concrete, one can even consider $X={\rm Spec}(R)$ to be the space of ...
0
votes
0answers
15 views

multidimensional curve fitting (regression) [migrated]

I have a data set extracted from a bunch of experiments. Imagine a matrix: x1 x2 x3 x4 x5 y Observation ...
0
votes
0answers
25 views

Gordan's Theorem usiing Duality Theorem [on hold]

I'm looking for a demostration of Gordan's Theorem just using duality. I have seen demostrations usign convexity or Farkas' Theorem but I'm interesting on aply duality. Thank you so much
0
votes
1answer
112 views

Why do we not lose any generality by proving it only for finitely generated groups

In the proof of following theorem, in a paper by Farkas- Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring ...
0
votes
1answer
12 views

smooth minimization of piecewise linear convex function

Is it possible to apply Nesterov's smooth minimization of non smooth function on a problem of the form $$\mathop {\min }\limits_{\lambda \in {R^m}} \mathop {\max }\limits_{\sigma \in {{\left\{ ...
0
votes
1answer
36 views

Combinatorial polynomials from general diagram fillings?

There is a plethora of polynomials defined on partition shaped Young diagrams, (Schur, Jack, Grothendieck,...), and skew Young diagrams. There are also composition shaped diagrams that are responsible ...
-4
votes
0answers
40 views

What are the limitations of the Fourier Transform and Fourier series? [on hold]

I am fond of Fourier series & Fourier transform. In Fourier domain, we can come to know what frequency components are present and the contribution of each component in forming the given signal. ...
2
votes
1answer
74 views

Kleinian groups containing an isomorphic copy of itself

Is there any example of a Kleinian group (acting on $\mathbb{H}^n$, $n \ge 3$) that contains a finite index isomorphic copy of itself? Here I don't consider Kleinian groups that only have parabolic ...
19
votes
3answers
449 views

Is every abelian group a colimit of copies of Z?

More precisely, is every abelian group a colimit $\text{colim}_{j \in J} F(j)$ over a diagram $F : J \to \text{Ab}$ where each $F(j)$ is isomorphic to $\mathbb{Z}$? Note that this does not follow ...
11
votes
0answers
122 views

Whatever happened to $L(j)$?

So this question probably shows my inner model theoretic ignorance, but: In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...
0
votes
0answers
13 views

Second derivative estimates for gradient dipendent elliptic equations

Caffarelli in the article " Interior a priori estimates for solutions of Fully non linear Equations" Ann. Math 130,No.1, 1989 proved that a continuous viscosity solution $ u $ of a uniformly elliptic ...
0
votes
0answers
30 views

Proximal point avoids a subvariety

I have a seemingly easy problem about the proximal point in a variety of a general point in the ambient space, which I don't have a proof: Let $X \subset \mathbb{C}^N$ be the affine cone over some ...
3
votes
0answers
105 views

Example of a genus-1 degree-7 plane curve

I am wondering if anyone knows how to construct an explicit example of an irreducible plane curve of degree 7 with 14 double points. Such a curve would have genus 1. One can show that for a general ...
1
vote
0answers
17 views

Solving IBVP of First Order Hyperbolic Semi-Linear Equation

I've been trying to figure out this problem for a while and was wondering what you thought. I have the PDE: \begin{equation*} \frac{\partial c}{\partial t}+\frac{K}{r^2}\frac{\partial c}{\partial ...
0
votes
0answers
24 views

Does every connected vertex transitive graph on $n$ vertices (except for $C_n$) have feedback vertex set of size $\Omega(n)$?

Feedback vertex set is a set of vertices whose removal leaves an acyclic graph. It is known that every vertex transitive graph on $n$ vertices has vertex cover of size $\Omega(n)$. It is also not ...
4
votes
3answers
249 views

Analogue of Cayley Hamilton theorem for operators on Hilbert space

Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
8
votes
0answers
74 views

When does Hochschild homology commute with infinite products?

Let $A$ be an associative algebra. Its zeroth Hochschild homology $\mathrm{HH}_0(A)$ is the cokernel of the linear map $A^{\wedge 2} \to A$, $a \wedge b \mapsto ab - ba$. I.e. you quotient the ...
-4
votes
0answers
33 views

Are mappings f(a, x) and f(b, x)-c conjugate for some parameter values a, b, and c? [on hold]

More precisely, what assumptions on f must hold for the above maps to be conjugate? We know that functions $f$ and $g$ are topologically conjugate, if there exists a homeomorphism $h$ such that ...
1
vote
0answers
31 views

A holomorphic vector bundle structure for $\Omega^{(0,1)}(M)$

For a complex manifold $M$, the complexified tangent space $\Omega^1(M)$ splits into a direct sum $\Omega^1(M) = \Omega^{(1,0)}(M) \oplus \Omega^{(0,1)}(M)$. As is well-known $\Omega^{(1,0)}(M)$ is ...
1
vote
0answers
30 views

Pertubations of self-adjoint first order operators

If we consider the self-adjoint operator $L= L_0 + h$ on the appropriate Sobolev spaces of maps from $S^1$ to $\mathbb{R}^N$, say, where $L_0$ is a first order self-adjoint operator and $h$ is a ...
3
votes
2answers
182 views

n torsion groups of quadratic twists of elliptic curves

If $E$ is an elliptic curve over a number field $K$ and $E^{F}$ is a quadratic twist of $E$. Then it is stated in ``Ranks of twists of elliptic curves and Hilbert’s tenth problem" due to Mazur and ...
1
vote
0answers
71 views

Which irrationals yield bounded sets of iterates?

For $r > 0$, define $f(n) = \lfloor {nr}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/r}\rfloor$ if $n$ is even. For which irrationals $r$ is the set $\{n,f(n), f(f(n)),\dots\}$ bounded for every ...
3
votes
1answer
152 views

Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...
0
votes
0answers
16 views

Information theoretic common sequence agreement (not secret key)

Supposing Alice and Bob share $\rho$-correlated sequences in $\{0,1\}^n$, what coding theory based schemes are available for Alice and Bod to extract sequences $A,B\in\{0,1\}^n$ respectively such that ...
-4
votes
0answers
50 views

Smallest Multiple in terms of other numbers [on hold]

Let $z$ be a positive integer and $z_0, \dotsc, z_{n-1}$ be non-negative integers. What is an algorithm that, in time $\operatorname O(z(n + \log(z)))$, finds the smallest multiple of $z$ that can be ...
-1
votes
0answers
38 views

Degree of interpolating polynomials [on hold]

$a\in(0,1)$ is fixed. $M\in\Bbb Z_{>1}$ is fixed. What is $\mathsf{deg}(f(x))$ given that $$f(0)=0\mbox{, }f(M)=1+a\mbox{, }f(1)=1-a$$$$\mbox{ }f(x)\in(1-a,1+a)\mbox{, }\forall x\in(1,M)?$$ ...
-3
votes
0answers
45 views

possibility of graph with specific degree for each vertice [on hold]

can we have a graph that has three vertices of degree 1, two vertices of degree 2, and four vertices of degree 3? I really don't know how I can answer this kind of questions!
0
votes
1answer
101 views

Union of connected sets

$\forall \beta \in I$, $A_{\beta }$ is connected, and $\left ( \bigcup_{\alpha < \beta }A_{\alpha } \right )\cap A_{\beta }\neq \varnothing$ . Is $\bigcup_{\alpha \in I}A_{\alpha } $connected? For ...
3
votes
1answer
80 views

Regular and extremal monomorphisms in the category of graphs

Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for ...
3
votes
1answer
185 views

Are essentially smooth schemes noetherian?

Let $k$ be a field. I am unable to find a precise definition of essentially smooth $k$ schemes, but I will stick to this definition below, since this is exactly what I need: Definition: A $k$-scheme ...
0
votes
0answers
13 views

Strong Markov Property of the joint process $(B_t,L_t)_{t\ge 0}$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion and $L=(L_t)_{t\ge 0}$ be its local time in zero. Given two strictly increasing functions $\phi_1$, $\phi_2: \mathbb R_+\to\mathbb R$ such that ...
1
vote
0answers
92 views

Reference request: Flipping the factors in the Künneth formula

I would like to know if there is a reference for the fact that the following diagram commutes: $$ \begin{array}{ccccccccc} 0 & \to & H_*(X) \otimes H_*(Y) & \to & H_*(X\times Y) & ...
1
vote
0answers
52 views

are extensions of flat connections flat? [migrated]

Let $U$ be a smooth complex variety and $X$ a compactification by a normal crossings divisor $D$. Let $E$ be a vector bundle on $U$ (i.e. locally free $\mathcal{O}_U$-module), together with a ...
0
votes
0answers
31 views

Orthogonal complements of intersections of closed subspaces

Let $H$ be a Hilbert space and $H_1, \cdots, H_n$ be closed subspaces of $H$. $\mathbf{Question}:$ Is it always true that the orthogonal complement $(H_1\cap\cdots\cap H_n)^\bot$ of the intersection ...
0
votes
0answers
29 views

Finding a fixed size summarizing structure for average angle calculation [on hold]

I asked this question also on math.SE ( http://math.stackexchange.com/questions/1264676/finding-a-summarizing-vector-for-average-angle-calculation ). Let $L$ and $R$ be two multisets of positive ...
1
vote
1answer
59 views

Null geodesic congruence

I came across a statement in Chandrasekhar's "Mathematical Theory of Black Holes" that I don't understand (rather say disagree): Assume we have a Newman Penrose tetrad $\lbrace l, ...
0
votes
0answers
114 views

Distribution of composite numbers

Motivation: I want to give an abstract formulation of Eratosthenes's Sieve and try to conject a property of Eratosthenes's Sieve. If this property are right, then I can use it to find a low bound of ...
0
votes
0answers
80 views

a problem about ideals of polynomial rings

Let $\{f_n\}_{n=1}^\infty\in \mathbb{C}[x,y]$ be a sequence of polynomials given by the following expressions $$ f_n(x,y)=\sum_{i=0}^{[\dfrac{n}{2}]}(-1)^{n-i}{{n-i}\choose i}x^{n-2i}y^i. $$ Let ...
-1
votes
0answers
25 views

Relationship between mass and magnitude? [on hold]

Why is the mass of an object considered a magnitude when magnitude is defined as the length of vector V from the origin? I mean that when you're going to solve for the x and y component of a vector ...
0
votes
0answers
12 views

Angle at self-intersection points of a curve in hyperbolic surface

Let $F$ be a hyperbolic surface of finite type. Let $\alpha$ be a closed oriented geodesic with more than one self intersection. Suppose all the self-intersections are double points. Let $\angle_p$ ...

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