All Questions

2
votes
0answers
40 views

example in $L^p_{s}-$Sobolev spaces

We define $L^p-$ Sobolev spaces as follows: $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$ where $\langle \...
0
votes
1answer
29 views

Minimizing the set of “faulty” edges in a map between the vertex sets of $2$ graphs

The starting point of this question is the fact that for some simple, undirected graphs $G, H$ there is no graph homomorphism $f:G\to H$. This is the case for instance if $\chi(G)>\chi(H)$. ...
6
votes
1answer
281 views

A question about Poincare duality

Let $k$ be a field. Let $C$ be a small category and assume that for every $i\geq 0$ we have a functor $H^i:C\rightarrow FinDimVect_k$. Assume that there is a function $dim:Obj(C)\rightarrow \mathbb{Z}...
-6
votes
0answers
34 views

Trigonometry Functions Help! [on hold]

I'm having some trouble trying to figure out the amplitude, period, phase shift, and vertical displacement of the trigonometric function: f(x)=3sin[2(x+2pi/3)].
0
votes
0answers
59 views

How many lattices require exactly 3 elements to generate them?

This question by Moshe Newman: How many different lattices are there on n points, that require exactly 3 elements to generate them? This sequence seems to start 0,0,1,0,4,3 (for n = 1 to 6) and seems ...
5
votes
0answers
117 views

Can one use forcing as a step to prove the Keisler-Shelah isomorphism theorem?

Can one use forcing (perhaps at the expense of using larger ultrafilters) to prove the Keisler-Shelah isomorphism theorem? My idea is to use an ultrafilter $U$ on a set $I$ such that if $M=V^{I}/U$, ...
0
votes
0answers
60 views

What matrix has only negative or zero real part for all the eigenvalues?

Say $X \in \mathbb{R}^{m\times m}$, Is it possible to have a constraint on $X$, such that all the eigenvalues has negative or zero real part? What I conjecture The following $X$ has only negative ...
2
votes
1answer
112 views

Leray-Serre spectral sequence for projective bundles

Let $\mathcal{E} \rightarrow X$ be a complex vector bundle of rank $r+1$ and let $F=\mathbb{P}^r \rightarrow E = \mathbb{P}\mathcal{E}\rightarrow X$ be the associated projective bundle. We know that ...
6
votes
0answers
62 views

Morita equivalence for graded von Neumann algebras

I am interested in understanding Morita equivalence of $Z_2$-graded von Neumann algebras. In the ungraded case, Rieffel showed that all Type I factors are Morita-equivalent, while for Type III factors ...
0
votes
0answers
45 views

switching the order of composition of two functions [on hold]

Let $f_a(x)=x+\exp(-ax)/a$ be a function, defined on $R_+$ for $a$ positive. Fix $a$ and $b$ positive, can we show that the sign of $f_a\circ f_b - f_b\circ f_a$ does not change on $R_+$, with $\circ$ ...
8
votes
0answers
120 views

Does there exist a non-trivial elementary embedding from an ultrapower $V^{I}/U$ to $V^{I}/U$?

Does there exist a set $I$ and an ultrafilter $U$ on $I$ and a non-trivial elementary embedding $j:V^{I}/U\rightarrow V^{I}/U$? So the Kunen inconsistency result states that there does not exist a ...
3
votes
1answer
186 views

Hochschild Homology and Formal Geometry

My question concerns the degeneration of the spectral sequence computing Hochschild homology of differential operators on a smooth affine variety $X$. The spectral sequence arises from the ...
1
vote
0answers
81 views

questions about a finite solvable group

These questions are by Moshe Newman Let G be a finite solvable group of derived length d, with the property that every proper subgroup and every proper quotient of G has derived length less than d. ...
1
vote
1answer
118 views

Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?

Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
3
votes
1answer
81 views

Group cohomology with coefficients in a chain complex

Let us suppose that I'm in the following situation: I have a chain complex $(C,\partial)$ and say a finite group $G$ acting over $C$ up to homotopy, meaning that for each $g \in G$ I have a self ...
11
votes
1answer
342 views

Are Hausdorff measures on the real line Haar measures for some locally compact topology?

For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...
2
votes
0answers
98 views

Berthelot’s comparison theorem and functoriality

Let $A$ be a noetherian $p$-adically complete ring with an ideal $I$ equipped with a PD structure and such that $p$ is nilpotent on $A/I$. Let $S = \text{Spec}(A)$, $S_0 = \text{Spec}(A/I)$, $Y\to S$ ...
3
votes
1answer
211 views

Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

Let $K$ be a field endowed with a rank (height) one valuation with completion $\hat{K}$, which is not discrete. Let $R$ be the valuation ring of $K$. Let $L \subset \hat{K}$ be a separable finite ...
1
vote
1answer
112 views

Principal Symbol for the Ricci-DeTurck Flow

I am following some lecture notes on Ricci flow and reached the section where we linearize the Ricci tensor and obtain the principal symbol for the resulting operator. We have $T \in \: \Gamma(Sym^2 ...
0
votes
0answers
167 views

Parity of number of partitions of $n!/6$ and $n!/2$

The parities of the number of partitions of $n!/6$ and $n!/2$ appear to be non-random initially, as follows — is there an explanation for this other than chance? With $p$ being the partition ...
1
vote
0answers
27 views

Convergence estimates for approximation with Gaussians / radial basis functions

tl;dr: Are there known convergence estimates for approximating a function with a radial basis family? Details: Let $\mathcal{G}$ be a family of radial basis functions, e.g. $\mathcal{G}=\{\exp(-\...
1
vote
0answers
35 views

Homeomorphism type of the horofunction boundary for nilpotent Lie groups

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form $$\beta_y(x)=d(x,y)-d(w,y).$$ The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...
1
vote
1answer
49 views

Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector

Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector. Question $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$ Observation This paper allows us to ...
15
votes
0answers
268 views
+100

Maximizing the expectation of a polynomial function of iid random variables

Let $f \colon \mathbb R^N \to \mathbb R$ be a smooth function. Let $\mu$ be a probability measure on $[0,1]$ and $X_1, \ldots , X_N$ be i.i.d. random variables on $\mathbb R$. Question 1. What is ...
1
vote
0answers
69 views

Requirement for weak pullback to be a Lie groupoid (Moerdijk)

Let $\phi:\mathcal{G}\rightarrow \mathcal{K}$ and $\psi:\mathcal{H}\rightarrow \mathcal{K}$ be morphisms of Lie groupoids. Is it necessary for $\phi_1:\mathcal{G}_1\rightarrow \mathcal{H}_1, \...
3
votes
2answers
150 views

Packing vertices on a hypercube graph?

Imagine a graph where the vertices and edges model an n dimensional hypercube (a line, a square, a cube and so on). A red vertex must have a minimum distance of 3 from every other red vertex. The ...
1
vote
0answers
88 views

Doubts related Shifting from Pure to Applied math [on hold]

I am a second year (Pure) Math and (Theoretical) Physics undergraduate in India. I want to do a masters in Applied/Computational Science or Math, for which I have apply after next 7 months. I have ...
0
votes
0answers
72 views

Bound of size $X\subset \mathbb{Z}/N\mathbb{Z}$ which satisfies $X+X=\mathbb{Z}/N\mathbb{Z}$

(Sorry for my poor english skill..) Let $N$ be a large integer and the set $X$ be the subset of $\mathbb{Z}/N\mathbb{Z}$. For two sets $A$ and $B$, we define \begin{equation} A+B:=\{a+b : a\in A, b\...
-3
votes
0answers
32 views

Low rank integral equations [on hold]

My question is about how to prove k(x,t)=i(x-t) is symmetric also k(x,t)=i(x+t) is not symmetric
5
votes
1answer
150 views

Multiple mirrors phenomenon from SYZ and HMS perspective

There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties ...
7
votes
0answers
216 views

Is it true that $\sum_{k=1}^\infty\frac{\binom{2k}k^2}{k16^k}(H_{2k}-H_k)=\frac23\sum_{k=1}^\infty\frac{\binom{2k}k^2H_{2k}}{(2k+1)16^k}$?

On Jan. 27, 2012, I conjectured the identity $$\sum_{k=1}^\infty\frac{\binom{2k}k^2}{k16^k}(H_{2k}-H_k)=\frac23\sum_{k=1}^\infty\frac{\binom{2k}k^2H_{2k}}{(2k+1)16^k},\tag{$*$}$$ where $H_n$ denotes ...
1
vote
0answers
43 views

Differences of $\omega$-plurisubharmonic functions

Let $X$ be a complex manifold, and $\omega$ a Kähler form on $X$. A smooth function $\phi$ on $X$ is $\omega$-plurisubharmonic ($\omega$-psh for short) if the form $\omega+\sqrt{-1}\partial\bar{\...
-1
votes
0answers
20 views

Independence of events [on hold]

I have a question about probability. Imagine we have a deck of 40 cards with 4 different suits of ten cards. Define two events: $A \equiv $ We take a card and its an ace $\Rightarrow P(A)=\frac{1}{10}...
0
votes
0answers
117 views

Normal subgroups of $p$-groups

I was reading Professor Yukov Berkovich' paper "On Subgroups of Finite $p$-groups" when I stumled upon the following theorem: Let $G$ be a nonabelian $p$-group with cyclic Frattini subgroup, $|\Phi(...
-1
votes
1answer
50 views

On spectral multiplicity of left shift operators

Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...
0
votes
0answers
100 views

Definition of the surface measure in some books

I am studying PDEs and in some books (Folland, Introduction to Partial Differential Equations and Evans, Partial Differential Equations), I found an integral integrated by the surface measure on a $C^...
1
vote
0answers
37 views

Name for a pair of lattices one of which having theta series with coefficients a subsequence of another lattice's theta series coefficients

Is there a name for a pair of lattices which have the property given in the title? The following example of a pair captures the property mentioned above: $$(i)\ 1 + 80q^3 + 270q^4 + 432q^5 + 960q^6 + ...
7
votes
1answer
280 views

Do any finite predictions of Quantum Mechanics depend on the set theoretic axioms used?

I was wondering if any of the finite predictions of Quantum Mechanics depend on what set theoretic axioms are used. We will say that Quantum Mechanics makes a finite prediction about an experiment if,...
1
vote
1answer
138 views

On the 2002 paper “Dynamics of polynomial automorphisms of $\mathbb{C}^k$” by Guedj and Sibony

I desperately need to read the paper [1] before meeting a would-be supervisor, but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, not even one ...
2
votes
0answers
176 views

Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Please let me know whether this question is suitable for Mathoverflow. Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...
0
votes
0answers
17 views

Find local coordinates of end points of a time varying line (w.r.t. two other objects) to minimize line-length variance

Line A'A rotates cyclically around point A' in a fixed time varying pattern and coordinates of the point A' are fixed (i.e. line A'A has only one degree of freedom). Line BB' can move along x-axis and ...
16
votes
2answers
509 views

Motivation behind Analytic Number Theory

I am an undergraduate student of mathematics and recently took an introductory course in analytic number theory, where the instructor roughly followed Apostol's first text on the subject. I have now ...
7
votes
2answers
172 views

Measure of the numbers with length of $n$ for a nonstandard number $n$

Is there any nonstandard model of $PA$ with the following properties? There exists a nonstandard number $n\in M$ such that $M\upharpoonright n$ is countable, Let $|x|=\lceil\log_2x\rceil$, ...
1
vote
3answers
59 views

Is there a process for finding an implicit representation for an approximation to an arbitrary plane curve?

There are a number of plane curves listed on mathworld, described by implicit algebraic equations, including the butterfly curve, ampersand curve, and bow curve. These all loosely resemble ...
2
votes
2answers
159 views

What do the eigenvectors of the $n$th roots of $I_n$ look like?

This was asked at math stackexchange a long time ago with no answers but some upvotes. Let $A^n=I,$ where $A$ is $n\times n,$ and assume that $A^k\neq I,$ for all $1\leq k<n.$ Since its ...
16
votes
1answer
432 views

Commutative rings : Topoi = Fields :?

The following is probably a bad question, but hopefully, it might have a very good answer. In category theory there is a quite famous analogy between topoi and commutative rings, I was never ...
-2
votes
0answers
37 views

Decompose geometric mean [on hold]

Sorry if this is a super-dumb question. I have three data series: organic growth, volume growth and price growth. Organic growth = volume growth + price growth. I have converted them into indices and ...
2
votes
0answers
117 views

Taylor Expansion on a Riemannian Manifold in Normal Coordinates

Let $\phi: (M,g)\hookrightarrow (N,\tilde{g})$ be an isometric embedding of a Riemannian manifold $M$ of dimension $m$ into a Riemannian manifold $N$ of dimension $n$. I am interested in trying to do ...
1
vote
1answer
61 views

Does every uncountably categorical theory have a $\varnothing$-definable strongly minimal imaginary?

An uncountably categorical theory always has a strongly minimal set definable over its prime model, but sometimes this set needs parameters to define. By a $\varnothing$-definable imaginary I mean ...
4
votes
1answer
122 views

Decidability: Presentations vs. Groups

This question is just a curiosity for me as a non-expert. Quite often we ask about decidability of various properties in a group. Often the answer is that the property is undecidable in general. ...

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