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29 views

Boundary regularity of rectifiable multiplicity 1 hypercurrents

Background. I have just recently started studying this aspect of geometric measure theory (and I am also by no means well versed in the latter) and I really can not seem to get the slightest hang of ...
1
vote
1answer
95 views

Presentations of groups of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
2
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0answers
71 views

Who classified varieties that are commutative groups?

Who are the authors of the theorems asserting that connected varieties/manifolds which are abelian groups are isomorphic to ${\bf R}^k \times {\bf T}^n$? In the smooth setting, I guess this is due to ...
2
votes
1answer
49 views

Hermite polynomial after rotation

When we consider the $n$-dimensional standard normal distribution, the orthogonal basis is $\{H_S(x)\}_{S}$ where $H_S(x) = \prod_{k=1}^n H_{s_k}(x_k)$. Here $H_*(x)$ is the normalized probabilist's ...
1
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0answers
89 views

pro-commutative group schemes

When $k$ is field, Demazure and Gabriel defined and worked with the category of commutative pro-algebraic groups over $k$. In their book, they proved that $Ext^n(\varprojlim G_i, H)= \varinjlim Ext^n(...
4
votes
2answers
193 views

How to determine the coefficient of the main term of $S_{k}(x)$?

Let $k\geqslant 2$ be an integer, suppose that $p_1,p_2,\dotsc,p_k$ are primes not exceeding $x$. Write $$ S_{k}(x) = \sum_{p_1 \leqslant x} \dotsb \sum_{p_k \leqslant x} \frac{1}{p_1+\dotsb +p_k}. $$...
1
vote
1answer
93 views

Bound for multinomial expansion involving Poisson random variables

Let $x_i, i=1, \ldots n$ be Poisson random variables with parameters $\lambda_i$ correspondingly with condition that $\sum_{i=1}^nx_i=T$. Due to linearity of the expectation one can write: $$ E\left(\...
2
votes
1answer
227 views

Run-away functions

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function. We say that f has the run-away property if for every compact subset $K\subseteq \mathbb{R}$ there is some positive integer N such ...
3
votes
1answer
204 views

Seminorm which is zero on dense subset

Let $X$ be a Banach space and let $\hat{X}$ be a dense subset of $X$. If $p$ is a seminorm on $X$ such that $p(x) =0 $ for all $x \in \hat{X}$, does $p(x) =0$ for all $x\in X$ (is $p$ the trivial ...
5
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1answer
196 views

On a quasi-separated assumption in a lemma for the homotopy exact sequence of the etale fundamental group

Background: I've seen two versions of the homotopy exact sequence for etale fundamental groups. One from Stacks: Stacks 0BTX: Let $k$ be a field with algebraic closure $\overline{k}$. Let $X$ be a ...
3
votes
1answer
101 views

Are “strongly finite dimensional” homotopy invariant sheaves with transfers (locally) constant?

Let $k$ be an algebraically closed field. Let $S$ be a homotopy invariant $\mathbb{Q}$-linear sheaf with transfers in the sense of Voevodsky–Suslin, and assume that the dimension of $S(U)$ (over $\...
-1
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0answers
43 views

Assumptions to map geodesics into geodesics

I'm trying to model a problem I have and I want to show that in such setting I can map geodesics into geodesic I hope you can help me to check if I can actually prove this given some hypotesis I'm ...
7
votes
0answers
137 views

Riemann hypothesis for the motivic zeta function

To repair the failure of rationality in general (as shown by Larsen and Lunts for products of two curves of genus > 1) of M. Kapranov's zeta function defined for a variety over a field $k$ and ...
7
votes
1answer
754 views

Are categories special, foundationally?

Some folks over at nLab want to use categories as a foundation for all of mathematics, I'm guessing as an alternative to sets. Sets work fine, and so do categories, so I have started wondering what ...
4
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0answers
61 views

On Glaeser's result for the square-root of a smooth non-negative function

One of the results due to Georges Glaeser is the following: there exists a non-negative $C^\infty$ function $f$ on the real line, flat at its zeroes, such that $\sqrt{f}$ is not $C^2$. On the other ...
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0answers
13 views

Find number of triangles formed by lines( given:angle along x-axis) [closed]

i came across this problem in a competitive coding class : A number of lines (extending infinity) in both directions are drawn on a plane. the lines are specified by the angle (positive or negative) ...
4
votes
1answer
112 views

Upper bounds on the sectional curvature of the real Grassmannian

Consider the real Grassmannian as the symmetric space $\operatorname{Gr}(n,k) \cong \operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$ for $n \geq 3$, $k \geq 2$, where the metric ...
2
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0answers
241 views

Elliptic curves: about a passage in J. Silverman's “Advanced topics of elliptic curves”

Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage ...
3
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0answers
71 views

Concavity of a rational function

Let $n\in\mathbb{N}$ arbitrary but fixed. Consider the polynomial function $Q_n:\mathbb{R}\to\mathbb{R}$ given by $$ Q_n(x):=(x^2-1^2)^2(x^2-2^2)^2...(x^2-n^2)^2-1^42^4...n^4. $$ I would like to prove ...
1
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1answer
98 views

Fixed locus of a Kahler $S^1$-action

Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold. It splits $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
3
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0answers
148 views

Why doesn't the Manin obstruction work for quadratic forms?

The Manin obstruction explains why the Hasse principle doesn't work for non-quadratic forms. Let's write the notation first; $V(\mathbb{Q})$ is variety for rational numbers. $V(A_\mathbb{Q})$ is ...
3
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0answers
71 views

Constructing functorial homotopies in derived infinity-category

I'm interested in the following problem : let $\mathcal{C}$ be an $\infty$-category and $\mathcal{D}:=D_\infty(\mathbb{Z})$ the derived $\infty$-category of abelian groups. Consider functors $A, B, C ,...
4
votes
1answer
68 views

Probability in Chromatic number upper bound of induced subgraph

Let $G=(V, E)$ be a graph with chromatic number $\chi(G)=1000 .$ Let $U \subset V$ be a random subset of $V$ chosen uniformly from among all $2^{|V|}$ subsets of $V$. Let $H=G[U]$ be the induced ...
5
votes
1answer
146 views

How to apply Hahn-Banach to the convex hull?

I am trying to understand the proof of Lemma 4.1.2 in Michel Talagrand's publication from 1995 on concentration inequalities (see below for the precise question statement): A bit of context: ...
2
votes
1answer
88 views

Lower bound on $L^2$ norm of a strongly convex function

Let $f\colon[0, 1] \to \mathbb R$ be an $m$-strongly convex function and $\mu$ be a probability measure on $[0,1].$ For any $t<1$, the goal is to find a lower bound on $\int_{0}^t f^2(x) d\mu(x)$ ...
0
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0answers
24 views

Confusion on the proof of the “block splitting theorems” of Kneading invariants

Assume $f:[0,1]\longrightarrow [0,1]$ be a unimodal map with the unique turning point $c$ such that $f(0)=f(1)=0$ and $f^{2}(c)<c<f(c)$. Then, the Kneading Invariant of such a unimodal map $f$ ...
1
vote
1answer
72 views

If $(κ_t)$ is a semigroup with invariant measure $\mu$ and $ν$ is singular to $\mu$, then $νκ_t$ might not converge to $\mu$ in total variation norm

Let $E$ be a Polish space, $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$, $\mu$ be a probability measure on $(E,\mathcal B(E))$ invariant with respect to $(\kappa_t)_{t\ge0}$ and $\...
3
votes
1answer
67 views

When is the intersection of two determinantal ideals equal to the product?

Let $S = k[x_{i,j}\mid 1\leq i\leq n, 1\leq j\leq m]$ be a polynomial ring over an arbitrary field $k$. Let $M$ be a generic $n\times m$ matrix of indeterminates in the ring $S$ where $n\leq m$. For ...
3
votes
1answer
130 views

On the automorphisms of the unitary group in the strong operator topology

Let $H$ be an infinite dimensional complex (or real) Hilbert space, and let $U(H)$ be the unitary (or orthogonal) group. We equip $U(H)$ with the strong topology. Now, suppose that $\phi: U(H) \...
2
votes
0answers
75 views

When does this limiting ratio give a real root $x$ to the equation of the form $\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0$?

By searching the Inverse Symbolic Calculator, we appear to be able to make the following conjecture about a real root to the equation: $$\sum\limits_{k=0}^d \frac{x^k a_{k+1}}{k!}=0 \tag{1}$$ Let the ...
1
vote
0answers
145 views

Why the name `Lipschitz-Free Banach spaces'?

There are many names for the same objects that is known as the Arens--Eells spaces, transportation cost spaces, free Banach spaces over a (pointed) metric space, and Lipschitz-free Banach spaces. The ...
-1
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0answers
81 views

Question regarding Cauchy sequence [closed]

I am working on a problem in sequences. The theorem I am trying to prove requires the following implication. But I am not sure how to prove this. Kindly share your thoughts. Let $(y_n)$ be a sequence ...
2
votes
0answers
119 views

Interpreting the Bockstein lemma?

I am reading through "Cohomology Operations and Applications in Homotopy Theory" by Mosher and Tangora and I had a little bit of confusion with the Bockstein lemma. All cohomology will be ...
5
votes
0answers
194 views

Galois action on torsion in homotopy groups not induced by homotopy equivalences

Let $V$ be a simply connected smooth projective complex variety defined over the rationals. Then for any integer $n\geq 2$ the group $\pi_n(V)$ is finitely generated abelian so profinite completion ...
0
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0answers
67 views

Inverse image of simplex

Let $M=B\times S^{1}$ be the solid torus where $\partial M=X\times F= S^{1}\times S^{1}$. We consider the projection $\pi : \partial M \longrightarrow X$ which induces the simplicial map $$\pi_{*} : ...
0
votes
1answer
88 views

Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space

For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that $$\int_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures ...
0
votes
1answer
321 views

How to solve equation $e^x \log x=2$

I want to know how to solve the equation $$e^x\log x=2.$$ We can get a numerical solution but it seems difficult to get an exact solution. I know the Lambert W function but unable to use it for the ...
1
vote
0answers
41 views

Maximizing distance between points on the positive surface of the unit hyper-sphere

Suppose we want to place $k$ ($k \geq 3$) points on the positive surface of a unit hyper-sphere in $\mathbb{R}^n$ ($n \geq 3$), where all coordinates of a point are positive, such that the minimum ...
-2
votes
0answers
58 views

Question regarding Algebra problem [closed]

"T.L. and Quina plan to add a 6-inch thick layer of gravel to their driveway. The driveway is 9 yards long and 3 yards wide. What volume of gravel in cubic yards is required?" My approach to ...
8
votes
0answers
157 views

Some computational results and goals of stable motivic homotopy theory of schemes

I am trying to learn ($\mathbb{P}^1$-)stable motivic ($\mathbb{A}^1$-)homotopy theory of schemes from the Cisinski-Deglise book, Triangulated Categories of Mixed Motives. In order to keep myself going ...
0
votes
0answers
30 views

Quasi-concavity of minimum of function

Consider a differentiable function $F(x,y,z)$ defined on $[0,1]\times[0,1]\times[0,1]$, which is increasing and quasi-concave in (x,y). That is, the partial derivatives of $F$ with respect to $x$, $y$ ...
4
votes
0answers
46 views

The least distance of $f\in\ell_\infty(K,X)$ to $C_b(K,X)$

Let $X$ be a paracompact space and consider a bounded function $f:K\to\mathbb R$ not necessarily continuous, that is, $f\in\ell_\infty(K,\mathbb R)$. It's a well-known fact that the least distance of $...
4
votes
1answer
101 views

Exterior algebra of normed spaces

This question is related to my prior question, but this one is aimed, even though it's more general. If $V$ is a vector space, we define the exterior algebra of $V$ do be: $$\bigwedge V := \bigoplus_{...
3
votes
0answers
172 views

A homotopy equivalence from a variety to itself that is not homotopic to a homeomorphism

Let $V$ be a simply connected smooth projective complex variety. Can there be a homotopy equivalence $V\to V$ that is not homotopic to a homeomorphism?
3
votes
1answer
89 views

Problem about two elastic ropes in equilibrium

I have an elementary geometric problem that has thus far resisted all efforts from my end. The problem concerns "elastic ropes" which I model as a sequence of points $\gamma=(x_1,x_2,\dots,...
3
votes
0answers
57 views

Definition of Iwahori subgroup independently of the Bruhat-Tits building

Let $G$ be the points of a connected, semisimple algebraic group over a $p$-adic field $k$. To make life easy, let's assume the underlying group scheme is simply connected. The Bruhat-Tits building $...
2
votes
2answers
113 views

Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients

Consider equation $$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$ with initial condition $u(0, x) = g(x).$ Suppose that $c(t, x)$ and $...
5
votes
0answers
322 views

Complex conjugation inducing a trivial map on the fundamental group

Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by ...
1
vote
1answer
86 views

Simultaneous diagonalization in Matlab [closed]

Crossposted from StackOverflow. The generalised diagonalization of two matrices $A$ and $B$ can be done in Matlab via [V,D] = eig(A,B); where the columns of $V$ ...
1
vote
0answers
102 views

Isomorphic objects in the derived category

Let $\mathcal{A}$ be an abelian category, $D(\mathcal{A})$ be its derived category and $X,Y$ be complexes with morphisms in $\mathcal{A}$. I am trying to understand what does it mean to say that $X$ ...

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