All Questions
153,392
questions
2
votes
1
answer
139
views
Two PDE for one unknown?
Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions.
My ...
3
votes
3
answers
548
views
Solving diagonal simultaneous quadratic equations
A problem I am trying to solve has led to me to the following system of equations:
$$A(x^2) + Bx + c = 0$$
Where $A$ and $B$ are known matrices, $c$ is a known vector, $x$ is the vector of unknowns ...
4
votes
0
answers
363
views
What is the analogy between the moduli of shtukas and Shimura varieties?
I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this ...
1
vote
1
answer
607
views
Hanson-Wright inequality (quadratic form concentration inequality) for bounded random vectors
Is there a concentration inequality for quadratic forms of bounded random vectors $X \in [-1, 1]^n$ with zero mean and given covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$ but otherwise ...
3
votes
1
answer
79
views
Concerning the Spanier group relative to an open cover
Let $\mathcal{U} = \{ U_i \; |\; i\in I \}$ be an open covering of $X$. Spanier defined $\pi (\mathcal{U}, x)$ to be the subgroup of $\pi_1 (X, x)$ which contains all homotopy classes having ...
1
vote
0
answers
103
views
Ricci flow preserves locally symmetry along the flow
Let $(M,g_0)$ be a closed locally symmetric Riemannian manifold and let $g(t)_{t\in[0,T)}$ be a solution to the Ricci flow on $M$ with $g(0)=g_0$. How one can prove that Ricci flow preserves locally ...
1
vote
1
answer
107
views
Surfaces extending modified geodesic paths
What happens if the usual geodesic equation on an n-manifold is directly modified from a source dimension 1 space (giving a path) to a dimension 2 space (giving a surface). I suspect that this gives a ...
3
votes
1
answer
236
views
Locally trivial fibration over a suspension
For $X$ a paracompact space, I am trying to classify all locally trivial fibration with base the suspension $SX = X \times [-1,1]\, /\, (X \times \{-1\} \cup X \times \{1\})$, and fiber-type a space $...
10
votes
1
answer
1k
views
Homology of the fiber
Let $f:X\rightarrow Y $ be a fibration (with fiber $F$) between simply connected spaces such that
$H_{\ast}(f):H_{\ast}(X,\mathbb{Z})\rightarrow H_{\ast}(Y,\mathbb{Z})$ is an isomorphism for $\ast\...
0
votes
1
answer
147
views
Finitely additive measure on Cartesian square of countable set
Let $\mu$ be a probability measure on $(\omega, 2^\omega, \mu)$ measure space which is finitely additive and $\mu(A)=0$ for finite sets.
We can define as usual $\mu^2$ on semiring $\mathcal{G}=\{A\...
10
votes
1
answer
454
views
Is there a pattern for the irreducible factors and degrees of a characteristic polynomial?
Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
1
vote
1
answer
223
views
Formal justification of the Chaos game in the Sierpinski triangle
I want to justify why the Chaos game works to produce Sierpinski triangle. I use a theorem taken from Massopust Interpolation and Approximation with Splines and Fractals.
Suppose that $(X,d)$ is a ...
0
votes
1
answer
109
views
Reference request: Markoff type equations
Consider the equation $x^2+ay^2+bz^2=(1+a+b)xyz.$ If there are infinitely many integral solutions, then up to permutations $(a,b)=(1,1),(1,2),(2,3).$ I have found a presentation by Waldschmidt (https:/...
2
votes
0
answers
73
views
Generalized definition of integrable condition on rough complex subbundle
Assume object are smooth at first. If we consider real subbundle, we can define integrability in terms of parameterization or coordinate.
A rank $r$ real subbundle $\mathcal V\le TM$ is called ...
13
votes
3
answers
1k
views
Does the set $\{\binom x3+\binom y3+\binom z3:\ x,y,z\in\mathbb Z\}$ contain all integers?
The Gauss-Legendre theorem on sums of three squares states that
$$\{x^2+y^2+z^2:\ x,y,z\in\mathbb Z\}=\mathbb N\setminus\{4^k(8m+7):\ k,m\in\mathbb N\},$$ where $\mathbb N=\{0,1,2,\ldots\}$.
It is ...
-2
votes
1
answer
117
views
Contracting non-adjacent points in the icosahedron [closed]
Are there $2$ non-adjacent points in the icosahedron graph $G$ such that contracting them leaves the Hadwiger number unchanged?
2
votes
1
answer
291
views
$X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a co-H-Space
I have asked the below question on MathSE (with a 200 point bounty) but have yet to receive an answer there, and so am trying here. I am happy to remove it if it is nevertheless decided that this ...
3
votes
0
answers
160
views
Functoriality of base change morphisms
Consider a commutative diagram of morphisms of schemes:
$$\begin{array}{ccccccccc}
X_2 & \xrightarrow{j_2} & Y_2 \\
f'\downarrow & & \downarrow f \\
X_1 & \xrightarrow{j_1} &...
2
votes
0
answers
258
views
Geometric irreducibility of fiber product of geometric irreducible schemes
Given three geometrically irreducible normal $k$-curves, $X$, $Y$, $Z$, and two morphisms $f \colon\ X \to Z$, $g \colon\ Y \to Z$. Assume that $X \times_Z Y$ is irreducible. Does $X \times_Z Y$ is ...
1
vote
0
answers
138
views
Power series ring $R[[X_1,\ldots,X_d]]$ over a domain $R$
Let $R$ be a domain and
\begin{align*}
T \,\colon= R[[X_1,\ldots,X_d]].
\end{align*}
Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\...
8
votes
0
answers
252
views
Structural Stability on Compact $2$-Manifolds with Boundary
I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary.
Let $M^2$ be a compact connected 2-manifold and $\...
7
votes
2
answers
255
views
Box dimension of the graph of an increasing function
This Hausdorff dimension of the graph of an increasing function shows that:
Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...
7
votes
1
answer
256
views
Hausdorff dimension of the boundary of fibres of Lipschitz maps
Let $f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$ be a Lipschitz map.
Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for
...
0
votes
1
answer
136
views
Generating Machin Type formulas with inverse hyperbolic tangents for logarithms
Machin Type formulas for $\pi$ have the following general form:
$$c_{0} \frac{\pi}{4}=\sum_{n=1}^{N} c_{n} \arctan \frac{a_{n}}{b_{n}}$$
Recently browsing through this question here, I really became ...
9
votes
0
answers
241
views
Interesting geometric flow of space curves with non-vanishing torsion
Recently, while thinking about CMC surfaces, I came up with an interesting geometric flow for curves in $\mathbb{R}^3$ given by
\begin{equation}
\partial_t \gamma = \tau^{-\frac{1}{2}} n,
\end{...
5
votes
1
answer
158
views
A question about integration of spherical harmonics on $(S ^ 2, can)$
Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that
$$ \int_{\mathbb{...
1
vote
0
answers
125
views
Convergence to a $C^\infty$ function
For every $\epsilon \in [0, 1)$ and $n$ in $\mathbb N$, let $g(n, \epsilon)$ be a function $\mathbb R \to \mathbb R$.
Suppose for all $\epsilon \in [0, 1)$, $g(n, \epsilon)$ is $C^n$ smooth and that ...
1
vote
0
answers
42
views
Calculations of residue homomorphisms in cycle modules
In the proof of Proposition 2.2 and Theorem 2.3 in Chow groups with coefficients https://eudml.org/doc/233731 written by M. Rost, he wrote
$\mathbb{A}^{1}={\rm Spec}F[u], \mathbb{A}^{2}={\rm Spec}F[...
3
votes
0
answers
437
views
De Rham cohomology and extension of scalars
Let $K$ be a field of characteristic zero and let $X$ be a smooth variety over $K$. Given a field extension $L/K$, I denote by $X_L$ the variety $X \times_{Spec(K)} Spec(L)$.
What is the easiest way ...
3
votes
0
answers
227
views
Reference for calculating definite integral involving sines
Recently I accidentally discovered a simple, elementary derivation of the following identity, valid for any $n,k \in \mathbb N$:
\begin{align*}
\frac1\pi \int_0^\pi {\rm d}x \left(\!\frac{\sin nx}{\...
7
votes
2
answers
3k
views
Embeddings of flag manifolds
Consider the flag manifold $\mathbb{F}(a_1,\dots,a_k)$ parametrizing flags of type $F^{a_1}\subseteq\dots\subseteq F^{a_k}\subseteq V$ in a vector spaces $V$ of dimension $n+1$, where $F^{a_i}$ is a ...
3
votes
0
answers
75
views
Functional characterization of local correlation matrices?
Definition: A matrix $C\in\mathbb R^{m\times n}$ is local correlation matrix iff there exists real random variables $x_1,\dots,x_m,y_1,\dots,y_n$ defined on a common probability space which takes ...
0
votes
1
answer
92
views
Can time-scale calculus be used to derive a counterpart theorem of discrete-time dynamic systems directly from continuos-time dynamics systems?
From what I read of time-scale calculus literature, most results of continuous-time and discrete-time systems can be generalized to arbitrary time-scales by considering the generalized derivative ...
8
votes
2
answers
317
views
What is the total square on the dual Steenrod algebra?
The dual Steenrod algebra ($p=2$) has generators $\xi_n$ and these have conjugates that are often labeled $\zeta_n$. I am curious about the left and right actions of the Steenrod algebra on its dual, ...
22
votes
2
answers
866
views
Videos of Gian-Carlo Rota lectures
I apologize if this is off topic.
I think most of his listeners would agree with me that Gian-Carlo Rota had a wonderful style of lecture delivery. I have heard him lecture, both as an undergraduate ...
2
votes
1
answer
190
views
Bounding integral arising from expectation of a random variable satisfying Bernstein's inequality
Let $X$ be a random variable s.t. for $v, b > 0$ and $C \geq 1$:
$$ P(X \geq t) \leq C\exp\left(-\frac{t^2}{2(v^2 + bt)} \right) $$
I am trying to show that $\mathbb{E}X \leq 2v(\sqrt{\pi} + \...
4
votes
0
answers
714
views
Feynman-Kac formula for domains with boundary
As far as I know (I am not an expert), any solution of the heat equation on $\mathbb{R}^n$ or on a closed Riemannian manifold with the given initial condition can be presented in terms of stochastic ...
5
votes
0
answers
75
views
Hypothetical uniqueness of an embedding of a Riemannian manifold to a compact Kähler one
Inspired by this question (Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold) I ask the following:
Suppose $X$ is a real analytic Riemannian manifold with a ...
4
votes
0
answers
121
views
Coordinate-free B.Feix's construction of a hyperkähler metric
In the 2001's paper 'Hyperkähler metrics on cotangent bundles' B.Feix gives a construction of a hyperkähler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic Kähler ...
6
votes
1
answer
222
views
Asympotic density of a very simple sequence
Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.
I'm actually even more ...
0
votes
1
answer
426
views
Weil restriction of a projective variety to a finite extension over $\mathbb{Q}$
My question maybe a bit stupid, but I can't quite grasp what the Weil restriction actually does... In particular:
Given a projective variety $V$ defined over $L$ algebraically closed, of ...
10
votes
2
answers
592
views
Examples of transfinite towers
I am looking for examples of constructions for transfinite towers $(X_{\alpha})_{\alpha}$ generated by structures $X$ where the problem of determining whether the tower $(X_{\alpha})_{\alpha}$ stops ...
6
votes
2
answers
954
views
Properties of heat equation
** I simplified the question: **
On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive.
I ...
0
votes
0
answers
285
views
Defining a graded ring structure on $\bigoplus_{i} \text{Ext}^i (A,A)$
I read that, using the fact that $\text{Tot}^{\prod}(\text{Hom}(P_{\bullet} ,Q_{\bullet}))$ can by used to compute $\text{Ext}^* (A,B)$ (which I understand), we can give $\bigoplus_{i} \text{Ext}^i (A,...
0
votes
0
answers
284
views
Measurability of the heat semigroup in $L^\infty$
Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...
5
votes
0
answers
146
views
Uniform versus non-uniform group stability
Group stability considers the question of whether "almost"-homomorphisms are "close to" true homomorphisms. Here, "almost" and "close to" are made rigorous using a group metric.
More precisely, ...
3
votes
1
answer
453
views
Extending section of étale morphism of adic spaces
This question is related to Lifting points via étale morphism of adic spaces.
Fix a complete non-archimedean field $k$. Let $(A,A^+)$ be a complete strongly noetherian Huber pair over $(k,k^\...
5
votes
2
answers
267
views
Is a set of nuclear functionals equicontinuous in compact-open topology if it is equicontinuous on each compact set?
Let $H$ be a Hilbert space and $B(H)$ be its space of all (bounded) operators. A nuclear functional on $B(H)$ is a linear functional $f:B(H)\to{\mathbb C}$ that can be represented in the form
$$
f(A)=\...
4
votes
0
answers
130
views
Orthogonality and 2-filtered 2-categories
Let $C$ be a category. It is well known that $C$ is $\omega$-filtered if and only if it is weakly right orthogonal to every $A\to A^\rhd$, where $A^\rhd$ is the right cone of a finite category $A$.
...
1
vote
0
answers
88
views
Examples of three-dimensional non-nilpotent Leibniz or Lie algebras
Can one give me some examples of three dimensional Non Nilpotent Leibniz algebras? Any references to the classification of three dimensional Non Nilpotent Leibniz or Lie algebras will also be helpful.