# All Questions

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108 views

### $0$-“norm” minimization with least-squares regularization

I have the following optimization problem in $\mathbf{x} \in \mathbb{R}^{K \times 1}$ $$\min_{\mathbf{x}>0} \quad \|\mathbf{A}\mathbf{x}\|_0 + \alpha \|\mathbf{B}\mathbf{x}-\mathbf{c}\|_2^2$$ ...
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### Point distributions in unit square which minimize E[1 / distance]

Choose $n$ points $p_1,\ldots,p_n$ in the unit square $[0,1]^2\subset\mathbb{R}^2$ such that $D:=\mathop{\sum}\limits_{1\le i<j\le n}\frac{1}{dist(p_i,p_j)}$ is minimized, where $dist(p_i,p_j)$ is ...
263 views

### A purely algebraic argument for existence of a section of a smooth projective morphism to the projective line

If I am reading this post correctly, any smooth projective $\mathbb{C}$-morphism of schemes $X\rightarrow \mathbb{P}^1$ admits a section. I am afraid of the topological argument presented there. Is ...
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### Lattices with no roots and spread out shells

I am looking for lattices with the following properties: The lattice has no roots. The norm (squared length) of the second shortest vectors should be at least twice as large as the norm of the ...
56 views

### Weight spaces of modules over Lie algebras

I know that an irreducible infinite-dimensional weight module over the Virasoro algebra in which it has a non-zero finite-dimensional weight space, then all its weight spaces have finite dimension. ...
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### A non-Kaehler manifold complex and symplectic in exactly one way

Does there exist a closed connected smooth manifold that admits exactly one (up to biholomorphism) integrable complex structure and exactly one (up to symplectomorphism and rescaling) symplectic ...
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### Solution of nonlinear second-order ODE $y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0$

Is there any way of solving the following second-order ODE $$y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0,$$ where $a$ and $b$ are some constant? If we know that one solution exists, how would it ...
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### Question about Local Henselian Rings

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces": Here the relevant excerpt: Remark: ...
157 views

### reductive groups in algebraic geometry

presumably this is a quite broad question but up to now I haven't found a discussion treating following question: in a lot of fields in algebraic geometry (e.g. GIT or topics on etale cohomology) ...
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### Stationary Points Possible Values [on hold]

I am having trouble with part d) of this question. It follows on with other parts of a question which I have attached. I have written that 'p' can indeed have stationary points but am not sure what ...
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### Theorem from Deformation Theory

My question refers to some steps it the proof of Theorem 3.3 part (b) in Christensen's paper treating Deformation theory (see pages 9-11): https://mathematics.stanford.edu/wp.../A.-Christensen-Draft....
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### CoCartesian vs. locally CoCartesian fibrations

Say $\pi: C\to J$ is an inner fibration of $\infty$-categories. Then "morally", $\pi$ corresponds to a diagram indexed by $J$ in the "category of categories with correspondences", and if $\pi$ is ...
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### Wildness of codimension 1 submanifolds of euclidean space

This question arose out of this stack exchange post. I am wirting a thesis about the $s$-cobordism theorem and Siebenmann's work about end obstructions. Combined they give a quick proof of the ...
117 views

### Does the strong multiplicity one theorem imply the isomorphy of these two automorphism groups?

Defining the notion of "Galois class of L-functions" as a set of automorphic L-functions belonging to the Selberg class closed under the usual product and the Rankin-Selberg convolution and containing ...
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### De Rham cohomology of Lie groupoid

Let $G$ be a Lie group acting on a manifold $M$. Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by ...
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### Have new conjectures generated by the Ramanujan machine been proven?

Recently the following preprint was published with new automatically generated conjectures on generalizes continuous fractions, e.g., for the Euler constant: Raayoni, Pisha, Manor, Mendlovic, Haviv, ...
49 views

### Infinitely generated PID algebra with infinitely many prime ideals

Given a field, is there a functorial construction of a PID algebra over it that has infinitely many prime ideals and is not finitely generated? This excludes the ring of univariate polynomials and the ...
25 views

### Integral of exponential + limit [closed]

Let $f:\mathbb{R} - \{-5\}->\mathbb{R}$, $f(x)=(x-1)e^{-(1/(x+5))}$. I have to calculate $lim_{(n->\infty)}=n^2\int_{0}^{1}x^nf(x)dx$. I've tried using integration by parts, but i'm still stuck....
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### Finding a cover of the Hilbert functor, that corresponds to a cover of the Grassmannian

Let $\mathrm{Grass}_{k+1,n+1}:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ be the Grassmann functor, which maps a scheme $S$ to the set: \left\{\mathscr{U}\subseteq\mathscr{O}_S^{n+1}:\,\,\mathscr{...
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### When is $C\text-\mathsf{dg\text-mod}$ determined by the connective base changes?

I'm using cohomological gradings. For $C\in k\text-\mathsf{cdga}$ (where $k$ can be taken of characteristic 0), a morphism $C\to A$ to a connective dg-algebra $A\in k\text-\mathsf{cdga}_{\leq0}$ ...
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### Strichartz estimates for the inhomogeneous wave equation

In the Blair, Smith and Sogge's paper Strichartz estimates for the wave equation on manifolds with boundary, the authors study integrability estimates for solution of the following problem: \begin{...
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### A class of Grothendieck categories

Is there a name and some work on Grothendieck categories in which each non-zero object has a simple subquotient? Every locally finitely generated Grothendieck category has this property, but I guess ...
381 views

### Reference request: The unit of an adjunction of $\infty$-categories in the sense of Riehl-Verity is a unit in the sense of Lurie

I'm looking for a reference (or proof) for the statement given in the title: that when we have an adjunction between quasicategories in the sense of Riehl and Verity (defined e.g. in Section 4 of ...
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### Symmetric monoidal category with trivial switch morphisms

Is there a specific terminology for a symmetric monoidal category in which for any object $x$ the switch map $x\otimes x\to x\otimes x$ is the identity ? (Or alternatively the action of the symmetric ...
76 views

### are spectral measures characterized by their moments?

On a Hilbert space $\cal H$, consider an essentially self-adjoint operator $A\colon Dom(A)\to {\cal H}$, and a vector $\psi\in\bigcap_{n=1}^\infty Dom(A^n)$. Without further assumptions, can we say ...
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### Embedding a graph in $\mathbb{R}^3$ with partial geometric information

I have a connected, sparse, graph (a molecule to be specific) and I'm interested in associating 3D coordinates with the vertices. Here's the kicker: I already have coordinates for none/some/all ...
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### Is ZFC interpretable in a kind of an extended form of second order arithmetic?

Informally the following theory is a kind of extension of second order arithmetic, where numbers are not limited to naturals, instead here we have formation of further numbers by setting limits on ...
88 views

### Kuenneth short exact sequence for K-homology

Atiyah proved a Kuenneth short exact sequence for K-theory. I need one for K-homology, but can not find any reference in the literature. Do you know one? Using general spectra stuff, one gets a ...
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### On semicontinuity of Hilbert function for a zero dimensional scheme

Let, $X$ be a zero dimensional subscheme of $\Bbb P^n$ and let us define a function as follows: $h_{\Bbb P^n}(X ,d) = \binom {n+d}{n}$ - dim $I_{X}(d)$ , where dim $I_{X}(d)$ = the dimension of ...
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The first order linear non-instantaneous impulsive evolution equations is given as; $u'(t)=Au(t)~~ t\in[s_i,t_{i+1}],\,i\in\mathbb{N_0}:=\{0,1,2,...\}$ $u(t_i^+)=(E+B_i)u(t_i^-),\,\,i\in\mathbb{N}:=... 1answer 256 views ### To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need? To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that$9g-9$curves suffice, by a standard pair-of-pants argument given in,... 1answer 73 views ### The mean E(X) of negative binomial distribution [closed] What I know about the mean of the negative binomial distribution is E(x)=r(1-p)/p. but there are some questions use E(x)=r/p as the mean. Very confusing and I don't understand at all. For example: ... 1answer 63 views ### Coloring in Combinatorial Design Generalizing Latin Square I have a question about a combinatorial design very similar to a Latin Square, which is arising out of an open problem in graph theory. The design is an$n \times n$matrix whose entries we want to ... 1answer 137 views ### Stochastic Calculus vs Stochastic Processes in Finance [on hold] I'm a second year student, interested in financial mathematics, who's trying to plan out his degree path currently. There's a stochastic processes unit offered in year 3 and a stochastic calculus unit ... 1answer 145 views ### For$G$an adjoint Chevalley group, are all of$G(\mathbb Z)$'s finite-index subgroups congruence subgroups? Let$G$be an adjoint Chevalley group. Are all of$G(\mathbb Z)$'s finite-index subgroups congruence subgroups? I read a theorem that states: When$G$is the universal Chevalley group and it's not of ... 0answers 52 views ### Stable m-Calabi Yau property for Frobenius categories Let$C$be a Frobenius category. The stable category$\underline{C}$is called$m$-Calabi Yau in case it is Hom-finite and there is a functorial duality$D \underline{Hom}(X,Y)=\underline{Hom}(Y,\...
Definitions: A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal). Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...