0
votes
0answers
2 views

Compute Faber polynomials in Matlab

I want to compute some faber polynomials associated to an ellipse centered at a point (u,v) (in the complex plane: u+iv) in Matlab. Say the ellipse has minor axis a along the x coordinates (real part) ...
0
votes
0answers
9 views

Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...
3
votes
0answers
9 views

Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...
5
votes
2answers
36 views

Introductory texts to mathematics

I am interested in texts recomendations for a 14 years old boy who wants to study more mathematics than he does at school. He seems quite talented, but his knowledge of maths is rather low. I would ...
8
votes
2answers
78 views

Mysterious identity between numbers of odd/even meander systems

Definitions: An upper arch system of order $n$ is a subset of the plane consisting of $n$ non-intersecting semicircles in the upper half-plane whose endpoints belong to the set $\{(k,0)\mid ...
0
votes
0answers
23 views

Proof for existence of isoperimetric minimizer by compactness theorem

Let $X$ be an $n$-dim closed Riemannian manifold, then given a number $0 <v < vol(X)$, there exists a Borel subset $A\subset X$ attaining $I(v)$, $I$ is the isoperimetric profile. The existence ...
1
vote
0answers
9 views

Approximating the norm of an operator-valued linear function with operator inputs via a matrix-valued linear function

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional Hilbert spaces. Let $B_1, \ldots, B_k \in B(\mathcal{H}).$ Define $L: B(\mathcal{K})^k \rightarrow B(\mathcal{H}\otimes \mathcal{K})$ via ...
1
vote
2answers
54 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
1
vote
0answers
38 views

first chern class versus compactifying divisor in Ramanujam's surface

I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...
-4
votes
0answers
26 views

Can anybody help me for this counting question? [on hold]

Peter has 12 pairs of socks and 6 pairs of gloves in different colors. His socks are in green, yellow, black, and grey (3 pairs each). Peter's gloves are either blue, black, or red (2 pairs each). ...
1
vote
0answers
42 views

Lower bound on class number of binary quadratic forms of discriminant of the form n^2+4

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...
7
votes
3answers
237 views

Random Diophantine polynomials: Percent solvable?

Suppose one generates a random polynomial of degree $d$ with integer coefficients uniformly distributed within $[-c_\max,c_\max]$. For example, for $d=8$, $|c_\max|=100$, here is one random ...
1
vote
1answer
57 views

Inner product spaces without symmetry/hermitian axiom

Consider a vector space $X$ over $\mathbb R$ and a bilinear form $ \langle \cdot, \cdot \rangle : X \times X \rightarrow \mathbb R$. We assume furthermore that for any $x \in X$ there exists $y \in ...
0
votes
1answer
55 views

Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...
3
votes
0answers
68 views

Koopman representation, weakly compact action, Ozawa Popa

Given a weakly compact action (Ozawa-Popa) of a discrete group $\Gamma$ on p.m space $X$, consider the Koopman representation $\pi$ on $L^2(X)$. Compose this representation with the Calkin projection. ...
4
votes
1answer
164 views

Order theory as a foundation of mathematics?

I know the followings kinds of formalization of mathematics: based on set theory (e.g. ZFC) based on type theory (e.g. the formalism of Coq proof assistant, as an advanced example) based on category ...
11
votes
2answers
332 views

A combinatorial question about orthonormal bases

Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. ...
1
vote
0answers
74 views

Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory

I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces. I got quite stuck in Corollary 3.27 ...
-2
votes
0answers
10 views

Partition on a Closed Set A= [2,3] [migrated]

Is it possible to define a partition on a closed set,such that the union of the partitions will give [2,3] and their intersection to be empty?
1
vote
1answer
38 views

Computionally efficient vertex enumeration for (convex) polytopes

Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...
1
vote
0answers
27 views

Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...
8
votes
0answers
80 views

Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed. Say that a coloring of the dots of a Ferrers ...
1
vote
0answers
96 views

All non-split Cartan subroups of $GL_2(\mathbb{Z}/n\mathbb{Z})$ are conjugate

Let $n>1$ be a positive integer and let $R$ be an order in an imaginary quadratic field with discriminant prime to $n$. Let $A=R/nR$ and let $\lbrace 1, \alpha \rbrace$ be a ...
3
votes
0answers
82 views

SubGROUPs of Banach spaces, when are they dense in a vector subspace?

It’s relatively easy to show that if $J$ is a closed subgroup of a finite-dimensional real Banach space, $B$, then it is a vector subspace iff for all bounded linear functionals $\sigma$ of $B$, ...
11
votes
0answers
230 views

Decidability of $x^3+y^3+z^3 = c$

I wondering if it is known whether the following problem is algorithmically decidable or undecidable by Turing machines: given an integer c, determine if there are integers $(x,y,z)$ such that ...
1
vote
1answer
35 views

Is an arbitrary Brownian-motion path a viscosity solution of every differential equation?

Is an arbitrary Brownian path a viscosity solution of every differential equation? My intuition is that a path of Brownian motion is so ill-behaved that it not only does not have derivatives ...
1
vote
0answers
27 views

For a non-convex function f, how to find a function g such that $g\circ f$ is strictly convex? [migrated]

The following function $f(x)={1\over (1+e^{-x})}$ is non-convex but $\ln(f(x))$ is convex. Given a non-convex function $f$, can we find a function $g$ such that $g\circ f$ is strictly convex? If yes, ...
5
votes
1answer
145 views

A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...
8
votes
1answer
85 views

Are the failure of SCH and “$cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular” equiconsistent?

Is it true that the following two statements are equiconsistent? (1) $2^\mu>\mu^+$ for some strong limit singular cardinal $\mu$ (2) $cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular ...
-4
votes
0answers
46 views

how should one locate ambulance stations so as to best serve the needs of the community..tnx [on hold]

how should one locate ambulance stations so as to best serve the needs of the community i don't know what algorithm to use, any suggestion/s?
0
votes
0answers
22 views

Gaussian gabor frame

It is widely known that $\phi(x)=e^{-\frac{x^2}{2}}$ does not define a Gabor frame if we consider translations by units of $1$ and multiplication by $e^{2 \pi inx}$for $n \in \mathbb{N}.$ A way to ...
8
votes
0answers
166 views

End of the Ext functors

Let $R$ be a ring, and consider the hom functor $\hom\colon Mod(R)^\text{op}\times Mod(R)\to Mod(R)$; the end of $\hom$ is well-known to be the set of endomorphisms (endonatural transformations) of ...
4
votes
0answers
36 views

Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path. A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...
-1
votes
0answers
32 views

Contour Integration [on hold]

I am trying to integrate $\frac{\sin x dx}{x(x-1)}$ over the real line except at an arbitrarily small neighborhood around 1, where the function has a singularity. My idea is to do an contour ...
3
votes
0answers
256 views

Do those manifolds atrached to L-functions give rise naturally to motives? [on hold]

Edited after Will Sawin's comment: Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product ...
1
vote
1answer
80 views

Frame-bundle reduction from spinor-bundle reduction

Let $(M,g)$ be a $d$-dimensional Riemannian oriented, spin manifold, and let us denote by $F(M)$ its frame bundle, by $SP(M)$ its spin bundle and by $S = P(M)\times_{\rho}\Delta$ its spinor bundle, ...
1
vote
1answer
85 views

Tangent cone of a complete intersection

Let $X$ be a quasi projective variety over $\mathbb{C}$. By the tangent cone of $X$ at a point $p \in X$, I mean the subvariety of the tangent space of $X$ at $p$ as it is defined in Harris' ...
1
vote
0answers
105 views

Unusual generalization of the law of large numbers

I have seen in physical literature an example of application of a very unusual form of the law of large numbers. I would like to understand how legitimate is the use of it, whether there are ...
2
votes
0answers
135 views

Is this diagram of sheaves actually Cartesian as claimed?

The question is about Corollary 1.6.2 (b) in the book by Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$ and morphisms $X \xrightarrow{f} Y \xrightarrow{g} Z$ of sheaves on a ...
0
votes
0answers
56 views

the inverse of Trace theorem when $p = 2$

I can see there is an answer the inverse for the trace theorem and Image of the trace operator My question is that given $f\in H^{1/2}(\partial\Omega)$, is it possible to extended it into $\Omega$ ...
10
votes
1answer
334 views

Can Shor's Algorithm be modified to run efficiently on a classical computer?

Shor's algorithm is an algorithm which factors integers in polynomial time on a quantum computer. If one tries to run it on a classical computer, one runs into the problem that the state vector that ...
-3
votes
0answers
36 views

construction of nonocommutative division rings [on hold]

I am studing "A first course in noncommutative rings by T.Y.Lam " Please introduce books or articles to better understand the contents of section 14 (noncommutative division rings) of this book. thank ...
-1
votes
0answers
24 views

Solving el Gammal given D can solve DDH [on hold]

I have a crypto final in 2 days and I am reviewing past finals but the prof does not give solutions. There is 1 question I cannot solve it and I spend a good 2-3 hours on it. Here is the question: ...
0
votes
0answers
56 views

What is the relation between the $K_0$ of a singular curve and its normalization?

Let $X$ be a singular curve over a field $k$. We define $K_0(X)$ to be the Grothendieck group of the category of coherent sheaves on $X$. For $X$ we have its normalization $\widetilde{X}$ and hence ...
6
votes
0answers
279 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
-2
votes
0answers
23 views

Determining odds of a slot machine given a payout value of the icon [on hold]

So most slot machines base the payout on the probability of the combination coming up. What I would like to do is flip that and set a payout and then have the probability based off of that if ...
3
votes
2answers
151 views

Density of polynomials which are soluble with respect to a set of primes

Suppose that $p$ is a prime, and $f(x)$ is a polynomial with integer coefficients and positive degree. Then there exists an integer $n_p$ such that $p | f(n_p)$ if and only if $f(x)$ has a linear ...
4
votes
0answers
114 views

On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...
6
votes
0answers
151 views

Are curves over imperfect fields defined over a smaller field?

Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
5
votes
1answer
271 views

Are reduced residue systems relative primorials an active area of research? If not, why not?

As a math amateur, I am finding the study reduced residue systems relative a primorial a very interesting way to understand the distribution of primes. For example, it is fascinating to me that it is ...

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