1
vote
1answer
21 views

Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact: If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...
3
votes
0answers
44 views

Adjunction algebra - is there anything similar to this in abstract algebra?

I call adjunction algebra a universal algebra with one binary operation denoted as the punctuation sign (;) "semicolon" (but I will be using only one space after it, not on both sides - to avoid going ...
4
votes
1answer
31 views

Subgroups of Nilpotent groups with prescribed center

Let $G$ be a torsion-free, finitely-generated, nilpotent group of nilpotency class at least 3. Does there exist a normal subgroup $N\leq G$ such that $G/N\cong \mathbb{Z}$ and $Z(G)=Z(N)$? (By ...
0
votes
0answers
12 views

Cubic graphs whose 2-factors all have the same cycle type

Let $G$ be a bridgeless cubic graph. I am interested in such graphs where all 2-factors are isomorphic (as graphs), i.e. have the same partition as cycle type. We'll say that this partition is ...
0
votes
0answers
40 views

Curvature and the area

Assume that $\Omega$ is a Jordan domain with $C^2$ boundary $\gamma$ with $2\pi$ length and let $\kappa_z$ be the curvature of $\gamma$ at a point $z$. Is this formula known: $$Area(\Omega)=2\pi ...
0
votes
0answers
26 views

Unbounded difference on multiples of any irrational number [on hold]

For some irrational r consider the sequence of integral multiples as r, 2r, ..., ...
5
votes
1answer
64 views

continuum many mutually generic filters

Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic ...
2
votes
0answers
37 views

Topology for bounded operators quotiented by Schatten ideal

I saw this particular question on stackexchange. Since there has been zero answers and since I've been interested in this question myself I want to ask it here. Given the $C^{\ast}$-algebra of bounded ...
0
votes
0answers
24 views

The class of uniformly accelerated curves and surfaces

Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math. Let's consider two examples of curves: 1)The curve defined by the ...
5
votes
1answer
103 views

Geodesics on $SU(4)$

Are the geodesics of the following metrics on $SU(4)$ known or easy (in a way not known to me!) to find? In the adjoint representation, one can express the Killing form as a matrix and consider it as ...
2
votes
1answer
84 views

Cyclotomic character in class field theory

Let $K$ be an extension of $\mathbb{Q}_p$. By local class field theory, the $p$-adic cyclotomic character $\mathrm{Gal}_K \rightarrow \mathbb{Z}_p^\times$ corresponds to a character $\chi : K^\times ...
0
votes
0answers
33 views

Re-expressing the Integral $\int h\left(t-\tau\right)x\left(t-\tau\right)x\left(\tau\right)^2\,\text{d}\tau$

Given the integral: $$ \int^{t}_{0} h(t-\tau)x(t-\tau)x(\tau)^2\,\text{d}\tau = \int^{t}_{0} h(t-\tau)F(\tau)\,\text{d}\tau + K $$ Can you find $F(\tau)$ so it is not a function of $t$ and find $K$ ...
1
vote
1answer
41 views

Explicitly relating two functions containing exponential terms [on hold]

This is an extremely basic question for a forum like this, but I am unable to think of any workable approaches myself. I have two functions related to the distribution of administered drugs in the ...
0
votes
0answers
66 views

Help estimating a number

I m lost estimating a number. I m looking for a tool to apply in this situation. Given $A_0=\begin{pmatrix}1 & 1 \\ 1 &0 \end{pmatrix}$ and $A_1=\begin{pmatrix}1 & 1 \\ 0 &1 ...
0
votes
0answers
17 views

Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
7
votes
0answers
84 views

Easiest example where field of definition is not field of moduli

There are many examples of varieties over $\overline{\mathbb Q}$ whose field of moduli is $\mathbb Q$ but which can't be defined over $\mathbb Q$. What is the easiest such example? It should be a ...
1
vote
0answers
42 views

Question on the consistency of Zermelo set theory minus specification and extensionality

Let $W=Z^{-}-Specification$ where $Z^{-}=Z-Extensionality$ and Z is Zermelo set theory. What is known about models of $W$ or $W^{+}=W+Extensionality$?
3
votes
2answers
182 views

(reference request) Chaitin's constant is incompressible

I've been looking for a full, detailed proof that Chaitin's constant is incompressible, i.e. there is a universal constant $c$ such that every program writing first $n$ digits of $\Omega$ has length ...
0
votes
1answer
40 views

Integral over Kronecker product

Let $A : [0,T] \to \mathbb R^{n \times n}, t \mapsto A(t)$ be smooth with the property that $$ \int_{0}^T A(t) dt $$ is invertible. Does it then follow that the matrix $$ \int_{0}^T A(t) \otimes ...
1
vote
0answers
63 views

Comonads and the category of Sets

In Vicary's paper, after eq 15, he talks about how the category of internal comonoids $C_\times$ has many properties of the category of sets. We know that a comonad on a category has the same axioms ...
2
votes
1answer
168 views

Can a closure make the index finite?

Let $F$ be a free finitely generated group, $H \leq F$ of infinite index. Let $c : F \rightarrow \hat{F}$ be the embedding in the profinite completion. Denote by $\tilde{F}, \tilde{H}$ the closure of ...
3
votes
0answers
30 views

Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...
-3
votes
0answers
29 views

Notation for rolling median [on hold]

I am writing a paper where a stack of images is processed using, among others, a rolling median. This takes ten images and calculates, pixel by pixel, the median value. What notation do you suggest to ...
0
votes
0answers
10 views

Characterization of the optimal solution in relative entropy minimization

The following optimization problem is related to relative entropy and to the limit of the iterative proportional fitting procedure. For $1 \leq i,j \leq n$ and fixed $w_{ij} \geq 0$, and fixed $a_i, ...
0
votes
0answers
60 views

Hilbert function of points in $\mathrm{P}^2$

Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If $$ ...
-2
votes
0answers
96 views

What's the second cohomology group of the tangent bundle of $CP^n$ [on hold]

I want to know what is the second cohomology of tangent bundle of $CP^n$. Especially, for $CP^1$. And is there any general method to compute cohomology for tangent bundle?
0
votes
0answers
53 views

What is a “normal crossings divisor relative to S”?

I'm reading SGA 1 Expose XIII, as given here: http://arxiv.org/pdf/math/0206203v2.pdf and I'm trying to understand the given definition of a "diviseur a croisements normaux relativement a S" for some ...
7
votes
1answer
102 views

Existence of solutions of a polynomial system

Fix $k \in \mathbb{N}$, $k \geq 1$. Let $p \in [0,1]$ and $x = (x_0, \ldots, x_k)$ be a $(k+1)$-dimensional real vector, and define $$S(p,x) = -x_0^2 + \sum_{i=0}^k {k \choose i} p^i (1 - p)^{k - i} ...
4
votes
0answers
30 views

Bloch group, hyperbolic manifolds and rigidity

I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to ...
5
votes
0answers
46 views

Factor a sum of products of cofactors

Let $M$ be an $n\times n$ matrix whose first column consists entirely of 1s. We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row ...
1
vote
1answer
86 views

Weak convergence of a sequence

I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : ...
1
vote
1answer
269 views

Are all linear transformations measurable?

Let $V$ and $W$ be topological vector spaces over $\mathbb{F}$ (with $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$), and let $T:V \to W$ be a linear transformation. It is well-known that $T$ is not ...
7
votes
2answers
77 views

Convexity of a certain sublevel set

Consider the polynomial of degree $4$ in the variable $r$ : $$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$ The discriminant of this polynomial in $r$ is the following expression (obtained using ...
0
votes
1answer
103 views

Function that dominates everything in little o

I have a function $f(n)$ that satisfies the following property: for any function $g(n) = o(n^{-2})$, we have $f(n) = \Omega(g(n))$ (the implied proportionality constant in the $\Omega$ expression ...
8
votes
3answers
255 views

Circles avoiding rational points of height $\le h$

Q. Which origin-centered circles $C(r)$ (or spheres in dimension $d$) of radius $r < 1$ avoid all rational points of height $\le h$? A rational point is a point all of whose coordinates ...
1
vote
1answer
84 views

Equivalent Killing vector fields via an isometry

Suppose that $(M,g)$ is a complete semi-Riemannian manifold. We say that two Killing vector fields $V$ and $W$ are equivalent if there is $\Phi:M\rightarrow M$ an isometry such that $\Phi_*(V)=W$. ...
0
votes
0answers
18 views

transfer function, Laplace transform of second order equation [on hold]

I have met such problem,I have the second order equation, as Au(t) + By(t) = C*y''(t).where A,B,C are paramters, and u(t) is input, the output is y(t). what is the transfer function of Y'(s)/U(s)? ...
-2
votes
0answers
47 views

Volume of revolving y=sin(x) about a line y=c [on hold]

Consider the surface formed by revolving $y=sin(x)$ about the line $y=c$ from some $0\le{c}\le{1}$ along the interval $0\le{x}\le{\pi}$. [![graph][1]][1] Set up and evaluate an integral to calculate ...
0
votes
0answers
19 views

Compatability of depth for elements in p-adic groups under base change

Suppose $F$ is a non-archimedean local field and $E$ is a tame extension of it. Let $G$ be any connected reductive group over $F$. For any $r>0$, let $G(E)_r$ be the set of elements with depth $\ge ...
1
vote
0answers
37 views

extending to bimeromorphic maps

A meromorphic map of complex spaces (in the sense of Remmert) f:X→Y is a multivalued map such that its graph Γ is an analytic subset of X×Y and off some analytic subset Z⊂Γ, the projection on the ...
4
votes
1answer
73 views

Reference or proof for the fact that $J(X_0(N))$ splits inabelian varieties with real multiplication

It´s known that $J_0(N) = J(X_0(N))= \bigoplus_f E(f)$ splits as a sum of abelian varieties parametrized by the Hecke eingenfunctions and that it´s an elliptic curve iff the Hecke eingenvalue is an ...
1
vote
2answers
194 views

decomposition of Hilbert space into tensor product $L^2([0,\tfrac{1}{2}]) \otimes L^2([\tfrac{1}{2},1]) \simeq L^2([0,1])$

The definition of entanglement entropy in Quantum Field Theory involves decompositing a Hilbert space into a tensor product $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$. As an example, is it ...
6
votes
1answer
102 views

Smallest Connected Graph for Given Degree Sequence

For a given integer sequence $(d_1, d_2,...,d_n)$, a natural question is if such a sequence is graphical, i.e. is a degree sequence of some graph. According to Erdős–Gallai theorem, A sequence of ...
0
votes
0answers
30 views

When does a stochastic process have its sample paths a.s. in the reproducing kernel hilbert space (RKHS) induced by its covariance function?

Let $T$ be a compact metrizable space. Consider a centered second order measurable process $(X_t\colon t\in T)$ with continuous covariance function $c(t,s):= \mathbb{E}X_t X_s$. Are there any known ...
3
votes
0answers
49 views

Non Borel Spaces: Gauge Integral

Question Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology? I'm wondering since it seems as the gauge crucially uses ...
1
vote
1answer
51 views

softening probability distribution function

I am working on ECG signals and I want to fit it's probability distribution function with gaussian mixture model (sum of 2 or 3 gaussians) to extract features but it has a very sharp pdf around zero. ...
3
votes
0answers
91 views

mod $p$ Jacquet-Langlands correspondence

Let $F$ be a local field of characteristic $0$. Let $D$ be division algebra over $F$ of dimension $n^2$. The construction of irreducible complex representations of $D^*$ is known by Howe, Zink, and ...
0
votes
0answers
130 views

Structure theorem for linear algebraic groups

I was wondering whether there is a structure theorem for linear algebraic groups over a number field $k$, i.e. something which tells us that any linear algebraic $k$-group is made up of, say, ...
1
vote
1answer
89 views

Empty real conic containing two pairs of conjugate points in the projective plane?

Given two conjugate pairs of points in general position in $\mathbb{CP}^2$, there is a pencil of real conics containing these four points. Is there a real empty conic in this pencil?
0
votes
1answer
148 views

Projection of a hypersurface from a point

Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...

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