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62 views

$A^{*}A=B^{*}B$ [closed]

Let $A,B$ be two invertible $n\times n$ matrices of complex numbers. If $$ AA^{*}=BB^{*},$$ does it follow that $$A^{*}A=B^{*}B$$ if not, then under what conditions this would be true? ($A^{*}$ is the ...
1
vote
1answer
57 views

Integrability condition for flat connections

I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5. Kobayashi is trying to prove that if $E$ is a vector ...
-2
votes
0answers
31 views

Proving independence of events

The measure space $(X,\mathcal{A},\mu)=(\Omega, \mathcal{E}, \mathbb{P})$ is the Bernoulli probability space, where $\Omega=\{(\omega_j)_{j \in \mathbb{N}} | \omega_j \in \{u,d\} \forall j \in \mathbb{...
2
votes
1answer
86 views

Hecke eigenform with integer Fourier coefficients

Is it true that for any even $k$ and $N,$ there always exist a Hecke eigenform with integer Fourier co-efficient of weight $k$ and level $N$ ?
2
votes
0answers
67 views

Extension of a theorem of Bisch to cyclotomic integers of fixed degree

Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such ...
3
votes
0answers
70 views

Given a chain of commuting matrices over the complex numbers, can we build one over the real numbers?

Suppose we have two $n\times n$ matrices $A$ and $B$ with entries in $\mathbb{R}$, and two non-scalar matrices $X$ and $Y$ with entries in $\mathbb{C}$, such that $AX=XA$, $XY=YX$, and $BY=YB$. Is it ...
5
votes
0answers
66 views

Is there a conjectured uniform Lindelof hypothesis for Hurwitz zeta functions

Consider $\zeta(s, a) = \sum_{n=1}^{\infty} (n+a)^{-s}$ (alternatively, consider its functional equation Dirichlet series $\sum_{n=1}e(a n) n^{-s}$). What is the expected growth-rate of $\zeta(1/2 + ...
2
votes
0answers
40 views

Going from a class of path functions to a topology

Someone asked a version of this 10 years ago, but no satisfactory answer was given, so I want to try again. I have a class of functions F from the reals to a set S (the specifics are complicated and ...
-2
votes
0answers
52 views

If (a+b):(b+c):(c+a)=2:3:4 and a+b+c=27, then how much is c? [closed]

I found this question in an old mathematics book and I'm stuck. Does anyone know the answer?
0
votes
0answers
16 views

Discrete control matrix - Upper limit

I have several questions regarding control theory which might be of interest on the mathematics community. One might recall a continuous-time linear system has the form $\dot{x} = A x + B \, u$. For ...
1
vote
0answers
63 views

Cellular chain complex of $G$-CW-complexes & their differentials

I not completly understand EXAMPLE 2.31 (page 19) dealing with homology of $G$-CW-complexes. Source: http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/LTCC-notes-Lecture3-2019.pdf ...
1
vote
0answers
241 views

Paradox in additive combinatorics

Let $S$ be an infinite set of positive integers. Let us define the following quantities: $N_S(z)$ is the number of elements of $S$, less or equal to $z$ $r_S(z)$ if the number of positive integer ...
3
votes
0answers
40 views

Finding a non-negative similar matrix

Let $B$ be some matrix. Is there a way to decide whether there exists an invertible matrix $P$, such that the matrix: $$ A = P^{-1}BP $$ Is non-negative? (That is - all the entries in the matrix are ...
2
votes
0answers
119 views

Holomorphic map from a punctured affine line to $M_g$ with Zariski-dense image

Denote by $M_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. Is there a holomorphic map $U\to M_g$ with Zariski-dense image where $U$ is the affine line with ...
9
votes
1answer
237 views

Cusp forms with integer Fourier-coefficients

Let $S_k(\Gamma_1(N))(\mathbb{Z})$ be the set of modular forms of weight $k$ and level $N$ with integer Fourier coefficients. Then is true that any cusp form can be written as $\mathbb{Q}$ linear ...
4
votes
1answer
158 views

An explicit negative solution to the Lüroth problem for non-algebraically closed fields

Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$. If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \...
-4
votes
0answers
45 views

Is (y-x)dx + (4xy)dy = 0 , a linear differential equation? [closed]

Dennis G. Zill A First Course in Differential Equation says in Chapter 1 Example 2 that we can write (y-x)dx + (4xy)dy = 0 as 4xy' + y = x but I get 4xyy' + y = x . Am I missing something?
1
vote
0answers
93 views

Cohomology of a family of twisted cubic curves (Hartshorne III, 12.9.2)

I'm trying to understand following Example in Hartshorne (Chapter III, Example 9.8.2 & Example 12.9.2): Let $X_1 \subset \mathbb{P}^{3}$ be a twisted cubic curve not containing the point $(0:...:1)...
3
votes
0answers
67 views

Let $T$ be a maximal torus of $SU(k+1)$. Who is the normalizer $N(T)$ of $T$ in $O(2k+2)$?

I'm reading an article which I cannot understand a paragraph very well. $T$ is a maximal torus of $SU(k+1)$ acting linearly on $\mathbb{C}^{k+1}$. And here is what is written that I cannot fully ...
1
vote
2answers
67 views

Difference between semilinear and fully nonlinear

I'm confused why the Hamilton Jacobi Bellman equation: $$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$ is considered fully nonlinear, but not semilinear. By ...
-4
votes
0answers
39 views

Formula to calculate 500 million base with an increase of 5% a year for X amount of years [closed]

I'm starting with 500 million user base with an average increase of 5% a year. I can get the result by doing: 500 million + 5% = A + 5% = B + 5% = C + 5% = D + 5% = E E will be the result of 5 years ...
3
votes
0answers
44 views

Algebras Morita equivalent with the Weyl Algebra and its smash products with a finite group

My question os motivated, naturally, by the problem of classifying symplectic reflection algebras up to Morita equivalence (a classical reference for rational Cherednik algebras is Y. Berest, P. ...
-2
votes
0answers
107 views

On the Dirichlet divisor problem. Proof that $\Delta(n) = O(n^{\frac{1}{4} + \epsilon})$?

Hello dear mathematicians! I have a few questions regarding my current work (paper) on counting the number of lattice points under a hyperbola $\frac{n}{xy},\; 1 \leq x \leq n,\; 1 \leq y \leq n$. ...
3
votes
0answers
86 views

Is identification of double points of an immersion smooth?

Let $f:M^m\to N^n$ be a generic map between smooth manifolds $n>m$. Depending on the pair $(m,n)$ generic maps will have a singular set of double points $\Sigma_2\subset M$. Let $\phi:\Sigma_2\to \...
1
vote
0answers
28 views

Minimax theorems in nonconvex setting

Let $X$ be a topological space, $Z$ be a compact convex subset of $\mathbb R^m$, and let $f:X \times Z \to \mathbb R$ be a continuous function (w.r.t the product topology on $X \times Z$). Question. ...
-7
votes
0answers
113 views

Verification of the Collatz problem [closed]

This post is to let the MO community know that the limit has been raised to which Collatz problem is computationally verified. Although the conjecture has not been proven, there is experimental ...
-5
votes
0answers
79 views

Origin of the “here comes a/the X” phrase [closed]

Sit through enough math lectures and you’ll notice a lot of folks say “here comes a definition/theorem/proof.” Is there some history behind this turn of phrase? If so, I’d love to know.
0
votes
1answer
97 views

Gauss curvature derived from unit normal vector

I want to know more about the differential geometry of surfaces, especially Gaussian curvature. Obviously, we can get the mean curvature of a surface from the divergence of the unit normal vector of ...
2
votes
3answers
235 views

Alternating sum over collections of sets

Let $\mathbf{P}$ be a collection of subsets of a finite set $X$. Let $\mathscr{S}$ be the set of all subsets $\mathbf{S}\subset \mathbf{P}$ such that $\bigcup_{S\in \mathbf{S}} S = X$. Can one give a ...
0
votes
1answer
110 views

Example of a representation of a finite group where Weyl's unitary trick is necessary?

Is there an example of a representation $\rho: G \rightarrow GL(V)$ for some finite group $G$ where say $W \subset V$ is a $G$-invariant subspace for $\rho$ but the orthogonal complement (in the ...
2
votes
0answers
96 views

Kan liftings and projective varieties

Regard the following two bicategories: $\operatorname{dg-\mathcal{B}imod}$, with objects dg categories, and morphisms categories from $C$ to $D$ being the categories of $C$-$D$-bimodules. Composition ...
-3
votes
0answers
40 views

Why is this integral taking the second reciprocal? [closed]

I am studying integrals and while integrating x^n, I am coming across situations where the fraction's second reciprocal is printed in the answer but when I work it out, I get the opposite. An example ...
2
votes
1answer
120 views

Estimate on Mobius function

Let $\mu(n)$ be the Mobius function, how to estimate $$\sum_{1\le i<j\le x}\mu(i)\mu(j) $$ as $x$ goes to $\infty$? Are there some references on this?
0
votes
0answers
88 views

Is a countable infinite union of $\Sigma_1$ sets is $\Sigma_1$? [migrated]

I’m reading Kunen’s book Foundations of mathematics. My question is whether a countable union of $\Sigma_1$ sets in $HF$ is also $\Sigma_1$ or not. I wonder if we can think $\Sigma_1$ sets as open ...
4
votes
0answers
84 views

Freys elliptic curves and Hilbert spaces?

Consider the Frey-Hellegouarch curve given $a,b$ positive rational numbers: $$y^2= x\left(x-\frac{a}{\gcd(a,b)}\right)\left(x+\frac{b}{\gcd(a,b)}\right)$$ The j-invariant is given by: $$j(a,b) = \frac{...
2
votes
0answers
34 views

An eigenvalue upper bound for 1-walk-regular graphs

Let $G$ be a graph and suppose that $G$ is 1-walk-regular (or, if you prefer, vertex- and edge-transitive, or distance-regular). Let $\theta_1>\theta_2>\cdots>\theta_m$ be the distinct ...
4
votes
0answers
47 views

Continuity vectors of log-concave measures

Let me begin by recalling some definitions and setting some notation I'll be using. As a complete reference I point to the book Bogachev, Differentiable measures and Malliavin Calculus (essentially ...
2
votes
0answers
66 views

Acyclic extensions of acyclic simplicial complexes

Say an abstract simplicial complex $X$ is acyclic if its reduced integral simplicial homology groups $\tilde{\mathrm{H}}^{\Delta}_p(X)$ vanish for all $p\geq 0$. Is it the case that, for any $n>0$, ...
1
vote
0answers
16 views

Antipodal vertices in spectral graph embeddings

Suppose your are given an antipodal graph $G=(V,E)$, that is, for every vertex $v\in V$ there is a unique maximally distant vertex $v'\in V$. Under which condistions does the following hold: If $\...
2
votes
0answers
22 views

Diagonalization of the generalized 1-particle density matrix

Let $\mathscr{H}$ be a complex separable Hilbert space and $\mathscr{F}$ be the corresponding fermionic Fock space generated by $\mathscr{H}$. Let $\rho: \mathscr{L}(\mathscr{F}) \to \mathbb{C}$ be a ...
-1
votes
0answers
38 views

What branch of mathematics studies rubber band balls [migrated]

I'm making a rubber band ball and am trying to not make any of the rubber bands flip. This is possible with one revolution (simplest), three revolutions, five, etc. Are there more than one way to wrap ...
4
votes
0answers
37 views

What are algorithms to compute the Ito representation of a functional?

Given an $L^2$ functional of Brownian motion $\Phi$ there exists a measurable process $u(t)$ so that $\Phi=E[\Phi]+\int_0^T u(t) dB(t)$. We know by Clark Ocone that $u(t)=E[D_t \Phi |\mathcal F_t]$ ...
1
vote
0answers
32 views

Reference request : Convergence of radial basis function interpolation or spline interpolation as points become dense, for a continuous function

Is there any proof for this. Kindly request a reference in case available or any related documents towards this. PS : I am specifically interested in the case of scattered data (irregularly placed), ...
1
vote
0answers
36 views

Dimensions of the intersection of 8 quadrics

Suppose $e_i,q_i \in \mathbb{R}^3$, $1\leq i \leq 3$ with $\Vert e_i \Vert=1$ are known. Define the projection on the plane orthogonal to $e_i$ $P_i= I-e_i e_i^T$ where $I$ is the $\mathbb{R}^{3\times ...
0
votes
1answer
55 views

Bound on the chromatic number of square of bipartite graphs

In continuation of the previous question, what is a strict upper bound on the chromatic number of the square of a bipartite graph? I think the chromatic number number of the square of the bipartite ...
-3
votes
0answers
58 views

Can you help me with these two integrals? [closed]

these are three functions that I try to integrate by hand, however, I try many methods and they do not work. Can you help me with that? The first function is: $$\int_{0}^{2\pi} \frac{1}{\sqrt{1- a\cos(...
5
votes
0answers
84 views

Federer's questions on the mass and comass norms

In Federer's book "Geometric measure theory", he says in section 1.8.3 (where $\|\cdot\|$ is the comass norm): Very little appears to be known about the structure of the convex sets $\wedge^...
5
votes
0answers
48 views

Is the cohomology of Hilbert modular surfaces spanned by special cycles?

We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
7
votes
1answer
510 views

Is there a collection of evidence and heuristic arguments against the Riemann hypothesis? [closed]

There is undoubtedly an overwhelming collection of evidence for the Riemann hypothesis. However, is there any evidence against it ?
7
votes
0answers
115 views

Realizing Stiefel-Whitney classes via vector bundles

Let $X$ be a CW complex. If $E$ is a vector bundle over $X$, then it's well-known that the Stiefel-Whitney classes $w_j(E) \in H^j(X,\mathbb F_2)$ of $E$ are determined from the classes $w_{2^k}(E)$ (...

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