# All Questions

115,063
questions

**-3**

votes

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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 $w_{2^k}(E)$ (...