All Questions
153,398
questions
0
votes
1
answer
108
views
Could someone help me to prove or disprove the following inequality?
Let $(c_{nr})$ be an $N\times R$ complex matrix, then $\forall z_n \in \mathbb{C}$, we have
$$
\sum_r \Big|\sum_n c_{nr}z_n\Big|^2 \geq \frac{1}{\sigma_{max}} \sum_n |z_n|^2
$$
where $\sigma_{max}$ is ...
- 395
0
votes
1
answer
234
views
Comparing sets of twin primes with other sets. Why is there a max and min value?
I have taken 2 sets: The first is a consecutive list of the first prime of twin pairs. The second is a consecutive list of numbers as follow 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5 ....
I have then compared ...
- 55
0
votes
1
answer
454
views
Bounds on variance of sum of dependent random variables
Let $x_1, \ldots, x_n$ be possibly dependent random variables, each taking values $x_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. ...
- 153
0
votes
1
answer
193
views
Is it true that $g-t$ is divisible by $f$?
Assume $f\in k[x_1,\ldots, x_n]$ is irreducible. Let for $g\in k[x_1,\ldots, x_n]$, $\partial(g)$ is divisible by $f$ for each derivation $\partial$ with $f\in\ker\partial$. Is it true that $g-t$ is ...
- 319
0
votes
2
answers
209
views
Intrinsically defining smooth/continuous/analytic functions
In mathematics, the notion of a continuous/smooth/analytic function $\mathbb{R}\to\mathbb{R}$ is introduced by defining the general set-theoretic function $\mathbb{R}\to\mathbb{R}$ and then imposing ...
user158636
0
votes
1
answer
176
views
A limit calculation [closed]
I wonder if the limit below $$\lim_{x\rightarrow +\infty} e^{-x}\sum_{j=0}^{\infty}\frac{x^{j+a}}{\Gamma(j+a+1)}$$
equals 1, for real constant $a>0$, and how shall we get this result?
- 13
0
votes
1
answer
119
views
How may a largest fixed-point be defined in second order logic?
Adapting from Anil Gupta and & Nuel Belnap, Revision theory of truth, MIT 1993, p. 194, in the context of a second order logic, where $A(x.G)$ is a formula where $G$ only occurs positively, a ...
- 3,292
0
votes
1
answer
88
views
Maximum number of edges in "square" hypergraph
For any set $X$ and any cardinal $\kappa$, let $[X]^\kappa$ denote the subsets of $X$ having cardinality $\kappa$.
A linear hypergraph is a hypergraph such that for all $e\neq e_1 \in E$ we have $|e\...
- 45.5k
0
votes
1
answer
108
views
Can we say that $f_\infty(t)=g_\infty(t) \text{ a.e}$?
Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f_n\}$ and $\{g_n\}$ two uniformly integrable sequences such that:
$$
\qquad\forall n\geq 1~:~ f_n(t)=g_n(t)~~ \text{ a.e.}\tag1
$$
$$
\...
- 199
0
votes
1
answer
105
views
Breaking up dense subset in non-separable space
Let $X$ be a not necessarily separable (infinite-dimensional) Banach space and $D\subseteq X$ be dense linearly independent subset. Then does there exist a set of infinite-dimensional separable ...
- 4,969
0
votes
1
answer
263
views
Anti-symmetric operators for the Dirac or Majorana spinors
In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) ...
- 2,137
0
votes
1
answer
138
views
Is $\int_M\Delta u = 0$ if $u$ is not $C^2$ on a set of measure zero?
Suppose that $M^2$ is a closed Riemannian manifold and that $u$ is a $C^2(M\setminus S),$ where $S$ is a closed measure set consisting possibly on a enumerable amount of points. Can we still conclude ...
- 1,774
0
votes
1
answer
246
views
Generalized Erdős multiplication table problem
Consider multiplication operation $$f(x_1,\dots, x_k)=\prod_{i=1}^kx_i$$ where $x_i\in\{1,\dots, n_i\}$ with $n_1,\dots, n_k\in\{1,\dots,\infty\}$.
What is the cardinality of the range?
At $k =2$ ...
- 1,816
0
votes
1
answer
389
views
Conjugacy in the quaternion group
Let $G$ be a non-commutative group, and suppose we are given two elements $x, y \in G$ which are conjugate, i.e. we know there exists some $z \in G$ such that $zxz^{-1} = y$. Can we find $z$ given $x$ ...
- 1,703
0
votes
1
answer
191
views
Fractional values in linear programming
Consider the linear programming problem
\begin{align}
f^* = \max_{x}&~p^Tx~\\~&A^Tx\leq b~,~0\leq x_i\leq 1
\end{align}where $p$ is a $n\times 1 $ vector, $A$ is a $n\times c$ matrix and $b$ ...
- 1,371
0
votes
1
answer
133
views
What are the exceptional properties of Mersenne exponent for known largest prime? [closed]
It is a clear that largest known primes are Mersenne prime. It is well known that $2^p - 1$ is prime only if $p$ is prime; however, the converse is not true - take $p = 11$. My question is: is there ...
user147204
0
votes
1
answer
126
views
Maximize function on rotation matrices [closed]
Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...
0
votes
2
answers
494
views
The union of two cuts is a cut?
Every poset $\langle P, \leq \rangle$ has a Dedekind-MacNeille completion, a complete lattice that embeds $\langle P, \leq \rangle$.
For $A \subseteq P$, the upset $U(A) = \{p \in P\ |\ \forall a \in ...
0
votes
2
answers
218
views
Left translation of automorphic form satisfies $K$-finiteness?
Does a left translation of an automorphic form satisfy left $K$-finiteness?
Let $F$ be a number field and $G$ is an algebraic group. Let $\phi$ is an automorphic form on $G$. Let $K$ be a maximal ...
- 1,719
0
votes
1
answer
418
views
Separability of an algebra is equivalent to separability of its spectrum
Let $A$ be a commutative C*-algebra.
I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.
Notes ...
- 143
0
votes
1
answer
159
views
does the ratio of the count of rational numbers on an $n\times n$ grid to $n^2$, converge as $n$ tends to infinity [closed]
Suppose we order the rational numbers using the diagonal method (used to prove they are countable) using an $n\times n$ grid. Now suppose we count the distinct rational numbers (those points on the ...
- 363
0
votes
1
answer
324
views
Eigenvalues of an integral operator
Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by
$$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$
What kind of assumption might I impose on $K$ such that $\lambda=1$ will be ...
- 545
0
votes
1
answer
170
views
Do we know absolute bounds for the norm of Satake parameters?
If we consider the set of all unramified Satake parameters $S$ of all automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ as $n$ varies, do we know absolute (that is, ...
- 6,932
0
votes
1
answer
178
views
Limit of a ratio of harmonic numbers?
Is there any way to find the following limit
$$R(n,m)=\lim_{N\to\infty}\frac{H_{nN,m}}{H_{N,m}}$$
which involves harmonic numbers (generalized if $m\neq 1$)
$$H_{N,m}=\sum_{k=1}^N k^{-m}\qquad ?$$
...
- 368
0
votes
1
answer
273
views
Explicit examples of (probability) measures on $\prod \mathbb{R}$
Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
- 4,969
0
votes
1
answer
110
views
Can we have cyclic generalized positive comprehension?
In positive set theory, the axiom scheme of generalized positive comprehension in $GPK^+_\infty$ [of Olivier Esser] is stated in a manner as to forbid the symbol of the asserted set to occur in the ...
- 9,266
0
votes
1
answer
118
views
Characterization of bounded variation
For a function $f:[0,1]\to\mathbb{R}$, define
$$ V(f)=\sup_{0=x_0<x_1<\ldots<x_n=1}\sum_{i=1}^{n}|f(x_n)-f(x_{n-1})|.
$$
For $f$ with integrable derivative, the definition coincides with
$V(f)...
- 6,061
0
votes
2
answers
177
views
PDF of $g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)}$?
Given the following function of random variables
$$g = \frac{1}{n} \sum_{k=1}^{n}{|h_k|\exp\left( j \theta_k \right)},$$
where $h_1, \cdots, h_n$ are i.i.d. random variables following the complex ...
0
votes
2
answers
241
views
Generalisation of modular forms
I am looking for a generalisation of a modular form that transforms as something like:
$f(\frac{a \tau+b}{c \tau+d}) = (c \tau+d)^k c^k f(\tau)$
I understand this cannot be literally true, as the ...
- 303
0
votes
1
answer
283
views
Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function
For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...
0
votes
1
answer
202
views
Non-constructible bijections in various set theories, do they exist?
Do there exist theorems of the form: "For every set theory there always exist at least two sets $M$ and $N$ between which there exists bijections and at least one bijection of those bijections cannot ...
user147968
0
votes
1
answer
444
views
Roots of unity, in the completion of the ring of integers of a number field, at a prime ideal
Let $p \in \mathbb{Z}$ be a prime and consider the number field $k = \mathbb{Q}[x]/(x^{(p^2 -1)/2} - p)$. We shall denote by $O_k$ the ring of integers of $k$. Let $\beta \in O_k$ be such that $\beta^{...
- 617
0
votes
1
answer
93
views
Recognition of a graph as a product of its quotients
Is there an algorithm to determine whether a given simple graph $G$ is a product graph, typically, say a cartesian product graph of two smaller simple graphs $G_1, G_2$, such that the two simple ...
- 2,027
0
votes
1
answer
199
views
Is it theoretically possible to find a factoring algorithm that runs in polynomial time? [closed]
Given that we don't know if P=NP, what's to stop someone from finding tomorrow an algorithm that makes prime factoring, or any other trap-door function reversing for that matter, computationally ...
- 1
0
votes
1
answer
82
views
Can global failure of Extensionality in fragments of NFU permit existence of singleton relation set?
Let $SF$ be the schema of stratified comprehension.
Take the theory $SF + Infinity + Choice + \text {Extensionality fails everywhere}$
Is the following consistent with this theory?
$\exists \iota \...
- 9,266
0
votes
1
answer
260
views
Axioms for the real numbers [closed]
Suppose we have a field with a total order which verifies
If $x,y\geq 0$, then $x+y\geq 0$ (here we relax the compatibility property for the addition).
If $x,y\geq 0$, then $xy\geq 0$.
...
- 29
0
votes
1
answer
159
views
Category of Frechet Spaces is Topological?
Let $sFre_{\mathbb{R}}$ (resp. $Fre_{\mathbb{R}}$) denote the category of (resp. separable) Fr\'{e}chet spaces over $\mathbb{R}$ as objects, and bounded linear operators as morphisms.
Is this a ...
- 4,969
0
votes
1
answer
618
views
Baker, Gill, Solovay - Relativization [closed]
How am I supposed to read the P=?NP relativization proof? I am reading
Theodore Baker, John Gill, and Robert Solovay. Relativization of the P=?NP problem. Siam Journal of Computing, 4:432-442, 1975 [...
- 9
0
votes
1
answer
1k
views
Definition of the Gauss symbol [closed]
Tahara refers to the "Gauss symbol" in the article, On the second cohomology groups of semidirect products, Math. Z. 129 (1972) 365--379. For a fixed $n$, let $S_{ij}$ be the expression
\begin{...
- 461
0
votes
1
answer
798
views
What are the eigenvalues and eigenvectors of a composition of two arbitrary linear transformations? [closed]
Let $S$ and $T$ be two linear transformations on $\mathbb{R}^n$. We can find the eigenvalues and eigenvectors of $S$ and $T$. I am trying to find out the relation(s) among the eigenvalues and ...
- 870
0
votes
1
answer
288
views
Formalizing ontological optimism
Inform speaking ontological optimisms means that everything that possibly exists in the abstract reality actually exists. From this principle we (again informally) get the Axiom of infinity, the Power ...
- 1,630
0
votes
1
answer
138
views
Chromatic Polynomial when two disjoint graphs are joined at $2$ distinct points [closed]
Consider a graph with chromatic polynomial $P(x)$ joined to a clique of order $k$ in two distinct points (joining here just means interesection of points). Then, what is the chromatic polynomial of ...
- 2,027
0
votes
1
answer
114
views
Proof of $\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx<\infty$ for Schwartz function $f$
For a function $f$ from the Schwartz space $\mathcal{S}(\mathbb{R})$ do we have that
$$\sum\limits_{k\in\mathbb{Z}}\int_{\mathbb{R}}|f(x+k)f'(x)|dx$$
converges?
- 123
0
votes
1
answer
147
views
Elusive groups and vertex-transitive graphs
This question is pertaining to finite connected vertex-transitive graphs.
I recently read "Transitive permutation groups without semiregular subgroup" by Cameron, Giudici, Jones, Kantor, Klin, ...
- 121
0
votes
1
answer
267
views
Bilinear Strichartz estimates for the Schrodinger equation
Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial_t+\Delta)u=(i\partial_t+\Delta)v=0.$ Let ...
- 923
0
votes
1
answer
309
views
Log Calabi-Yau surfaces without maximal boundaries
Let $X$ be a smooth projective surface over $\mathbb{C}$, $D\subset X$ is an effective divisor. $(X,D)$ is a log Calabi-Yau pair if $K_X+D$ is a principal divisor. The complement $M=X\setminus D$ is a ...
- 3,157
0
votes
1
answer
154
views
A property of convex cones in Euclidean spaces
EDIT: Let $K$ be a closed convex cone in a Euclidean space of finite dimension. Assume it is non-zero and not the whole space.
Does there exist a non-zero point $x\in K$ such that
$$(x,y)\geq 0 ...
- 21.1k
0
votes
2
answers
321
views
subspace topology and strong topology
Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...
- 71
0
votes
2
answers
203
views
Newton's minimizing method converge to local maximum [closed]
I have to minimize $f(x)=x^4-24x^2$ starting on the point $x_o=1$. The method converge to $x=0$, but i know that the solution is $x=+-2\sqrt{3}$. The hessian and the derivate of the function are $C_2$-...
- 3
0
votes
1
answer
261
views
Continuity of $\arg\min$
Let $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}$ be a continuous function.
Let $A = \{(x,\arg\min_x f(x,y)) | x \in \mathbb{R}^n\}$.
Is there necessarily a continuous function $g: \mathbb{R}^...
- 101