All Questions
152,618
questions
0
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6
views
Quotient of estimators
Say $A$ is a set of a finite number of samples, and $\hat{\mu}_A$ and $\hat{\sigma}_A$ are unbiased estimators (computed over A) of $\mu$ and $\sigma$ which are some distinct population statistics. ...
-1
votes
0
answers
25
views
A Near Closed-Form Expression of Strict Partition Function Inquiry
I am an independent researcher working in various fields of mathematics and sciences. I am working on a strict partition problem. I believe I have found a very fast exact solution that is a near-...
0
votes
0
answers
26
views
Solving system of linear diophantine equations over the integers
In general, solving a system of linear diophantine equations over the integers is polynomial time solvable on the size of the coefficients of the equations.
I am interested in an extension of this ...
0
votes
0
answers
23
views
Are the coefficients in the stationary phase approximation computed explicitly somewhere
In Stein's "Harmonic analysis" book, page 334, one can find
the asymptotic expansion
An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
-4
votes
0
answers
61
views
Goldbach's conjecture with Claude 3 [closed]
I wanted to test the mathematical abilities of Claude 3, so I typed a prompt leading to the following conjecture:
Assume Goldbach's conjecture and denote by $r_{0}(n):=\inf\{r\geq 0\mid(n-r,n+r)\in\...
0
votes
0
answers
13
views
Find transseries from difference equation
I want to find a method to solve equations of the form
$f(x+1)=f(x)+g(x)$ for a given function $g$ and $f(x)=0$.
The paper here has solutions for $f(x+1)=\lambda(x)f(x)+g(x)$, which is more general ...
0
votes
0
answers
35
views
Is this reduction process always commutative?
There was a question posted on this website earlier today by a new Account "Nick". He asked,
...
11
votes
0
answers
146
views
Proofs of the valence formula that avoid tricky contours?
$\DeclareMathOperator\ord{ord}\DeclareMathOperator\Im{Im}$The valence formula for a modular form asserts that if $f: \mathbf{H} \to \mathbf{C}$ is a modular form of weight $k$ on the upper half-plane $...
-6
votes
0
answers
38
views
Is the Square-free Test A Polynomial Time Algorithm?
The paper
"MILLER’s PRIMALITY TEST",
Volume 8, number 2 INFORMATION PROCESSING LETTERS February 1979
by H.W. LENSTRA, Jr.,
claims to have a square-free algorithm of polynomial time ...
1
vote
0
answers
86
views
Classification of complex irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ [duplicate]
Is there a classification of complex irreducible representations of the group $\operatorname{GL}_n(\mathbb{F}_q)$, where $\mathbb{F}_q$ is a finite field with $q$ elements?
0
votes
0
answers
40
views
On partitions into distinct parts and binary
Let $a(n)$ be A000009 (i.e., number of partitions of $n$ into distinct parts or number of partitions of $n$ into odd parts).
Let
$$
b(n) = \sum\limits_{i=0}^{n} a(i)
$$
Let
$$
\ell(n) = \left\lfloor\...
4
votes
2
answers
164
views
Prime differences and zero multiplicity
Paul Erdős conjectured, for consecutive primes, that:
$$\sum_{p_n < x} (p_n -p_{n-1})^2 = O(x \log x)$$
Let's call this hypothesis EH. Assuming the Riemann hypothesis (RH), Selberg showed that a ...
1
vote
1
answer
133
views
Is the irreducible $ \mathrm{SU}(3) $ subgroup of $ \mathrm{SU}(6) $ maximal?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\O{O}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\S{S}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\PSL{PSL}\...
-2
votes
0
answers
123
views
The precise meaning of Algebra, what it is? [closed]
My feeling is that the meaning of the word "algebra", at its origin, is not known(?). First of all it appeared in the title "Al-Jibr wa Al-Moqabala", by Al-Khawarizmi. So, what ...
1
vote
1
answer
60
views
Irreducible subspaces in the space of functions on Grassmannian acted by $\mathrm{GL}_n(\mathbb{F}_q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}_q$ be a finite field with $q$ elements. Let $\Gr_{i,n}(\mathbb{F}_q)$ denote the Grassmannian of linear $i$-dimensional ...
-2
votes
0
answers
16
views
Population-level measurable function from samples
Consider a set of points $\{(x_i, y_i)\}_{i=1}^\infty$, where $x_i \sim X$ for some random variable $X$. Let’s say they are all in $[0, 1]$. We may assume $x_i$ are dense in $[0,1]$. Is it generally ...
0
votes
0
answers
43
views
Monotonicity of ratio of symmetric polynomials
The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by
\begin{equation*}
h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
0
votes
0
answers
34
views
$f(x) = \sum_{j=1}^{4}c_{j}e^{r_{j}x} \quad \text{and} \quad g(x) = \sum_{j=1}^{4}d_{j}e^{s_{j}x}$. How to determine $c_{j},d_{j}$?
Let $\alpha_{i},\beta_{i} \in \mathbb{C}$, $\forall i= 1,2$ and $0 < a < b < \infty$.
\begin{align}
&f^{\prime \prime} + \alpha_{1} f = \alpha_{2} g^{\prime} \quad \text{in} \quad [a,b] \\...
2
votes
0
answers
51
views
Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
16
votes
0
answers
271
views
A conjecture about inclusion–exclusion
$\newcommand\calF{\mathcal{F}}
\def\cupdot {\stackrel{\bullet}{\cup}}
\def\minusdot {\stackrel{\bullet}{\setminus}}$This post presents a conjecture that we have with some colleagues. It is about ...
4
votes
0
answers
56
views
Quotients in categories of metric spaces
There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (=non-expansive or contractive) ...
0
votes
0
answers
48
views
How to distinguish birth and death bifurcations?
Let $f : \mathbb{R} \to \mathbb{R}$ have a degenerate critical point at $x = 0 \, ($ie, $f(0) = f'(0) = f''(0) = 0)$.
Perturbing $f$ locally around $0$ may cause multiple scenarios:
Birth: the ...
2
votes
0
answers
85
views
About pushforward of a sheaf of divisor
Let $X$ be a normal variety over an algebraically closed field of arbitrary characteristic, $f:X'\to X$ a log resolution, $L$ a Cartier divisor on $X$, and suppose $L\sim_{\mathbb{Q},f}E$, where $E$ ...
2
votes
0
answers
48
views
When is a coherent sheaf on an algebraizable space algebraizable?
Let $\mathcal{X}$ denote an algebraizable rigid analytic space over $\text{Spm}(\mathbb{Q}_p)$,
i.e. there exists a finite type $\mathbb{Q}_p$-scheme $X$ such that $\mathcal{X}$ is its rigid ...
0
votes
0
answers
16
views
Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?
Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
1
vote
0
answers
51
views
Mirror of a local K3 surface
Is there any description of a mirror manifold of a (non-compact) Calabi-Yau threefold given by the total space of the trivial line bundle on a K3 surface? If yes, in what way is it a mirror?
Thanks ...
4
votes
1
answer
246
views
Subgroup of p-adic units
Let $\smash{\widehat{\mathbb Z}}^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
-4
votes
0
answers
42
views
Non-isomorphic geometric objects obtained by cutting the Möbius band [closed]
How many non-isomorphic geometric objects can be obtained by cutting a Möbius band while keeping parallel on the substrate?
-1
votes
0
answers
45
views
Positiveness of a measure in the unit circle [closed]
Assume that $\mu$ is a positive measure in the unit circle and assume that $\mu = \eta +\delta$, where $\delta$ is Dirac measure. Whether in general $\eta$ is a positive measure. I see this argument ...
0
votes
0
answers
40
views
Convergence in probability of quadratic form with positive mean
Let $\boldsymbol{X}_n\in\mathbb{R}^n$ be a sequence of Gaussian random vectors with independent entries, such that $X_{n,i}\sim \mathcal{N}(\mu_i,\sigma^2)$ (that is, all entries of the $n$th vector ...
2
votes
0
answers
50
views
Projective resolution of a quiver with relations
How do we compute the projective resolution of a representation of a quiver with relations.
For example consider the Beilinson quiver $B_4$
$.
with the relations $\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
0
votes
0
answers
21
views
Prove the NP-hardness of the following problem: Whether there exists a partition for a set of data points
Can anybody help me prove the NP-hardness of the following question:
Given $x_0, x_1, ..., x_m \in \mathbb{R}^n$, determine whether there exists a partition $S\cup [m]\backslash S$, such that $x_0 \in ...
0
votes
0
answers
69
views
Verifying the Cauchy behavior of a sequence
Let me consider the iteration $x_{n+1}=Tx_n$ and $T$ is a self-map from a non-empty subset $K$ of a smooth Banach space $X$ to itself, satisfying $W(Tx, Ty) \leq W(x, y)$, where $W(x, y)=\Vert x \Vert^...
1
vote
1
answer
66
views
Minimal norm problem whose unknown is an operator
Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that
$$
f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2
$...
0
votes
0
answers
35
views
Bound on a two-dimensional recursive series
For $n,k\in\mathbb{N}$, let $f(n,k)$ be defined as follows.
If $n \geq k$ and $n > 2$, then
$$
f(n,k) = \frac{k(n-k)}{n(n-1)}f(n-2,k-1) + \frac{k(k-1)}{n(n-1)}f(n-2,k-2) + \frac{n-k}{n}f(n-1,k) + \...
0
votes
0
answers
17
views
The existence of convergent subsequences
Considering an optimization problem on an infinite-dimensional Euclidean space, the sequence of objective functions is $J_{n}(\theta_{n})=||f(X,\theta_{n})-Y||^{2} $, where X and Y are datas.
This can ...
3
votes
1
answer
272
views
Cohomology of the partial flag variety associated with the minimal nilpotent orbit
Let $G$ be a semi-simple group over complex number; for simplicity let us assume that it is simply laced. Let $X$ be the orbit of the highest root line in the adjoint representation of $G$. This is a ...
2
votes
1
answer
108
views
Relationship between Frobenius theorem, curvature, and integrability
In this answer to References for "modern" proof of Newlander-Nirenberg Theorem John Hubbard alluded to something called the "Frobenius integrability form" $\phi\mapsto\bar\partial\...
7
votes
0
answers
110
views
Cohomology dimensions are well approximated by Gaussian for multiply-fibered manifolds ? (Topological central limit theorem)
Consider some manifold $M$ say compact smooth. Let $b_i$ be its Betti numbers (non-zero), i.e. its cohomology dimensions.
Assume $M$ can be subsequently fibered by many manifolds, i.e. there is $ M_{...
2
votes
0
answers
199
views
Is there a version of 3-SAT that is NP-complete but grows like $2^n$ instead of $2^{n \choose 3}$?
If I have $n$ variables and I want to write down all 3-SAT problems, the number of problems is $2^{8{n \choose 3}}$, since each clause has 3 variables and each variable can be negated or not.
But ...
0
votes
1
answer
76
views
Convex sets via fixed point equations
I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
0
votes
0
answers
45
views
Reference request for equivalent Lipschitz smoothness conditions
For an open set $Z\subseteq\mathbb{R}^n$, let $f: Z\mapsto \mathbb{R}$ be a continuously differentiable function on $Z$, and let $L>0$ be fixed. Also, suppose that (a) $f$ is nonconvex and (b) $f$ ...
4
votes
1
answer
335
views
Hyperbolic three-manifolds that fiber over the circle
Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...
3
votes
2
answers
350
views
Monotonicity of matrix conjugation
Let $A$ and $B$ be positive-definite matrices such that $A \le B.$
By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$
I am now curious under ...
0
votes
0
answers
29
views
A question about regularity results in the Elliptic case which are given by Schauder theory
I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he ...
-1
votes
0
answers
54
views
Cube containing nine unit cubic lattices [closed]
What is the volume of the smallest cube containing nine unit cubes?
0
votes
0
answers
133
views
Fourier series but different waveform
Given a nondegenerate smooth simple closed curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$. Let $f_n: [0,2\pi]\to \mathbb C \setminus \{0\}$ be such that $f_n(...
2
votes
0
answers
82
views
Multiplicative structure on Čech–Alexander complexes
I have the following basic question on Čech–Alexander complexes.
Let $R$ be a ring and $A$ be an $R$-algebra. To this datum one can attach a cosimplicial ring which assigns to an object $[n]=\{0,1,\...
2
votes
0
answers
177
views
What are the Hodge and log Hodge groups of $M_{g,n}$?
I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
1
vote
1
answer
135
views
Isn't every algebraic operad equipped with a trivial weight?
In Loday–Vallette "Algebraic Operads" they state the following result (Theorem 6.6.2, Operadic twisting morphism fundamental theorem):
Let $P$ be a connected weight graded differential ...