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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. ...
CWC's user avatar
  • 351
-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-...
jables's user avatar
  • 1
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 ...
user1868607's user avatar
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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 ...
Medo's user avatar
  • 676
-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\...
Sylvain JULIEN's user avatar
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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 ...
opfromthestart's user avatar
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, ...
Snared's user avatar
  • 109
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 $...
Terry Tao's user avatar
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-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 ...
user524928's user avatar
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?
asv's user avatar
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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\...
Notamathematician's user avatar
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 ...
Felixson's user avatar
  • 152
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}\...
Ian Gershon Teixeira's user avatar
-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 ...
Al-Amrani's user avatar
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 ...
asv's user avatar
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-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 ...
Athere's user avatar
  • 89
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} \...
Rachid Ait-Haddou's user avatar
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] \\...
user253963's user avatar
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 ...
Mikhail Borovoi's user avatar
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 ...
M.Monet's user avatar
  • 261
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) ...
Jochen Wengenroth's user avatar
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 ...
Azur's user avatar
  • 101
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 $E$ ...
nariri's user avatar
  • 226
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 ...
kindasorta's user avatar
  • 1,373
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 $R$ ...
Alex's user avatar
  • 313
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 ...
Cranium Clamp's user avatar
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 ...
Nandor's user avatar
  • 289
-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?
asa's user avatar
  • 3
-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 ...
Pikala's user avatar
  • 1
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 ...
Student88's user avatar
  • 503
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^...
user52991's user avatar
  • 159
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 ...
Robeto Leo's user avatar
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^...
PPB's user avatar
  • 75
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 $...
user8469759's user avatar
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) + \...
macat's user avatar
  • 135
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 ...
xingye zhan's user avatar
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 ...
Alexander Braverman's user avatar
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\...
level1807's user avatar
  • 467
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_{...
Alexander Chervov's user avatar
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 ...
Logan 's user avatar
  • 21
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 ...
rimu's user avatar
  • 749
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) $f$ ...
William Kong's user avatar
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 ...
user524868's user avatar
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 ...
António Borges Santos's user avatar
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 ...
luyao's user avatar
  • 1
-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?
asa's user avatar
  • 3
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(...
Zhang Yuhan's user avatar
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,\...
Stabilo's user avatar
  • 1,469
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 ...
Leo Herr's user avatar
  • 1,084
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 ...
groupoid's user avatar
  • 197

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