All Questions
153,422
questions
2
votes
0
answers
104
views
On a real part of a series with complex numbers
Let $P(z)=\sum_{m=0}^na_mz^m $ be a polynomial of degree $n$ having all its zeros in $|z|\leq 1.$ Then what is the best value for 'L' in
$$\Re\left(\sum_{k=1}^nP(zw_k)\frac{w_k}{(w_k-1)^2}\right)\geq ...
- 21
7
votes
1
answer
283
views
A commutative variant of the exterior algebra
Consider $A = \mathbb{R}[t_1,\ldots,t_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t_1,\ldots,t_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = ...
- 193
13
votes
1
answer
457
views
One question about the $\eta$ invariant
This question is from the paper, The Analysis of Elliptic Families
II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8.
Suppose ...
- 1,905
3
votes
4
answers
350
views
References request: representations of classical groups
Are there some good references about representations of classical groups? What are the fundamental representations of classical groups of type $B, D$?
I would like to know explicit formulas of the ...
- 6,121
5
votes
2
answers
231
views
Can we stay invertible while approximating linear maps in Sobolev spaces?
Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.
Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
- 6,621
2
votes
4
answers
833
views
Complex differential equations
I'm looking for a gentle an concise introduction to complex-variable differential equations. Eventually, I need to look at complex PDEs, but I assume one starts with complex ODEs.
Mostly, I'm just ...
- 3,554
5
votes
0
answers
618
views
Cadlag and adapted (usual conditions assumed) imply progressively measurable (related to Protter's Stochastic Calculus theorem 6)
Hi maybe someone on here can help me. I have been stuck on showing this fact for several months. I asked this question in the stack exchange and it has floated around for a while but to no avail.
...
- 151
0
votes
1
answer
110
views
Is the numerable product of finite abelian groups a cantor set? [closed]
Today I have heard that statement but I can't find the reference.
Can somebody know a reference and/or a proof?
- 319
5
votes
2
answers
418
views
Improving equi-integrability for a family $\mathcal F$ in $L^1(\Omega)$
Let $\mathcal F$ be a weakly compact subset of $L^1(\Omega)$. Dunford–Pettis theorem says that $\mathcal F$ is uniformly integrable. Also, by de la Vallée-Poussin theorem we can find an increasing ...
- 1,245
3
votes
1
answer
315
views
Wavefront set and Duhamel's principle
Consider the Cauchy problem:
$$
\frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0,
$$
where $A$ has real principal ...
- 43
2
votes
0
answers
126
views
Expressing modular functions of level 9 and 32 as rational functions
Let
$$\gamma_2(\tau)=j(\tau)^{1/3},\qquad \mathfrak f_1(\tau)=\frac{\eta(\tau/2)}{\eta(\tau)},$$
where $j$ is the modular invariant and $\eta$ is the Dedekind eta function. Cox in his Primes of the ...
- 2,335
3
votes
0
answers
63
views
Analytic continuation of a Dirichlet series with several complex variables
For $w_1,w_2,z_1,z_2\in\mathbb{C}$ with $\operatorname{Re}(w_1)>0$ and $\operatorname{Re}(w_2)>0$, define
\begin{equation*}
U(w_1,w_2;z_1,z_2):=\prod_{p}\left(1-\frac{e^{z_1}}{p^{1+w_1}}-\frac{e^...
- 131
10
votes
1
answer
204
views
homeomorphisms induced by composant rotations in the solenoid
Let $S$ be the dyadic solenoid.
Let $x\in S$, and let $X$ be the union of all arcs (homeomorphic copies of $[0,1]$) in $S$ containing $x$.
$X$ is called a composant of $S$.
It is well-known ...
- 955
1
vote
1
answer
270
views
Subschemes of projective varieties
I'm studying the article Introduction to Lawson Homology of Peters and Kosarew. In their exposition of Lawson homology they call projective variety any zero-locus $X$ of homogeneous polynomials in ...
- 1,252
1
vote
1
answer
189
views
Giving Uniform Bound on Differences of Sums of Converging Polynomials
The title does not quite capture the essence of the difficulty, please allow me to be more explicit here.
I thought of this question when I was trying out an open problem by Ovidiu Furdui(See problem ...
- 89
7
votes
1
answer
282
views
Finite group with a character having one nonzero absolute value
Let $G$ be a finite group. Assume that $\chi$ is a complex irreducible character of $G$ of degree $n\geq 2$, with the property that for each element $g\in G$ either $\chi(g)=0$ or $|\chi(g)|=n$.
...
- 167
4
votes
0
answers
248
views
Symmetric power contained in tensor power?
Let $V$ be an $R$-module. Traditionally the symmetric algebra $S(V)$ is defined as a quotient of the tensor algebra $T(V)$, by the ideal generated by all $a\otimes b-b\otimes a$.
Can $S^n(V)$ also be ...
- 2,489
1
vote
1
answer
138
views
Cryptography with general RSA type integers?
Denote $\mathcal N_r=\{n\in\mathbb Z:\exists\mbox{ distinct equal bit primes }p_1,\dots,p_r:n=p_1p_2\dots p_{r-1}p_r\}$.
$\mathcal N_1$ refers to primes and $\mathcal N_2$ referes to balanced ...
- 13.7k
2
votes
0
answers
46
views
Must bounded, closed, smooth curves with long straights have sharp bends?
Consider the family of bounded, closed, and continous curves $\Gamma$, i.e. for all $\gamma \in \Gamma$, we have $\gamma : [0, 1) \to [0, 1]^2$. Within this family, I am interested in curves that ...
9
votes
0
answers
279
views
Frey-Mazur for abelian varieties
Let $K$ be a number field. The Frey-Mazur conjecture asserts the existence of a constant $N_K$ such that for all primes $p>N_K$, and all pairs of elliptic curves $E_1$, $E_2/K$, if $\overline{\rho}...
- 3,132
4
votes
1
answer
249
views
Interesting (non) examples of singular support
I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $...
- 41
5
votes
1
answer
548
views
The Ungraded Milnor-Moore Theorem
Let $k$ be a field of characteristic $0$.
There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\...
user30211
10
votes
1
answer
3k
views
Books with exercises to learn Langlands program, Galois representations, modular forms
I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do ...
- 775
9
votes
1
answer
347
views
Special Cases of Duistermaat-Heckman Formula
The Duistermaat Heckman localization formula states how integrals over symplectic spaces with Hamiltonian $U(1)$ group actions.
$$ \int_M \frac{\omega^n}{n!} e^{-\mu} = \sum_{x_i \text{ fixed}} \frac{...
- 22.6k
4
votes
1
answer
651
views
Shafarevich's theorem about solvable groups as Galois groups
I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.
- 101
3
votes
0
answers
66
views
How to solve this linear Cauchy Problem
within my thesis, I am struggeling with the following PDE:
$u_t+a(x,y)u_{xx}+b(x,y)u_{xy}+c(x,y)u_{yy}+d(x,y)u_{x}+e(x,y)u_{y}+f(x,y)u=0$
$u(T,x,y)=1,$
where $a,b,c,d,e,f$ are polynomials and the ...
- 31
18
votes
1
answer
1k
views
A linear algebra problem in positive characteristic
Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...
- 4,454
18
votes
1
answer
356
views
Proof as a Σ₁ approximation to truth: what about higher degrees?
Let us posit that the goal of mathematics is to study mathematical truths, and let us stick to arithmetic for simplicity: so let $\mathscr{T} \subseteq \mathbb{N}$ be the set of (Gödel codes of) true ...
- 30.2k
1
vote
0
answers
43
views
Random solute transport equation
After reading the article Probability density functions for solute transport in random fields, by M. Shvidler, K. Karasaki, see https://pubarchive.lbl.gov/islandora/object/ir%3A116072/datastream/PDF/...
- 141
1
vote
1
answer
93
views
Solve nonlinear equation
Suppose that $f:E\to F$(between Banach spaces), is of the form
$$f(x)=f(0)+D(x)+N(x).$$
Here $D$ is a linear term, whose kernel is of finite dimension, and admits a right inverse $G$, i.e. $D(G)(\...
- 1,905
6
votes
1
answer
173
views
Smallest set of nonzero vectors in $\mathbb F_2^n$ which intersects every 2-dimensional subspace
What is the smallest set of nonzero vectors in $\mathbb F_2^n$ which intersects every 2-dimensional subspace?
For example, for n = 3, the set {001, 010, 011} does the job, and is minimal. For n = 4, {...
- 61
1
vote
0
answers
121
views
Is there only one meaningful definition of product of games?
Work in the context of combinatorial games as introduced by Conway.
For surreals, the definition of the product is forced by the requirement that surreals should form an ordered field.
Say, if $s' <...
- 183
2
votes
2
answers
950
views
Calculate percentage of symmetry of a given matrix
Is it possible to calculate a percentage that quantifies how symmetric a given matrix is?
For example, even if a given matrix is not symmetric, instead of just classifying it as "not symmetric", one ...
- 21
4
votes
0
answers
162
views
Constant in trace theorem for balls
Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$
The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
user127450
3
votes
1
answer
157
views
Identity for $Ext^1$ for special algebras
Let $A$ be a finite dimensional algebra and assume all modules are also finite dimensional. A module $M$ is said to have dominant dimension at least $n$ in case the term $I_i$ for $i=0,1,...,n-1$ are ...
- 26.1k
2
votes
2
answers
296
views
Random walk and isoperimetric constant
I assume that a result of the following kind is known, and I would really appreciate a reference for it... Or at least, some hints as to where to start looking.
Theorem(?): Let $\varepsilon>0$ ...
- 11.2k
3
votes
1
answer
289
views
Basic questions on spectra
I have a basic question on Voevodsky's stable homotopy category of spectra $\mathbf{SH}(S)$, where $S$ is a finite dimensional noetherian scheme.
Let $E$ be an $\Omega$-spectrum and $\varphi \colon ...
- 2,741
8
votes
1
answer
136
views
A relative Kuiper theorem
Let $(H_0, \langle \,,\,\rangle_0)$ be a real separable Hilbert space,
and let $(H_1, \langle \,,\,\rangle_1)$ be a Hilbert space such that $H_1 \subset H_0$ is dense and such
that the inclusion $(...
1
vote
0
answers
74
views
Nonlinear maps in Riesz Thorin theorem
The Riesz Thorin theorem allows us to interpolate between $L^p$ spaces and the usual assumption is that the map $T$ is linear.
What I was wondering about is whether this is because otherwise you do ...
user127450
1
vote
0
answers
301
views
Derivative of complex matrix pseudo inverse with respect to real and imaginary components
I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$.
I am interested in evaluating the ...
1
vote
0
answers
74
views
Proof of Lemma 7.1 Bonsall and Duncan
In the proof of Lemma 1 in section 7 (A functional calculus for single Banach algebra element) of the book Complete normed algebras by Bonsall and Duncan, the last line says
$$\phi\left(\frac{1}{2\pi ...
- 41
0
votes
1
answer
133
views
A weaker version of Dirac's theorem
This is related to Dirac's theorem.
For any finite, simple, undirected graph $G=(V,E)$ let $\delta(G)$ denote the minimal degree of all vertices.
Are there positive integers $n,c\in\mathbb{N}$ with ...
- 45.5k
11
votes
1
answer
942
views
Proof of Green's formula for rectifiable Jordan curves
$\newcommand{\Ga}{\Gamma}$
I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
- 118k
9
votes
2
answers
856
views
What did the Intuitionists want to do with applied mathematics?
Oversimplification: Newton & Leibnitz &c build the calculus and other methods that solve a vast number of practical problems. Weierstrass, Dedekind, Cantor &c build a foundation under it ...
- 309
3
votes
1
answer
145
views
variant of Motzkin-Rabin Theorem
The Motzkin-Rabin Theorem says:
If $A$ and $B$ are finite disjoint sets of points in the plane and $A \cup B$ is noncollinear, then there exists a line that contains at least two points from one of ...
- 1,125
9
votes
0
answers
218
views
On a paper of Formanek about $PGL_4$
In his paper "The center of the ring of 4 × 4 generic matrices" (Journal of Algebra, 1980) Formanek shows that, if $V$ is the representation of $PGL_4$ given by simultaneous conjugation of pairs of ...
- 91
10
votes
2
answers
757
views
Is the boundary of a manifold topologically unique? [duplicate]
Let $X$ be a manifold without boundary and let $Y$ and $Z$ be two manifolds with boundary such that $X$ is homeomorphic to their interiors: $X \cong Y^\circ \cong Z^\circ$. Does it follow that $Y \...
- 5,933
2
votes
0
answers
156
views
theories where angles exist without a metric
The underlying basic question, which I'm sure I'm not the first to ask, is what are the possible exotic/nonintuitive models of Euclid's axioms/postulates, outside the one where "lines" are interpreted ...
- 2,031
1
vote
0
answers
140
views
When does the Jacobian of a smooth curve contains an unique principal polarization
Let $X$ be a smooth, projective curve of genus at least $4$ and $X$ non-hyperelliptic. I am looking for additional conditions on $X$ such that the Jacobian $J(X)$ of $X$ contains an unique principal ...
- 2,022
-2
votes
1
answer
575
views
Can this criterion to indicate the randomness some numbers? [closed]
John Derbyshire in his book PRIME OBSESSION says on page 366:
CHAPTER 3
10.
"Here is an example of e turning up unexpectedly. Select a random number
between 0 and 1. Now select another and add ...
- 1,305