Highest scored questions
159,036 questions
-2
votes
1
answer
255
views
Schemes over $K_s$ and over $\bar{K}$
Let $K$ be a field. Let $X$ be a scheme over $K$. We denote by $K_s$ and by $\bar{K}$ the separable closure and the algebraic closure of $K$ respectively.
By base change we have the schemes $X_{K_s}$ ...
- 147
-2
votes
1
answer
73
views
On a characterization of some subsets [closed]
Let $X$ be a Banach space. $A$ is a linear closed and densely defined operator and $S$ is a bounded invertible operator. What' s the relation between $\sigma_{e,S}(\lambda S - A)$ and $\sigma_{e,S}(A)...
-2
votes
1
answer
131
views
SHPS and SPHS inequality using monounary algebra
Let $A_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n $ otherwise $f(n) = 1$.
This describes a mono unary algebra.
The proof for $HPS \neq SPHS$ I know uses metabelian groups and was ...
- 155
-2
votes
1
answer
105
views
Monotonic sequence (edited) [closed]
For any two n-dim vector $v$ and $v'$ define $v\leq v'$ iff for each $1\leq i\leq n$, $v_i\leq v_i'$.
Suppose further that the entry of vectors can only take values from $m$ distinct values $\{a_1, ...
- 502
-2
votes
1
answer
185
views
Order of the zero of a meromorphic function under the action of $Gal(\mathbb{C},\mathbb{Q})$
Let's take $X$ a Riemann surface as an algebraic curve in $\mathbb{P}^n$.
The group $Gal(\mathbb{C},\mathbb{Q})$ (automorphisms of $\mathbb{C}$ which act as the identity on $\mathbb{Q}$) acts on $X$ ...
-2
votes
1
answer
202
views
Natural constructions (not depending on parameters) [closed]
Consider graph clusterings as a prototypical example of (logical) constructions.
Let a clustering of a graph $(V,E)$ be any covering of $V$, i.e. a set $C$ with $\bigcup C = V$.
I am looking for a ...
- 9,720
-2
votes
1
answer
601
views
the number of connected components [closed]
I am finding a solution of the following problem. I would like you to give a short proof.
Let A be the adjacency matrix of a d-regular graph G. Prove that d is an eigenvalue of A
with multiplicity at ...
-2
votes
1
answer
442
views
Relation of Hodge Theorem to Eigenfunction Basis of Laplacian
The classical Hodge theorem I know of relates the de Rham cohomology groups isomorphically to the space of harmonic forms and shows that $Id=\pi+\Delta G$, where $\pi$ is the harmonic projection of $k$...
-2
votes
1
answer
366
views
Equivalence relations on powerset of R^2
Let A and B be two subsets of R^2. I define the relation T(A,B) to hold between A and B iff there exists a translation f on R^2 such that the image set of A under f is B. It is easy to prove that T is ...
- 2,023
-2
votes
1
answer
139
views
Equivalence left invertible and injective, right invertible and surjective? [closed]
Does it hold in every (concrete) category?
for the category of set: ok but that's the only proof we usually find on internet.
- 189
-2
votes
1
answer
395
views
non-trivial convergent sequence [duplicate]
I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)
can you give me a example of ...
- 147
-2
votes
1
answer
502
views
Why is any maximal minor of the Bezoutian matrix divisible by the resultant?
I'm referring to Emiris and Mourrain's paper "Matrices in Elimination Theory," Theorem 3.13. Toward the end of the proof, it says that, just because $(f_1,\ldots,f_{n+1})$ is dense in ${\cal Z}({\rm ...
- 551
-2
votes
1
answer
513
views
Sheaf cohomology and double covers
Let $\pi:X\rightarrow Y$ be a double cover of complex varieties and take $L$ holomorphic line bundle on $Y$.
I read that there are the isomorphisms
1) $H^p(X,\mathcal{O}_X)\simeq H^p(Y,\pi_*\mathcal{...
- 175
-2
votes
1
answer
169
views
Maximal torus and application in prime graph [closed]
I am studying papers about " Prime graph" , for example "Prime graph components of finite groups" [ williams], " Groups with complete prime graph connected components" [ Lucido and moghaddanfar]. ...
- 81
-2
votes
2
answers
492
views
On a inequality [closed]
I hope the following kind of inequality holds: let $a_i,b_i\in R$ with $b_i>0$, $\sum _{i=1}^mt_i=1$ with $t_i>0$, then
$$\frac{t_1a_1+\cdot+t_ka_k}{t_1b_1+\cdot+t_kb_k}\le\frac{a_1}{b_1}+\cdot+...
- 19
-2
votes
1
answer
191
views
Composition factor of a group which isomorphic to the alternating group of order 7 [closed]
I want to find groups whose composition factor is isomorphic to the alternating group of order 7, which groups have this condiction?
best regards
-2
votes
1
answer
295
views
When does the adjoint operator map closed convex subsets to closed convex subset?
Let $T:X\rightarrow Y$ be a linear continuous map between Banach spaces $X$ and $Y$ and denote by $T':Y'\rightarrow X'$ the norm adjoint of $T$. Let $M\subseteq U'$ be a subset of
the unit sphere $U'$ ...
- 101
-2
votes
2
answers
7k
views
Uniform continuity and boundedness [closed]
Hi.
I have come across a proof which I understand almost completely, except for one part:
THEOREM: If $f$ is uniformly continuous on a bounded interval $I, [a,b]$ then $f$ is also bounded on $I$.
...
- 99
-2
votes
1
answer
314
views
holomorphic equation
hi,
i am working for some time on a problem and at some point i cant go further. here the critical part: Let $U \subset \mathbb{C}^{n}$ be a open set and consider $c : U \rightarrow \mathbb{R}$ a ...
- 11
-2
votes
2
answers
162
views
Proof of edge-order inequality [closed]
I was just going through a past exam paper for my intro graphs module and the following question came up, and I can't find any notes on it:
Let G = (V,E) be a simple graph. Show that:
$2|E| \leq |V|^...
-2
votes
1
answer
248
views
for examples in probability [closed]
Give an example satisfying the following conditions:
give out a sequence of random variables defined on a probability space, and a sub sigma algebra: the sequence converges almost surely to a limit ...
- 105
-2
votes
1
answer
792
views
How about this book Topological Methods in Group Theory [closed]
Topological Methods in Group Theory witten by Ross Geoghegan
What about this book?
- 105
-2
votes
3
answers
700
views
Good books on fields and Galois theory [closed]
What are some good books on field and Galois theory?
-2
votes
1
answer
438
views
how to prove that the the unitary group SUn(q) is generated by transvections?
how to prove the simplicity of unitary group over finite fields
- 1
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votes
1
answer
476
views
Countable open subgroup
In a Hausdorff topological group, how can I show that every infinite topological group has a countable open subgroup?
- 7
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votes
1
answer
3k
views
Algorithm for determining if a path exists in a graph or if not, the closest edit distance. [closed]
Given a directed acyclic graph G and a path made up from its set of nodes N, what is the closest approximate match to N, ...
- 387
-2
votes
1
answer
679
views
Terminology: Lexicographical order [closed]
I would like to order a group of things by a set of rules of decreasing precedence. Please critique this sentence to help illustrate that:
We define a lexicographical ordering for sheep by looking at ...
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-2
votes
1
answer
271
views
how to weigh the conditions given in a proposition
As we can see,there are some conditions given in a proposition.
If there are 2 propositions having approximate conclusions.Usually，we can name one propositions gives stronger conditions than the ...
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votes
1
answer
236
views
How to construct/characterize "Thermal" sections ?
There were errors in the way I framed the question last time. So doing a major revision this time.
Consider $SU(2)$ as a homogeneous space $SU(2)\times SU(2)/SU(2)$ and sections of this principle ...
- 3,541
-2
votes
1
answer
840
views
The convergence of Eisenstein series of weight zero [closed]
Consider Eisenstein series of weight zero, i.e.
$ E_{\mathfrak{a}}(z,\ s,\ \chi) = \sum_{ \gamma \in \Gamma_{\mathfrak{a}} \backslash \Gamma } \bar{\chi}(\gamma) (Im\sigma_{\mathfrak{a}}^{-1}
\gamma ...
- 361
-2
votes
2
answers
245
views
Evaluate a fair game [closed]
I'm not a mathematician, so my question may be not so clear, sorry.
Let's say we toss up an ideal coin and win 1 dollar on heads and lose 1 dollar on tails. So, expected value is M = 1×0.5 &...
- 1
-2
votes
0
answers
89
views
Can this prime pyramid reveal deeper insights into prime distribution? Has someone seen this pattern before? [closed]
Definition
The Burz Prime-Number Pyramid is a triangular arrangement of consecutive integers, structured such that the length of each row corresponds to a prime number, except the first row which ...
- 1
-2
votes
0
answers
30
views
Convergence of $ \vec{x_{n+1}} = A^{-1} f(\vec{x_{n}},\vec{y_{n}}), \:\:\: \vec{y_{n+1}} = A^{-1} g(\vec{x_{n}},\vec{y_{n}}) $ [closed]
I have two systems
$$ A \vec{x} = f(\vec{x}, \vec{y}), \:\:\: A \vec{y} = g(\vec{x}, \vec{y}) $$
Both have the same constant, square, invertible matrix $A$. I implemented an iterative algorithm with ...
- 97
-2
votes
0
answers
72
views
There is a typo in Stall's textbook on Set Theory: unable to prove the trichotomy of sets (m ∈ n or m = n, or n ∈ m) [migrated]
Here is the textbook, chapter 7, page 300. This lemma seems very of important, and I've spend about 8 hours trying to figure it out, but I'm unable to prove even the weaker version of the lemma (only ...
- 105
-2
votes
0
answers
82
views
Every well-ordered set is isomorphic to an unique ordinal? [closed]
Every well-ordered set $W$ is isomorphic to a unique ordinal
Proof: We start with some well-ordered set $W$. We attempt to construct a bijective map from $W$. Let us consider this class
$$\{(x, \...
- 1
-2
votes
0
answers
24
views
Conditions for a cubic function to be quasiconcave or quasiconvex [closed]
I would like to understand under what conditions a cubic function $f(x)=ax^3+bx^2+cx+d$ can be considered quasiconcave or quasiconvex. Specifically, I am interested in finding conditions on the ...
- 13
-2
votes
0
answers
54
views
Density of squared bessel process
I was trying to find a transition density function for a squared Bessel process. In the book "Continuous martingale and Brownian motion" by Revuz and Yor, I find a Corollary on page 441 that ...
- 11
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votes
0
answers
64
views
A Problem using Limits of Sequences of Functions
Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
- 1
-2
votes
1
answer
151
views
Averaged measure in integrations
Consider
\begin{align}
& F(n,x)\equiv \int_0^x \cdots g (x_5)\int_0^{x_5} ~\int_0^{x_4} g (x_3)~~\int_0^{x_3} ~\int_0^{x_2} g (x_1)~~A(x_1)\,dx_1\cdots dx_n
\end{align}
where $g(x)$ is a measure. ...
- 141
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votes
1
answer
112
views
How to prove that increasing the number of constant symbols of a first-order logic by the number of formulas keeps the number of formulas the same [closed]
Let $S$ be a set of theory symbols for a first-order logic, and let $C$ be a set of constant symbols in $S$ such that $|C| = |L(S)|$, where $L(S)$ is the set of all formulas generated by $S$ in the ...
- 1
-2
votes
1
answer
183
views
Property of positive semi-definite
Let $A$ is a positive semi-definite matrix like this:
$$ A = \begin{bmatrix}
1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\
\alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\
\...
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votes
1
answer
159
views
Can having no more than countably many classes, be inferred from, having every class being countable?
In a theory having proper classes it won't be that easy to phrase how many classes we have in comparison to the number of elements of some class. Here, I'll adopt the following method:
We'd say that: ...
- 11.3k
-2
votes
1
answer
66
views
The number of decompositions of $2n-1$ into a difference of two squares? [closed]
Is there a exact formula for number of decompositions of $2n-1$ into a difference of two squares?
Examples:
...
- 197
-2
votes
1
answer
223
views
Gradient Descent for Markov Dynamics [closed]
The closed-loop dynamics of a linear optimal controller are simple but have interesting properties. From a starting state $\mathbf{v}(0)$ the dynamic can be iterated to reach a final state $\mathbf{v}(...
-2
votes
1
answer
353
views
Can we attain the maximum and minimum of a Rayleigh quotient over any subspace? [closed]
Let $M\in\mathbb{C}^{n\times n}$ be a Hermitian matrix and let $E$ be a subspace of $\mathbb{C}^n$.
$$\mbox{Are } \sup_{x\in E\\
x\neq0}\dfrac{x^*Mx}{x^*x}\mbox{ and }\inf_{x\in E\\
x\neq0}\dfrac{x^*...
- 596
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votes
1
answer
118
views
In which cases $E(e^{t S_n S_m})$ converges to $E(e^{t X Y})$
Suppose that $S_n$ and $S_m$ are two random binomial variables, which are independent and with the same distribution parameter $p$. I am wondering, in which cases $E(e^{t S_n S_m})$ converges to $E(e^...
-2
votes
1
answer
138
views
Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?
I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
-2
votes
1
answer
174
views
What is known about iterated matching as a TSP heuristic
A fairly wellknown heuristic for TSP that is based on matching is described in the 2003 paper Match twice and stitch: a new TSP tour construction heuristic by Andrew B. Kahng and Sherief Reda.
Its ...
- 13.2k
-2
votes
1
answer
105
views
What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]
sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there:
How can $\sqrt{2x^2}$ become $4x^2$ ?
- 99
-2
votes
1
answer
121
views
Brownian motion and Durret book [closed]
I have a problem to understand the following simple definition in Durrett book: Brownian motion and martingales in analysis. What does the following mean: $T = \inf \{t: B_t \in A\}$. It seems to ...
- 719