1
vote
0answers
45 views

Physical meaning of the Lebesgue measure

Question (informal) Is there an empirically verifiable scientific experiment that can empirically confirm that the Lebesgue measure has physical meaning beyond what can be obtained using just the ...
0
votes
0answers
44 views

Functional equation: Can we find a function that satisfies this equation?

$f^{\alpha }\left( \overrightarrow {0}\right) +f^{\beta }\left( \overrightarrow {0}\right) =f^{\alpha \beta +\alpha +\beta }\left( \overrightarrow {0}\right)$ Is there a function $f$ from ...
0
votes
0answers
11 views

Proof of partial derivative of a distribution

I wanted to proof the formula for the partial derivative of a distribution, but I have an error on my result and I don't understand where I am wrong. Here is my proof : I assume that my function f ...
0
votes
0answers
7 views

Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps. In the decomposition $$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f ...
6
votes
1answer
50 views

Union-closed family generated by n 2-sets

I asked this question on Stackexchange, but I got no answer, so I ask it here. Let us define a $2$-set as a set with exactly $2$ elements. For a natural number $n$, let $l(n)$ denote the least ...
-1
votes
0answers
37 views

Connected Lie subgroups of SU(2) and SU(3) [on hold]

Two questions: Why is every connected subgroup of SU(2) closed? Can we find a non closed connected subgroup of SU(3)?
0
votes
0answers
10 views

One problem about tower stability

Some years ago i asked myself a question that i still can not answer. Here it is. Let be given a tower consisting of finite homogeneous equal to each other cubic blocks staying one on another. What is ...
1
vote
0answers
28 views

A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...
3
votes
0answers
63 views

What are the sense and reference of the propositions $R$$\notin$$R$,$R$$\in$$R$, where $R$={x|x$\notin$x} in Frege's Grundgesetze?

In their paper, "Frege's New Science" (Notre Dame Journal of Formal Logic, Vol. 41,No. 3, 2000), Antonelli and May give the following quote of Frege, from his paper "Uber die Grundlagen der ...
0
votes
0answers
20 views

A question about the definition of complete dg Lie algebras in a paper of Lazarev and Markl

In their paper Disconnected Rational Homotopy Theory, Lazarev and Markl give the following definition (page 23): Definition: A complete differential graded Lie algebra is an inverse limit of ...
2
votes
0answers
32 views

Effective divisor in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-classes

Does anybody knows an effective class in $\overline{\cal{M}}_{g,n}$ with negative $\psi$-coefficients? The standard references; Logan, Farkas or Brill-Noether divisors have all non-negative ...
2
votes
0answers
36 views

Socle of tilting modules in the BGG category $\mathcal{O}$ over a semisimple Lie algebra

Suppose that $\mathfrak{g}$ is a finite dimensional, complex, semisimple Lie algebra. Let $\mathcal{O}$ be the BGG category over $\mathfrak{g}$. Tilting module theory play an important role in the ...
0
votes
0answers
18 views

Experimental Investigations on the Statistics of Infinite, Discrete, Evenly Distributed Pointsets in the Euclidean Plane

I am trying to estimate the distribution of certain planar polygons in the Euclidean plane; to accomplish that, I generate finite set of points, that are evenly distributed in w.l.o.g. the ...
0
votes
0answers
45 views

Invariance of Gauss-Bonnet theorem with respect to connection?

I am stuck with a basic understanding of the generalized (and even the ordinary version of) Gauss-Bonnet theorem. For a compact 2-dimensional Riemannian manifold $M$ with boundary $\partial M$, let ...
-2
votes
1answer
33 views

Hamiltonian path in countable connected graph such that $\text{deg}(v)=\omega$ for all $v$

Is there a countable connected graph $G=(\omega, E)$ such that $\text{deg}(v)=\omega$ for all $v\in\omega$, but there is no Hamiltonian path in $G$?
1
vote
0answers
36 views

Game Theory Cake Cutting

I'm familiar with Game Theory concepts (I took one course at College, but it was rather superficial), but my mathematical skills aren't at the best level though. However, I'd like to hear from more ...
1
vote
0answers
33 views

What is an upper bound for $|E(X|\mathcal{A})-E(X)|$?

Let $X$ be a random variable with $|X|\le1$, and $\mathcal{A}$ be a $\sigma$-algebra. What is an upper bound for $|E(X|\mathcal{A})-E(X)|$? Existing results: It has been known that ...
1
vote
0answers
18 views

On the numerical range of non-self adjoint Gaussian matrix

For a complex $n \times n$ matrix $A$, its numerical range is the set $$W(A) = \left\{\mathbf{x}^*A\mathbf{x} \mid \mathbf{x}\in\mathbb{C}^n,\ \|x\|_2=1\right\} .$$ We can further define the ...
5
votes
0answers
115 views

Semi-continuity of intersection numbers

I always trusted the following quite vague statement: If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say ...
2
votes
1answer
202 views

Field of definition of an algebraic set

I find this definition in Silverman's book, The Arithmetic of Elliptic Curves: an algebraic set(in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in ...
0
votes
0answers
33 views

Simulated Annealing algorithm for MINLP [on hold]

In the objective function of a mathematical programming model,we have an expression like this: $$ \biggl(\biggl|X\biggl| \biggl) . Q $$ in which both X and Q are continuous variables, and $||$ ...
3
votes
0answers
40 views

Solving algebraic recurrence relations on a cyclic graph

I have a set of $n$ variables $p_1, \ldots p_n$ with $0 \leq p_i \leq 1$ and a defining equation for each of one of the forms: $p_i = 0$. $p_i = 1$ $p_i = p_j p_k$ for some $j, k$ with $i, j, k$ all ...
0
votes
1answer
54 views

Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...
2
votes
1answer
105 views

sum over all integer partitions, of the product of the factorials of the terms

I'm looking for something making tractable the sum, over all partitions into k terms of an integer n, of the product of the factorials of all the terms. Thanks,
24
votes
1answer
686 views

Complex manifold with subvarieties but no submanifolds

I previously asked this question on MSE and offered a bounty but received no responses. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. ...
4
votes
0answers
125 views
+100

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite $k$-algebras ($k$ is a (preferably, but not necessarily finite) field) with one ...
20
votes
1answer
503 views

Does $E_8$ know $Spin(7)$?

One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For ...
5
votes
1answer
120 views

Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where \begin{equation} Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}. \end{equation} To ...
3
votes
0answers
134 views

For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]: If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free ...
1
vote
2answers
862 views

Encoding vectors of size $n$ in matrices which less than $2n$ rows

I have a set of vectors and each has $n$ nonnegative entries. Moreover, each entry of a vector has a quality: (1) or (2). It makes $2^n$ different possible patterns. For example, let's take two ...
4
votes
1answer
138 views

Extension of functions from geodesically convex compact sets in a Riemannian manifold

In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
5
votes
1answer
170 views

Spin structures on schemes

This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and ...
8
votes
0answers
133 views

Two transfers for ramified or branched covers

Let $\pi: X \rightarrow Y$ be a 2-fold branched cover of complex varieties. I know of (at least) two types of pushforwards associated to this situation: If I'm not mistaken, there is a pushforward ...
0
votes
1answer
162 views

Normed space between $H^{0+}$ and $L^2$

In the space $\in L^2(\mathbb{R}^3)$, consider the following condition. $$\int_{\mathbb{R}^3}\frac{|\widehat{f}(x)|}{1+|x|^{3/2}}dx<+\infty\qquad\mbox{(*)}$$ Of course if $f\in H^s(\mathbb{R}^3)$ ...
48
votes
17answers
11k views

Parodies of abstruse mathematical writing

Perhaps under the influence of a recent question on perverse sheaves, in conjunction with the impending $\pi$-day (3/14/15 at 9:26:53), I recalled a long-ago parody of abstruse mathematical language ...
0
votes
0answers
41 views

On joint equidistribution of residues

Given $N\in\Bbb N$ we vary $\alpha$ over integers in interval $(N,2N)$ and $\beta,\gamma$ over integers in $(N^{1/c},2N^{1/c})$ where $c>1$ and fix $\delta$ an integer randomly picked from ...
0
votes
0answers
5 views

Probability of disjoint cycles

Let $c_1,c_2\in S_n$ be two disjoint cycles of length $|c_1|$ and $|c_2|$ respectively. Let $I(c_i)$ be the coordinates on which permutation $c_i$ acts at $i\in\{1,2\}$. Note by choice we have ...
3
votes
2answers
496 views

When a smooth algebra is regular?

Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, ...
31
votes
10answers
5k views

de Rham cohomology and flat vector bundles

I was wondering whether there is some notion of "vector bundle de Rham cohomology". To be more precise: the k-th de Rham cohomology group of a manifold $H_{dR}^{k}(M)$ is defined as the set of closed ...
4
votes
0answers
95 views

Integer sum of distinct reciprocals with no integer subset sum

Question $\def\nn{\mathbb{N}}$ For any $n \in \nn^+$, is there a finite set $S \subset \nn^+$ such that $\sum_{k \in S} \frac{1}{k} = n$ but $\sum_{k \in T} \frac{1}{k} \notin \nn^+$ for any $T ...
19
votes
7answers
3k views

Asymptotic density of k-almost primes

Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is $$\pi_k(x)\sim\frac{x(\log\log ...
4
votes
1answer
235 views

function space in comma category

Let TOP be a category of topological spaces and B be an object of TOP. Is there a notion of function space in the comma category TOP/B.
4
votes
2answers
540 views

Is the functor of open subschemes representable?

First some simple observations in order to motivate the question: The functor $Set^{op} \to Set, X \to \{\text{subsets of }X\}, f \to (U \to f^{-1}(U))$ is representable. The representing object is ...
0
votes
0answers
7 views

Passing motivic decompositions from rational to algebraic equivalence

It is well known that there are several adequate equivalence relations for algebraic cycle (see https://en.wikipedia.org/wiki/Adequate_equivalence_relation for a list including definitions). The ...
0
votes
0answers
36 views

How to solve these simultaneous equations? [on hold]

Find the all possible pairs of $x,y,z$ so that 1) $y^3 = x^3 + 9x^2 - 9x +8$ 2) $y^2 = z^3 + 17$ Note that, $x,y,z$ are all positive integers. I've tried many ways to solve this problem but ...
-4
votes
0answers
18 views

significance level from 0.10 to 0.05, increase or decrease? [on hold]

A trend test show the trend is downward (alpha<0.10). Then the trend is still downward, but alpha<0.05. Then I describe as follows: 1. The statistical significance is increasing. 2. The ...
3
votes
1answer
73 views

How to write the map $ℂ[G/U]↪ℂ[B]$ explicitly?

Let $G$ be a reductive algebraic group and $B$ a Borel subgroup of $G$. Let $T$ be a maximal torus of $G$ contained in $B$. The $B=UT=TU$ for some unipotent subgroup $U$ of $G$. We have Bruhat ...
4
votes
0answers
32 views

Points of failure in definition of X- and A-moduli spaces for arbitrary G

In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. ...
3
votes
0answers
165 views

Cohomology theory “from” Grothendieck's six operations?

How, precisely, (as suggested by grothendieck) cohomology theory naturally follows from the Grothendieck's six operations associated to the category derived from the topos? I would like some ...
0
votes
0answers
13 views

Densely defined symmetric operators having no self-adjoint extensions

Does there exist a complex Hilbert space H and a densely defined symmetric operator A on H such that Range(A*+i) is not equal to H?

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