4
votes
2answers
39 views
An operation on binary strings
Consider the “product” $\gamma = \alpha \times \beta$ of two binary strings $\alpha$, $\beta$ $\in \lbrace 0,1\rbrace^+$ which one gets by replacing every 1 in $\beta$ …
3
votes
0answers
175 views
On Perelman’s paper
In section 5 in "The entropy formula for the Ricci flow and its geometric applications" Grisha Perelman has written:
Fix a closed manifold $M$ with a probability measure $m$, and …
0
votes
0answers
39 views
can we say that $(p^2+1)/2\ne p_0^2$ where $p$ is a Mersenne prime
Let $p=2^a-1>7$ be a Mersenne prime and so $a$ is an odd prime.
Can we say that $(p^2+1)/2$ is not equal to the square of a prime number?
Many thanks for your help
BHZ
0
votes
0answers
15 views
Hyperbolic pair of pants.
Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential …
0
votes
0answers
74 views
Help me on proof of an equation.
I wanna prove following equation
$ \sum_{i=1}^n \prod_{k=1,k\neq i}^n \prod_{j=1,j\neq k}^{n+1}(x_j - x_k) = -\prod_{i=1}^n \prod_{j=1,j\neq i}^n (x_j - x_i) $
I have verified sev …
2
votes
1answer
31 views
$f^{-1}\mathcal I \cdot \mathcal O_X$ vs $f^\ast \mathcal I$
Let $X$ ad $Y$ be (noetherian) schemes and let $\mathcal I \subseteq \mathcal O_Y$ be a sheaf of ideals on $Y$. Let $f \colon X \to Y$ be a morphism of schemes. In general the shea …
29
votes
11answers
1k views
Why don’t more mathematicians improve Wikipedia articles?
Wikipedia is a widely used resource for mathematics. For example, there are hundreds of mathematics articles that average over 1000 page views per day. Here is a list of the 500 mo …
0
votes
0answers
11 views
Natural Isomorphism of $S(V[1])$ and $(\bigwedge V)[n]$
Let $V:=\oplus_{j\in\mathbb{Z}}V_j$ be a graded $\mathbb{F}$-vector space over
the field $\mathbb{F}$. The graded tensor product of graded vector spaces is given
by
$V \otimes W: …
2
votes
3answers
133 views
Group action on the real line
Hi,
I was wondering about the following question:
if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct …
15
votes
2answers
181 views
Order type of the smallest set containing the identity function and closed under exponentiation
Let $E$ be the smallest set of functions $\mathbb{N}^+\to\mathbb{N}^+$ containing the identity function $n \mapsto n$ and closed under exponentiation $(f,g) \mapsto \left(n \mapst …
3
votes
3answers
70 views
Subquotients in ZF
In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surje …
2
votes
1answer
109 views
Growth of Thompson’s group $F$
EDIT: Mark Sapir pointed a reference (in the comments) giving a lower bound of $2^{1/4}$ for the minimal rate. Is this the state of art? The third question remains unanswered. EN …
4
votes
1answer
99 views
Effective Chebotarev without Artin’s conjecture
Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form
of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Arti …
0
votes
0answers
47 views
Phase transition in dynamical systems
There are several occasions in the study of dynamical systems that are called phase transitions. For example
the parameters $t$'s where the pressure $P(f,t\phi)$ fail to be $C^k$ …
11
votes
2answers
232 views
How closed-form conjectures are made?
Recently I posted a conjecture at Math.SE:
$$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$
where $J_\mu( …
5
votes
0answers
87 views
Do operations generate well-ordered sets only?
I've read @TauMu's question about the set of functions $\mathbb N\rightarrow\mathbb N$ generated from the identity map by repeatedly applying exponentia …
1
vote
0answers
41 views
Almost orthogonal vectors in subsets of euclidean space
Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of almost …
6
votes
2answers
91 views
Integer lattice points on a hypersphere
Is the following statement true?
For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ i …
-1
votes
0answers
38 views
Find Total number of ways out of N Number taking K numbers every M interval [closed]
Hello Topcoders, I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Pro …
6
votes
2answers
101 views
How many triangulations of the genus $g$ surface on $n$ vertices?
By "a triangulation of $X$", I mean a simplicial complex whose geometric realization is homeomorphic to $X$. Tutte showed that the number of combinatorially distinct triangulations …
5
votes
1answer
108 views
Sheaves on Contractible Analytic Spaces
Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is cont …
1
vote
2answers
68 views
Eigenvalues of Symmetric Tridiagonal Matrices
Suppose I have the symmetric tridiagonal matrix:
$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\
b_{1} & a & b_{2} & & ... \\
0 & b_{2} & a …
0
votes
0answers
52 views
Equivariant $K$-theory, singular vectors, and flag manifolds
For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_ …
0
votes
1answer
138 views
$P^1$ minus k points
For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write
$$ P^1 \setminus S \cong \mathbb{H}/G $$
where $\mathbb{H}$ is the upper-half plane …
3
votes
0answers
59 views
Homotopy left-exactness of a left derived functor
Let
$$
F: \mathcal{C} \leftrightarrows \mathcal{D} :G
$$
be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors
$$
\mathb …
0
votes
1answer
63 views
What is the Bahadur-Anderson Algorithm?
What is the Bahadur-Anderson Algorithm, and which book could one read to learn it?
0
votes
0answers
20 views
decidability of matrix generating group
For a given set $S$ of complex square matrices $M1,M2\cdots,Mn$, one can obtain a matrix group $G$ generated by matrx multiplication. For any $i$, we can define a matrix space $Gi$ …
1
vote
1answer
89 views
General Orthogonal Group and its properties
I know that exist a Lie Group Called the Orthogonal Group $O(n)$.
That correspond to all matrix of $n \times n$ in the real numbers such that the columns are a orthogonal basis for …
0
votes
0answers
97 views
Avoiding reflexive paradox in set theory
I am an amateur mathematician, and certainly not a set theorist, but there seems to me to be an easy way around the reflexive paradox: Add to set theory the primitive $A(x,y)$, whi …
3
votes
1answer
113 views
Fields whose embeddings into the complex numbers are invariant under complex conjugation
Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an inv …
2
votes
2answers
188 views
Did Oresme know the zeroth power?
Working on a contribution for a festschrift I touched the introduction of powers. Unfortunately I have no access to the original works of Oresme who was among the first, if not the …
15
votes
3answers
366 views
Is deciding whether a Turing machine *provably* runs forever equivalent to the halting problem?
Assume for this question that ZF set theory is sound.
Now consider the language "PROVELOOP," which consists of all descriptions of Turing machines M, for which there exists a ZF p …
5
votes
2answers
175 views
Langlands product
In his 'Märchen' Langlands considers for a local field $F$ a certain abelian category $\Pi(F)$ whose objects are given by isomorphisms classes of irreducible admissible representat …
4
votes
1answer
96 views
Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$
Hi.
Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I …
3
votes
0answers
18 views
unitary structures on fusion categories
A unitary fusion category is a fusion category with a $C^*$-tensor structure.
Hence, in principle, a fusion categoriy could have more than one unitary structure. Does exist a fusio …
-1
votes
0answers
96 views
Why is it that Wikipedia has no coverage of Quantum stochastic calculus [closed]
Why is there no coverage in wikipedia on Quantum stochastic calculus.
The biographies of mathematicians like KR Parthasarathy, Robin Lyth Hudson, VP Belavkin, and others. Is it no …
-1
votes
0answers
65 views
Solution of Equation [closed]
Can anyone show me, how to solve these system of Equations:
x+y+z = 2
(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y) = 1
X^2(y+z)+Y^2(z+x)+Z^2(x+y) = -6
3
votes
0answers
116 views
Sperner’s lemma and Tucker’s lemma
In their article "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" (American Mathematical Monthly, April 2013), Nyman and Su write "[W]e are unaware of a direct proo …
0
votes
1answer
59 views
common roots of bivariate polynomial equations
Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but differ …
5
votes
1answer
142 views
Constructing Polynomial Count Varieties
I have some naive questions about polynomial-count affine varieties over $\mathbb{C}$:
Are all reductive algebraic groups strongly polynomial-count?
Are products of strongly poly …
0
votes
1answer
168 views
8 queens puzzle
In the 8 queen puzzle, if we use the incremental approach, i.e. put the queen one by one on the board, the number of possible sequences would be 2057. How is that number calculated …
2
votes
2answers
198 views
Motivation for Frankl’s conjecture?
Frankl's conjecture, open since 1979, says that if $F$ is a union-closed family of subsets of $X$, then there is some $x \in X$ such that $x$ appears in at least half the sets in $ …
0
votes
1answer
77 views
What is the corresponding version in the complex space of this proposition got in the real space real
How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x_1,...,x_n)\in C^n …
3
votes
1answer
44 views
Matrix norms / eigenvalues / singular values / another thing
OK, here is what is probably a stupid question.
Let $M$ be a non-symmetric real matrix: for example, the shear matrix
$\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} …
0
votes
1answer
65 views
A group G acting Properly Discontinuously and Cocompactly on a Proper geodesic space X [closed]
Let G be a group acting properly discontinuously and cocompactly on a proper geodesic space X. How can I show that:
(a) G is finitely generated;
(b) G is quasi-isometry to X?
3
votes
1answer
91 views
What is the name of this measure of matrix “degenerateness”
Given a spanning set, consider the minimum number of vectors that you must remove in order to make it no longer span. What is this number called?
If the vectors are columns in a …
1
vote
2answers
87 views
simple explaination of simplicial volume=4g-4 when genus $\ge 1$
In Gromov's famous book ,it says "simplical volume of every oriented surface of genus $ \ge 1$ satisfies${\left\| {\left[ S \right]} \right\|_\Delta } = 4g - 4 = - 2\chi \left( S …
0
votes
0answers
66 views
Chern Character of a Symmetric Power
Simple question: I was trying to find a formula for the Chern character $ch(S^mE)$ in terms of $ch(E)$ but couldn't find a reference too easily. It can be worked out using symmetri …
1
vote
0answers
39 views
Why are the left exact functors from an abelian category to abelian groups cocomplete and have a injective generator?
Let $\mathcal{C}$ be an abelian category, $\mathcal{Ab}$ the category of abelian groups and $Lex(\mathcal{A}, \mathcal{B})$ the category of left exact functors between abelian cate …
17
votes
14answers
846 views
objects which can’t be defined without making choices but which end up independent of the choice
It happens a lot of times that when one defines a new object (ring, module, space, group, algebra, morphism, whatever) out of given data one first chooses some additional structure …
