This tag is used if a reference is needed in a paper or textbook on a specific result.

**0**

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5 views

### Is wave operator bounded on $L^{2}(\mathbb R^{n})$?

Consider the linear Shr\"odinger equation,
$$ (LS)
\begin{cases}
\partial_{t}u= i\triangle u, t\in \mathbb R,\\
u(x,0)=u_{0}(x),
\end{cases}
$$
$x\in \mathbb R^{d}.$
In terms of Fourier transform, we ...

**0**

votes

**0**answers

36 views

### Kan extensions in the $2$-category of monoidal categories

Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of ...

**1**

vote

**1**answer

81 views

### Paper of Denis Simon on quadratic equations in dimensions 4, 5?

In several places I have come across references to a 2005-6 preprint of Denis Simon entitled
Quadratic equations in dimensions 4, 5, and more
This paper gives fast algorithms to find isotropic ...

**6**

votes

**0**answers

46 views

### Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” ...

**0**

votes

**1**answer

46 views

### Combinatorial polynomials from general diagram fillings?

There is a plethora of polynomials defined on partition shaped Young diagrams, (Schur, Jack, Grothendieck,...), and skew Young diagrams.
There are also composition shaped diagrams that are responsible ...

**14**

votes

**0**answers

162 views

### Whatever happened to $L(j)$?

So this question probably shows my inner model theoretic ignorance, but:
In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...

**1**

vote

**0**answers

18 views

### Second derivative estimates for gradient dipendent elliptic equations

Caffarelli in the article " Interior a priori estimates for solutions of Fully non linear Equations" Ann. Math 130,No.1, 1989 proved that a continuous viscosity solution $ u $ of a uniformly elliptic ...

**1**

vote

**0**answers

103 views

### Reference request: Flipping the factors in the Künneth formula

I would like to know if there is a reference for the fact that the following diagram commutes:
$$
\begin{array}{ccccccccc}
0 & \to & H_*(X) \otimes H_*(Y) & \to & H_*(X\times Y) & ...

**1**

vote

**0**answers

49 views

### When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?

I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...

**1**

vote

**1**answer

136 views

### Non-Pythagorean proof for the square root of 2 and solution to YBC7289 [on hold]

My name is J. Frederic Teubner I am an independent researcher. I wish to publish a proof for the non-Pythagorean solution to the Babylonian tablet YBC7289 and am currently inquiring as to whether or ...

**0**

votes

**0**answers

103 views

### The following is a necessary condition for a number to be prime, from its digit expansion. Is it already known? [on hold]

Concerning a numbers’ digits we know some neccessary conditions on them for the number to be prime, besides the last digit having to be odd (except for prime 2). For example in decimal representation ...

**13**

votes

**3**answers

539 views

### Inverse problem of Chern Classes

For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work ...

**2**

votes

**1**answer

150 views

### Open problems books [on hold]

As the title might indicate , I would like to look for recommendations for mathematical book that present open problems in depth with commentary.
The only book of this type that I've come across is ...

**5**

votes

**1**answer

213 views

### Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface

It is well known that
Grayson's dumbbell neck-pinch1,2 separates
into disconnected pieces under
mean curvature flow:
Image ...

**9**

votes

**1**answer

227 views

### Assuming AD, is every infinite cardinal closed under power set in a choice model?

Assume AD+DC. Assume $\kappa$ is an infinite cardinal and $N$ is a (set or class) transitive model of ZFC containing $\kappa$.
Is it true that for all $\alpha<\kappa$, $N$ thinks that the power ...

**5**

votes

**0**answers

138 views

### Heisenberg group: function without vertical derivative

Let $\mathbb H$ - Heisenberg group,
$$
X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t
$$
- vector fields, and $U\subset\mathbb H$ - open set.
I am ...

**0**

votes

**1**answer

142 views

### Redundancy of the Cantor Enumeration of the Rationals

What is the cardinality of the set of values corresponding to the first $n$ rationals generated in Cantor's enumeration scheme for proofing of their countability?
Edit:
following the suggestion of ...

**1**

vote

**1**answer

219 views

### When can we write fundamental units explicitly

Given a number field $K$, the Dirichlet Unit Theorem tells us about the structure of the unit group $O_K^\times$. However, the proofs do not seems to give any way to explicitly write out a set of ...

**2**

votes

**3**answers

322 views

### A table for irreducible integral representation of finite cyclic groups

Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...

**0**

votes

**0**answers

110 views

### Heat asymptotics

Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...

**1**

vote

**1**answer

36 views

### Fractional Sobolev spaces and interpolation in unbounded Lipschitz domains

I am not really familiar with the topic, thus I am looking for some references about the following problem.
Let $s>0$ be a positive real and let $p\in(1,+\infty)$. We define the Bessel Potential ...

**2**

votes

**0**answers

73 views

### Quantitative estimate of heat dispersion - off diagonal estimates

Consider the heat equation $\partial_t u - \Delta u = 0$ on a compact manifold $M$ (if $M$ has a smooth boundary, then we assume either Dirichlet or Neumann boundary condition). Consider $u_0 (x) = ...

**1**

vote

**0**answers

76 views

### Fixed point theorem in ordered spaces

Can someone provide a proof or a source containing a proof of the following theorem
Theorem: Let $D$ be a subset of the cone $K$ of partially
ordered space $E,$ $F:D\rightarrow E$ be nondecreasing. ...

**2**

votes

**2**answers

189 views

### Could we extend any line bundle on the smooth part of a singular curve to a line bundle on the whole curve?

Let $X$ be a singular curve over an algebraic closed field $k$ with characteristic zero. Let $Z$ be the closed subset of singular points on $X$ and $U=X-Z$ be the smooth part, which is an open subset ...

**2**

votes

**2**answers

128 views

### constant rank theorem for banach spaces

Is there a similar statement to the constant rank theorem for finite dim real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dim Banach space ...

**1**

vote

**1**answer

91 views

### Collections in direct products and freeness

I am looking for references about the following type of questions:
Let $G$ and $H$ be two groups,
let $(g_i:i\in I)\subset G$ and $(h_i:i\in I)\subset H$ be collections of group elements,
and ...

**2**

votes

**0**answers

217 views

### Programming workbooks in C++ and Research Math [closed]

I know the basics of C++ by taking a few courses and going through "C++ Primer" by Lippman. As a math graduate student, I would love to get my hands on some programming-math exercises geared towards ...

**5**

votes

**2**answers

270 views

### References for the moduli space of complex structures

I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...

**1**

vote

**1**answer

87 views

### Iinterchanging limits for doubly indexed random sequences

I've encountered the following problem which seems to be quite standard but for which I can't find any proper references (asking on mathematics SE didn't bring up any answers so I'm reposting my ...

**1**

vote

**1**answer

69 views

### Derivative of a time evolution operator w.r.t. a parameter

Let $N\geq1$ be an integer and let $H:[0,1]^2\to\mathbb C^{N\times N}$ be a pointwise hermitean matrix valued function.
For $y\in[0,1]$ and $0\leq a\leq b\leq 1$, let $U_y(b,a)$ be the time evolution ...

**6**

votes

**1**answer

215 views

### 3-manifolds with isomorphic fundamental groups

There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces ...

**0**

votes

**0**answers

30 views

### A source for integral operators in the context of Arthur-Selberg trace formula

Could you suggest a textbook for integral operators
(in the context of Arthur-Selberg trace formula)?

**0**

votes

**0**answers

74 views

### Boundedness of heat semigroup on $L^1(\Omega)$

On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...

**20**

votes

**0**answers

230 views

### Guess that group via product queries

Suppose someone (person B) knows a finite group $G$ of order $n$.
You (person A) know only the order $n$,
and that $1$ is the name of the identity element.
The group elements are named $1,2,\ldots,n$ ...

**2**

votes

**2**answers

358 views

### Exact sequences of pointed sets - two definitions

It seems to me that there are (at least) two notions of exact sequences in a category:
1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite ...

**2**

votes

**0**answers

68 views

### Sources on evolution of submanifolds subject to Ricci flow

I am seeking any textbook or paper addressing the evolution of submanifolds of a manifold undergoing Ricci Flow. Please, any pointer towards this topic is more than welcome.
This old MO post may be ...

**10**

votes

**2**answers

328 views

### Group theory in machine learning

I'm a Machine Learning researcher who would like to research applications of group theory in ML.
There is a term "Partially Observed Groups" in machine learning theory which has been popularized by ...

**1**

vote

**3**answers

266 views

### Has this formula for $G_{k}:=\lim\inf_{n\to\infty}p_{n+k}-p_{n}$ been conjectured?

I give here a heuristics that suggests that the quantity $\displaystyle{G_{k}:=\liminf_{n\to\infty}p_{n+k}-p_{n}}$ should be approximately equal to $k(1+H_{k})$, where $H_{k}$ is the $k$-th harmonic ...

**1**

vote

**0**answers

75 views

### Generalization of Schur polynomials

I am making a list of generalizations of Schur polynomials and other closely related polynomials that appear in representation theory. My motivation is to eventually make a nice poster of ...

**0**

votes

**1**answer

80 views

### Question about measure lemma?

"Let (u_j) be a bounded sequence from $W^{1,p}(\Omega)$ how to prove that there exists a subsequence such that $u_j\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ and $|\nabla u_j|\rightharpoonup d\mu,$ ...

**7**

votes

**1**answer

124 views

### Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?

Let us denote the Riesz potential in $\mathbb R^d$ by
$$
I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
By the classical Hardy-Littlewood-Sobolev theorem ...

**2**

votes

**0**answers

115 views

### A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
...

**-1**

votes

**0**answers

51 views

### Generalization of a class of sets [closed]

In topological space, we start with open set, which serves as fundamental set. We know that union of finite disjoint open sets is the smallest set amongst any kind of unions of open sets, so we have a ...

**0**

votes

**0**answers

88 views

### Is it possible to find an explicit definition of the “universal” (co)tangent bundle?

Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$
maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point
$y \in \mathbb{P^1} $ ...

**1**

vote

**1**answer

65 views

### Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...

**4**

votes

**0**answers

124 views

### Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...

**1**

vote

**1**answer

90 views

### Inner product spaces without symmetry/hermitian axiom

Consider a vector space $X$ over $\mathbb R$ and a bilinear form
$ \langle \cdot, \cdot \rangle : X \times X \rightarrow \mathbb R$.
We assume furthermore that for any $x \in X$ there exists $y \in ...

**5**

votes

**1**answer

217 views

### A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...

**4**

votes

**0**answers

116 views

### Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path.
A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...

**0**

votes

**0**answers

78 views

### What is the relation between the $K_0$ of a singular curve and its normalization?

Let $X$ be a singular curve over a field $k$. We define $K_0(X)$ to be the Grothendieck group of the category of coherent sheaves on $X$.
For $X$ we have its normalization $\widetilde{X}$ and hence ...