# Questions tagged [reference-request]

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

9,216 questions

**15**

votes

**5**answers

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### Sum of the reciprocals of radicals

Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$.
For a paper, I need the result that
$$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\...

**5**

votes

**0**answers

115 views

### A relation concerning the “sum of squares” counting function $r_2(n)$

This is a re-post from MSE as I did not get any response there.
Let $r_2(n)$ denote the number of ways in which a positive integer $n$ can be expressed as the sum of squares of two integers. Here ...

**1**

vote

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96 views

### Error correcting codes via random matrices: How close to the Shannon bound?

I have a vague and probably rather naive question on error correcting codes.
Suppose we want to encode binary vectors of length $k$ as binary vectors of length $n$ in such a way that differences of ...

**6**

votes

**0**answers

99 views

### The automorphism group of the fibered cylinder

My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...

**0**

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46 views

### linear fractional laplacian problem

If $U(x)$ is a classical solution of
\begin{equation}
\ \ \left\{\begin{aligned}
(-\Delta)^s U+ U &=m(r)U &&\text{in } B \\
U &= 0 &&\text{in } \mathbb R^N \...

**-1**

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80 views

### Can we have Levy area for N dimensional process?

Consider a two dimension Brownian motion $(X_t,Y_t)$ and we can consider Levy's area as $\int_0^t X_sdY_s-\int_0^t Y_sdX_s$. Is there a equivalent area for N dimensional Brownian motion, if so what ...

**10**

votes

**1**answer

331 views

### Looking for “Set theory for a small universe” by Ketonen

In the paper Partition theorems for systems of finite subsets of integers, Pudlák and Rödl show a Ramsey-type result. The main feature of this result is that the sizes of sets in such systems are not ...

**19**

votes

**1**answer

940 views

### Is there an accessible exposition of Gelfand-Tsetlin theory?

I'm hoping to start an undergraduate on a project that involves understanding a bit of Gelfand-Tsetlin theory, and have been tearing my hair out looking for a good reference for them to look at. ...

**7**

votes

**1**answer

319 views

### Reference request for K-Theory linearization

I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that.
In Waldhausen's paper Algebraic K ...

**2**

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**0**answers

53 views

### Volume of a double class of a parahoric subgroup

Let $F$ be a non-archimedean local field with residue field $F_q$. Let $G$ be the group of $F$-rational points of a connected reductive group defined and split over $F$. Fix a maximal split torus $T$ ...

**1**

vote

**0**answers

43 views

### Is there a transient graph whose spectral dimension two?

Let $G = (V(G), E(G))$ be an infinite connected simple graph.
Let $((S_n)_n, (P^x)_{x \in V(G)})$ be the simple random walk on $G$.
Let $p_n (x,y) = P^x (S_n = y)$.
A spectral dimension of $G$ is ...

**3**

votes

**1**answer

179 views

### Locally presentable categories

Under category
Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...

**4**

votes

**1**answer

87 views

### Spin groups in terms of matrices and/or linear operators

Thus far, the books and articles I have read dealing with spin groups $\mathbf{Spin}(n)$ and $\mathbf{Spin}(p,q)$ consider them only in terms of either Clifford algebras or topologically as the double ...

**1**

vote

**0**answers

75 views

### Are these kinds of “crossed product” studied?

Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...

**1**

vote

**0**answers

50 views

### If $P \ll Q$, are the regular conditional probabilities a.s. absolutely continuous?

Let $P$ and $Q$ be probabilities on $(\Omega, \mathcal{A})$, and let $\mathcal{F}$ be a sigma-subalgebra of $\mathcal{A}$. Assume $P \ll Q$. Assume that $P(\cdot \mid \mathcal{F})$ and $Q(\cdot \mid \...

**1**

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**0**answers

150 views

### Skew-symmetric multi-derivations of $k[x_1,…,x_n]/I$

Let $I = \langle f_1, \ldots f_r \rangle$ be an ideal in $R=k[x_1,\ldots,x_n]$ where $k$ is a field, and put $A = R/I$.
(If $I$ is prime then $A$ is the coordinate ring of an irreducible affine ...

**7**

votes

**1**answer

118 views

### Tensor product of a DGA and an $A_\infty$ algebra

In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge ...

**1**

vote

**1**answer

121 views

### The quotients of double cosets $P_\theta \backslash P_\theta w P_\Omega$ are algebraic varieties over $k$

Let $k$ be a $p$-adic field, $G$ a connected reductive group over $k$ with minimal parabolic $P_0$ containing a maximal split torus $A_0$. Let $W = N_G(A_0)(k)/Z_G(A_0)(k)$ be the Weyl group, and $S \...

**3**

votes

**0**answers

81 views

### $L$-functions for quadratic orders and Siegel's solution of the class number problem

Let $K$ be an imaginary quadratic field and $D_K$ its discriminant. Further let $\mathcal O$ be an order in $K$ with conductor $f$ and
$$L(\chi,s)=\sum_{\mathfrak a}\chi(\mathfrak a)N(\mathfrak a)^{-...

**0**

votes

**0**answers

44 views

### References for formal powers of measures

In Information geometry, Ay et al. define the space of formal $r$th powers* of signed measures as the limit $\mathcal S^r(\Omega) = \injlim L^{1/r}(\Omega, \mu)$ of maps $\phi\mapsto (\mu/\nu)^r\phi :$...

**1**

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**0**answers

60 views

### Unique continuation from the boundary for inhomogeneous elliptic pde

Let $Lu = f$ be satisfied on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, where $L$ is a strongly elliptic second order differential operator with real ...

**12**

votes

**1**answer

427 views

### Schur's Theorem about immanants

$\DeclareMathOperator\Imm{Imm}$I am looking for a proof in English or French of Schur's theorem that, for every $H$ in the space $\mathbb H_n^+$ of positive semi-definite Hermitian matrices, and every ...

**1**

vote

**1**answer

61 views

### Linear relations between volume of a polytope and its faces

Let $P$ be a polytope. Is anything known about the set of linear relations that hold between the volumes of the (not-necessarily proper) faces of $P$ as $P$ “varies slightly”? By varies slightly I ...

**3**

votes

**0**answers

75 views

### Jacobson radical of a tensor product

Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...

**4**

votes

**1**answer

434 views

### Prerequisites for reading papers of arithmetic such as Ribet, Mazur, Faltings, Wiles

I've studied some fundamentals of algebraic geometry and number theory, and now I want to read papers which seem to be the "main stream" of frontier research on arithmetic.
I've heard that Mazur's "...

**1**

vote

**1**answer

43 views

### Translation of Fakeev's Optimal Stopping Rules for Stochastic Processes with Continuous Parameter

I am looking for a translation of Fakeev's "Optimal Stopping Rules for Stochastic Processes with Continuous Parameter" from 1970.
I can only find it in Russian. Does anyone know where to find this?

**4**

votes

**0**answers

109 views

### Group scheme with isomorphic fibers

Let $X$ be a smooth irreducible algebraic curve over $\mathbb C$. Let $\mathcal G\rightarrow X$ be a smooth affine group scheme over $X$ such that for any closed points $p\in X$, we have $\mathcal G_p\...

**2**

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**0**answers

45 views

### Comparison of length functions on Weyl groups

Let $G$ be a connected reductive group over an algebraically closed field $k$ (with nice enough characteristic), and let $\sigma:G\to G$ be a finite order automorphism of $G$. The connected component $...

**0**

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**0**answers

45 views

### English reference text on Snell envelopes of càdlàg processes w.o negative jumps

I am looking for an English reference on the theory of Snell envelopes of càdlàg processes with and without negative jumps. In particular which contain results on existence of Snell envelope and ...

**0**

votes

**0**answers

57 views

### English translation of Schmidt's work on nonlinear integral equations. Part III

Does anybody know about the English translation of Erhard Schmidt's paper
Zur Theorie der linearen und nichtlinearen Integralgleichungen. III Teil. Über die Auflosing der nichtlinearen ...

**2**

votes

**0**answers

96 views

### About relation between Kostka numbers and Littlewood-Richardson coefficient

The fact that Kostka numbers equals to Littlewood-Richardson coefficients for some partitions is already known $\colon$
\begin{align}
K_{\lambda \mu} = c_{\sigma \lambda}^\tau
\end{align}
where $\...

**2**

votes

**0**answers

48 views

### General SLLN-like asymptotic mean concentration

Disclaimer: As I am not very knowledgeable of the field to which this question pertains, I will introduce a temporary terminology to convey the idea of my question at the risk of conflicting with ...

**2**

votes

**0**answers

246 views

### Asymptotics of Littlewood polynomials

Littlewood in [L] states several conjectures regarding asymptotics of polynomials with $\pm1$ coefficients.
He considers the class $\mathscr F$ of polynomials of form $\sum^n\pm z^m$ and asks whether ...

**0**

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**0**answers

146 views

### Convergent sequences in projective varieties

It's very well known that if $X$ is an irreducible projective variety (feel free to assume that the base-field is $\mathbb{C}$), then any two points $x,y\in X$ can be connected by the image of a non-...

**6**

votes

**1**answer

452 views

### Physical (GR) Differential Geometry?

I am looking for problem lists or books which contain open problems in the area of mathematics motivated by physics. Ideally, I am looking for questions asking about which reduce to some calculation ...

**8**

votes

**3**answers

405 views

### How should I think about the Grothendieck-Springer alteration?

Given a simple complex Lie algebra $\mathfrak{g}$, recall the Springer resolution of its nilpotent cone $\widetilde{\mathcal{N}}\to \mathcal{N}$. Several times I have seen someone explaining Springer ...

**3**

votes

**0**answers

57 views

### Integration on a family of differential forms

Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...

**0**

votes

**1**answer

80 views

### Looking for a reference by Moree and Niklash

A while ago I saved an internet reference to a work by P. Moree and G. Niklasch, published exclusively on a website, related to high-precision computations of constants related to prime numbers. I ...

**5**

votes

**1**answer

154 views

### Need software for subject classification to stick to, for personal library purposes?

This question obviously applies to not only mathematics, so maybe I should post it on academia.SE or somewhere else; but then again, mathematical literature has its own specifics related to existing ...

**3**

votes

**4**answers

467 views

### A generalization of Landau's function

For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$
the least common multiple of all ...

**2**

votes

**1**answer

88 views

### Why Triangle of Mahonian numbers T(n,k) forms the rank of the vector space?

I am looking for an explanation of why Triangle of Mahonian numbers T(n,k) form the rank of the vector space $H^k(GL_n/B)$?
With respect to the property of Kendall-Mann numbers where the statement ...

**3**

votes

**1**answer

114 views

### Congruence of normalized eigenforms at two primes

Let $f_i\in S_{k_i}(\Gamma_0(N_i))$ be normalized cuspidal eigenforms for $i=1,2$ and let $K$ be the composite of the fields of Fourier coefficients generated by $f_1$ and $f_2$ and let $\mathfrak{p}...

**1**

vote

**1**answer

48 views

### Conditions for a pushforward of a involutive vector bundle to be involutive

I know that the following statement is true, but I would like to find a reference for it so I don't have to write the proof. Do you guys have a reference?
Let $\Omega$ and $\Omega'$ be smooth ...

**10**

votes

**2**answers

286 views

### Almost graceful tree conjecture

The graceful tree conjecture is the following statement: for any tree $T = (V, E)$ with $|V| = n$ there is a bijective map $f: V \to [n]$ such that $D = \{|f(x) - f(y)| \mid xy \in E\} = [n - 1]$.
...

**0**

votes

**0**answers

47 views

### Theta Summable operator with bounded trace

Let $D$ be an unboudned self-adjoint operator on the Hilbert space $H$. We assume that all spectrum of $D$ are eigenvalues and $D$ is theta-summable, i.e. $e^{-tD^2}$ is of trace class for all $t>...

**3**

votes

**1**answer

180 views

### Foliation with a compact leaf

Let $M$ be a closed oriented manifold, and $F$ be a fixed foliation of $M$. We assume the dimension and codimension of $F$ are both greater than $1$.
Q Under what condition, we can say that $F$ ...

**11**

votes

**1**answer

386 views

### Is there a name of semidirect product of a group with its automorphism group?

Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name ...

**5**

votes

**1**answer

324 views

### Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...

**3**

votes

**0**answers

62 views

### Examples of explicit computations of log-resolutions

I have been working with log-resolutions lately and learning more about them. I am aware that in general producing explicit log-resolutions is difficult, but I was wondering if this has been done in ...

**6**

votes

**1**answer

154 views

### The weight filtration on etale cohomology and Berkovich analytic geometry

If $X$ is a smooth projective curve over $\mathbb C_p$, then its first etale cohomology $\mathrm H^1_{et}(X,\mathbb Q_\ell)$ (with $\ell\neq p$) carries a certain weight filtration $W_\bullet$ -- also ...