**1**

vote

**1**answer

152 views

### Euler-Poincare equations with constraints

It is well known that the stationary curves (say $\Xi(t)$) of a regular Lagrangian $\mathcal{L}$ on a compact, semi-simple Lie group $G$ have the property that $\xi(t) = \frac{d \Xi(t)}{dt} ...

**11**

votes

**5**answers

1k views

### What is $\sum (x+\mathbb{Z})^{-2}$?

This is a simple question, but its been bugging me. Define the function $\gamma$ on $\mathbb{R}\backslash \mathbb{Z}$ by
$$\gamma(x):=\sum_{i\in \mathbb{Z}}\frac{1}{(x+i)^2}$$
The sum converges ...

**13**

votes

**27**answers

3k views

### Justifying a theory by a seemingly unrelated example

Here is a topic in the vein of Describe a topic in one sentence and Fundamental examples : imagine that you are trying to explain and justify a mathematical theory T to a skeptical mathematician ...

**6**

votes

**4**answers

751 views

### Visualizing functions with a number of independant variables

I need to graph real valued functions ( for exposition and analysis)
the issue is the independent variables are more so that the conventional graphing method cant be used and further i don't want to ...

**26**

votes

**1**answer

1k views

### Does a proof of Selberg's 3.2 inequality exist?

A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that
$$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ...

**2**

votes

**1**answer

119 views

### Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...

**15**

votes

**5**answers

3k views

### (Co)homology of the Eilenberg-MacLane spaces K(G.n)

Let $(G, n)$ be a pair such that $n$ is a natural number, $G$ is a finite group which is abelian if $n \geq 1$. It is well-known that $\pi_n((K,(G,n)) = G$ and $\pi_i (K(G,n)) = 0$ if $i \neq n$.
...

**33**

votes

**8**answers

9k views

### Is Galois theory necessary (in a basic graduate algebra course)?

By definition, a basic graduate algebra course in a U.S. (or similar) university with
a Ph.D. program in mathematics lasts part or all of an academic year and is taken
by first (sometimes second) ...

**9**

votes

**2**answers

406 views

### Decidability in Groups

This is not my area of research, but I am curious. Let $G=\left< X|R \right>$ be a finitely presented group, where $X$ and $R$ are finite. There are many questions which are undecidable for all ...

**4**

votes

**1**answer

1k views

### Choice of Lipschitz constant for proximal gradient optimization

I'm trying to use proximal gradient methods (Forward-Backwards Splitting and FISTA) to minimize a function
$f(B) = \frac{1}{2}|| XB - Y||_F^2 + \frac{\gamma}{2}||B C^T||_F^2$, where $X \in ...

**6**

votes

**2**answers

335 views

### showing the subgroup membership problem is undecidable for $F_2 \times F_2$

Let $F_2$ denote the free group of rank 2. Does anybody have a fast proof that the subgroup membership problem is undecidable for $F_2 \times F_2$? I saw a really fast proof last semester that ...

**9**

votes

**4**answers

853 views

### covering disks with smaller disks

How many disks with radius 1/2 are needed to cover a disk with radius 1? It certainly cannot be done with less than 5 small disks, and some non-rigorous drawings of mine suggest it can be done with 7 ...

**11**

votes

**4**answers

3k views

### Integer points of an elliptic curve

Where can I found some resources to learn how to determine the integer points of given elliptic curve? I would like to learn a method based on computing the rank and the torsion group of given curve. ...

**12**

votes

**1**answer

737 views

### Least prime in an arithmetic progression and the Selberg sieve

My question concerns a technical step in the proof of Linnik's theorem on the least prime in an arithmetic progression, as presented in Chapter 18 of Iwaniec-Kowalski: Analytic number theory.
The ...

**15**

votes

**2**answers

596 views

### injectivity of torsion submodules of injectives

Local cohomology with respect to an ideal $\mathfrak{a}$ is often studied over a Noetherian ring $R$. However, the proof of a lot of basic results does not rely on noetherianity of $R$, but rather on ...

**9**

votes

**1**answer

328 views

### Topological conditions forcing continuity

Let $X$, $Y$, and $Z$ be topological spaces. Let $f:X \rightarrow Y$. Further assume that for every continuous function $g:Y \rightarrow Z$, $g \circ f$ is continuous.
Question: Under what ...

**6**

votes

**1**answer

1k views

### Bijection from $\mathbb{R}$ to $\mathbb{R^2}$

Importantly, I am looking for a constructive proof (which does not rely on the Cantor–Bernstein–Schroeder theorem). Motivated by this discussion.

**3**

votes

**1**answer

336 views

### Injective modules and torsion functors

(This is a related question.)
Local cohomology is studied mostly over Noetherian rings. Parts of the machinery do in fact not rely on Noetherianness, but on some weaker properties, for example the ...

**7**

votes

**3**answers

3k views

### Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry

I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started ...

**3**

votes

**2**answers

334 views

### Can a zerodivisor reduce both the depth and the dimension?

In this question $R$ is a commutative noetherian local ring with unity.
One can construct examples of rings $R$ and zerodivisors $z$ such that $\dim R/(z)=\dim R-1$, e.g., $S\colon=k[a,b,c],\ ...

**3**

votes

**2**answers

632 views

### Integer partition and sum of squares

Hello,
The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics)
For all integers $n\geq 2$ denote by ...

**8**

votes

**7**answers

15k views

### Notation for the all-ones vector [closed]

What's the most common way of writing the all-ones vector, that is, the vector, when projected onto each standard basis vector of a given vector space, having length one? The zero vector is frequently ...

**1**

vote

**2**answers

305 views

### quotient of integral polynomials not being integral

So let $R$ be an integral domain and $K$ its fraction field. Let $f,g,h\in K[x]$
be 3 monic polynomials such that $f=gh$. So I would like to have a simple example
of a ring $R$ for which one has that ...

**2**

votes

**1**answer

112 views

### Star-transfer of powerset

What is the difference between ${^\sigma}\mathcal{P}(\mathbb{R})$, ${^\ast}\mathcal{P}(\mathbb{R})$, and $\mathcal{P}({^\ast}\mathbb{R})$? I know that $\mathcal{P}({^\ast}\mathbb{R})$ is the powerset ...

**5**

votes

**6**answers

1k views

### Reference request: Moduli spaces of bundles over singular curves

I would like to know some reference (articles, books...) about any kind of moduli spaces of any of the following objects:
vector bundles
torsion-free sheaves
principal bundles
parabolic bundles
...

**7**

votes

**1**answer

355 views

### If $A\in \mbox{Rings}\subset E_\infty\mbox{-rings}$, what is the equivalence between objects of $\mathcal{D}(\mbox{Mod}_A)$ and $A$-module spectra?

In Lurie's "A Survey of Elliptic Cohomology", he writes on page 14 that if $A$ is an ordinary commutative ring considered as an $E_\infty$-ring, then $A$-module spectra are the same thing as objects ...

**4**

votes

**0**answers

242 views

### Fibred manifolds with boundary

A fibred manifold is a triple $(E,\pi,M)$ where $E$ and $M$ are manifolds and $\pi : E \rightarrow M$ is a surjective submersion. (Saunders, The Geometry of Jet bundles)
A special case of this is the ...

**4**

votes

**1**answer

844 views

### Different definitions of Novikov ring?

Following, e.g., Wikipedia's definitions, the (small) quantum cohomology ring of $X$ is defined over a "Novikov ring" consisting of formal power series of the form $$ \sum_{\beta \in ...

**0**

votes

**0**answers

6 views

### Are Sobolev trace spaces equal from both sides of the boundary?

Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...

**-4**

votes

**0**answers

35 views

### Linear Independence for functions defined by integration [on hold]

Given that the set of functions $$f_i(x,y), \quad i=1,\dots,n$$
are linearly independent for $(x,y) \in [0,1]^2$.
Is the set of functions, $g_1,\ldots,g_n$, defined by
$$
g_i(x) = \int_{y\in [0,1] } ...

**-3**

votes

**0**answers

36 views

### Decomposition of orthogonal matrix into 2 orthogonal matrices [on hold]

Is there anyway to find a decomposition of orthogonal matrix $A$ into 2 orthogonal matrices $P$ and $Q$ such that $A = PQ^T$?

**-4**

votes

**0**answers

25 views

### Discrete time equivalent to ODE [on hold]

I'm reading a paper in which it is noted that
$$\frac{dv(t)}{dt} = f(t) - \varepsilon v(t)$$
has the discrete time equivalent
$$v(t+1) = v(t)\exp(-\varepsilon) + \frac{f(t)}{\varepsilon}[1 - ...

**-5**

votes

**1**answer

77 views

### is this formula correct [on hold]

I found this formula in a book:
for cylindrical coordinates where: $x=r\cos\theta$ and $y=r\sin\theta$, then:
$$\dfrac{\partial}{\partial x} = \cos\theta \dfrac{\partial}{\partial r} - ...

**-1**

votes

**0**answers

52 views

### Interpolating Product of two Polynomials

Consider we have two non-constant polynomials $A(x)$ and $B(x)$. We define the polynomials over field $\mathbb{Z}_p$, for a large prime number $p$.
We define Polynomial $A(x)$ as follows: ...

**7**

votes

**4**answers

323 views

### Lower bounding the multiplicative order of 2 modulo p

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.
Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?
...

**-4**

votes

**0**answers

86 views

### Why should “small” P be preferred? [on hold]

In contrast, of course, is the approach of finding an NP language of super-polynomial complexity.
But why the overwhelming, obvious yet implicit favoritism?
Has it anything to do with our ...

**-4**

votes

**0**answers

24 views

### Find limit of polynomial in detail [on hold]

I want to know what is the limit n->infinity for
(n + a)^ b/ n^b
Please provide a detailed answer. Regards

**0**

votes

**0**answers

99 views

### Quantities associated to deformed sheaves

I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...

**0**

votes

**1**answer

135 views

### An exercise in the Kaplansky's book

I saw the following exercise in the Kaplansky's book that is due to D. Lizard. Where can i find the main text for the proof of this exercise?
Let $P$ be a prime ideal of $R$, $I$ the ideal generated ...

**8**

votes

**1**answer

732 views

### Remark on Fermat's Last Theorem by Darmon, Diamond and Taylor

In their paper, Darmon, Diamond and Taylor remarked the following :
(the previous paragraph of Section 2.2 (p. 55), https://www.math.wisc.edu/~boston/ddt.pdf)
If $\rho : G \rightarrow ...

**3**

votes

**1**answer

73 views

### The growth of a subset of a group

Let $S$ be a symmetric subset of a group $G$ containing the identity, and let $S^n$ be the set of all products of $n$ elements of $S$. If $S^3\subset gS$ for some translate $gS$ of $S$ then it ...

**5**

votes

**0**answers

167 views

### Constructing a simple $A$-module

Let $n \ge 2$, and let $A$ be the (unital and associative, but noncommutative) $\mathbb{C}$-algebra with generators $x_1, \dots, x_n$ and relations $x_ix_j + x_j x_i = 2\delta_{ij}$. What is a ...

**-2**

votes

**0**answers

21 views

### where I can find an example of higher order nonlinear systems? [on hold]

where I can find an example of higher order nonlinear systems for example $6 \times 6 $ dimension ,? I want to applied a linearizion methods.

**5**

votes

**0**answers

105 views

### Derived global functions on (derived) stacks $BG$ and $G/G$

In Toen's Affine Stacks, he computes that $\mathcal{O}(B\mathbb{G}_a) = k[\epsilon]$ with $|\epsilon| = 1$ and trivial differential (where here $\mathcal{O}$ is computed in a derived sense, and we ...

**1**

vote

**0**answers

62 views

### Interpolating a Polynomial Given Multiplier of each $y_i$

It would be great even if you answer only one of the below questions.
We have polynomial $P(x)=(x-\beta)\cdot g(x)$, where degree of $P(x)$ is fixed n-1, $\beta$ chosen uniformly at random from the ...

**5**

votes

**0**answers

136 views

### Intrinsic definition of the weight filtration

Let $X$ be a smooth quasiprojective complex variety. Then Deligne (Theorie de Hodge II) defined a weight filtration on the Betti cohomology of $X$. The general philosophy is quite simple: express the ...

**-2**

votes

**0**answers

121 views

### $\mathsf{GCD}$ in arithmetic progression

Given $\mathsf{M\in\Bbb N}$, pick $\mathsf{r,s,A,B\in\Bbb N}$ randomly with $\mathsf{0<r<s<A<B<M}$ satisfying $\mathsf{gcd(A,B)=1}$.
Given $\mathsf{c\geq1}$, what is the probability ...

**0**

votes

**0**answers

116 views

### Radius of convergence of Taylor expansion of $z \mapsto (1 - z \cdot a)^{-1}$ [migrated]

Let $A$ be a Banach $\mathbb{C}$-algebra with norm $\text{N}(-)$ and let $a \in A$. Where can I find a reference to/can somebody supply a proof of the following posited equality?$$\max_{z \in ...

**12**

votes

**0**answers

189 views

### Research situation in the field of Information Geometry

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field.
I have read (4) and parts of (3). ...

**-3**

votes

**0**answers

45 views

### Does the language suggest hard average cases? [on hold]

\begin{equation*}
\begin{aligned}
\ \\
L & = \{ D \, | \, permuted \, C \, on \, its \, submatrices \, C_{i} \, \}
\ \\
\ \\
C & = [\,C_{1}\, C_{2}\, ...\, C_{k-1} \, C_{k} \, C_{k+1} \, ... ...