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0
votes
0answers
43 views

Find an analytic function [migrated]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
1
vote
4answers
201 views

Homology of infinite intersection

If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. ...
8
votes
1answer
414 views

What is $\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$

What is $$\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$$
15
votes
0answers
160 views

Finite limits in the category of smooth manifolds?

The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...
1
vote
2answers
191 views

Looking for a limit related to the series in a previous post

Can any one show that the following limit? $$ \lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1. $$ If one uses the ...
0
votes
0answers
31 views

Sequence of solutions of approximated problem converging to stationary solution of the original problem

Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...
1
vote
0answers
58 views

CLT for a Markov Renewal Process

Suppose $(X,T)=\{(X_n,T_n)\}_{n\geq0}$ is a Markov renewal process, where $X$ is a finite-state, discrete-time Markov chain with state space $\{1,2,...,R\}$. $T$ is the additive component, more ...
3
votes
0answers
198 views

Homotopy category of groupoids

The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description: iso-classes of functors. formally invert equivalence functors (i.e. ...
1
vote
0answers
57 views

Berry-Esseen result for triangular arrays

Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ ...
2
votes
1answer
322 views

What are the uses of Limits and Colimits of Category Theory in every day problems? [closed]

I am interested in knowing how we can use the concepts of Limits and Colimits in modeling problems in every day life? Could anyone provide (Software) engineering examples, perhaps? Or describe ...
10
votes
1answer
240 views

The geometric-mean factorial

Think of the factorial as $f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1$, where $\odot$ is the binary operator for multiplication, $\cdot$. This suggests exploring replacing $\odot$ with other ...
4
votes
1answer
144 views

Extending a Certain Result from Locally Convex Topological Vector Spaces to General Topological Vector Spaces

In this Math Stack Exchange post, I proved the following result. Theorem: Let $ X $ be a locally convex topological vector space. Let $ x \in X $ and suppose that $ (x_{n})_{n \in \mathbb{N}} $ is ...
3
votes
0answers
87 views

Inverting a matrix with entries equal to positive or negative infinity

I would like to define an inverse on matrices whose entries may be positive or negative infinity. To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I ...
7
votes
1answer
235 views

Is there a practical criterion to determine whether the limit of a diagram of real chain complexes is also a homotopy limit?

Consider a diagram D: I→ChR of real connective chain complexes. In the example I have in mind all chain complexes are concentrated in some fixed degree n. There is a canonical map lim D → holim D ...
1
vote
0answers
79 views

Limit wiht logarithm and dilogarithm functions [closed]

I'm trying to find out how to calculate the following limit: $$ \lim_{u\to \infty} \left\{\log(u) \left[\log\left(1 + \frac{2u}{i- 2 t}\right) - \log\left(1 - \frac{2 u}{i + 2 t}\right)\right] + Li_2 ...
1
vote
0answers
83 views

Bound a sum of a serie defined by a recursive integer function

I'm using a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$, that is defined as \begin{equation} f(n)=\lceil \log(f(n-1)) \rceil +f(n-1) \end{equation} where $f(1)=F\in \mathbb{N}$, and ...
3
votes
0answers
160 views

Ends and Coends - Analogues for higher arity - Horn Filling

Consider the setting of categories enriched over a suitable monoidal category $\mathbb V$. We define $$\mathrm{Dist}(X,Y):=\mathbb V−\mathrm{Cat}(X^ \mathrm{op}⊗Y,\mathbb V).$$ Recall the definition ...
1
vote
3answers
196 views

limit of a singular integral

Denote $f_{\gamma}(x) =\frac { (1+\gamma)}{2} |x|^{\gamma}$. We consider: $$I(\gamma) = \int_{-1}^1\int_{-1}^1 \ln (|x-y|) f_{\gamma}(x) f_{\gamma}(y) dx dy$$ I would like to know the limit of ...
10
votes
0answers
688 views

Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$ runs over the integers? The existence of the limes inferior follows from Dirichlet's approximation theorem, but the ...
19
votes
2answers
937 views

Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$?

For any $m\in\mathbb N$, let $S(m)$ be the digit sum of $m$ in the decimal system. For example, $S(1234)=1+2+3+4=10, S(2^5)=S(32)=5$. Question 1 :Is the following true? ...
-1
votes
1answer
90 views

Recursion equation convergence to sqrt [closed]

I'm new here. I have a recursion equation $$ a[n] = \frac{n \cdot a[n-1]}{n+a[n-1]} + k $$ $$ a[1] = 1 $$ $$ k \in \mathbb{R}, k > 0 $$ and from computing and ploting graphs I found that it can ...
1
vote
2answers
186 views

Asymptotic Expansion of Distribution in Central Limit Theorem for Non-Identically Distributed Random Variables

My question is related to the following theorem (e.g. Section XVI.4 of Feller's 1971 book): Let $Z_i$ $(i=1,\cdots,n)$ be independent and identically distributed random variables with mean zero, ...
2
votes
1answer
157 views

Limit involving regularlized gamma function and its inverse

Let $$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$ where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function ...
3
votes
1answer
155 views

Limits in span categories

What are the limits in the span categories? and what is known about them in the literature?
7
votes
1answer
173 views

Explicit description of the oplax limit of a functor to Cat?

The nCatLab Grothendieck construction page gives an explicit description of the oplax colimit of any functor to Cat. Can someone give me a similarly explicit description (the objects and morphisms) ...
3
votes
1answer
159 views

Does Ind-completion commute with finite limits?

The broad and vague question is in the title. The more precise question is: Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with ...
1
vote
1answer
134 views

Limiting probability question [closed]

Let $X$ denote an $m\times n$ matrix and suppose that each value $x_{ij}$ is an integer that is selected uniformly at random from ${1,\dots,n}$, independently of all other values. If we fix $m$ and ...
3
votes
0answers
67 views

integral against a gaussian density over an increasing space

Consider the following Gaussian density over $\mathbb{R}^{2^n}$ $$p_n(\underline{x}):=\frac{\exp(-\frac{1}{2n}\langle C_n^{-1}\underline{x},\underline{x}\rangle)}{\sqrt{(2\pi n)^{2^n} \det C_n}}$$ ...
2
votes
2answers
248 views

Projective limit construction of a semigroup

Let $\tilde{\mathbb N}$ be the Abelian semigroup (under addition) given by $\mathbb N\cup\{0,\infty\}$, and let $S_n$ be the Abelian monoid $\tilde{\mathbb N}^{2^n}$ under point-wise addition. ...
5
votes
0answers
264 views

Constructing pointwise Kan extensions as adjoints to some functor

Background I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because ...
7
votes
1answer
266 views

Is the evaluation of polynomial functors appropriately continuous?

I'd like a nice proof of the following fact. Let $C$ and $D$ be categories, and let $\mathbf{Cat}/(C\times D)$ be the usual (1-categorical) slice category whose objects are triples $(X,F\colon X\to ...
2
votes
1answer
136 views

Central Limit Theorem for additive function of permutations of sequences

Fix the finite sets $\mathcal{X}$ and $\mathcal{Y}$, and probability mass functions $P_X(x)$ and $P_Y(y)$ on these sets. Assume each value of $P_X(x)$ and $P_Y(y)$ is rational. For each $n$ such ...
4
votes
2answers
362 views

Central Limit Theorem (and Berry-Esseen theorem) for non-independent variables

Consider the triangular array $X_{n,k}$ such that, for each $n>0$, the variables $(X_{n,1},\cdots,X_{n,n})$ have the following properties: For any given $1 \le L \le n$, all subsets of ...
2
votes
1answer
461 views

Which limits does group cohomology commute with?

For a discrete group G, if $M$ is a direct/inverse limit of $M_i$, is $H^i(G, M)$ the direct/inverse limit of the $H^i(G, M_i)$? Of course, cohomology commutes with finite direct sums, but how about ...
22
votes
1answer
366 views

“Harmonacci” recurrence and identities for $\pi$

While playing with something totally irrelevant I stumbled upon the recurrence: $$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$ It turns out that given $a_0 = 1, a_1 = 1$, $$lim \frac{a_{2n}}{a_{2n-1}} = ...
0
votes
1answer
120 views

Residue at an integration border in case of a limit?

I am dealing with an integral in a limit of the following shape: $$\lim_{\epsilon \to 0} \int_0^{\frac{\pi}{2}} dx \frac{2 \epsilon}{1-(1-\epsilon^2)\sin^2(x)}$$ Formally, assuming that ...
1
vote
2answers
79 views

Finding Kuramoto Model coupling strength with limits?

The following is an equation describing the coupled phases of N oscillators according to the Kuramoto model: $$ 1 = K \int_{-\pi/2}^{\pi/2}\cos^{2}\left(\theta\right)\, {\rm ...
1
vote
0answers
243 views

The inductive and projective limits of compact Hausdorff topological groups

Are there conditions known under which the inductive or projective limit of a family of compact Hausdorff topological groups is compact? (For instance, such a result is valid for the projective limit ...
5
votes
1answer
343 views

Adjoint Functors as Initial Objects of Some Category

Just as universal arrows can be characterized as initial objects of some appropriate comma category, and (co)limits can be characterized as (initial) terminal objects of the appropriate (co)cone ...
0
votes
1answer
136 views

Limits to infinity of functions ratio [closed]

Thanks, sorry didn't read FAQ.
3
votes
0answers
119 views

Inverse limit of graded rings

Let $(I,\le)$ be a directed set and let $(\rho^{\beta\alpha}: R^\beta \to R^\alpha)_{\alpha \le \beta}$ be an $I$-directed system of $\mathbb{Z}$-graded rings whose multiplication is denoted by ...
1
vote
1answer
110 views

Question about exponential-type limit

Here's the setup: let $N(m)$, $m = 1,2, \cdots$ be a sequence of nonnegative integers for which $(*)~~~~~\sum_{m = 1}^{\infty} \frac{N(m)}{m} < \infty$ Now let $C>0$ be any constant. My ...
4
votes
2answers
272 views

Evaluating a limit similar to the Euler constant

In the course of studying a certain complex-valued functional equation, I have had a need to evaluate the following limit: $$\gamma_\mathcal{T}=\lim_{n\to\infty}\left(-\frac{i}{2}\sum_{k=1}^n ...
0
votes
0answers
96 views

Terminology question: a 2-variable function which converges to infinity in one variable, uniformly w.r.t the other

I am interested in two-variable functions f with the following property: $\forall M, \exists N$ such that $\forall x, \forall y>N, f(x,y)>M.$ (To be absolutely clear: the $\forall x$ ...
0
votes
1answer
139 views

A property of a quasiperiodic function

Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b. Does the limit of $\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$ Exist ...
1
vote
1answer
204 views

Asymptotic behaviour of a mean

Fix $x>0$ and $c\in\mathbb{N}$. Let $f(x):=\frac{c}{4c-2+2x^2}$ and $$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$ I'm pretty sure $m_N(x)\to\infty$ as $N\to\infty$. I ...
3
votes
1answer
392 views

asymptotic behaviour of a sum

I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum $$ Z_N(x) := 2^{-N/2} \sum_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$ in order to compute ...
8
votes
4answers
823 views

Categorical description of the restricted product (Adeles)

Background on the Adèles The Adèles $\mathbb{A}_K$ of a number field or function field $K$ are defined as a restricted product of the complete local fields $K_\nu$, where $\nu$ ranges over all places ...
2
votes
1answer
176 views

Pro-affine varieties over a local field

Let $K$ be a (perfect) local field, and let $S = \lim (\mathrm{Spec} A_i)_{i=0}^\infty$ be a pro-affine variety over $K$. This means that each $A_i$ is a finite type $K$-algebra and that the affine ...
0
votes
0answers
165 views

Convergence of $L^p$ means and/or norms

Dear all, I have a question about convergence of $L^p$-means. It can be shown (Inequalities, Theorem 193, Hardy, Littlewood, Polya) that $\forall f \in L^p(D,\mu)\cap L^\infty(D,\mu), M_p(f) = ...