The limits tag has no wiki summary.

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### 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. ...

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

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159 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 ...

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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 ...

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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 ...

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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 ...

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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. ...

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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 $E[X_{n1}]=\mu_n<\infty$ ...

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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 ...

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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 ...

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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 ...

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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 ...

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234 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 ...

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78 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 ...

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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 ...

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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 ...

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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 ...

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### 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 ...

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936 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?
...

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### 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 ...

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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, ...

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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 ...

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155 views

### Limits in span categories

What are the limits in the span categories? and what is known about them in the literature?

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### 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) ...

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158 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 ...

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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 ...

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### integral against a gaussian density over an increasing space

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

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### 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. ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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
...

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460 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 ...

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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}} = ...

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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 ...

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### 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 ...

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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 ...

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### 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 ...

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### 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
...

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### 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 ...

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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 ...

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### 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 $\forall x$ ...

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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 ...

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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 ...

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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 ...

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### 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 ...

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### 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 ...

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### 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) = ...

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277 views

### When are iterated limits of sets equal to a double limit?

Suppose $\mathcal{A}$ is a $\sigma$-algebra, and $A_{1,1},A_{1,2},...,A_{2,1},A_{2,2},... \in \mathcal{A}$ is a double sequence of measurable sets. Under what circumstances do we have the following?
...