## probability – “First principles” proof of the limit of the expected excess of the uniform renewal function

The closed form of the expected number of samples for $$sum_r X_r geqslant t, X_r sim text{U(0,1)}$$ is given by:

$$m(t) = sum_{k=0}^{lfloor t rfloor} frac{(k-t)^k}{k!}e^{t-k}$$

From this we can deduce the expected amount by which this sum exceeds $$t$$, namely:

$$varepsilon(t) = frac{m(t)}{2} – t$$

From knowing that $$m(t) to 2t+dfrac{2}{3}$$, we can easily see that $$varepsilon(t) to dfrac{1}{3}$$.

Is there a simple (“low tech”) way of proving that $$varepsilon(t) to dfrac{1}{3}$$ without first passing through proving $$m(t) to 2t+dfrac{2}{3}$$ ?

## dg.differential geometry – Uniform convergence of Eigenfunction decomposition on Riemannian sphere?

Let $${u_k}_{k=1}^infty$$ be a sequence of ($$L^2$$ normalized) mutually orthogonal eigenfunctions of the operator $$-Delta$$ on the sphere $$mathbb{S}^n$$ (here $$Delta$$ is the Laplace Beltrami operator). Let $$u$$ be a smooth (real valued) function on the sphere. It is a well-known result that we can write $$u=sum_{k=1}^infty c_k u_k$$ for some (real) constants $$c_k$$. My question is: Is the convergence of this sum uniform?

I am trying to prove that the optimal constant in the Poincare inequality is $$lambda_1=n$$. That is to say, I am trying to prove the inequlity $$int_{mathbb{S}^n} |nabla u|^2 geq n int_{mathbb{S}^n} |u|^2$$. Here is what I have done so far:

First, integrate by parts on the LHS so that it suffices to prove $$int_{mathbb{S}^n} -uDelta u geq n int_{mathbb{S}^n} |u|^2$$. Then use $$u=sum_{k=1}^infty c_k u_k$$ and assume that the convergence is uniform. Then we can switch the order of the sum with the derivative and integral (and use the fact that $${u_k}$$ are orthonomal) so that
begin{align*} int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)Delta left(sum_{j=1}^infty c_j u_jright)&= int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty c_j Delta u_jright)= int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty lambda_j c_j u_jright) \&= sum_{j,k}c_k c_j lambda_jint_{mathbb{S}^n} u_k u_j= sum_{j,k}c_k c_j lambda_j delta_{jk}=sum_{j}c_j^2 lambda_jgeq lambda_1 sum_j c_j^2. end{align*}
By the same logic, the last sum is equal to $$int |u|^2$$.

Now obviously, this proof requires some argument showing that the sum commutes with $$Delta$$ and the integral but I have not been able to find a reference that the sum converges uniformly. My thought is that this would follow from some basic facts in Harmonic analysis though I am no expert in that field. Would anyone be able to provide a reference for this?

## pr.probability – Comparison of the distribution of uniform r.v.s with $mathcal{N}(0, 1)$

$$​Given :\ 1. X_1, X_2, X_3, … are independent random variables \ X_n sim Uniform(-n , 3n) where n = 1, 2, ..\ 2. S_N = frac{1}{sqrt{N}}sum_{n=1}^{N} frac{X_n}{n} where N = 1, 2, ..\ 3. F_N be the distribution function of S_N.\ 4. phi sim mathcal{N}(0, 1)\\$$

$$How to prove the following:\ a. lim_{N to infty} F_N(0) leq phi(0) \ b. lim_{N to infty} F_N(1) leq phi(1)$$

## plotting – Uniform Color distribution with the command Show[]

I have the following definitions:

a=Sqrt(2 + 2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)));

a3=Sqrt(5 + 4 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)));

beta = 2 a1^2 + 2 a2^2 + 2 a3^2 + 2 a1^2 a2^2 + 2 a1^2 a3^2 +
2 a2^2 a3^2 - a1^4 - a2^4 - a3^4 - Sqrt(((a + a + a3)^2 - 1)*((a - a + a3)^2 -
1)*((a + a - a3)^2 - 1)*((a - a - a3)^2 - 1)) - 1;


then, I define the plots A and B as follows:

A = Plot3D({1/2 Log((beta/(8 a^2)))}, {r, -1.0, 1.0}, {(Theta),
0.01 (Pi), 1.99 (Pi)},
RegionFunction ->
Function({r, (Theta)},
0 < Sqrt(
2 + 2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) -
1/Sqrt(2) ((Sqrt)(((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
2 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r))
Sqrt(Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) +
Abs(-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r))) (Sqrt)((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
8 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r))
Sqrt(Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r)))))/(7 +
6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))))),
PerformanceGoal -> "Quality", AxesLabel -> Automatic,
PlotRange -> All, PlotPoints -> 30, Mesh -> 5, MaxRecursion -> 7,
ColorFunction -> "TemperatureMap");
(*-------------------------------------------*)

B=Plot3D({1/2 Log(((a^2 - a3^2)/(a^2 - 1))^2)}, {r, -1.0,
1.0}, {(Theta), 0.01 (Pi), 1.99 (Pi)},
PerformanceGoal -> "Quality", AxesLabel -> Automatic,
RegionFunction ->
Function({r, (Theta)},
0 >= Sqrt(
2 + 2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) -
1/Sqrt(2) ((Sqrt)(((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
2 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r))) +
Abs(-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r))) (Sqrt)((-3 -
2 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))^2 +
8 (7 + 6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r))
Sqrt(Cosh(2 r) +
Cos(2 (Theta)) Sinh(2 r)))))/(7 +
6 Sqrt(Cosh(2 r) - Cos(2 (Theta)) Sinh(2 r)) Sqrt(
Cosh(2 r) + Cos(2 (Theta)) Sinh(2 r)))))),
PlotRange -> All, PlotPoints -> 30, Mesh -> 5, MaxRecursion -> 8,
ColorFunction -> "TemperatureMap");


then, by the command Show() I join the two plots obtaining

Therefore, both 3Dplots match, as I expected; however, I have the question:

(1) There is a way to show a uniform color distribution for both plots by using the comand Show()?

that is, each plot appears with its own color distribution when I display both with Show(). This is logical since I define separately each function. On the other hand, it must be noted that the region function for plot A is of the form: RegionFunction -> Function({r, (Theta)}, 0 <f) and for B is RegionFunction -> Function({r, (Theta)}, 0 >=f) being f the function of $$r$$ and $$theta$$ displayed in the code, which could help to define a conditional to display a single plot without the need to use Show().

## banach spaces – Uniform smoothness and twice-differentiability of norms

To get to the simplest case, consider a norm $$|cdot|$$ over $$R^n$$ that is uniformly convex of power-type 2, that is, there is a constant $$C$$ such that $$frac{|x+y| + |x – y|}{2} le 1 + C |y|^2$$ for all $$x$$ with $$|x| = 1$$ and for all $$y$$.

Question: Does this guarantee that $$|cdot|$$ has a second-order Taylor expansion on $$R^n setminus {0}$$, that is, there is a vector $$g$$ and a symmetric matrix $$A$$ such that $$|x + y| = |x| + langle g, y rangle + frac{1}{2} langle Ay, y rangle + o(|y|^2)$$ for all $$x neq 0$$. (Apparently this is a weaker requirement than twice-differentiability of $$|cdot|$$ on $$R^n setminus {0}$$)

It is easy to see that $$|cdot|$$ is differentiable on $$R^n setminus {0}$$, and a classic result of Alexandrov guarantees that the above second-order Taylor expansion holds for any convex function on almost every point $$x$$. It is also known that the norm of any separable Banach space can be approximated arbitrarily well by a power-type 2 norm that is twice differentiable on $$R^n setminus {0}$$ (see Lemma 2.6 here). But I wonder if the original norm itself has a second-order Taylor expansion.

## gn.general topology – Uniform spaces as condensed sets

$$DeclareMathOperatorHom{Hom}DeclareMathOperatorUnif{Unif}DeclareMathOperatorCHaus{CHaus}DeclareMathOperatorSet{Set}DeclareMathOperatorop{op}DeclareMathOperatorInd{Ind}DeclareMathOperatorFin{Fin}$$In Barwick-Haine Example 2.1.10, they showed that the functor $$Hom_{Unif}(-,X)colonCHaus^{op}toSet$$ is a pyknotic set, i.e., a sheaf on the site $$CHaus$$ of compact Hausdorff spaces equipped with the coherent topology.

Question I wonder how much is known about realization of uniform spaces as condensed sets.

I did not check whether the sheaf above is accessible (when restricted to $$Ind(Fin^{op})$$, but I am slightly skeptical about this approach. It seems to me that the “correct” realization of a uniform space should be a condensed set which records its underlying topological space, along with an extra structure which records the uniform structure.

I suppose that this extra structure would be described by a certain kind of enriched groupoid. Indeed, the uniform structure on a topological space could be understood as a groupoid enriched in filters. See nLab page for a description of this sort.

This is motivated by an attempt to eliminate the restriction that the adjective “solid” only applies to condensed abelian groups.

Let $$M$$ be a topological abelian group. I was about to understand what it means for $$M$$ that the condensed abelian group $$underline M$$ is solid. Following Lecture II, for any sequence $$(m_n)_{ninmathbb N}in M^{mathbb N}$$ convergent to $$0$$, we associate a (continuous) map from the profinite set $$S:=mathbb Ncup{infty}$$ (there seems a bug in MathJax which renders { as C) to $$M$$ which maps $$n$$ to $$m_n$$ and $$infty$$ to $$0$$, or equivalently, a map $$Stounderline M$$ of condensed sets by, say, Yoneda’s lemma.

Suppose that $$underline M$$ is solid, then this map extends uniquely to a map $$mathbb Z(S)^blacksquaretounderline M$$ of condensed abelian groups. If I am not mistaken, $$mathbb Z(S)^blacksquareto M$$ further factors through $$underline{mathbb Z((t))}$$, the condensed abelian group associated to the topological abelian group $$mathbb Z((t))$$ with $$(t)$$-adic topology, and by full faithfulness, we get a factorization $$Stomathbb Z((t))to M$$ where the second map is additive.

Consequently, for every sequence $$(a_n)_{ninmathbb N}in{mathbb Z}^{mathbb N}$$ of integers, the series $$sum_na_nt^n$$ converges in $$mathbb Z((t))$$, therefore the series $$sum_na_nm_n$$ converges in $$M$$, which should imply, if I am not mistaken, that the uniform structure on $$M$$ is non-archimedean and complete, at least when $$M$$ is first countable (by the way, I don’t understand why it is claimed that it is not directly as any kind of limit of finite sums).

So the non-archimedean nature is rooted in the formalism. I suppose that a natural approach is to generalize the uniform structure to condensed sets, and to generalize the classical Cauchy-completeness. I don’t know whether it is convinced that this does not work. The current presentation separates non-archimedean and $$mathbb R$$-case, which covers neither non-abelian groups nor the general completeness of topological abelian groups.

## how can I generate a random number from 1e-9 to 1e9 with uniform probability in r?

I am trying to generate a random number from 1e-9 to 1e9. The very naive idea is to generate a sequence from 1 to 1e18, then divide by 1e9. as following.
but seems not working.

set.seed(100)
rand <- sort(runif(10000, min =1, max= 1e18))/1e9
result <- sample(rand, 1)
min(rand)
max(rand)

# result 664426274
# min(rand) 199051.1
# max(rand) 999853646


seems the min number is much higher than I expect.

## coding theory – Prove that probabilistic adaptive algorithm that can explore only k bits of n bit input can’t distinguish k-independent distribution from uniform

Definition: Distribution $$D$$ on $${0,1}^n$$ is called k-independent if for every random variable $$X$$ with distribution $$D$$ and for all $$i_1, dots, i_k in {1,2,dots,n}$$ random variable $$X_{i_1,dots, i_k}$$ has distribution $$U_k$$ (uniformal).

Problem: Consider probabilistic algorithm A that has an oracle access to input of length $$n$$. It means that algorithm $$A$$ can adaptively request $$k$$ bits of input (in more detail, $$A$$ can request one bit, then based on the oracle answer request another bit and repeat it not more than $$k$$ times).
Prove that if $$D$$ is k-independent distribution than $$Pr_{x sim D} (A(x) = 1) = Pr_{x sim U_n}(A(x) = 1)$$

First of all I don’t understand how adaptivity can potentially help Algorithm to distinguish uniformal distribution from k-independent. Second question is in the problem.

## ag.algebraic geometry – Uniform Łojasiewicz constant in 2D

Łojasiewicz inequality is a classical result in real algebraic geometry. In particular, for any given polynomial $$f:mathbb R^2to mathbb R$$ there is some $$C>0$$ and some $$alpha>0$$ such that for all $$|x|<1$$ we have
$$mathrm{dist}(x,Z(f))^alphaleq C|f(x)|,$$
where $$Z(f)$$ denotes the zero set of $$f$$. There has been abundant reseach into the optimal exponent $$alpha$$, and in the 2D case there is an elementary paper by Kuo (1974):

for an explicit computation of the exponent $$alpha$$.
However, I was wondering if there is a uniform estimate (in terms of the degree of $$f$$) of the constant $$C$$, even in 2D. Of course this question would make no sense if the coefficients of $$f$$ can be arbitrarily small; for this reason I require that at least one coefficient of $$f$$ has absolute value bounded below by 1.