Strong Approximation Theorem Research Articles

Optimal lifting for the projective action of SL3(ℤ)

Let $\epsilon>0$ and let $q$ be a prime going to infinity. We prove that with high probability given $x,y$ in the projective plane over the finite field $F_q$ there exists $\gamma$ in $SL_3(Z)$, with coordinates bounded by $q^{1/3+\epsilon}$, whose projection to $SL_{3}(F_q)$ sends $x$ to $y$. The exponent $1/3$ is optimal and the result is a high rank generalization of Sarnak's optimal strong approximation theorem for $SL_2(Z)$.

Zariski density and computing with S-integral groups

We generalize our methodology for computing with Zariski dense subgroups of SL(n,Z) and Sp(n,Z), to accommodate input dense subgroups H of SL(n,Q) and Sp(n,Q). A key task, backgrounded by the Strong Approximation theorem, is computing a minimal congruence overgroup of H. Once we have this overgroup, we may describe all congruence quotients of H. The case n=2 receives particular attention.

The nonlinear wave solutions and parameters discovery of the Lakshmanan-Porsezian-Daniel based on deep learning

We apply the deep learning approach to learn some nonlinear wave solutions of the Lakshmanan-Porsezian-Daniel (LPD) model characterizing the evolution of ultrashort optical pulse in optical fibers. Based on the strong universal approximation theorem, we give the initial-boundary value data and residual collocation points, choose the parameters initialization Xavier method and parameters optimization Adam and L-BFGS algorithms to construct the optimal neural network model. Then, we derive the data-driven solutions of the rogue wave, anti-dark soliton, multi-peak soliton, non-rational W-shaped soliton, rational W-shaped soliton as well as periodic-wave solutions for the LPD model. Finally, we study the parameters discovery of such model via the anti-dark soliton solution with 1% perturbation (or without perturbation).

Bayesian inference of a stochastic diffusion process for the dynamic of HIV in closed heterosexual population with simulations and application to Morocco case

A model of the HIV dynamic in a heterosexual population of fixed size is considered in this paper. We represent the dynamic by a multidimensional SIR model. We formulate the stochastic diffusion approximation process to describe the dynamic of HIV by using strong approximation theorems for density-dependent Markov chains. Our aim is to estimate the parameters of this model; to reach this goal, we use Bayesian inference with Markov chain Monte Carlo (MCMC) simulations. We prove that the posterior distributions of the parameters are shifted Generalized Inverse Gaussian (GIG). The obtained results are well illustrated by simulations and real application of Morocco’s case.

On class number relations and intersections over function fields

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question.

Strong Embeddings for Transitory Queueing Models

In this paper, we establish strong embedding theorems, in the sense of the Komlós-Major-Tusnády framework, for the performance metrics of a general class of transitory queueing models of nonstationary queueing systems. The nonstationary and non-Markovian nature of these models makes the computation of performance metrics hard. The strong embeddings yield error bounds on sample path approximations by diffusion processes in the form of functional strong approximation theorems.

Gaussian process approximations for multicolor Pólya urn models

AbstractMotivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Optimal strong approximation for quadratic forms

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in $d\geq5$ variables, we prove an optimal strong approximation theorem. Let $\Omega$ be a fixed compact subset of the affine quadric $F(x_1,\dots,x_d)=1$ over the real numbers. Take a small ball $B$ of radius $0<r<1$ inside $\Omega$, and an integer $m$. Further assume that $N$ is a given integer which satisfies $N\gg_{\delta,\Omega}(r^{-1}m)^{4+\delta}$ for any $\delta>0$. Finally assume that an integral vector $(\lambda_1, \dots, \lambda_d) $ mod $m$ is given. Then we show that there exists an integral solution $X=(x_1,\dots,x_d)$ of $F(X)=N$ such that $x_i\equiv \lambda_i \text{ mod } m$ and $\frac{X}{\sqrt{N}}\in B$, provided that all the local conditions are satisfied. We also show that 4 is the best possible exponent. Moreover, for a non-degenerate integral quadratic form in 4 variables we prove the same result if $N$ is odd and $N\gg_{\delta,\Omega} (r^{-1}m)^{6+\epsilon}$. Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square root cancellation in a particular sum that appears in Remark~\ref{evidence}, we conjecture that the theorem holds for any quadratic form in 4 variables with the optimal exponent $4$.

The strong approximation theorem and computing with linear groups

We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group H≤SL(n,Z) for n≥2. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of SL(n,Q) for n>2.

A spectral strong approximation theorem for measure-preserving actions

Let$\unicode[STIX]{x1D6E4}$be a finitely generated group acting by probability measure-preserving maps on the standard Borel space$(X,\unicode[STIX]{x1D707})$. We show that if$H\leq \unicode[STIX]{x1D6E4}$is a subgroup with relative spectral radius greater than the global spectral radius of the action, then$H$acts with finitely many ergodic components and spectral gap on$(X,\unicode[STIX]{x1D707})$. This answers a question of Shalom who proved this for normal subgroups.

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.

Strong approximation of the St. Petersburg game

ABSTRACTLet be i.i.d. random variables with and let . The properties of the sequence have received considerable attention in the literature in connection with the St. Petersburg paradox (Bernoulli 1738). Let be a semistable Lévy process with underlying Lévy measure . For a suitable version of and , we prove the strong approximation a.s. This provides the first example for a strong approximation theorem for partial sums of i.i.d. sequences not belonging to the domain of attraction of the normal or stable laws.

Constructive General Neron Desingularization for one dimensional local rings

An algorithmic proof of General Neron Desingularization is given here for one dimensional local rings and it is implemented in Singular. Also a theorem recalling Greenberg' strong approximation theorem is presented for one dimensional local rings.

The local Langlands correspondence in families and Ihara's lemma for U(n)

The goal of this paper is to reformulate the conjectural Ihara lemma for U(n) in terms of the local Langlands correspondence in families π˜Σ(⋅), as developed by Emerton and Helm. The reformulation roughly takes the following form. Suppose we are given an irreducible mod ℓ Galois representation r¯=r¯m, which is modular of full level (and small weight), and a finite set of places Σ, none of which divides ℓ. Then π˜Σ(rm) exists, and has a global realization as a localized module of ℓ-integral algebraic modular forms, where rm is the universal deformation of r¯ of type Σ. This is unconditional for n=2, where Ihara's lemma is an almost trivial consequence of the strong approximation theorem. We emphasize that throughout we will work with banal primes ℓ>n. That is, those for which #k(v)i≢1 (mod ℓ), where 1≤i≤n, for v∈Σ. A weakening of this assumption (quasi-banal) is ubiquitous in [CHT] (particularly in their Chapter 5 on the non-minimal case and Ihara), where the mod ℓ representation theory of GLn(Fv˜) plays a pivotal role. We will need actual banality, to ensure the functors we define are exact. Banality also removes many of the subtleties in the Emerton–Helm correspondence. For example, in yet unpublished work [Hel13], Helm shows that local Langlands in families do exist in the banal case, although we do not use his result. We should stress that our main result has been common knowledge among experts for some time, but its proof has never appeared in print.

On the Local Time of the Weighted Bootstrap and Compound Empirical Processes

This article is mainly concerned with the local times of the weighted bootstrap process. We prove a strong approximation theorem for the local time of the weighted bootstrap process by the local time of a Brownian bridge. We consider also the local time of the compound empirical processes that can be seen, asymptotically, as the local time of the convolution of two independent Gaussian processes.

Simultaneous inference of the mean of functional time series

For functional time series with physical dependence, we construct confidence bands for its mean function. The physical dependence is a general dependence framework, and it slightly relaxes the conditions of m-approximable dependence. We estimate functional time series mean functions via a spline smoothing technique. Confidence bands have been constructed based on a long-run variance and a strong approximation theorem, which is satisfied with mild regularity conditions. Simulation experiments provide strong evidence that corroborates the asymptotic theories. Additionally, an application to S&P500 index data demonstrates a non-constant volatility mean function at a certain significance level.

A note on the strong approximation for long memory processes and its application

Let {X k ; k≥1} be a long memory process defined by where {ϵ i ;−∞<i<∞} is a doubly infinite sequence of independent and identically distributed random variables and a i ∼ i −α l(i) for some 1/2<α<1 is a sequence of real numbers. Under some mild conditions, a general strong approximation theorem for partial sums of {X k ; k≥1} is derived, where the variances of the innovations may be infinite. As applications, we establish a general law of the iterated logarithm for the long memory processes and investigate the asymptotic properties of the heavy-tailed long memory model and the adjusted range of partial sums for a kind of long memory processes.

Uniform bounds on symbolic powers

We study the question of obtaining uniform bounds for the growth of symbolic powers of ideals in Noetherian rings. First we use a generalized version of Artinʼs strong approximation theorem to obtain uniform bounds for prime ideals in the integral closure of a finite Galois field extension of the fraction field of an excellent Henselian regular local ring. As a corollary, we obtain a similar result for a complete Noetherian local domains. We also obtain linear uniform bounds for the growth of symbolic powers in certain normal subrings of equicharacteristic regular rings under mild conditions.

Strong convergence theorems of strongly relatively nonexpansive semi-group in Banach spaces

In this paper, we prove some strong convergence approximation theorems of strongly relatively nonexpansive semi-group in the framework of Banach spaces. Using the concept of duality theorems, we obtain analogue results for strongly generalized nonexpansive semi-group and for semi-group of firmly generalized nonexpansive type. The results presented in this paper improve and extend some corresponding results announced by some authors. ∗

Integral points for groups of multiplicative type

We construct a finite subgroup of Brauer–Manin obstruction for detecting the existence of integral points on integral models of homogeneous spaces of linear algebraic groups of multiplicative type. As an application, the strong approximation theorem for linear algebraic groups of multiplicative type is established. Moreover, the sum of two integral squares over some quadratic fields is discussed.