相关论文: Some sum-product type estimates for two-variables …
Type III methods were introduced by SAS to address difficulties in dummy-variable models for effects of multiple factors and covariates. They are widely used in practice; they are the default method in several statistical computing…
We consider differences of one- and two-variable finite products and provide combinatorial proofs of the nonnegativity of certain coefficients. Since the products may be interpreted as generating functions for certain integer partitions,…
The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on…
We study product sets of finite arithmetic progressions of polynomials over a finite field. We prove a lower bound for the size of the product set, uniform in a wide range of parameters. We apply our results to resolve the function field…
We propose a two parameter ratio-product-ratio estimator for a finite population mean in a simple random sample without replacement following the methodology in Ray and Sahai (1980), Sahai and Ray (1980), Sahai and Sahai (1985) and Singh…
Let ${\mathcal H}$ be a multiplicative subgroup of $\mathbb{F}_p^*$ of order $H>p^{1/4}$. We show that $$ \max_{(a,p)=1}\left|\sum_{x\in {\mathcal H}} {\mathbf{\,e}}_p(ax)\right| \le H^{1-31/2880+o(1)}, $$ where ${\mathbf{\,e}}_p(z) =…
We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.
The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic…
We prove a Cherednik style $p$-adic uniformization theorem for Shimura varieties associated to certain groups of unitary similitudes of size two over totally real fields. Our basic tool is the alternative modular interpretation of the…
In this paper we prove local analytic hypoellipticity for a degenerate sum of squares of complex vector fields generalizing those of Kohn in "Hypoellipticity and Loss of Derivatives". Kohn's article is to appear in the Annals of Mathematics…
The concept of a composed product for univariate polynomials has been explored extensively by Brawley, Brown, Carlitz, Gao, Mills, et al. Starting with these fundamental ideas and utilizing fractional power series representation (in…
Daily internet communication relies heavily on tree-structured graphs, embodied by popular data formats such as XML and JSON. However, many recent generative (probabilistic) models utilize neural networks to learn a probability distribution…
Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian…
Singh et al (20009) introduced a family of exponential ratio and product type estimators in stratified random sampling. Under stratified random sampling without replacement scheme, the expressions of bias and mean square error (MSE) of…
We give a proposal for future development of the model theory of valued fields. We also summarize some recent results on p-adic numbers.
We generalize Bourgain's discretized sum-product theorem to matrix algebras.
In this thesis we give an in-depth introduction to the General Number Field Sieve, as it was used by Buhler, Lenstra, and Pomerance, before looking at one of the modern developments of this algorithm: A randomized version with provable…
Given a large set $U$ where each item $a\in U$ has weight $w(a)$, we want to estimate the total weight $W=\sum_{a\in U} w(a)$ to within factor of $1\pm\varepsilon$ with some constant probability $>1/2$. Since $n=|U|$ is large, we want to do…
We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form $$\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s,$$ where $p$ is an odd prime and $n, s$ are positive integers. Let us denote by…
A sum rule for determining |V_us| from a combination of hadronic tau decay and electroproduction data is discussed. Indications of problems with analogous, purely tau-decay-based analyses, most likely associated with slow convergence of the…