Distribution of points on the circle

In connection with the proof of his celebrated “2.4-Theorem”, Freiman proved that if α₁,…,αN are real numbers such that each interval [u,u+1/2) contains at most n of the αj mod 1, then . Freiman's result was extended by Moran and Pollington, and recently by Lev. This paper contains further extensions.     

Essentialité dans les bases additives

In this article we study the notion of essential subset of an additive basis, that is to say the minimal finite subsets P of a basis A such that A−P does not remains a basis. The existence of an essential subset for a basis is equivalent for this basis to be included, for almost all elements, in an arithmetic non-trivial progression. We show that for every basis A there exists an arithmetic progression with a biggest common difference containing A. Having this common difference a we are able to give an upper bound to the number of essential subsets of A: this is the radical's length of a (in particular there is always many finite essential subsets in a basis). In the case of essential subsets of cardinality 1 (essential elements) we introduce a way to “dessentialize” a basis. As an application, we definitively complete the result of Deschamps and Grekos about the majoration of essential elements of a basis by showing that for all basis A of order h, the number s of essential elements of A satisfy where , and we show that this inequality is best possible.     

On Atkin and Swinnerton-Dyer congruence relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the p-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigenforms. We also show that there is an automorphic L-function over whose local factors agree with those of the l-adic Scholl representations attached to the space of noncongruence cusp forms. The research of the second author was supported in part by an NSA grant #MDA904-03-1-0069 and an NSF grant #DMS-0457574. Part of the research was done when she was visiting the National Center for Theoretical Sciences in Hsinchu, Taiwan. She would like to thank the Center for its support and hospitality. The third author was supported in part by an NSF-AWM mentoring travel grant for women. She would further thank the Pennsylvania State University and the Institut des Hautes études Scientifiques for their hospitality.     

On Birkhoff interpolation (0;2) case

P. Turán and his associates^[2] considered in detail the problem of (0,2) interpolation based on the zeros of π_n(x). Motivated by these results and an earlier result of Szabados and Varma^[9] here we consider the problem of existence, uniqueness and explicit representation of the interpolatory polynomial R_n(x) satis fying the function values at one set of nodes and the second derivative on the other set of nodes. It is important to note that this problem has a unique solution provided these two sets of nodes are chosen properly. We also promise to have an interesting convergence theorem in the second paper of this series, which will provide a solution to the related open problem of P. Turán.     

On Atkin and Swinnerton-Dyer congruence relations (3)

It is now well known that Hecke operators defined classically act trivially on genuine cuspforms for noncongruence subgroups of SL₂(Z). Atkin and Swinnerton-Dyer speculated the existence of p-adic Hecke operators so that the Fourier coefficients of their eigenfunctions satisfy three-term congruence recursions. In the previous two papers with the same title ([W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148] by W.C. Li, L. Long, Z. Yang and [A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358] by A.O.L. Atkin, W.C. Li, L. Long), the authors have studied two exceptional spaces of noncongruence cuspforms where almost all p-adic Hecke operators can be diagonalized simultaneously or semi-simultaneously. Moreover, it is shown that the l-adic Scholl representations attached to these spaces are modular in the sense that they are isomorphic, up to semisimplification, to the l-adic representations arising from classical automorphic forms.In this paper, we study an infinite family of spaces of noncongruence cuspforms (which includes the cases in [W.C. Li, L. Long, Z. Yang, On Atkin and Swinnerton-Dyer congruence relations, J. Number Theory 113 (1) (2005) 117-148; A.O.L. Atkin, W.C. Li, L. Long, On Atkin and Swinnerton-Dyer congruence relations (2), Math. Ann. 340 (2) (2008) 335-358]) under a general setting. It is shown that for each space in this family there exists a fixed basis so that the Fourier coefficients of each basis element satisfy certain weaker three-term congruence recursions. For a new case in this family, we will exhibit that the attached l-adic Scholl representations are modular and the p-adic Hecke operators can be diagonalized semi-simultaneously.     

Weak convergence of approximate solutions of stochastic equations with applications to random differential and integral equations ∗

In this paper, we considerably extend our earlier result about convergence in distribution of approximate solutions: of random operator equations, where the stochastic inputs and the underlying deterministic equation are simultaneously approximated. As a by-product, we obtain convergence results for approximate solutions of equations between spaces of probability measures. We apply our results to random Fredholm integral equations of the second kind and to a random [nbar]onlinear elliptic boundary value problem.     

Central derivatives of <Emphasis Type="Italic">L</Emphasis>-functions in Hida families

We prove a result of the following type: given a Hida family of modular forms, if there exists a weight two form in the family whose L-function vanishes to exact order one at s = 1, then all but finitely many weight two forms in the family enjoy this same property. The analogous result for order of vanishing zero is also true, and is an easy consequence of the existence of the Mazur–Kitagawa two-variable p-adic L-function. This research was supported in part by NSF grant DMS-0556174.     

Extending Färe and Zelenyuk (2003)

In our recent work, Färe and Zelenyuk (2003) [On aggregate Farrell efficiency scores. European Journal of Operational Research 146 (3), 615–620], we have proposed a way of aggregating Farrell-type efficiency scores with weights (and aggregation function) derived from economic-type optimization behaviour. In this comment we correct a mathematical error present in that work as well as generalize the price-independent weights we have proposed earlier.     

离散参数集值（上下）鞅的停时定理

本文建立了更广泛的各种集值（上下）鞅的停时定理；推广并改进了N．S．Papageoriou[10]和张，汪，高[13]中的结果。     

Mixed sums of squares and triangular numbers (III)

In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p²=x²+8(y²+z²) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.     

On the linearity of ω-primality in numerical monoids

In an atomic, cancellative, commutative monoid, the ω-value measures how far an element is from being prime. In numerical monoids, we show that this invariant exhibits eventual quasilinearity (i.e., periodic linearity). We apply this result to describe the asymptotic behavior of the ω-function for a general numerical monoid and give an explicit formula when the monoid has embedding dimension 2.     

On a new circular summation of theta functions

In this paper, we prove a new formula for circular summation of theta functions, which greatly extends Ramanujan's circular summation of theta functions and a very recent result of Zeng. Some applications of this circular summation formula are given. Also, an imaginary transformation for multiple theta functions is derived.     

<Emphasis Type="Italic">p</Emphasis>-adic interpolation of Taylor coefficients of modular forms

In this paper we show that the Taylor coefficients of a Hecke eigenform at a CM-point, suitably modified, form a sequence of algebraic numbers that satisfy the Kubota–Leopoldt generalization of the Kummer congruences for primes that split in the imaginary quadratic field associated with a CM-point. More generally, we show that these numbers are moments of a certain p-adic measure. In addition, we write down explicitly the “Euler factor” at p in terms of the p th Hecke eigenvalue of the modular form in question and certain data of the CM-point. P. Guerzhoy is supported by NSF grant DMS-0700933.     

Pattern sequences in ‹q, r›- numeration systems

Let q ≥ 2 and 0 ≤ r ≤ q − 2 be integers. In this paper, we study pattern sequences for patterns in ‹q, r›-numeration systems through their generating functions. Our result implies that any nontrivial linear combination over ? of pattern sequences chosen from different ‹q, r›-numeration systems cannot be a linear recurrence sequence. In particular, pattern sequences in different ‹q, r›-numeration systems are linearly independent over ?, while within one ‹q, r›-numeration system they can be linearly dependent ?.     

Normal integral bases for cyclic Kummer extensions

Let K/F be a Kummer cyclic extension of number fields. In the case when the degree is a prime number, Gómez Ayala gave an explicit criterion for the existence of a normal integral basis. More recently Ichimura proposed a generalization of that result for cyclic extensions of arbitrary degree, but we have found that Ichimura’s result is incorrect. In this paper we present a counter-example to Ichimura’s result as well as the correct generalization of Gómez Ayala’s result.     

Square classes of totally positive units

In this article, we investigate some conditions for a real cyclic extension K over Q to satisfy the property that every totally positive unit of K is a square. As an application, we give a partial answer to Taussky's conjecture. We then extend our result to real abelian extensions of certain type.     

Coproducts in the category $${{\mathcal {S}}}(B)$$ S ( B ) of Segal topological algebras,revisited

Periodica Mathematica Hungarica - In this paper we generalize the result (obtained by the author earlier in the paper titled “Products and coproducts in the category $${{\mathcal {S}}}(B)$$...     

Asymptotic normality of additive functions on polynomial sequences in canonical number systems

The objective of this paper is the study of functions which only act on the digits of an expansion. In particular, we are interested in the asymptotic distribution of the values of these functions. The presented result is an extension and generalization of a result of Bassily and Kátai to number systems defined in a quotient ring of the ring of polynomials over the integers.     

Exact formulas for traces of singular moduli of higher level modular functions

Zagier proved that the traces of singular values of the classical j-invariant are the Fourier coefficients of a weight 3/2 modular form and Duke provided a new proof of the result by establishing an exact formula for the traces using Niebur's work on a certain class of non-holomorphic modular forms. In this short note, by utilizing Niebur's work again, we generalize Duke's result to exact formulas for traces of singular moduli of higher level modular functions.