Prime Producing Quadratic Polynomials and Class Number One or Two

Let d≡ 5 mod 8 be a positive square-free integer and let h(d) be the class number of the real quadratic field ℚ(√d). Let p be a divisor of d = pq and let . Assume that is prime or equal to 1 for all integers x with 0≤x<W. Under the assumption that the Riemann hypothesis is true, we prove that if , then h(d) < 2. Furthermore we show that h(d)< 2 implies d < 4245. In the case when there exists at least one split prime less than W, we prove the following results without any assumptions on the Riemann hypothesis. If then h< 2 or h = 4. If , then h≤ 2, h = 4 or h = 2^t−2, where t is the number of primes dividing d. In the case when h = 2^t−2 we have , where φ = 2 or 4. 2000 Mathematics Subject Classification: Primary–11R29     

Refinement of Convergence Rates for Tail Probabilities

Let X₁, X₂,... be, i.i.d. random variables, and put . We find necessary and sufficient moment conditions for , where δ≥ 0 and q>0, and with a_n>0 and b_n is either or The series f(x) we deal with are classical series studied by Hsu and Robbins, Erdős, Spitzer, Baum and Katz, Davis, Lai, Gut, etc     

On Some Sets with Even Valued Partition Function

Let IN be the set of positive integers, = {b₁ < ⋅s < b_h}⊂IN, N∊IN and N≥ b_h. = ₀( ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy) such that {1,..., N} = and p( ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( ,n) denotes the number of partitions of n with parts in . Let us denote by σ ( ,n) the sum of the divisors of n belonging to . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( ,2^k n))_{n≥ 1}mod 2^k+1 is periodic with an odd period q_k. In this paper, we will characterize for any fixed odd positive integer q, the sets and the integers N such that q₀ = q, and those for which q_k = q for all k≥ 0. Moreover, a set = ₀( ,N) is constructed with the property that its period, i.e. the period of (σ( ,n))_{n≥ 1}mod 2, is 217, and for which the counting function is asymptotically equal to that of ₀({1,2,3,4,5},5) which is a set of period 31.Dedicated to Professor J.-L. Nicolas on the occasion of his 60th birthday2000 Mathematics Subject Classification: Primary—11P81, 11P83Research supported by MIRA 2002 program n^o 0203012701, Number Theory, Lyon-Monastir.     

Spherical Monogenics: An Algebraic Approach

The space of spherical monogenics in can be regarded as a model for the irreducible representation of Spin(m) with weight . In this paper we construct an orthonormal basis for . To describe the symmetry behind this procedure, we define certain Spin(m − 2)-invariant representations of the Lie algebra (2) on . Received: October, 2007. Accepted: February, 2008.     

Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

Irrationality measures are given for the values of the series , where and W_n is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality of all the numbers

A singular Moser-Trudinger embedding and its applications

Let Ω be a bounded domain in , we prove the singular Moser-Trudinger embedding: if and only if where and . We will also study the corresponding critical exponent problem.     

Variations on a Theme of Hardy’s

Let θ > 1 and let ϕ : [0,1] → ℂ be such that the two-sided series converges for all x ∊ [0,1], (then necessarily φ(0) = φ(1) = 0).Suppose Define For different classes of functions φ we show that À notre ami, Jean-Louis Nicolas2000 Mathematics Subject Classification: Primary—11B83, 11B99     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Pointwise Estimates for Linear Combinations of Gamma Operators

For linear combinations of Gamma operators G_n,r(f,x), we give an equivalent theorem with (f,t), where (f,t) are the Ditzian-Totik moduli of smoothness ( ).AMS Subject Classification (1991): 41A25, 41A36.Supported by NSF of Hebei ProvinceSupported by NSF of Hebei Province     

Cubic elliptic functions

The function

Spectral Shift Function in the Large Coupling Constant Limit

In the large-coupling constant limit we obtain an asymptotic expansion in powers of of the derivative of the spectral shift function corresponding to . Here the potential W(x) is positive and near infinity for some δ> n and ω₀ ∈C^∞ (S^n-1). submitted 18/05/05, accepted 19/10/05     

Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Let A be a separable unital nuclear simple C＊-algebra with torsion K0 （A）, free K1 （A） and with the UCT. Let T ： A→M（K）/K be a unital homomorphism. We prove that every unitary element in the commutant of T（A） is an exponent, thus it is liftable. We also prove that each automorphism α on E with α ∈ Aut0（A） is approximately inner, where E is a unital essential extension of A by K and α is the automorphism on A induced by α.     

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of and to characterize the elements of in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals of , j = 0,..., , such that is the linear span of sums of permutations with a common set of interior peaks and is the peak algebra. We extend the above results to , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation     

Discrete Periodic Sampling with Jitter and Almost Periodically Correlated Processes

The zero-mean process is said to be almost periodically correlated whenever its shifted covariance kernel is almost periodic in t uniformly with respect to . Then it admits a Fourier–Bohr decomposition: . This paper deals with the estimation of the spectral covariance a(λ,τ) from a discrete time observation of the process , when jitter and delay phenomena are present in conjunction with periodic sampling. Under mixing conditions, we establish the consistency and the asymptotic normality of empirical estimators as the sampling time step tends to 0 and the sampling period tends to infinity.      

Lower bounds of some singular integrals and their applications

Let and . We are interested in the lower bounds of the integral:

Growth Estimates for Cauchy Integrals of Measures and Rectifiability

We show that if μ is a finite Borel measure on the complex plane such that

Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.     

The Polynomial Degree of the Grassmannian {\mathcal G_{\bf 1,}{\bf n,}{\bf 2}}

For a subset ψ of PG(N, 2) a known result states that ψ has polynomial degree ≤ r, r≤ N, if and only if ψ intersects every r-flat of PG(N, 2) in an odd number of points. Certain refinements of this result are considered, and are then applied in the case when ψ is the Grassmannian , to show that for n <8 the polynomial degree of is .