Series of tail distributions of finitely inhomogeneous random walks

Let Sn=X₁+?+Xn be a random walk, where the steps Xn are independent random variables having a finite number of possible distributions, and consider general series of the form

(∗)     

Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

不同分布ρ混合序列乘积和的强收敛性

研究了ρ混合序列的收敛性质,利用得到的结果和ρ混合序列的矩不等式讨论了ρ混合序列乘积和的强收敛性质.     

On a general class of trigonometric functions and Fourier series†

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented.     

Threefield identities and simultaneous representations of primes by binary quadratic forms

Kaplansky [2003] proved a theorem on the simultaneous representation of a prime p by two different principal binary quadratic forms. Later, Brink found five more like theorems and claimed that there were no others. By putting Kaplansky-like theorems into the context of threefield identities after Andrews, Dyson, and Hickerson, we find that there are at least two similar results not on Brink?s list. We also show how such theorems are related to results of Muskat on binary quadratic forms.     

Conjugacy Classes,Class Sums and Character Tables for Hopf Algebras

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d ^?2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS ₃), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.     

Walker's Cancellation Theorem

Walker's cancellation theorem says that, if B ⊕ Z is isomorphic to C ⊕ Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the original theorem does not have a constructive proof even if B and C are subgroups of the free abelian group on two generators. Both of these results contrast with a group whose endomorphism ring has stable range one, which allows a constructive proof of cancellation and also a proof in any diagram category.     

A nullstellensatz and a positivstellensatz for ordered differential fields

We use the model completeness and axiomatisation of the theory of closed ordered differential fields to give a differential version of Dubois, Krivine and Risler's nullstellensatz and Stengle's positivstellensatz for ordered fields.