Irreducibility of A-hypergeometric systems

We give an elementary proof of the Gel'fand-Kapranov-Zelevinsky theorem that non-resonant A-hypergeometric systems are irreducible. We also provide a proof of a converse statement In this second version we have removed the condition of saturatedness in Theorems 1.2 and 1.3. In Theorem 1.3 it is replaced by the condition of Cohen-Macaulayness of the toric ideal

Irrationality of some p-adic L-values

We give a proof of the irrationality of the $p$-adic zeta-values $\zeta_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show irrationality of some other $p$-adic $L$-series values, and values of the $p$-adic Hurwitz zeta-function

Sequences, modular forms and cellular integrals

It is well-known that the Ap\'ery sequences which arise in the irrationality proofs for $\zeta(2)$ and $\zeta(3)$ satisfy many intriguing arithmetic properties and are related to the $p$th Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.Comment: 26 pages, to appear in Mathematical Proceedings of the Cambridge Philosophical Societ

On B.Dwork's accessory parameter problem

Let P ε C [z] be a monic quadratic polynomial with non-zero discriminant and P(0) ≠ 0. Let λ ε C. Consider the linear differential equation zP(z) d2u/dz2 + (zP(z)) ,du/dz + (z-λ)u=0. Note that this is the general shape of a Fuchsian differential equation on P1 with singularities in four points, including ∞, having local exponents 0,0 at the nite points and 1; 1 at ∞. By scaling z if necessary we can assume that P has the form P(z) =z2+az-

Vakantiecursus 1999: P=NP?

De eerste electronische computer, de ENIAC(="Electronic Numeric Integrator and Calculator")) werd in 1946 in gebruik genomen en luidde het tijdperk van electronisch rekenen in. Deze kolos bevatte maar liefst 18.800 radiobuizen en was in staat om zo'n 5000 elementaire bewerkingen per seconde uit te voeren. Voor die tijd een onvoorstelbare snelheid. Maar de tijd heeft niet stilgestaan

Становлення і розвиток православних духовних семінарій на Правобережній Україні (кінець ХVІІІ – перша половина ХІХ ст.)

In this note we prove results of the following types. Let be given distinct complex numbers satisfying the conditions for and for every there exists an such that . Then If, moreover, none of the ratios with is a root of unity, then The constant −1 in the former result is the best possible. The above results are special cases of upper bounds for obtained in this paper

Zeeman's monotonicity conjecture

In this paper we prove a conjecture of Zeeman about the monotonicity of the rotation number of a family of dieomorphisms φ of the first quadrant Q of R

Cyclotomic points on curves

We show that a plane algebraic curve f = 0over the complex numbers has on it either at most 22V (f) points whose coordinates are both roots of unity, or innitely many such points. Here V (f) is the area of the Newton polytope of f: We present an algorithm for nding all these points