We give a simplified derivation of the expression of instanton numbers and of mirror map in terms of Frobenius map on p-adic cohomology and use this expression to prove integrality theorems. Modifying this proof we verify that the Aganagic-Vafa formulas for the number of holomorphic disks can be expressed in terms of Frobenius map on p-adic relative cohomology; this expression permits us to pro...

FEASIBILITY OF INTEGER KNAPSACKS∗ ISKANDER ALIEV† AND MARTIN HENK‡ Abstract. Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider the set F(A) of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ R≥0 : Ax = b} contains an integer point. When m = 1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number ass...

where a1, . . . , an are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a1, . . . , an, find the largest value of t (the Frobenius number) such that m1a1 + · · · + mnan = t has no solution in positive integers m1, . . . , mn. This is equivalent to the problem of finding the largest dilate tP such that ...

We give further results for Perron-Frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. We indicate two techniques for establishing the main theorem ofPerron and Frobenius on the numerical range. In the rst method, we use acorresponding version of Wielandt's lemma. The second technique involves graphtheory.

ion is analogous to the role of Galois categories in Galois theory or monoids in the geometry of log schemes. This abstract category-theoretic framework preserves many of the important features of the classical theory of divisors and line bundles on models of finite separable extensions of a function field or number field such as the global degree of an arithmetic line bundle over a number fiel...

We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality ax mod b 6 x, with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of P1 with ramification to order ei at general points Pi, in the case that all ei are less than the characteristic. We also develop a new, more functorial construction for the basic theory of limit linear series, which works transparently in posi...

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and p-genus of generalized Fibonacci numerical semigroups. Here, p-numerical semigroup Sp is defined as set integers whose integral linear combinations given positive a1,a2,…,ak are expressed in more than p ways. When p=0, S0 with 0-Frobenius 0-genus original Frobenius genus. In this paper, consider involving Jaco...