Related papers: Arithmetic of (independent) sigma-fields on probab…
Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities.
In this note, we establish some new results on some special types of function algebras and also give new proofs to some existing ones
We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is…
The article investigates several questions about field generated by certain formal group laws. This is continuation of a study initiated by the author in prior article titled 'Field Generated by Division Points of Certain Formal Group…
In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known results, and also extend the range of such…
In this paper, we provide a general framework for counting geometric structures in pseudo-random graphs. As applications, our theorems recover and improve several results on the finite field analog of questions originally raised in the…
We prove that several results in different areas of number theory such as the divergent series, summation of arithmetic functions, uniform distribution modulo one and summation over prime numbers which are currently considered to be…
We construct a computable, computably categorical field of infinite transcendence degree over the rational numbers, using the Fermat polynomials and assorted results from algebraic geometry. We also show that this field has an intrinsically…
The work is devoted to the construction of a new interval arithmetic which would combine algorithmic efficiency and high quality estimation of the ranges of expressions.
Recent attempts at studying the Fermat equation over number fields have uncovered an unexpected and powerful connection with $S$-unit equations. In this expository paper we explain this connection and its implications for the asymptotic…
In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher…
Correlation functions can be calculated on Riemann surfaces using the operator formalism. The state in the Hilbert space of the free field theory on the punctured disc, corresponding to the Riemann surface, is constructed at infinite genus,…
By elaborating on the recent progress made in the area of Feynman integrals, we apply the intersection theory for twisted de Rham cohomologies to simple integrals involving orthogonal polynomials, matrix elements of operators in Quantum…
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by…
We offer some summation formulas that appear to have great utility in probability theory. The proofs require some recent results from analysis that have thus far been applied to basic hypergeometric functions.
For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial…
We take a general approach to uncertainty on product spaces, and give sufficient conditions for the independence structures of uncertainty measures to satisfy graphoid properties. Since these conditions are arguably more intuitive than some…
Simple constructions are given for finite semifields that include as special cases both old semifields and recently constructed semifields.
For a metric space $(A,d)$, and a set $\Sigma$ of equations, some quantities are introduced that measure the size of discontinuities that must occur in operations satisfying $\Sigma$ (identically) on $A$. We are able to evaluate these…
We introduce the notion of "quasi-symmetric" polynomials, which is a generalization of the notion of symmetry, and is particularly suited to the setting of polynomial rings over finite fields. The properties of this new class of functions…