We study the asymptotics of marginal likelihood integrals for discrete models using resolution of singularities from algebraic geometry, a method introduced recently by Sumio Watanabe. We briefly describe the statistical and mathematical foundations of this method, and explore how Newton diagrams and toric modifications help solve the problem. The approximations are then compared with exact com...

Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang-Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions.

This paper explores the algebraic matching approach for detection of vulnerabilities in binary codes. The algebraic programming system is used for implementing this method. It is anticipated that models of vulnerabilities and programs to be verified are presented as behavior algebra and action language specifications. The methods of algebraic matching are based on rewriting rules and techniques...

Algebraic Independent Component Analysis (AICA) is a new ICA algorithm that exploits algebraic operations and vectordistance measures to estimate the unknown mixing matrix in a scaled algebraic domain. AICA possesses stability and convergence properties similar to earlier proposed geometic ICA (geo-ICA) algorithms, however, the choice of the proposed algebraic measures in AICA has several advan...

We tackle several problems related to a finite irreducible crystallographic root system $\Phi$ in the real vector space $\mathbb E$. In particular, we study combinatorial structure of subsets cut by affine subspaces E$ and their projections. As byproducts, obtain easy algebraic proofs refinements Oshima's Lemma result Kostant, partial towards resolution problem Hopkins Postnikov, new enumerativ...

A ‘resolution’ of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a point-particle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and, in extenso, Differential Calculus-free purely algebraic (:sheaf-theoretic) conceptual and tec...

lthough fuzzy set theory and  sheaf theory have been developed and studied independently,  Ulrich Hohle shows that a large part of fuzzy set  theory  is in fact a subfield of sheaf theory. Many authors have studied mathematical structures, in particular, algebraic structures, in both  categories of these generalized (multi)sets. Using Hohle's idea, we show that for a (universal) algebra $A$, th...

Algebraic integral equations is a special category of Volterra integral equations system,  that has many applications in physics and engineering. The principal aim of this paper is to serve the numerical solution of an integral algebraic equation by using the Taylor expansion method. In this method, using the Taylor expansion of the unknown function, the algebraic integral equation system becom...

Beams of microscopic particles penetrating scattering background matter play an important role in several applications. The parameter choices made here are motivated by the problem of electron-beam cancer therapy planning. Mathematically, a steady particle beam penetrating matter, or a configuration of several such beams, is modeled by a boundary value problem for a Boltzmann equation. Grid-bas...

The algebraic formalism of Q E D is expounded in order to demonstrate both the resolution of constraints and to verify gauge equivalence between temporal gauge and Coulomb gauge on the quantum level. In the algebraic approach energy eigenstates of Q E D in temporal gauge are represented in an algebraic G N S basis. The corresponding Hilbert space is mapped into a functional space of generating ...