Arithmetic Applications Research Articles

Applications of interval arithmetic in solving polynomial equations by Wu's elimination method

Wu's elimination method is an important method for solving multivariate polynomial equations. In this paper, we apply interval arithmetic to Wu's method and convert the problem of solving polynomial equations into that of solving interval polynomial equations. Parallel results such as zero-decomposition theorem are obtained for interval polynomial equations. The advantages of the new approach are two-folds: First, the problem of the numerical instability arisen from floating-point arithmetic is largely overcome. Second, the low efficiency of the algorithm caused by large intermediate coefficients introduced by exact compaction is dramatically improved. Some examples are provided to illustrate the effectiveness of the proposed algorithm.

Fibonacci Arithmetic Expressions

In this paper, we extend the arithmetic (AR) expressions for functions on finite dyadic groups to functions used in Fibonacci interconnection topologies. We have introduced the Fibonacci-Arithmetic (FibAR) expressions for representation of these functions. We discussed the optimization of FibARs with respect to the number of non-zero coefficients through the Fixed-Polarity FibARs defined by using different polarities for the Fibonacci variables. In this way, we provide a base to extend the application of ARs and related powerful CAD design tools for switching functions to functions in Fibonacci interconnection topologies.

Parallel arithmetic numerical simulation and application of secondary migration-accumulation of oil resources

From such actual conditions as the effects of characteristics of miltilayer petroleum geology and permeation fluid mechanics, a new numerical model is put forward and coupling splitting-up implicit interactive scheme is formed. For the actual situation of Dongying hollow (four-layer) and Tanhai region (three-layer) of Shengli Petroleum Field, the numerical simulation test results and the actual conditions are coincident.

Uniform bounds for Fourier coefficients of theta-series with arithmetic applications

Uniform bounds for Fourier coefficients of theta-series with arithmetic applications

Geometric classification tests using interval arithmetic in b-rep solid modeling

In this work, the use of interval arithmetic is considered to increase robustness of geometric classification algorithms in B-Rep solid modeling systems. The classification algorithms, also known as incidence tests, are important to keep the consistency between topology and geometry in a solid model during the application of Boolean operations. An error in the incidence test has deep impact over the steps that follow the Boolean operations and can damage the result, generating an inconsistent solid. The interval arithmetic incorporates approximation errors, so that, it eliminates the need of defining a fixed tolerance to do the comparison between floating-point numbers. However, it is not possible to directly convert floating-point algorithms to interval arithmetic, so that, it is necessary to reformulate the entire algorithm. Another important step in the Boolean operation is the determination of intersection points where the use of interval arithmetic can have side effects as intervals with large dimensions, and may cause incidence tests failures. It is necessary to control the growth of the intervals based on the geometry and topology. This work will introduce the application of interval arithmetic to a B-Rep solid modeler.

Applications of interval arithmetic in non-smooth global optimization

In this paper, an expansion scheme for general functions is presented, which can be used to achieve better enclosure of function ranges. In the context of interval branch-and-bound methods for non-smooth global optimization, a pruning technique is given via the expansion. An interval pruning test is established. This pruning test offers the possibility to cut away a large part of the currently investigated box by the optimization algorithm, which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. Numerical computation shows that the proposed method is reliable and can find all solutions of global optimization problem.

The Brauer group of torsors and its arithmetic applications

Soit X une variete algebrique definie sur un corps de caracteristique zero k. Soit Y un X-torseur sous un tore. Nous calculons le groupe de Brauer de Y et nous en deduisons des consequences arithmetiques pour X quand k est un corps de nombres.

ZETA REGULARIZATIONS AND <i>q</i>-ANALOGUE OF RING SINE FUNCTIONS

So called the zeta regularization is one of the most effective methods to carry out necessary renormalization calculations in a variety of situations such as the determinant expressions of elliptic operators [KV, Vo] and certain arithmetic applications [D] (see also [KKSW]). In the present paper we focus our interest on a particular class of functions which are defined in forms of the zeta regularized products. Let us recall first the formula essentially due to Lerch [L] as a typical example we deal with:

On the structure theory of the Iwasawa algebra of a p-adic Lie group

This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, 6 of a p-adic analytic group G. For G without any p-torsion element we prove that 6 is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null 6-module. This is classical when G=Êkp for some integer kS1, but was previously unknown in the non-commutative case. Then the category of 6-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the Êp-torsion part of a finitely generated 6-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.

Zero-sumfree sequences in cyclic groups and some arithmetical application

Nous montrons que dans un groupe cyclique d'ordre n, toute suite S de longueur |S| > sans sous-suite non vide de somme nulle contient un element d'ordre n ayant une grande multiplicite.

The Flagged Prefix Adder and its Applications in Integer Arithmetic

This paper shows how a parallel prefix adder computing S e A + B may be slightly modified to yield a new adder structure, called a “flagged prefix adder”, capable of returning the related computation pairs A + B and A + B + 1, or, if the bits of B are inverted, A − B and B − A. This adder is of use in digital communications applications and video compression, as well as arithmetic processor designs. The new adder uses 25% less transistors than the conditional-sum adder, which has been used previously in such situations.

Integration on ℋ p × ℋ and Arithmetic Applications

This article describes a conjectural p-adic analytic construction of global points on (modular) elliptic curves, points which are defined over the ring class fields of real quadratic fields. The resulting conjectures suggest that the classical Heegner point construction, and the theory of complex multiplication on which it is based, should extend to a variety of contexts in which the underlying field is not a CM field.

Partial Zeta Functions of Algebraic Varieties over Finite Fields

Motivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. We then explain two approaches to the general structural properties of the partial zeta function in the direction of the Weil-type conjectures. The first approach, using an inductive fibred variety point of view, shows that the partial zeta function is rational in an interesting case, generalizing Dwork's rationality theorem. The second approach, due to Faltings, shows that the partial zeta function is always nearly rational.

An Addition Theorem and Its Arithmetical Application

We study almost arithmetical multiprogressions, which are defined as certain unions of arithmetical progressions. We prove an addition theorem stating that arbitrary sumsets of such sets are of the same type again. Almost arithmetical multiprogressions appear as sets of lengths in rings of algebraic integers, and the addition theorem will be applied to an arithmetical situation again.

Reducibility mod p of Hypersurfaces in Projective Spaces—An Application of Arithmetic Bézout

Given an absolutely irreducible horizontal hypersurface Z in a projective space over the ring of integers R of a number field, we give an explicit bound for the product of the norms of the prime ideals of R over which the fibre of Z becomes reducible. This bound is given as a function of a projective height of Z and is obtained using arithmetic intersection theory, in particular, an arithmetic Bézout theorem.

Mollification of the fourth moment of automorphic L-functions and arithmetic applications

We compute the asymptotics of twisted fourth power moments of modular L-functions of large prime level near the critical line. This allows us to prove some new non-vanishing results on the central values of automorphic L-functions, in particular those obtained by base change from GL 2(Q) to GL 2(K) for K a cyclic field of low degree.

Interval analysis: theory and applications

We give an overview on applications of interval arithmetic. Among others we discuss verification methods for linear systems of equations, nonlinear systems, the algebraic eigenvalue problem, initial value problems for ODEs and boundary value problems for elliptic PDEs of second order. We also consider the item software in this field and give some historical remarks.

νMOS-based Sorter for Arithmetic Applications

The capabilities of the conceptual link between threshold gates and sorting networks are explored by implementing some arithmetic demonstrators. In particular, both an (8 × 8)-multiplier and a (15, 4) counter which use a sorter as the main building block have been implemented. Traditional disadvantages of binary sorters such as their hardware intensive nature are avoided by using νMOS circuits. It allows both an improving of previous results for multipliers based on a similar architecture, and to obtain a new type of counter which shows a reduced delay when compared to a conventional implementation.

Worst case analysis of weight inaccuracy effects in multilayer perceptrons

We derive here a new method for the analysis of weight quantization effects in multilayer perceptrons based on the application of interval arithmetic. Differently from previous results, we find worst case bounds on the errors due to weight quantization, that are valid for every distribution of the input or weight values. Given a trained network, our method allows to easily compute the minimum number of bits needed to encode its weights.

Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve

Let M be an overconvergent isocrystal on a smooth affine curve X/ k over a perfect field of characteristic p > 0, realized as a module on a suitable lifting of X with connection. We give a topological condition on the connection which guarantees that the rigid cohomology of M is finite-dimensional. As a result, one sees that M has finite-dimensional cohomology if it satisfies an analogue of Grothendieck's local monodromy theorem. Some arithmetic applications are given.