Zolotarev's proof of Gauss reciprocity and Jacobi symbols

2000 Mathematics Subject Classification: Primary 11A15.

Coincidence of Vietoris and Wijsman Topologies: A New Proof

Let (X, d) be a metric space and CL(X) the family of all nonempty closed subsets of X. We provide a new proof of the fact that the coincidence of the Vietoris and Wijsman topologies induced by the metric d forces X to be a compact space. In the literature only a more involved and indirect proof using the proximal topology is known. Here we do not need this intermediate step. Moreover we prove that (X, d) is boundedly compact if and only if the bounded Vietoris and Wijsman topologies on CL(X) coincide.

Visualization of Problems’ Solution

Джурджица Такачи - В доклада се разглеждат дидактически подходи за решаване на задачи, упражнения и доказване на теореми с използване на динамичен софтуер, по-специално – с вече широко разпространената система GeoGebra. Въз основа на концепция-та на Пойа се анализира използването на GeoGebra като когнитивно средство за решаване на задачи и за обсъждане на техни възможни обобщения.

Essential Arity Gap of Boolean Functions

In this paper we investigate the Boolean functions with maximum essential arity gap. Additionally we propose a simpler proof of an important theorem proved by M. Couceiro and E. Lehtonen in [3]. They use Zhegalkin’s polynomials as normal forms for Boolean functions and describe the functions with essential arity gap equals 2. We use to instead Full Conjunctive Normal Forms of these polynomials which allows us to simplify the proofs and to obtain several combinatorial results concerning the Boolean functions with a given arity gap. The Full Conjunctive Normal Forms are also sum of conjunctions, in which all variables occur.

On the Exponential Bound of the Cutoff Resolvent

A simpler proof of a result of Burq [1] is presented.

The Development of the Generalization Algorithm Based on the Rough Set Theory

This paper considers the problem of concept generalization in decision-making systems where such features of real-world databases as large size, incompleteness and inconsistence of the stored information are taken into account. The methods of the rough set theory (like lower and upper approximations, positive regions and reducts) are used for the solving of this problem. The new discretization algorithm of the continuous attributes is proposed. It essentially increases an overall performance of generalization algorithms and can be applied to processing of real value attributes in large data tables. Also the search algorithm of the significant attributes combined with a stage of discretization is developed. It allows avoiding splitting of continuous domains of insignificant attributes into intervals.

Automatic Translation of MSC Diagrams into Petri Nets

Development-engineers use in their work languages intended for software or hardware systems design, and test engineers utilize languages effective in verification, analysis of the systems properties and testing. Automatic interfaces between languages of these kinds are necessary in order to avoid ambiguous understanding of specification of models of the systems and inconsistencies in the initial requirements for the systems development. Algorithm of automatic translation of MSC (Message Sequence Chart) diagrams compliant with MSC’2000 standard into Petri Nets is suggested in this paper. Each input MSC diagram is translated into Petri Net (PN), obtained PNs are sequentially composed in order to synthesize a whole system in one final combined PN. The principle of such composition is defined through the basic element of MSC language — conditions. While translating reference table is developed for maintenance of consistent coordination between the input system’s descriptions in MSC language and in PN format. This table is necessary to present the results of analysis and verification on PN in suitable for the development-engineer format of MSC diagrams. The proof of algorithm correctness is based on the use of process algebra ACP. The most significant feature of the given algorithm is the way of handling of conditions. The direction for future work is the development of integral, partially or completely automated technological process, which will allow designing system, testing and verifying its various properties in the one frame.

A Geometrical Construction for the Polynomial Invariants of some Reflection Groups

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.

The Graves Theorem Revisited II: Robust Convergence of the Newton Method

AMS subject classification: 65J15, 47H04, 90C30.

A solution of the trigonometric moment problem via Tagamlitzki's "Theorem of the Cones"

In 1952 Y. Tagamlitzki gave an elegant proof of the classical Bochner’s theorem on the positively definite functions. Unfortunately, he never published his proof. In this paper we consider a related but simpler problem, the trigonometric moment problem, by using Tagamlitzki’s approach.

Subvarieties of the Hyperelliptic Moduli Determined by Group Actions

2000 Mathematics Subject Classification: 14Q05, 14Q15, 14R20, 14D22.

Corrigendum for "Weierstrass Points with first Non-Gap four on a Double Covering of a Hyperelliptic Curve"

In the proof of Lemma 3.1 in [1] we need to show that we may take the two points p and q with p ≠ q such that p+q+(b-2)g21(C′)∼2(q1+… +qb-1) where q1,…,qb-1 are points of C′, but in the paper [1] we did not show that p ≠ q. Moreover, we hadn't been able to prove this using the method of our paper [1]. So we must add some more assumption to Lemma 3.1 and rewrite the statements of our paper after Lemma 3.1. The following is the correct version of Lemma 3.1 in [1] with its proof.