Serdica Mathematical Journal

Volume 31, Numbers 1-2, 2005

C O N T E N T S

Survey

• Kumchev, A. V., D. I. Tolev. An invitation to additive prime number theory (pp. 1-74)

  Articles

• Aamri, M., D. El Moutawakil. Weak compatibility and common fixed point theorems for A-contractive and E-expansive maps in uniform spaces (pp. 75-86)
• Muneer, Y. E. On the convergence of (0, 1, 2) interpolation (pp. 87-94)
• Kirchev, K. P., G. S. Borisova. Triangular models and asymptotics of continuous curves with bounded and unbounded semigroup generators (pp. 95-174)

A B S T R A C T S

AN INVITATION TO ADDITIVE PRIME NUMBER THEORY
A. V. Kumchev kumchev@math.utexas.edu, D. I. Tolev dtolev@pu.acad.bg

2000 Mathematics Subject Classification: 11D75, 11D85, 11L20, 11N05, 11N35, 11N36, 11P05, 11P32, 11P55.
Key words: Goldbach problems, additive problems, circle method, sieve methods, prime numbers.

The main purpose of this survey is to introduce the inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

WEAK COMPATIBILITY AND COMMON FIXED POINT THEOREMS FOR A-CONTRACTIVE AND E-EXPANSIVE MAPS IN UNIFORM SPACES
M. Aamri, D. El Moutawakil d.elmoutawakil@math.net

2000 Mathematics Subject Classification: 47H10, 54E15.
Key words: Uniform spaces, common fixed point, contractive maps, expansive maps, compatible maps, weakly compatible maps.

The purpose of this paper is to define the notion of A-distance and E-distance in uniform spaces and give several new common fixed point results for weakly compatible contractive or expansive selfmappings of uniform spaces.

ON THE CONVERGENCE OF (0, 1, 2) INTERPOLATION
Y. E. Muneer muneeralnour@yahoo.co.uk

2000 Mathematics Subject Classification: 41A05.
Key words: Zeros, modulus of continuity, interpolation process, approximation.

For the Hermite interpolation polynomial, H_m (x) we prove for any function f Î C^(2q)([-1,1]) and any s = 0,1,2,¼, q, where q is a fixed integer that

| Hm(s) (x) - f(s)(x)| = O(1)w( 1 m ,f(2q)) logn n2q - 2s .

Here m is defined by m = 3n-1.

If f Î C^(q)([-1,1]), then

| Hm(s) - f(s)(x)| = O(1)w( 1 m ,f(q)) logn (1 - x2)q / 2

for x Î (-1, 1).

TRIANGULAR MODELS AND ASYMPTOTICS OF CONTINUOUS CURVES WITH BOUNDED AND UNBOUNDED SEMIGROUP GENERATORS
Kiril P. Kirchev kpkirchev@abv.bg, Galina S. Borisova g.borisova@fmi.shu-bg.net

2000 Mathematics Subject Classification: Primary 47A48, Secondary 60G12.
Key words: Unbounded operator, operator colligation, characteristic function, nondissipative curve, correlation function, wave operator, scattering operator.

In this paper classes of K^r-operators are considered - the classes of bounded and unbounded operators A with equal domains of A and A^*, finite dimensional imaginary parts and presented as a coupling of a dissipative operator and an antidissipative one with real absolutely continuous spectra and the class of unbounded dissipative K^r-operators A with different domains of A and A^* and with real absolutely continuous spectra. Their triangular models are presented. The asymptotics of the corresponding continuous curves with generators from these classes are obtained in an explicit form. With the help of the obtained asymptotics the scattering theory for the couples (A^*,A) when A belongs to the introduced classes is constructed.