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Based on Mertens theorem, this paper presents and proves the iterative theorem of prime number distribution. A more accurate prime number theorem is obtained by using the iterative theorem of prime number distribution for transformation.

Keywords: prime number; Prime number theorem, prime number distribution iteration theorem, Mertens theorem.

This paper discusses algorithms adapted for a construction of three types of wavelets. We build in the rst case scaling functions and associated wavelets which are fast decreasing. In the second case, we construct wavelets which are exponential decreasing. In the last one, we construct scaling functions which are regular and have compact support and we prove that the associated wavelets have compact support and preserve the original regularity of the scaling functions. As applications, we prove that the wavelet bases constructed in this paper are adapted for the study of regular functional spaces C s(????) and Hs(????) (s ???? ????) and are easy to implement.

Keywords: Scaling function, Wavelet, Riesz basis, Dual basis, Multiresolution analysis, Sobolev space.

Mathematics Subject Classication : 42C15, 44A15.

A plane, passing through the center and orthogonal to a diagonal, slices a cube into two identical halves each having three triangles, three pentagons and a hexagon. If you label the ten vertices of a half-cube with numbers, then each face is said to receive an induced face label given by the sum of all vertices around it. Label the ten vertices of a half-cube with digits 0 through 9 so that the induced labels of the three triangles and the three pentagons constitute two distinct values.

Keywords: Puzzle, solution, rotation, reflection, permutation.

AMS Subject Classification: 05C78

This paper is a continuation of study of I0 *-modules. We provide several characterization of module which it is endomorphism ring is a principal right (left) I0 *-ring. New results obtained include necessary and sufficient conditions of module to be a principal right (left) I0 *-module (I0 *?module). Connectio between a principal right (left)module (module) and its endomorphism ring studied. Several basic properties of subsets ????*[M, N], ????*[M, N], Tr *[M, N], Jr *[M, N] of bi-module [M, N] are proved, for every two modules M, N include when Tr *[M, N] is equal Jr *[M, N].

Keywords: I0-Ring and Module, I0-Ri ng and Module, (Co)retractable module, Semi-injective (projective) modules.

We present in this paper Riesz bases and dual Riesz bases. Next, we define and study general biorthogonal multiresolution analyses on the real line and we prove commutation properties between derivation and projectors. As applications, we prove that the wavelet bases constructed in this paper are adapted for the study of the Sobolev spaces Hs(????) and HRs(R)(s R N).

Keywords: Multiresolution analysis, Riesz and Dual basis, Wavelet, Sobolev space.

Mathematics Subject Classication: MSC Code: 42C15; 44A15.

In an optical network, a Scheduled Lightpath Demand (SLD) is a connection demand between two nodes, during a certain time and with a certain wavelength. We consider the following NPhard Routing and Wavelength Assignment (RWA) problem dealing with SLDs: given a set of SLDs and a number W of wavelengths, maximize the number of SLDs to which we can assign a lightpath (i.e. a routing path and a wavelength) without exceeding the number W of available wavelengths. The constraints are: a same wavelength must be assigned all along the routing path of any SLD; at any time, a given wavelength on a given edge of the network cannot be used to satisfy more than one SLD. To solve this problem, we study an approach stating the problem as the successive searches of independent sets in some conflict graphs. Moreover, we improve this approach thanks to a post-optimization method. The experimental results show that this model and the post-optimization method are quite efficient to provide a large number of routed SLDs.

Keywords: WDM Optical Networks, Routing and Wavelength Assignment (RWA) Problem, Scheduled Lightpath Demands (SLD), Combinatorial Optimization, Conflict Graphs, Independent Sets, Post-Optimization.