We consider the computation of Bernoulli, Tangent (zag), and Secant (zig or Euler) numbers. In particular, we give asymptotically fast algorithms for computing the first n such numbers in O(n2(logn)2+o(1)) bit-operations. We also give very short in-place algorithms for computing the first n Tangent or Secant numbers in O(n2) integer operations. These algorithms are extremely simple, and fast fo...

Let X (n,m) (1,d) denote the Segre-Veronese embedding of Pn×Pm via the sections of the sheaf O(1, d). We study the dimensions of higher secant varieties of X (n,m) (1,d) and we prove that there is no defective sth secant variety, except possibly for n values of s. Moreover when (m+d d ) is a multiple of (m+ n+ 1), the sth secant variety of X (n,m) (1,d) has the expected dimension for every s.

<p><span>The paper outlines the simulation of various pulse generators for enhancement optical fiber access transmission networks within flow rate 10 Gbps and range 100 km. The are gaussian, hyperbolic secant, triangle, sine, raised cosine in stage. Proposed mixed with both electro-absorption modulator (EAM) mach-zehnder (MZM) efficient transmission. We have compared max. quality fa...

We consider a multiqubit system of atoms or ions with two computational ground states and an interacting excited state in the so-called blockade regime, such that only one qubit can be at any time. Examples systems are rare-earth-ion-doped crystals neutral trapped tweezer arrays. present simple laser excitation protocol yields complex phase factor on desired product state, which used to impleme...

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an ...

We derive the properties of self-induced transparency ~SIT! solitary waves in a one-dimensional periodic structure doped uniformly with resonance two-level atoms. In our model, the electromagnetic field is treated classically and the dopant atoms are described quantum mechanically. The resulting solitary waves take the form of ultrashort ~picosecond! laser pulses which propagate near the band e...

FIGURE 1 Two-dimensional representation of a spherical structure presenting a morphological particularity on a portion of its surface . AB, arc corresponding to the cross-section of the cap of the sphere presenting the characteristic ; OA = OB = R; 0, angle defining the cap ; v, vector defining the secant planes. S I and S2 are two-dimensional representation of secant planes which intersect (S2...

Many in the mathematical community believe that the secant method arose from Newton’s method using a finite difference approximation to the derivative, most likely because that is the way that it is taught in contemporary texts. However, we were able to trace the origin of the secant method all the way back to the Rule of Double False Position described in the 18th-century B.C. Egyptian Rhind P...

A well known theorem of Helly [l] asserts that if $ is a finite family of convex subsets of En in which each re + 1 members have a common point, then there is a point common to all members of g. In this connection, Vincensini [5; 6] has raised the following question: For «^2 and Ogrgn-1, is there an integer k = k(n, r) such that if § is an arbitrary finite family of convex subsets of En in whic...

We analyze the transfer function of a three-dimensional atomic Bragg beamsplitter formed by two counterpropagating pulsed Gaussian laser beams. Even for ultracold ensembles, efficiency depends significantly on residual velocity particles as well losses into higher diffraction orders. Additional aberrations are caused spatial intensity variation and wavefront curvature beam envelope, studied wit...