E-polynomial of the SL(3,C)-character variety of free groups.

We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3,C). Expanding upon techniques of Logares, Muñoz and Newstead (Rev. Mat. Complut. 26:2 (2013), 635-703), we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give a new proof of results of Cavazos and Lawton (Int. J. Math. 25:6 (2014), 1450058).

High-speed polynomial basis multipliers over GF(2(m)) for special pentanomial.

Efficient hardware implementations of arithmetic operations in the Galois field are highly desirable for several applications, such as coding theory, computer algebra and cryptography. Among these operations, multiplication is of special interest because it is considered the most important building block. Therefore, high-speed algorithms and hardware architectures for computing multiplication are highly required. In this paper, bit-parallel polynomial basis multipliers over the binary field GF(2(m)) generated using type II irreducible pentanomials are considered. The multiplier here presented has the lowest time complexity known to date for similar multipliers based on this type of irreducible pentanomials.

On the uniform approximation of Cauchy continuous functions

In the context of real-valued functions defined on metric spaces, it is known that the locally Lipschitz functions are uniformly dense in the continuous functions and that the Lipschitz in the small functions - the locally Lipschitz functions where both the local Lipschitz constant and the size of the neighborhood can be chosen independent of the point - are uniformly dense in the uniformly continuous functions. Between these two basic classes of continuous functions lies the class of Cauchy continuous functions, i.e., the functions that map Cauchy sequences in the domain to Cauchy sequences in the target space. Here, we exhibit an intermediate class of Cauchy continuous locally Lipschitz functions that is uniformly dense in the real-valued Cauchy continuous functions. In fact, our result is valid when our target space is an arbitrary Banach space.

Dual self-image technique for beam collimation

We propose an accurate technique for obtaining highly collimated beams, which also allows testing the collimation degree of a beam. It is based on comparing the period of two different self-images produced by a single diffraction grating. In this way, variations in the period of the diffraction grating do not affect to the measuring procedure. Self-images are acquired by two CMOS cameras and their periods are determined by fitting the variogram function of the self-images to a cosine function with polynomial envelopes. This way, loss of accuracy caused by imperfections of the measured self-images is avoided. As usual, collimation is obtained by displacing the collimation element with respect to the source along the optical axis. When the period of both self-images coincides, collimation is achieved. With this method neither a strict control of the period of the diffraction grating nor a transverse displacement, required in other techniques, are necessary. As an example, a LED considering paraxial approximation and point source illumination is collimated resulting a resolution in the divergence of the beam of σ φ = ± μrad.

Equivalent norms in polynomial spaces and applications.

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy Littlewood constants for 2-homogeneous polynomials on l(p)(2) spaces, 2 < p <= infinity. We also provide lower estimates for the Hardy-Littlewood constants for polynomials of higher degrees.

E-polynomial of SL(2,C)-character varieties of complex curves of genus 3.

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g = 3 into SL(2, C), and also of the moduli space of twisted representations. The case of genus g = 1, 2 has already been done in [12]. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.