Francesco Dalla Piazza

Genus Four Superstring Measures

A main issue in superstring theory are the superstring measures. D’Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are the product of modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently, S. Grushevsky published this result as part of a more general approach.

Two Loop Superstring Amplitudes and S 6 Representations

In this paper we describe how representation theory of groups can be used to shorten the derivation of two loop partition functions in string theory, giving an intrinsic description of modular forms appearing in the results of D’Hoker and Phong (Nucl Phys B639:129–181, 2002). Our method has the advantage of using only algebraic properties of modular functions and it can be extended to any genus g.

Multi-layer perceptrons with discrete weights

The feasibility of restricting the weight values in multilayer perceptrons to powers of two or sums of powers of two is studied. Multipliers could be thus replaced by shifters and adders on digital hardware, saving both time and chip area, under the assumption that the neuron activation function is computed through a lookup table (LUT) and that a LUT can be shared among many neurons. A learning procedure based on back-propagation for obtaining a neural network with such discrete weights is presented. This learning procedure requires full real arithmetic and therefore must be performed offline. It starts from a multilayer perceptron with continuous weights learned using back-propagation. Then a weight normalization is made to ensure that the whole shifting dynamics is used and to maximize the match between continuous and discrete weights of neurons sharing the same LUT. Finally, a discrete version of BP algorithm with automatic learning rate control is applied up to convergence. Some test runs on a simple pattern recognition problem show the feasibility of the approach

Perturbative photon production in a dispersive medium

We investigate photon pair-creation in a dispersive dielectric medium induced by the presence of a spacetime varying dielectric constant. Our aim is to examine the possibility to observe new phenomena of pair creation induced by travelling dielectric perturbations e.g. created by laser pulses by means of the Kerr effect. In this perspective, we adopt a semi-phenomenological version of the Hopfield model in which a space-time dependent dielectric susceptibility appears. We focus our attention on perturbation theory, and provide general formulas for the photon production induced by a local but arbitrarily spacetime dependent refractive index perturbation. As an example, we further explore the case of a uniformly travelling perturbation, and provide examples of purely time-dependent perturbations.

An adaptive spline non-linear function for blind signal processing

A new adaptive non linear function for blind signal processing is presented. It is based on a spline approximation whose control points are adaptively changed using information maximization techniques. The monotonously increasing characteristic is obtained using suitable B-spline functions imposing simple constraints on its control points. In particular, the problem of adaptively maximizing the entropy of the output is considered for flattening (make uniform) the probability density function (pdf) of a random signal. We derive a simple form of the adaptation algorithm and present some experimental results that demonstrate the effectiveness of the proposed method.

Low complexity adaptive nonlinear function for blind signal separation

An adaptive nonlinear function for blind signal separation is presented. It is based on a spline approximation whose control points are adaptively changed using information maximization techniques. The monotonously increasing characteristic is obtained using suitable B-spline functions imposing simple constraints on its control points. In particular, the problem of adaptively maximizing the entropy of the output is considered in the context of blind separation of independent sources. We derive a simple form of the learning algorithm which allows us not only to adapt the separation matrix coefficients but also the shape of the nonlinear functions. A comparison with the mixture-of-densities approach is also presented on some experimental data that demonstrates the effectiveness and efficiency of the proposed method.

Classical theta constants vs. lattice theta series, and super string partition functions

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus g = 3, 4, 5. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant.

More on superstring chiral measures

Abstract In this paper we study the expressions of the superstring chiral measures for g ⩽ 5 . In genus three and four we obtain certain new equivalent expressions for the measures which are functions of higher powers of theta constants. For g = 3 we show that the measures can be written in terms of fourth power of theta constants and for g = 4 in terms of squares of theta constants. In both cases the forms Ξ 8 ( g ) [ 0 ( g ) ] appearing in the expression of the measures are defined on the whole Siegel upper half space. Instead, for g = 5 we find a form Ξ 8 ( 5 ) [ 0 ( 5 ) ] which is a polynomial in the classical theta constants, well defined on the Siegel upper half space and satisfying some suitable constraints on the moduli space of curves (and not on the whole Siegel upper half space) that could be a candidate for the genus five superstring measure. Moreover, we discuss the problem of the uniqueness of this form in genus five. We also determine the dimension of certain spaces of modular forms and reinterpret the vanishing of the cosmological constant in terms of group representations.

Improved evoked potential estimation using neural network

The possibility of using the multilayer perceptron (MLP) neural network for the processing of the evoked potentials (EPs) is analyzed. In this case, the process can be conceived as deterministic low amplitude signals (damped sine waves), corresponding to the brains response to stimuli, embedded in strongly colored noise, the EEG background activity. Typical values of the signal-to-noise ratio are less than 0 dB. The network, used as a nonlinear filter, is trained using iteratively as the input signal one of a set of available EP ensembles and as the target signal another ensemble of the same set. Experimental results, both on synthetic and real data, show that the method provides good results with very few EP ensembles. Therefore, it allows a noteworthy reduction of the signal nonstationarity and the patients annoyance

Design of multi-layer neural networks with powers-of-two weights

The feasibility of restricting the weight values to powers-of-two or sums of powers-of-two in multilayer neural networks is discussed. A learning procedure based on back-propagation to obtain a neural network with such weights is presented. This learning procedure requires full real arithmetic, and therefore must be performed offline. These neural networks do not require multipliers, and are well suited for high-speed and high-integration digital neural circuits. To show the effectiveness of the approach, tests on a pattern recognition problem are presented.<<ETX>>