on the complexity of computing and learning with multiplicative neural networks in a great variety of neuron models neural inputs are combined using the summing operation we introduce the concept of multiplicative neural networks that contain units that multiply their inputs instead of summing them and thus allow inputs to interact nonlinearly the class of multiplicative neural networks comprises such widely known and well studied network types as higher order networks and product unit networks we investigate the complexity of computing and learning for multiplicative neural networks in particular we derive upper and lower bounds on the vapnik chervonenkis vc dimension and the pseudodimension for various types of networks with multiplicative units as the most general case we consider feedforward networks consisting of product and sigmoidal units showing that their pseudo dimension is bounded from above by a polynomial with the same order of magnitude as the currently best known bound for purely sigmoidal networks moreover we show that this bound holds even when the unit type product or sigmoidal may be learned crucial for these results are calculations of solution set components bounds for new network classes as to lower bounds we construct product unit networks of fixed depth with superlinear vc dimension for sigmoidal networks of higher order we establish polynomial bounds that in contrast to previous results do not involve any restriction of the network order we further consider various classes of higher order units also known as sigma pi units that are characterized by connectivity constraints in terms of these we derive some asymptotically tight bounds multiplication plays an important role in both neural modeling of biological behavior and computing and learning with artificial neural networks we briefly survey research in biology and in applications where multiplication is considered an essential computational element the results we present here provide new tools for assessing the impact of multiplication on the computational power and the learning capabilities of neural networks

diophantine problems and p adic period mappings we give an alternative proof of faltings s theorem mordell s conjecture a curve of genus at least two over a number field has finitely many rational points our argument utilizes the set up of faltings s original proof but is in spirit closer to the methods of chabauty and kim we replace the use of abelian varieties by a more detailed analysis of the variation of p adic galois representations in a family of algebraic varieties the key inputs into this analysis are the comparison theorems of p adic hodge theory and explicit topological computations of monodromy by the same methods we show that in sufficiently large dimension and degree the set of hypersurfaces in projective space with good reduction away from a fixed set of primes is contained in a proper zariski closed subset of the moduli space of all hypersurfaces this uses in an essential way the ax schanuel property for period mappings recently established by bakker and tsimerman

acoustic bullets transient bessel beams generated by planar apertures acoustic bullets are defined to be waves which maintain their peak amplitude and or general shape as they propagate in free space a general theoretical approach is presented to investigate the space time properties of a new class of acoustic bullets which are defined here as acoustic transient bessel beams the approach encompasses both baseband and modulated carrier signal wave fields closed form space time expressions are developed using fourier and hilbert transform methods for several special cases of transient bessel beams the space time source distributions for generating such wave fields from planar apertures are developed from the free space wave fields via the use of the rayleigh surface integral solution for time dependent fields finally numerical results are presented to illustrate the general space time characteristics of the acoustic transient bessel beams

explicit chabauty kim for the split cartan modular curve of level 13 we extend the explicit quadratic chabauty methods developed in previous work by the first two authors to the case of non hyperelliptic curves this results in an algorithm to compute the rational points on a curve of genus g 2 over the rationals whose jacobian has mordell weil rank g and picard number greater than one and which satisfies some additional conditions this algorithm is then applied to the modular curve x s 13 completing the classification of non cm elliptic curves over q with split cartan level structure due to bilu parent and bilu parent rebolledo

comment on analytical results for a bessel function times legendre polynomials class integrals a result is obtained stemming from gegenbauer where the products of certain bessel functions and exponentials are expressed in terms of an infinite series of spherical bessel functions and products of associated legendre functions closed form solutions for integrals involving bessel functions times associated legendre functions times exponentials recently elucidated by neves et al j phys a math gen 39 l293 are then shown to result directly from the orthogonality properties of the associated legendre functions this result offers greater flexibility in the treatment of classical heisenberg chains and may do so in other problems such as occur in electromagnetic diffraction theory

dyson s lemma with moving parts in its original form d dyson s lemma bounds the order of vanishing of a polynomial in two variables at a finite set of points and was used in studying rational approximations to a fixed algebraic irrational number bombieri b revived interest in dyson s lemma and subsequently esnault and viehweg ev and vojta v1 vastly extended the scope of the result to polynomials in several variables and to a section of a line bundle on a product of two algebraic curves respectively the theorems of vojta and of esnault and viehweg are highly refined geometric results which are used to prove the mordell conjecture and roth s theorem respectively the main theorems of both ev and up to a factor of two v1 were recovered in n as special cases of a more general result before n had been written vojta v3 formulated a result which would vastly improve the main result of n the goal of this note is to prove vojta s statement on a product of an arbitrary number of projective lines and to give a counterexample on a product of three or more curves of genus 1 in order to state our main result we need to fix some notation let

unsupervised feature evaluation a neuro fuzzy approach demonstrates a way of formulating neuro fuzzy approaches for both feature selection and extraction under unsupervised learning a fuzzy feature evaluation index for a set of features is defined in terms of degree of similarity between two patterns in both the original and transformed feature spaces a concept of flexible membership function incorporating weighted distance is introduced for computing membership values in the transformed space two new layered networks are designed the tasks of membership computation and minimization of the evaluation index through unsupervised learning process are embedded into them without requiring the information on the number of clusters in the feature space the network for feature selection results in an optimal order of individual importance of the features the other one extracts a set of optimum transformed features by projecting n dimensional original space directly to n dimensional n n transformed space along with their relative importance the superiority of the networks to some related ones is established experimentally

schmidt s subspace theorem with moving targets in recent years due to work of vojta lang osgood etc people have started to realize that there is a close relationship between nevanlinna theory and diophantine approximation p vojta has compiled a dictionary see vojta v1 providing correspondences for the concepts fundamental to the two theories the second main theorem for meromorphic functions is the most important result in nevanlinna theory and its counterpart in number theory via vojta s dictionary is roth s theorem in 1929 r nevanlinna conjectured that the second main theorem remains correct if distinct points in p1 c are replaced by distinct holomorphic maps ai c p1 c with tai r o tf r where tf r is the nevanlinna characteristic function of the holomorphic map f c p1 c see rs 1 for the definition after many attempts by various people osgood proved the conjecture in 1985 which is now known as the second main theorem with moving targets in 1986 steinmetz gave another simple and elegant proof recently vojta v 2 and v 3 formulated and proved its counterpart in number theory theorem a roth s theorem with moving targets let k be a number field and let s be a finite set of places of k let q z 0 and let 0 then there does not exist an infinite sequence of distinct tuples

pattern recognition properties of various feature spaces for higher order neural networks the authors explore alternatives that reduce the number of network weights while maintaining geometric invariant properties for recognizing patterns in real time processing applications this study is limited to translation and rotation invariance the primary interest is in examining the properties of various feature spaces for higher order neural networks honns in correlated and uncorrelated noise such as the effect of various types of input features feature size and number of feature pixels and effect of scene size the robustness of honn training is considered in terms of target detectability the experimental setup consists of a 15 20 pixel scene possibly containing a 3 10 target each trial used 500 training scenes plus 500 testing scenes results indicate that honns yield similar geometric invariant target recognition properties to classical template matching however the honns require an order of magnitude less computer processing time compared with template matching results also indicate that honns could be considered for real time target recognition applications

the ax schanuel conjecture for variations of hodge structures we extend the ax schanuel theorem recently proven for shimura varieties by mok pila tsimerman to all varieties supporting a pure polarizable integral variation of hodge structures in fact hodge theory provides a number of conceptual simplifications to the argument the essential new ingredient is a volume bound for griffiths transverse subvarieties of period domains

intrinsic limits of dimensionality and richness in random multipath fields we study the dimensions or degrees of freedom of farfield multipath that is observed in a limited source free region of space the multipath fields are studied as solutions to the wave equation in an infinite dimensional vector space we prove two universal upper bounds on the truncation error of fixed and random multipath fields a direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension for circular and spherical spatial regions we show that this finite dimension is proportional to the radius and area of the region respectively we use the karhunen loegraveve kl expansion of random multipath fields to quantify the notion of multipath richness the multipath richness is defined as the number of significant eigenvalues in the kl expansion that achieve 99 of the total multipath energy we establish a lower bound on the largest eigenvalue this lower bound quantifies to some extent the well known reduction of multipath richness with reducing the angular power spread of multipath angular power spectrum

unlikely intersections with isogeny orbits in a product of elliptic schemes fix an elliptic curve e 0 without cm and a non isotrivial elliptic scheme over a smooth irreducible curve both defined over the algebraic numbers consider the union of all images of a fixed finite rank subgroup of arbitrary rank of e 0 g also defined over the algebraic numbers under all isogenies between e 0 g and some fiber of the g th fibered power mathcal a of the elliptic scheme where g is a fixed natural number as a special case of a slightly more general result we characterize the subvarieties of arbitrary dimension inside mathcal a that have potentially zariski dense intersection with this set in the proof we combine a generalized vojta remond inequality with the pila zannier strategy

analytical results for a bessel function times legendre polynomials class integrals when treating problems of vector diffraction in electromagnetic theory the evaluation of the integral involving bessel and associated legendre functions is necessary here we present the analytical result for this integral that will make numerical quadrature techniques or localized approximations unnecessary the solution is presented using the properties of the bessel and associated legendre functions

spherical coordinate descriptions of cylindrical and spherical bessel beams this paper derives a generalized spherical harmonic description of bessel beams the spherical harmonic description of the well known cylindrical bessel beams is reviewed and a family of spherical bessel beams are introduced which can provide a number of azimuthal phase variations for a single beam radial amplitude the results are verified by numerical simulations

the motivic fundamental group of p1 0 1 and the theorem of siegel in this paper we establish a link between the structure theory of the pro unipotent motivic fundamental group of the projective line minus three points and diophantine geometry in particular we give a p adic proof of siegel s theorem