constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

on infinite goldie dimension abstract two elements x and y of a partially ordered set p are said to be disjoint if there is no z p such that z x and z y denote by p the supremum of the cardinals such that p contains a subset of pairwise disjoint elements with cardinal number p erdos and a tarski ann of math 44 1943 315 329 proved that unless p is weakly inaccessible p contains a subset of pairwise disjoint elements with cardinal number p j dauns and l fuchs j algebra 115 1988 297 302 defined the goldie dimension of a module m denoted by gd m as the supremum of all cardinals such that m contains the direct sum of nonzero submodules they proved that unless gd m is weakly inaccessible m contains a direct sum of gd m submodules in this paper a unified proof of these two results is given it is also shown that similar results hold in the context of modular lattices and abelian categories

triangular prisms for edge based vector finite element analysis of conformal antennas this paper deals with the derivation of the edge based shape functions for the distorted triangular prism and their applications for the analysis of doubly curved conformal antennas in the context of the finite element method fem although the tetrahedron is often the element of choice for volume tessellation mesh generation using tetrahedra is cumbersome and the central processing unit cpu intensive on the other hand the distorted triangular prism allows for meshes which are unstructured in two dimensions and structured in the third dimension this leads to substantial simplifications in the meshing algorithm and many conformal printed antenna and microwave circuit geometries can be easily tessellated using such a mesh the new edge based shape functions are first validated by computing the eigenvalues of three different cavities rectangular cylindrical and pie shell we then proceed with their application to computing the input impedance of conformal patch antennas on planar spherical conical and other doubly curved ogival platforms where the fem mesh is terminated using an artificial absorber applied conformal to the platform use of artificial absorbers for mesh termination avoids introduction of green s functions and in contrast to absorbing boundary conditions a knowledge of the principal radii of curvature of the closure s boundary is not required

regular languages defined with generalized quantifiers we study an extension of first order logic obtained by adjoining quantifiers that count with respect to an integer modulus it is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups we obtain an analogous result for regular languages and establish some connections with complexity theory for fixed depth families of circuits

logic networks with a minimum number of nor nand gates for parity functions of n variables design of logic networks in single rail input logic with a minimum number of nor gates for parity functions of an arbitrary number of variables is described this is partly based on minimum networks for parity functions of a small number of variables which are designed by the integer programming logic design method although it is generally difficult to design minimum networks for functions of an arbitrarily large number of variables we have previously designed minimum networks for adders of an arbitrary number of variables the minimum networks for parity functions of an arbitrary number of variables discussed in this paper is another case many unique properties of minimum nor networks for parity functions are shown minimum networks with nand gates for parity functions can be easily obtained from those with nor gates because of duality relationship between nand and nor

a hybrid fem mom method for 3 d electromagnetic scattering in layered medium accurate and efficient prediction of electromagnetic scattering from inhomogeneous objects in layered medium is one of the most challenging issues in engineering applications this paper presents the first 3 d higher order hybrid finite element method fem and method of moments mom for the accurate modeling of inhomogeneous dielectric objects in multilayered medium the main challenges of this paper include 1 the integration of these algorithms for layered medium and 2 the higher order computational approach involved in layered medium for high efficiency in the proposed method the mom with the layered medium dyadic green s function is used as the exact radiation boundary condition in an inhomogeneous background and the fem is applied to model the inhomogeneous objects furthermore the higher order maximally orthogonal basis functions with curl conforming and divergence conforming properties are used in the fem and mom respectively to improve the modeling capability of this algorithm for 3 d inhomogeneous objects scattering in multilayered medium this new method requires a much more tightly truncated simulation domain than the traditional fem and provides much higher flexibility than the pure surface integral equation method finally some numerical results are provided to validate the accuracy efficiency and flexibility of this method

the complexity of the parity function in unbounded fan in unbounded depth circuits abstract almost everything is known on the complexity of the parity function in fan in 2 circuits over various bases also the minimal depth of polynomial size unbounded fan in dl circuits for the parity function has been studied here the complexity without any depth restriction is considered for the basis dl almost optimal bounds and for the basis of nor gates and the basis of all threshold functions optimal bounds on the number of gates are obtained for the basis dl the minimal number of wires is determined for threshold circuits an exponential gap between synchronous and asynchronous circuits is proved the results not only answer open questions in complexity theory but also have implications for the real life circuit design

a novel grid robust higher order vector basis function for the method of moments a set of novel grid robust higher order vector basis functions is proposed for the method of moments mom solution of integral equations for three dimensional 3 d electromagnetic em problems these basis functions are defined over curvilinear triangular patches and represent the unknown electric current density within each patch using the lagrange interpolation polynomials the highlight of these basis functions is that the lagrange interpolation points are chosen to be the same as the nodes of the well developed gaussian quadratures as a result the evaluation of the integrals in the mom is greatly simplified additionally the surface of an object to be analyzed can be easily meshed because the new basis functions do not require the side of a triangular patch to be entirely shared by another triangular patch which is a very stringent requirement for traditional vector basis functions the proposed basis functions are implemented with point matching for the mom solution of the electric field integral equation the magnetic field integral equation and the combined field integral equation numerical examples are presented to demonstrate the higher order convergence and the grid robustness for defective meshes using the new basis functions

accurate solution of square scatterer as benchmark for validation of electromagnetic modeling of plate structures an infinitesimally thin square scatterer of size spl lambda spl times spl lambda excited normally by an incident plane wave which is polarized along a scatterer edge is analyzed the accurate solution of its current distribution is found in the form of a double series of basis functions which automatically satisfy the continuity equation at the plate edges and include the edge effect the coefficients that multiply basis functions are determined starting from the electric field integral equation by using the galerkin method the solution obtained for the order of approximation n 8 is adopted as a benchmark the corresponding coefficients are tabulated and graphs of such obtained current distribution are given the solution adopted as a benchmark is applied for comparison of rooftop basis functions and polynomial entire domain basis functions the relative error of the mean absolute value of current deviation is used as an error metric

hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements new vector finite elements are proposed for electromagnetics the new elements are triangular or tetrahedral edge elements tangential vector elements of arbitrary polynomial order they are hierarchal so that different orders can be used together in the same mesh and p adaption is possible they provide separate representation of the gradient and rotational parts of the vector field explicit formulas are presented for generating the basis functions to arbitrary order the basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning matrix assembly for the frequency domain curl curl equation is conveniently carried out by means of universal matrices application of the new elements to the solution of a parallel plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient for tetrahedral elements to order 4 in particular the full order elements have only the same asymptotic convergence rate as elements with a reduced gradient space such as the whitney element however further tests reveal that the optimum balance of the gradient and rotational components is problem dependent

the expressiveness of a family of finite set languages abstract in this paper we characterize exactly the complexity of a set based database language called srl which presents a unified framework for queries and updates by imposing simple syntactic restrictions on it we are able to express exactly the classes p and logspace we also discuss the role of ordering in database query languages and show that the hom operator of machiavelli language in ohori et al 1989 does not capture all the order independent properties

approximating threshold circuits by rational functions motivated by the problem of understanding the limitations of threshold networks for representing boolean functions we consider size depth trade offs for threshold circuits that compute the parity function using a fundamental result in the theory of rational approximation we show how to approximate small threshold circuits by rational functions of low degree we apply this result to establish an almost optimal lower bound of n2 ln2n on the number of edges of any depth 2 threshold circuit with polynomially bounded weights that computes the parity function we also prove that any depth 3 threshold circuit with polynomially bounded weights requires n1 2 ln5 3n edges to compute parity on the other hand we give a construction of a depth d threshold circuit that computes parity with n1 1 d edges where 1 5 2 is the golden ratio we conjecture that there are no linear size bounded depth threshold circuits for computing parity

spatially large domain and temporally entire domain electric field integral equation method of moments for 3 d scattering analysis in time domain a novel spatially large domain and temporally entire domain method of moments mom is proposed for surface integral equation sie modeling of 3 d conducting scatterers in the time domain td the method uses higher order curved lagrange interpolation generalized quadrilateral geometrical elements higher order spatial current expansions based on hierarchical divergence conforming polynomial vector basis functions and temporal current modeling by means of orthogonal weighted associated laguerre basis functions it implements full temporal and spatial galerkin testing and marching on in degree mod scheme for an iterative solution of the final system of spatially and temporally discretized mom td equations numerical examples demonstrate excellent accuracy efficiency convergence and versatility of the new mom mod method the results also demonstrate very effective large domain mom td sie models of scatterers using flat and curved patches of electrical sizes of up to about 1 7 wavelengths at the maximum frequency in the frequency spectrum of the pulse excitation higher order current expansions of spatial orders from 2 to 8 in conjunction with entire domain laguerre temporal bases and minimal numbers of unknowns

on the goldie dimension of rings and modules we find a bound for the goldie dimension of hereditary modules in terms of the cardinality of the generating sets of their quasi injective hulls several consequences are deduced in particular it is shown that every finitely generated hereditary module with countably generated quasi injective hull is noetherian it is also shown that every right hereditary ring with finitely generated injective hull is right artinian thus answering a long standing open question posed by dung gomez pardo and wisbauer

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

optimal modeling parameters for higher order mom sie and fem mom electromagnetic simulations general guidelines and quantitative recipes for adoptions of optimal higher order parameters for computational electromagnetics cem modeling using the method of moments and the finite element method are established and validated based on an exhaustive series of numerical experiments and comprehensive case studies on higher order hierarchical cem models of metallic and dielectric scatterers the modeling parameters considered are electrical dimensions of elements h refinement polynomial orders of basis functions p refinement orders of gauss legendre integration formulas integration accuracy and geometrical curvature orders of elements in the model the goal of the study which is the first such study of higher order parameters in cem is to reduce the dilemmas and uncertainties associated with the great modeling flexibility of higher order elements basis and testing functions and integration procedures this flexibility is the principal advantage but also the greatest shortcoming of the higher order cem and to ease and facilitate the decisions to be made on how to actually use them by both cem developers and practitioners

higher order hybrid fem mom technique for analysis of antennas and scatterers a novel higher order large domain hybrid computational electromagnetic technique based on the finite element method fem and method of moments mom is proposed for three dimensional analysis of antennas and scatterers in the frequency domain the geometry of the structure is modeled using generalized curved parametric hexahedral and quadrilateral elements of arbitrary geometrical orders the fields and currents on elements are modeled using curl and divergence conforming hierarchical polynomial vector basis functions of arbitrary approximation orders and the galerkin method is used for testing the elements can be as large as about two wavelengths in each dimension as multiple mom objects are possible in a global exterior region the mom part provides much greater modeling versatility and potential for applications especially in antenna problems than just as a boundary integral closure to the fem part the examples demonstrate excellent accuracy convergence efficiency and versatility of the new fem mom technique and very effective large domain meshes that consist of a very small number of large flat and curved fem and mom elements with p refined field and current distributions of high approximation orders the reduction in the number of unknowns is by two orders of magnitude when compared to available data for low order fem mom modeling

double higher order large domain volume surface integral equation method for analysis of composite wire plate dielectric antennas and scatterers a novel double higher order large domain galerkin type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for analysis of composite dielectric and metallic radiation scattering structures combining the volume integral equation vie approach for dielectric parts and the surface integral equation sie approach for metallic parts of the structure the technique employs lagrange type interpolation generalized hexahedra and quadrilaterals of arbitrary geometrical mapping orders for the approximation of geometry and hierarchical divergence conforming polynomial vector basis functions of arbitrary expansion orders for the approximation of currents within the elements the double higher order vsie vie sie method is extensively validated and evaluated against the analytical solutions and the numerical results obtained by alternative higher order methods

size energy tradeoffs for unate circuits computing symmetric boolean functions a unate gate is a logical gate computing a unate boolean function which is monotone in each variable examples of unate gates are and gates or gates not gates threshold gates etc a unate circuit c is a combinatorial logic circuit consisting of unate gates let f be a symmetric boolean function of n variables such as the parity function mod function and majority function let m 0 and m 1 be the maximum numbers of consecutive 0 s and consecutive 1 s in the value vector of f respectively and let l min m 0 m 1 and m max m 0 m 1 let c be a unate circuit computing f let s be the size of the circuit c that is c consists of s unate gates let e be the energy of c that is e is the maximum number of gates outputting 1 over all inputs to c in this paper we show that there is a tradeoff between the size s and the energy e of c more precisely we show that n 1 l m s e we also present lower bounds on the size s of c represented in terms of n l and m our tradeoff immediately implies that logn elogs for every unate circuit c computing the parity function of n variables

duffy method for evaluation of weakly singular sie potential integrals over curved quadrilaterals with higher order basis functions a duffy method for singularity cancellation is proposed for evaluation of weakly singular potential integrals defined on generalized curved parametric quadrilateral patches with polynomial basis functions such integrals arise in evaluation of galerkin matrix elements in method of moments analysis of antennas and scatterers in cases of coincident source and test elements examples demonstrate that the proposed duffy method is more accurate more rapidly converging with the increase of orders of gauss legendre integration formulas and faster to execute than four other methods for singularity treatment considered in the study

infinite goldie dimensions let r be a ring with identity and m a unital right r module by the finite goldie dimension of m is meant the largest integer m provided it exists such that m contains the direct sum of m nonzero submodules if no such m exists then m is said to have infinite goldie dimension until recently not much attention has been given to the case of infinite goldie dimensions though the definition is immediate the goldie dimension of m denoted gd m is the supremum 1 of all cardinals k such that m contains the direct sum of k nonzero submodules we believe that one of the main reasons why infinite goldie dimensions have been ignored is that suprema are difficult to handle and there is no guarantee that a module m of goldie dimension 1 contains the direct sum of exactly 2 nonzero submodules given a cardinal number k we say k is attained in m if m contains a direct sum of k nonzero submodules if k is not a limit cardinal i e if it is of the form tc h for some ordinal cl then k 2 gd m is attained in m recall that an infinite cardinal k is called regular if k for ie z with izl k implies c k k otherwise it is called singular an uncountable regular limit cardinal is said to be inaccessible hj p 1631 or l p 1373 the reader is reminded that the existence of inaccessible cardinals cannot be proved in zfc zermelo fraenkel set theory with axiom of choice added and that in the constructible universe there are no such cardinals all proofs here are within the framework of zfc our purpose here is to show that if gd m is not an inaccessible cardinal then gd m is attained in m our result is best possible for inaccessible cardinals see the remark at the end

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

higher order frequency domain computational electromagnetics a review of the higher order computational electromagnetics cem for antenna wireless and microwave engineering applications is presented higher order cem techniques use current field basis functions of higher orders defined on large e g on the order of a wavelength in each dimension curvilinear geometrical elements which greatly reduces the number of unknowns for a given problem the paper reviews all major surface volume integral and differential equation electromagnetic formulations within a higher order computational framework focusing on frequency domain solutions with a systematic and unified review of generalized curved parametric quadrilateral triangular hexahedral and tetrahedral elements and various types of higher order hierarchical and interpolatory vector basis functions in both divergence and curl conforming arrangements a large number of actual higher order techniques representing various combinations of formulations elements bases and solution procedures are identified and discussed the examples demonstrate the accuracy efficiency and versatility of higher order techniques and their advantages over low order discretizations the most important one being a much faster higher order convergence of the solution it is demonstrated that both components of the higher order modeling namely higher order geometrical modeling and higher order current field modeling are essential for accurate and efficient cem analysis of general antenna scattering and microwave structures

monomial cubature rules since stroud a compilation a bibliography of references to cubature rules which have appeared since the publication of stroud s book 1971 is presented the standard regions that are treated in this paper are the n cube the n simplex the n sphere and the entire space

comparison of a class of subdomain and entire domain basis functions automatically satisfying kcl accuracy number of unknowns and cpu time are compared for piecewise linear subdomain basis functions and polynomial entire domain basis functions both types of functions automatically satisfy a continuity equation at wire ends and junctions according to kirchoff s current law kcl the relative root mean square rms current deviation is chosen as the error metric an electrically short scatterer a crossed wire scatterer and an electrically long scatterer are used for comparison currents are obtained by solving the electric field integral equation efie by means of the galerkin method it was shown that in most cases for the same accuracy required the entire domain approximation uses three to five times less numbers of unknowns and 10 100 times less cpu time than the subdomain approximation generally such efficiency is achieved by using entire domain expansions the order of which is up to n 5 and cannot be significantly improved by using higher order expansions

higher order hierarchical legendre basis functions for electromagnetic modeling this paper presents a new hierarchical basis of arbitrary order for integral equations solved with the method of moments mom the basis is derived from orthogonal legendre polynomials which are modified to impose continuity of vector quantities between neighboring elements while maintaining most of their desirable features expressions are presented for wire surface and volume elements but emphasis is given to the surface elements in this case the new hierarchical basis leads to a near orthogonal expansion of the unknown surface current and implicitly an orthogonal expansion of the surface charge in addition all higher order terms in the expansion have two vanishing moments in contrast to existing formulations these properties allow the use of very high order basis functions without introducing ill conditioning of the resulting mom matrix numerical results confirm that the condition number of the mom matrix obtained with this new basis is much lower than existing higher order interpolatory and hierarchical basis functions as a consequence of the excellent condition numbers we demonstrate that even very high order mom systems e g tenth order can be solved efficiently with an iterative solver in relatively few iterations

maximally orthogonalized higher order basis functions in large domain finite element modeling in electromagnetics curl conforming max ortho basis functions mobfs are coupled with higher order large domain curved finite elements fes the performance of the functions is compared with that of the classical and near ortho basis functions through numerical experiments it is shown that max ortho fes yield highly orthogonal mass matrices for practically arbitrarily high orders of polynomial field approximations this facilitates the usage of iterative solvers and it significantly increases their efficiency accurate and fast computation of mobfs of arbitrarily high orders is enabled by the proposed two term recurrent formula

computing with noisy information this paper studies the depth of noisy decision trees in which each node gives the wrong answer with some constant probability in the noisy boolean decision tree model tight bounds are given on the number of queries to input variables required to compute threshold functions the parity function and symmetric functions in the noisy comparison tree model tight bounds are given on the number of noisy comparisons for searching sorting selection and merging the paper also studies parallel selection and sorting with noisy comparisons giving tight bounds for several problems

double higher order method of moments for surface integral equation modeling of metallic and dielectric antennas and scatterers a novel double higher order galerkin type method of moments based on higher order geometrical modeling and higher order current modeling is proposed for surface integral equation analysis of combined metallic and dielectric antennas and scatterers of arbitrary shapes the technique employs generalized curvilinear quadrilaterals of arbitrary geometrical orders for the approximation of geometry metallic and dielectric surfaces and hierarchical divergence conforming polynomial vector basis functions of arbitrary orders for the approximation of electric and magnetic surface currents within the elements the geometrical orders and current approximation orders of the elements are entirely independent from each other and can be combined independently for the best overall performance of the method in different applications the results obtained by the higher order technique are validated against the analytical solutions and the numerical results obtained by low order moment method techniques from literature the examples show excellent accuracy flexibility and efficiency of the new technique at modeling of both current variation and curvature and demonstrate advantages of large domain models using curved quadrilaterals of high geometrical orders with basis functions of high current approximation orders over commonly used small domain models and low order techniques the reduction in the number of unknowns is by an order of magnitude when compared to low order solutions