some applications of covariance identities and inequalities to functions of multivariate normal variables abstract we apply general covariance identities and inequalities to some functions of multivariate normal variables we recover in particular a recent covariance identity due to siegel and provide simple estimates on the variance of order statistics we also present some computations

a radial basis function algorithm with automatic model order determination we present a new radial basis function rbf algorithm for constructing nonlinear models from data that may be sparsely scattered in high dimensions we propose a simplified method for identifying function locations that is based on an approximation to the autocorrelation function local regions are now defined based on zero crossings of the modified autocorrelation contribution the resulting local spatiotemporal data appear to capture increased structure as evidenced by the reduced number of basis functions required to achieve algorithm convergence and the fact that the resulting approximations have significantly improved accuracy we prove a zero crossing lemma and convergence of the algorithm in the norm for continuous functions over a compact domain further we show that several hypotheses tests can be used to detect this convergence the model parameters and the number of basis functions are determined automatically from the given data the only user parameter is the confidence level of the hypothe

poincare type inequalities via stochastic integrals martingales and stochastic integrals are applied to prove poincare type inequalities involving probability distributions on the euclidean space these inequalities generalize and improve several results in the literature and are shown to yield weighted poincare inequalities on some special compact manifolds this leads to a new method of calculating all the eigenvalues and eigenfunctions of the laplacian on then sphere as a by product the eigenvalues are shown to be related to the moments of a probability distribution

subgroups of free idempotent generated semigroups need not be free abstract we study the maximal subgroups of free idempotent generated semigroups on a biordered set by topological methods these subgroups are realized as the fundamental groups of a number of 2 complexes naturally associated to the biorder structure of the set of idempotents we use this to construct the first example of a free idempotent generated semigroup containing a non free subgroup

divisor class groups of singular surfaces we compute divisors class groups of singular surfaces most notably we produce an exact sequence that relates the cartier divisors and almost cartier divisors of a surface to the those of its normalization this generalizes hartshorne s theor em for the cubic ruled surface in p 3 we apply these results to limit the possible curves that can be s et theoretic complete intersection in p 3 in characteristic zero on a nonsingular variety the study of divisors and linear systems is classical in fact the entire theory of curves and surfaces is dependent on this study of codimension one subvarieties and the linear and algebraic families in which they move this theory has been generalized in two directions the weil divisors on a normal variety taking codimension one subvarieties as prime divisors and the cartier divisors on an arbitrary scheme based on locally principal codimension one subschemes most of the literature both in algebraic geometry and commutative algebra up to now has been limited to these kinds of divisors more recently there have been good reasons to consider divisors on non normal varieties jaffe 9 introduced the notion of an almost cartier divisor which is locally principal off a subset of codimension two a theory of generalized divisors was proposed on curves in 14 and extended to any dimension in 15 the latter paper gave a complete description of the generalized divisors on the ruled cubic surface in p 3 in this paper we extend that analysis to an arbitrary integra l surface x explaining the group apicx of linear equivalence classes of almost cartier divisors on x in terms of the picard group of the normalization s of x and certain local data at the singular points of x we apply these results to give limitations on the possible curves that can b e set theoretic compete intersections in p 3 in characteristic zero in section 2 we explain our basic set up comparing divisors on a variety x to its normalization s in section 3 we prove a local isomorphism that computes the group of almost cartier divisors at a singular point of x in terms of the cartier divisors along the curve of singulari ties and its inverse image in the normalization in section 4 we derive some global exact sequences for the groups picx apicx and pics which generalize the results of 15 6 to arbitrary surfaces these results are particularly transparent for surfaces wi th ordinary singularities meaning a double curve with a finite number of pinch points and triple points

second order poincare inequalities and clts on wiener space we prove infinite dimensional second order poincare inequalities on wiener space thus closing a circle of ideas linking limit theorems for functionals of gaussian fields stein s method and malliavin calculus we provide two applications i to a new second order characterization of clts on a fixed wiener chaos and ii to linear functionals of gaussian subordinated fields

zero crossing rates of mixtures and products of gaussian processes formulas for the expected zero crossing rate of non gaussian mixtures and products of gaussian processes are obtained the approach we take is to first derive the expected zero crossing rate in discrete time and then obtain the rate in continuous time by an appropriate limiting argument the processes considered which are non gaussian but derived from gaussian processes serve to illustrate the variability of the zero crossing rate in terms of the normalized autocorrelation function p t of the process for gaussian processes rice s formula gives the expected zero crossing rate in continuous time as 1 spl pi spl radic spl rho 0 we show processes exist with expected zero crossing rates given by spl kappa spl pi spl radic spl rho 0 with either spl kappa spl gt 1 or spl kappa spl lt 1 consequently such processes can have an arbitrarily large or small zero crossing rate as compared to a gaussian process with the same autocorrelation function

undecidability of the word problem for one relator inverse monoids via right angled artin subgroups of one relator groups we prove the following results 1 there is a one relator inverse monoid mathrm inv langle a w 1 rangle with undecidable word problem and 2 there are one relator groups with undecidable submonoid membership problem the second of these results is proved by showing that for any finite forest the associated right angled artin group embeds into a one relator group combining this with a result of lohrey and steinberg j algebra 320 2 728 755 2008 we use this to prove that there is a one relator group containing a fixed finitely generated submonoid in which the membership problem is undecidable to prove 1 a new construction is introduced which uses the one relator group and submonoid in which membership is undecidable from 2 to construct a one relator inverse monoid mathrm inv langle a w 1 rangle with undecidable word problem furthermore this method allows the construction of an e unitary one relator inverse monoid of this form with undecidable word problem the results in this paper answer a problem originally posed by margolis et al in semigroups and their applications reidel dordrecht pp 99 110 1987

hnn extensions of inverse semigroups and groupoids abstract we use the isomorphism between the categories of inverse semigroups and inductive groupoids to construct hnn extensions of inverse semigroups where the associated inverse subsemigroups are order ideals of the base properties of groupoids then ensure that the base inverse semigroup always embeds in an hnn extension constructed in this way other properties of hnn extensions are determined including a description of the maximal subgroups and of the maximal group image and the structure of the inverse subsemigroup generated by the stable letters finally some examples are given

zero crossing rates of functions of gaussian processes formulas for the expected zero crossing rates of random processes that are monotone transformations of gaussian processes can be obtained by using two different techniques the first technique involves derivation of the expected zero crossing rate for discrete time processes and extends the result of the continuous time case by using an appropriate limiting argument the second is a direct method that makes successive use of r price s 1958 theorem the chain rule for derivatives and s o rice s 1954 formula for the expected zero crossing rate of a gaussian process a constant which depends on the variance of the transformed process and a second moment of its derivative is derived multiplying rice s original expression by this constant yields the zero crossing formula for the transformed process the two methods can be used for the general level crossing problem of random processes that are monotone functions of a gaussian process

bass serre theory for groupoids and the structure of full regular semigroup amalgams t e hall proved in 1978 that if s1 s2 u is an amalgam of regular semigroups in whichs1 s2 uis a full regular subsemigroup ofs1ands2 i e s1 s2 anduhave the same set of idempotents then the amalgam is strongly embeddable in a regular semigroupsthat containss1 s2 anduas full regular subsemigroups in this case the inductive structure of the amalgamated free produces1 us2was studied by nambooripad and pastijn in 1989 using ordman s results from 1971 on amalgams of groupoids in the present paper we show how these results may be combined with techniques from bass serre theory to elucidate the structure of the maximal subgroups ofs1 us2 this is accomplished by first studying the appropriate analogue of the bass serre theory for groupoids and applying this to the study of the maximal subgroups ofs1 us2 the resulting graphs of groups are arbitrary bipartite graphs of groups this has several interesting consequences for example ifs1ands2are combinatorial then the maximal subgroups ofs1 us2are free groups finite inverse semigroups may be decomposed in non trivial ways as amalgams of inverse semigroups

spectral zero crossings localization properties and applications we present an analysis of the rate of sign changes in the discrete fourier spectrum of a sequence the sign changes of either the real or imaginary parts of the spectrum are considered and the rate of sign changes is termed as the spectral zero crossing rate szcr we show that szcr carries information pertaining to the locations of transients within the temporal observation window we show duality with temporal zero crossing rate analysis by expressing the spectrum of a signal as a sum of sinusoids with random phases this extension leads to spectral domain iterative filtering approaches to stabilize the spectral zero crossing rate and to improve upon the location estimates the localization properties are compared with group delay based localization metrics in a stylized signal setting well known in speech processing literature we show applications to epoch estimation in voiced speech signals using the szcr on the integrated linear prediction residue the performance of the szcr based epoch localization technique is competitive with the state of the art epoch estimation techniques that are based on average pitch period

a new application of random matrices ext c _ mathrm red f_2 is not a group in the process of developing the theory of free probability and free entropy voiculescu introduced in 1991 a random matrix model for a free semicircular system since then random matrices have played a key role in von neumann algebra theory cf v8 v9 the main result of this paper is the following extension of voiculescui s random matrix result let x n 1 x n r be a system of r stochastically independent n i n gaussian self adjoint random matrices as in voiculescui s random matrix paper v4 and let x1 xr be a semi circular system in a c probability space then for every polynomial p in r noncommuting variables lim niaeia p x n 1 o x n r o p x1 xr for almost all o in the underlying probability space we use the result to show that the ext invariant for the reduced c algebra of the free group on 2 generators is not a group but only a semi group this problem has been open since anderson in 1978 found the first example of a c algebra a for which ext a is not a group

free products of inverse semigroups a structure theorem is provided for the free product s inv t of inverse semigroups s and t each element of s inv t is uniquely expressible in the form r a a where a is a certain finite set of left reduced words and either a i i or a a a is a reduced word with aa e a the word a1 l am in s sgp t is called reduced if no letter is idempotent and left reduced if exactly a is idempotent the notation a stands for fl aa a e a under a product remarkably similar to scheiblich s product for free inverse semigroups the corresponding pairs a a form an inverse semigroup isomorphic with s inv t this description enables various properties of s inv t to be determined for example s inv t s u t is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup of s or t if neither s nor t has a zero then s inv t is fundamental but in general fundamentality itself is not preserved

a generalization of chernoff inequality via stochastic analysis let be a probability measure on rdwith density c exp 2u x where u c2 rd left nabla u x right 2 delta u x to infty and u x as x by using stochastic analysis and theorems in schrodinger operators we have the following result there exists a constant c 0 such that var mu f underline leqslant c e mu left nabla f right 2 1 for any f l1 with a well defined distributional gradient f under our condition the operator frac 1 2 delta nabla u cdot nabla in l2 has discrete spectrum 0 1 2 m m 1 with corresponding eigenfunctions n which form a c o n s complete orthonormal system if the r h s of 1 is finite then equality holds iff f sum limits i 1 m b i phi i for some b1 bm r moreover the constant c can be taken as 2 2 1

multivariate inequalities of chernoff type for classical orthogonal polynomials abstract in this paper we present two sided chernoff type inequalities for the error of the best approximation of a smooth d variate function by polynomials of total degree less than k in the l w 2 norm it is supposed that the d variate weight function w has one dimensional classical component weights which satisfy pearson differential equations similarly as in the univariate case the leading coefficients of the multivariate classical polynomials orthonormal with respect to the weight function w play an important role in the presented estimates

asymptotics for high dimensional regression m estimates fixed design results we investigate the asymptotic distributions of coordinates of regression m estimates in the moderate p n regime where the number of covariates p grows proportionally with the sample size n under appropriate regularity conditions we establish the coordinate wise asymptotic normality of regression m estimates assuming a fixed design matrix our proof is based on the second order poincare inequality and leave one out analysis some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices we also show a counterexample namely an anova type design to emphasize that the technical assumptions are not just artifacts of the proof finally numerical experiments confirm and complement our theoretical results

structural information and dynamical complexity of networks in 1953 shannon proposed the question of quantification of structural information to analyze communication systems the question has become one of the longest great challenges in information science and computer science here we propose the first metric for structural information given a graph g we define the k dimensional structural information of g or structure entropy of g denoted by mathcal h k g to be the minimum overall number of bits required to determine the k dimensional code of the node that is accessible from random walk in g the k dimensional structural information provides the principle for completely detecting the natural or true structure which consists of the rules regulations and orders of the graphs for fully distinguishing the order from disorder in structured noisy data and for analyzing communication systems solving the shannon s problem and opening up new directions the k dimensional structural information is also the first metric of dynamical complexity of networks measuring the complexity of interactions communications operations and even evolution of networks the metric satisfies a number of fundamental properties including additivity locality robustness local and incremental computability and so on we establish the fundamental theorems of the one and two dimensional structural information of networks including both lower and upper bounds of the metrics of classic data structures general graphs the networks of models and the networks of natural evolution we propose algorithms to approximate the k dimensional structural information of graphs by finding the k dimensional structure of the graphs that minimizes the k dimensional structure entropy we find that the k dimensional structure entropy minimization is the principle for detecting the natural or true structures in real world networks consequently our structural information provides the foundation for knowledge discovering from noisy data we establish a black hole principle by using the two dimensional structure information of graphs we propose the natural rank of locally listing algorithms by the structure entropy minimization principle providing the basis for a next generation search engine

a class of inverse monoids acting on ordered forests inverse monoid actions on ordered forests are studied to generalize the bass serre theory to a certain class of inverse monoids we introduce the concepts of graphs of inverse monoids and their fundamental inverse monoids and discuss their basic properties using these concepts we characterize the inverse monoids acting on ordered forests satisfying some conditions as the groups acting on trees without inversion are characterized as the fundamental groups of graphs of groups we also investigate the local action of a maximal subgroup of the fundamental inverse monoid on a connected component of the universal cover and obtain a presentation of the maximal subgroup

hardy s inequality for dirichlet forms abstract we prove an abstract form of hardy s l2 inequality in which the dirichlet integral is replaced by the dirichlet form of a general symmetric markov process a number of examples are provided

generalized framework for the level crossing analysis of ordered random processes this paper investigates the following general problem relating to ordered random processes given n independent but not necessarily identical random processes how frequently on average does any given process become one of the pth p 1 2 n 1 largest processes this is a fundamental problem arising in the design and analysis of contemporary multidimensional wireless communication systems e g multiantenna multiuser employing opportunistic selection we formulate this problem as one involving the level crossing rate lcr of a carefully defined ordered random process across the zero threshold which we solve by developing a new mathematical framework based on the theory of permanents for the case where the processes correspond to time varying rayleigh fading channels we present exact closed form formulas for the lcr simplified tight upper bounds as well as asymptotic results for n and p approaching infinity but with fixed ratio these results reveal interesting fundamental limits for the lcr and are shown to given meaningful insight even for small values of n and p we further use our mathematical framework to characterize the required per branch and overall switching rate of a generalized selection combining diversity receiver allowing for different average powers for each branch with the aid of majorization theory we demonstrate that the overall switching rate is maximized when the power delay profile is uniform

a note on an inequality involving the normal distribution the following inequality is useful in studying a variation of the classical isoperimetric problem let x be normally distributed with mean 0 and variance 1 if g is absolutely continuous and g x has finite variance then e lbrack g x rbrack 2 geq operatorname var lbrack g x rbrack with equality if and only if g x is linear in x the proof involves expanding g x in hermite polynomials

fisher information inequalities and the central limit theorem we give conditions for an o 1 n rate of convergence of fisher information and relative entropy in the central limit theorem we use the theory of projections in l 2 spaces and poincare inequalities to provide a better understanding of the decrease in fisher information implied by results of barron and brown we show that if the standardized fisher information ever becomes finite then it converges to zero

discrete isoperimetric and poincare type inequalities we study some discrete isoperimetric and poincare type inequalities for product probability measures n on the discrete cube 0 1 n and on the lattice z n in particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube more generally we characterize those probability distributions on z which satisfy these inequalities on z n the class of these distributions can be described by a certain class of monotone transforms of the two sided exponential measure a similar characterization of distributions on r which satisfy poincare inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes

on the level crossing duration distributions of nakagami processes the problem related to the fade and non fade duration distributions fdd and nfdd of randomly distributed signals has been studied for more than forty years however for non gaussian processes closed form general solutions for these distributions are still unknown in this paper we use an orthogonal series expansion to analyze the fdd and nfdd also known as downcrossing and upcrossing duration distributions respectively for the rather general case of nakagami distributed random processes our results show that for asymptotic and non asymptotic threshold levels the crossing duration distributions can be approximated in a simple closed form way using the gamma distribution we back up this approximation with some goodness of fit metrics and show how the parameters of such distribution can be computed a priori analytically

some results on upper bounds for the variance of functions of the residual life random variables as a measure of maximum dispersion from the mean upper bounds on variance have applications in all areas of theoretical and applied mathematical sciences in this paper we obtain an upper bound for the variance of a function of the residual life random variable xt since one of the most important types of system structures is the parallel structure we give an upper bound for the variance of a function of this system consisting of n identical and independent components under the condition that at time t nr 1 r 1 n of its components are still working here we characterize the pareto distribution through cauchys functional equation for mean residual life it is shown that the underlying distribution function f can be recovered from the proposed mean and variance residual life function of the system for r 1 moreover we see that the variance residual lifetime of the components of the system is not necessarily a decreasing function of r and increasing of n for r 1 unlike their mean residual lifetime as an application the variance of xf1 p0 for all p0 0 1 is investigated and also a real data analysis is presented

dixmier groups and borel subgroups abstract in this paper we study a family g n n 0 of infinite dimensional ind algebraic groups associated with algebras morita equivalent to the weyl algebra a 1 c we give a geometric presentation of these groups in terms of amalgamated products generalizing classical theorems of dixmier and makar limanov our main result is a classification of borel subgroups of g n for all n we show that the conjugacy classes of non abelian borel subgroups of g n are in bijection with the partitions of n furthermore we prove an infinite dimensional analogue of the classical theorem of steinberg 52 that characterizes borel subgroups in purely group theoretic terms combined together the last two results imply that the g n are pairwise non isomorphic as abstract groups this settles an old question of stafford 51

amalgams of finite inverse semigroups abstract we show that the word problem is decidable for an amalgamated free product of finite inverse semigroups in the category of inverse semigroups this is in contrast to a recent result of m sapir that shows that the word problem for amalgamated free products of finite semigroups in the category of semigroups is in general undecidable

on upper and lower bounds for the variance of a function of a random variable chernoff 1981 obtained an upper bound for the variance of a function of a standard normal random variable using hermite polynomials chen 1980 gave a different proof using the cauchy schwarz inequality and extended the inequality to the case of a multivariate normal here it is shown how similar upper bounds can be obtained for other distributions including discrete ones moreover by using a variation of the cramer rao inequality analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions matrix inequalities are also obtained all these bounds involve the first two moments of derivatives or differences of the function

amalgamated free products of inverse semigroups abstract this paper is the first of two papers devoted to the study of amalgamated free products of inverse semigroups the subject of our second paper is the structure of amalgamated free products while this one is concerned with offering concrete descriptions of normal forms we define a lower bounded amalgam of inverse semigroups and present a procedure by which the schutzenberger automata of the amalgamated free product can be constructed those automata which can be such a schutzenberger automaton are characterized yielding a canonical form for the amalgamated free product our proof makes use of the graph theoretic ideas developed by jones margolis meakin and stephen for presentations of inverse semigroups

normal approximation on poisson spaces mehler s formula second order poincare inequalities and stabilization we prove a new class of inequalities yielding bounds for the normal approximation in the wasserstein and the kolmogorov distance of functionals of a general poisson process poisson random measure our approach is based on an iteration of the classical poincare inequality as well as on the use of malliavin operators of stein s method and of an integrated mehler s formula providing a representation of the ornstein uhlenbeck semigroup in terms of thinned poisson processes our estimates only involve first and second order difference operators and have consequently a clear geometric interpretation in particular we will show that our results are perfectly tailored to deal with the normal approximation of geometric functionals displaying a weak form of stabilization and with non linear functionals of poisson shot noise processes we discuss two examples of stabilizing functionals in great detail i the edge length of the k nearest neighbour graph ii intrinsic volumes of k faces of voronoi tessellations in all these examples we obtain rates of convergence in the kolmogorov and the wasserstein distance that one can reasonably conjecture to be optimal thus significantly improving previous findings in the literature as a necessary step in our analysis we also derive new lower bounds for variances of poisson functionals