sigma delta modulation with i i d gaussian inputs the response of a single loop sigma delta modulator to an independent identically distributed i i d gaussian input signal is analyzed a continuous time stochastic model is developed and the connection of the model to the system is described a condition is given so that the difference between the behavior of the model and that of the true system can be made arbitrarily small theories from renewal and wiener processes are applied to show the convergence and mixing properties of the output sequence also derived is the power spectrum of the quantization noise compared to the spectrum when the input is dc the i i d gaussian random process smears the discrete spectrum into band structures

cored hypergraphs power hypergraphs and their laplacian h eigenvalues abstract in this paper we introduce the class of cored hypergraphs and power hypergraphs and investigate the properties of their laplacian h eigenvalues from an ordinary graph one may generate a k uniform hypergraph called the kth power hypergraph of that graph power hypergraphs are cored hypergraphs but not vice versa sunflowers loose paths and loose cycles are power hypergraphs while squids generalized loose s paths and loose s cycles for 2 s k 2 are cored hypergraphs but not power graphs in general we show that the largest laplacian h eigenvalue of an even uniform cored hypergraph is equal to its largest signless laplacian h eigenvalue especially we find out these largest h eigenvalues for even uniform squids moreover we show that the largest laplacian h eigenvalue of an odd uniform squid loose path and loose cycle is equal to the maximum degree i e 2 we also compute the laplacian h spectra of the class of sunflowers when k is odd the laplacian h spectra of the loose cycle of size 3 and the loose path of length 3 are characterized as well

the first few unicyclic and bicyclic hypergraphs with largest spectral radii abstract a connected k uniform hypergraph with n vertices and m edges is called r cyclic if n m k 1 r 1 for r 1 or 2 the hypergraph is simply called unicyclic or bicyclic in this paper we investigate hypergraphs that attain largest spectral radii among all simple connected k uniform unicyclic and bicyclic hypergraphs specifically by using some edge operations the formula on power hypergraph eigenvalues the weighted incidence matrix and a result on linear unicyclic hypergraphs we determined the first five hypergraphs with largest spectral radius among all unicyclic hypergraphs and the first three over all bicyclic hypergraphs

on the spectral radius of uniform hypertrees abstract we determine the unique hypertrees with maximum spectral radius among k uniform non hyper caterpillars with m edges and diameter d for 4 d m 2 among k uniform non hyper caterpillars with m edges for m 6 among k uniform hypertrees with m edges and independence number for m k 1 1 k m among k uniform hypertrees with m edges and matching number for 1 m k 1 1 k and among k uniform hypertrees with m edges and s branch vertices for 0 s m 1 2 respectively

ordering of some uniform supertrees with larger spectral radii abstract a supertree is a connected and acyclic hypergraph we study some uniform supertrees with larger spectral radii we first define a new type of edge moving operation on hypergraphs and study its applications on the comparison of the spectral radii of hypergraphs by using this new operation on some supertrees together with the general edge moving operation introduced by li shao and qi in 2015 and using a result by zhou et al in 2014 about the relation between the spectral radii of an ordinary graph and its power hypergraph we are able to determine the first eight k uniform supertrees on n vertices with the largest spectral radii

quantization noise in single loop sigma delta modulation with sinusoidal inputs an exact nonlinear difference equation is derived and solved for a simple sigma delta modulator consisting of a discrete time integrator and a binary quantizer inside a single feedback loop and an arbitrary input signal it is shown that the system can be represented as an affine operation discrete time integration of a biased input followed by a memoryless nonlinearity an extension of the transform method for the analysis of nonlinear systems is applied to obtain formulas for first and second order time average moments of the binary quantization noise including the sample mean energy and autocorrelation the results are applied to the special case of a sinusoidal input signal to evaluate these time averages and the power spectrum in the limit of large oversampling ratios the marginal moments behave as if the quantization noise had a uniform distribution the spectrum is neither white nor continuous however even in the limit of large oversampling ratios

the maximum spectral radii of uniform supertrees with given degree sequences abstract this paper generalizes the breadth first search ordering on trees to k uniform supertrees by using 2 switch operation and edge moving operation on supertrees we determine the unique k uniform supertree with the maximum spectral radius among all k uniform supertrees with a given degree sequence moreover we generalize the majorization theorem on trees to k uniform supertrees

spectra of uniform hypergraphs abstract we present a spectral theory of uniform hypergraphs that closely parallels spectral graph theory a number of recent developments building upon classical work has led to a rich understanding of symmetric hyperdeterminants of hypermatrices a k a multidimensional arrays symmetric hyperdeterminants share many properties with determinants but the context of multilinear algebra is substantially more complicated than the linear algebra required to address spectral graph theory i e ordinary matrices nonetheless it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally we apply this notion to the adjacency hypermatrix of a uniform hypergraph and prove a number of natural analogs of basic results in spectral graph theory open problems abound and we present a number of directions for further study

complex valued analytic torsion for flat bundles and for holomorphic bundles with 1 1 connections the work of ray and singer that introduced analytic torsion a kind of determinant of the laplacian operator in topological and holomorphic settings is naturally generalized in both settings the couplings are extended in a direct way in the topological setting to general flat bundles and in the holomorphic setting to bundles with 1 1 connections which by using the newlander nirenberg theorem are seen to be the bundles with both holomorphic and antiholomorphic structures the resulting natural generalizations of laplacians are not always self adjoint and the corresponding generalizations of analytic torsions are thus not always real valued the cheeger muller theorem on equivalence in a topological setting of analytic torsion to classical topological torsion generalizes to this complex valued torsion on the algebraic side the methods introduced include a notion of torsion associated to a complex equipped with both boundary and coboundary maps 2009 wiley periodicals inc

a general product of tensors with applications abstract we study a general product of two n dimensional tensors a and b with orders m 2 and k 1 this product satisfies the associative law and is a generalization of the usual matrix product using this product many concepts and known results of tensors can be simply expressed and or proved and a number of applications of it will be given using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables we define the similarity and congruence of tensors which are also the generalizations of the corresponding relations for matrices and prove that similar tensors have the same characteristic polynomials thus the same spectra we study two special kinds of similarity permutational similarity and diagonal similarity and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors we also define the direct product of tensors in matrix case it is also called the kronecker product and give its applications in the study of the spectra of two kinds of the products of hypergraphs we also give applications of this general product in the study of nonnegative tensors including a characterization of primitive tensors the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors

some properties of the spectral radius for general hypergraphs abstract in this paper the adjacency tensor of a general hypergraph is investigated we study the perron frobenius theorem for the general hypergraphs and obtain some relevant results based on it in particular the techniques of weighted incidence matrix and moving edge are extended to general hypergraphs for determining the structure with the maximum spectral radius a nearly m uniform supertree is both connected and acyclic in which each edge contains either m 1 or m vertices to begin with the structures obtaining the maximum spectral radius in two classes of nearly uniform supertrees are determined where one class is with given number of edges and the other is with given number of vertices

two stage sigma delta modulation an exact analysis of the nonlinear difference equation describing a discrete time two stage sigma delta quantizer with dc and sinusoidal inputs is presented it is shown that for dc inputs the binary quantizer error of the second quantizer which appears at the output of the modulator as a second order difference is asymptotically white uniformly distributed and uncorrelated with the input for sinusoidal inputs this result holds with one exception the autocorrelation which is a sum of weighted bessel functions does not correspond to that of a white noise process at large oversampling ratios the autocorrelation is approximately white and hence the white additive noise model is a good approximation for evaluating the first and second order moments it is found that for both a sinc sup 3 decoder filter and an ideal low pass filter decoder the average quantization noise power is inversely proportional to the fifth power of the oversampling ratio r for both types of inputs this represents an improvement over the single loop sigma delta quantizer by a factor of r sup 2 simulation results that agree with the theoretical analysis are presented

on the spectral radius of a class of non odd bipartite even uniform hypergraphs abstract in order to investigate the non odd bipartiteness of even uniform hypergraphs starting from a simple graph g we construct a generalized power of g denoted by g k s which is obtained from g by blowing up each vertex into a s set and each edge into a k 2 s set where s k 2 when s k 2 g k s is always odd bipartite we show that g k k 2 is non odd bipartite if and only if g is non bipartite and find that g k k 2 has the same adjacency respectively signless laplacian spectral radius as g so the results involving the adjacency or signless laplacian spectral radius of a simple graph g hold for g k k 2 in particular we characterize the unique graph with minimum adjacency or signless laplacian spectral radius among all non odd bipartite hypergraphs g k k 2 of fixed order and prove that 2 5 is the smallest limit point of the non odd bipartite hypergraphs g k k 2 in addition we obtain some results for the spectral radii of the weakly irreducible nonnegative tensors

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

the first two largest spectral radii of uniform supertrees with given diameter abstract a supertree is a connected and acyclic hypergraph let s m d k be the set of k uniform supertrees with m edges and diameter d in this paper we determine the supertrees with the first two largest spectral radii among all supertrees in s m d k for 3 d m 1

oversampled sigma delta modulation oversampled sigma delta modulation has been proposed as a practical implementation for high rate analog to digital conversion because of its simplicity and its robustness against circuit imperfections to date mathematical developments of the basic properties of such systems have been based either on simplified continuous time approximate models or on linearized discrete time models where the quantizer is replaced by an additive white uniform noise source in this paper we rigorously derive several basic properties of a simple discrete time single integrator loop sigma delta modulator with an accumulate and dump demodulator the derivation does not require any assumptions on the correlation or distribution of the quantizer error and hence involves no linearization of the nonlinear system but it does show that when the input is constant the state sequence of the integrator in the encoder loop can be modeled exactly as a linear system in an appropriate space two basic properties are developed 1 the behavior of the sigma delta quantizer when driven by a constant input and its relation to uniform quantization and 2 the rate distortion tradeoffs between the oversampling ratio and the average mean squared quantization error

spectral analysis of quantization noise in a single loop sigma delta modulator with dc input an exact discrete time analysis of the moments and spectra of the quantization noise of a discrete time single loop sigma delta modulator with a dc input is presented an exact difference equation for the discrete time nonlinear system is used to evaluate the first and second order moments and power spectrum of the binary quantizer noise and the binary quantizer output for a single loop sigma delta encoder with a dc input it is shown that the sample mean and power of the binary quantization noise are consistent with the common uniform distribution assumption but that the autocorrelation and power spectrum are not consistent with the white noise assumption the results are used to evaluate the overall sample average mean squared quantization error as a function of the decimation filter used

on some properties of the α spectral radius of the k uniform hypergraph abstract in this paper we show how the spectral radius changes under the edge grafting operations on connected k uniform hypergraphs we characterize the extremal hypertree for spectral radius among k uniform non caterpillar hypergraphs with given order size and diameter we also characterize the supertree with the second largest spectral radius among all k uniform supertrees on n vertices

the p spectral radius of k partite and k chromatic uniform hypergraphs abstract the p spectral radius of an r uniform hypergraph g of order n is defined for every real number p 1 as p g max x 1 p x n p 1 r i 1 i r e g x i 1 x i r it generalizes several hypergraph parameters including the lagrangian the spectral radius and the number of edges this paper presents solutions to several extremal problems about the p spectral radius of k partite and k chromatic hypergraphs of order n two of the main results are i let k r 2 and let g be a k partite r graph of order n for every p 1 p g p t k r n unless g t k r n where t k r n is the complete k partite r graph of order n with parts of size n k or n k ii let k 2 and let g be a k chromatic 3 graph of order n for every p 1 p g p q k 3 n unless g q k 3 n where q k 3 n is a complete k chromatic 3 graph of order n with classes of size n k or n k the latter statement generalizes a result of mubayi and talbot

ordering trees by their largest eigenvalues abstract the set of trees with n vertices is denoted by t n hofmeister has determined the first five values of the largest eigenvalue of trees in t n and the corresponding trees for these values linear algebra appl 260 1997 43 in other words an order of the first five trees in t n by their largest eigenvalues has been given focus on the same purpose we shall give a partition for trees in t n first in the this paper and then extend this order to the eighth tree

uniform hypergraphs with the first two smallest spectral radii abstract the spectral radius of a uniform hypergraph g is the maximum modulus of the eigenvalues of the adjacency tensor of g for k 2 among connected k uniform hypergraphs with m 1 edges we show that the k uniform loose path with m edges is the unique one with smallest spectral radius and we also determine the unique ones with second smallest spectral radius when m 2

on the two largest eigenvalues of trees abstract very little is known about upper bounds for the largest eigenvalues of a tree that depend only on the vertex number starting from a classical upper bound for the largest eigenvalue some refinements can be obtained by successively removing trees from consideration the results can be used to characterize those trees that maximize the second largest eigenvalue this corrects a result from the literature and it includes a proof of a conjecture of neumaier the main tool for this endeavor is the theory of partial engenvectors

decimation for sigma delta modulation decimation is an important component of oversampled analog to digital conversion it transforms the digitally modulated signal from short words occurring at high sampling rate to longer words at the nyquist rate here we are concerned with the initial stage of decimation where the word rate decreases to about four times the nyquist rate we show that digital filters comprising cascades of integrate and dump functions can match the structure of the noise from sigma delta modulation to provide decimation with negligible loss of signal to noise ratio explicit formulas evaluate particular tradeoffs between modulation rate signal to noise ratio length of digital words and complexity of the modulating and decimating functions

a use of double integration in sigma delta modulation sigma delta modulation is viewed as a technique that employs integration and feedback to move quantization noise out of baseband this technique may be iterated by placing feedback loop around feedback loop but when three or more loops are used the circuit can latch into undesirable overloading modes in the desired mode a simple linear theory gives a good description of the modulation even when the quantization has only two levels a modulator that employs double integration and two level quantization is easy to implement and is tolerant of parameter variation at sampling rates of 1 mhz it provides resolution equivalent to 16 bit pcm for voiceband signals digital filters that are suitable for converting the modulation to pcm are also described

the effect on the spectral radius of r graphs by grafting or contracting edges abstract let h n r be the set of all connected r graphs with given size n in this paper we investigate the effect on the spectral radius of r uniform hypergraphs by grafting or contracting an edge and then give the ordering of the r graphs with small spectral radius over h n r when n 20

symmetric nonnegative tensors and copositive tensors abstract we first prove two new spectral properties for symmetric nonnegative tensors we prove a maximum property for the largest h eigenvalue of a symmetric nonnegative tensor and establish some bounds for this eigenvalue via row sums of that tensor we show that if an eigenvalue of a symmetric nonnegative tensor has a positive h eigenvector then this eigenvalue is the largest h eigenvalue of that tensor we also give a necessary and sufficient condition for this we then introduce copositive tensors this concept extends the concept of copositive matrices symmetric nonnegative tensors and positive semi definite tensors are examples of copositive tensors the diagonal elements of a copositive tensor must be nonnegative we show that if each sum of a diagonal element and all the negative off diagonal elements in the same row of a real symmetric tensor is nonnegative then that tensor is a copositive tensor some further properties of copositive tensors are discussed