local properties of intertwining operators on the sphere blaschke s original question regarding the local determination of zonoids or projection bodies has been the subject of much research over the years in recent times this research has been extended to include intersection bodies and it has been shown that neither zonoids nor intersection bodies have local characterizations however it has also been proved that both these classes of bodies admit equatorial characterizations in odd dimensions but not in even dimensions the proofs of these results were mostly analytic using properties of associated spherical integral transforms the cosine transform and the radon transform here we elaborate a general principle showing that such local or equatorial characterization problems are equivalent to corresponding support properties of the spherical operators we discuss this within a general framework for intertwining operators on c functions and apply the results to further geometric constructions namely to certain mean section bodies to lq centroid bodies and to k intersection bodies

matrices a such that as 1r ra with rk i abstract we study matrices a c n n such that a s 1 r r a where r k i n and s k are nonnegative integers with k 2 such matrices are called r s 1 k potent matrices the s 0 case corresponds to matrices such that a r a r 1 with r k i n and is studied using spectral properties of the matrix r for s 1 various characterizations of the class of r s 1 k potent matrices and relationships between these matrices and other classes of matrices are presented

monte carlo generation of self avoiding walks with fixed endpoints and fixed length we propose a new class of dynamic monte carlo algorithms for generating self avoiding walks uniformly from the ensemble with fixed endpoints and fixed length in any dimension and prove that these algorithms are ergodic in all cases we also prove the ergodicity of a variant of the pivot algorithm

generalized intersection bodies abstract we study the structures of two types of generalizations of intersection bodies and the problem of whether they are in fact equivalent intersection bodies were introduced by lutwak and played a key role in the solution of the busemann petty problem a natural geometric generalization of this problem considered by zhang led him to introduce one type of generalized intersection bodies a second type was introduced by koldobsky who studied a different analytic generalization of this problem koldobsky also studied the connection between these two types of bodies and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized busemann petty problem we show that these classes share many identical structural properties proving the same results using integral geometry techniques for zhang s class and fourier transform techniques for koldobsky s class using a functional analytic approach we give several surprising equivalent formulations for the equivalence problem which reveal a deep connection to several fundamental problems in the integral geometry of the grassmann manifold

a generalization of n matrices abstract ky fan defines an n matrix to be a matrix of the form a ti b b 0 t b where b is the spectral radius of b and is the maximum of the spectral radii of all principal submatrices of b of order n 1 in this paper we define the closure n 0 matrices of n matrices by letting t it is shown that if a z and a 1 a n certain inequalities of n matrices are shown to hold for n 0 matrices and a method for constructing an n matrix from an m matrix is given

characterizations and linear combinations of k generalized projectors abstract we extend generalized projectors introduced by gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 to k generalized projectors and we characterize them obtaining results in the aforesaid paper as a consequence moreover we list all situations when a linear combination of commuting k generalized projectors is a k generalized projector the method for obtaining this result permits to give a revisited version of the main result by baksalary and baksalary on linear combinations of generalized projectors linear algebra appl 388 2004 17 24 in addition the case of orthogonal projectors is also analyzed

distribution of the distance between opposite nodes of random polygons with a fixed knot we examine numerically the distribution function fk r of the distance r between opposite polygonal nodes for random polygons of n nodes with a fixed knot type k here we consider three knots such as 31 and 3131 in a wide range of r the shape of fk r is well fitted by the scaling form 1 of self avoiding walks the fit yields the gaussian exponents and k 1 furthermore if we re scale the intersegment distance r by the average size rk of random polygons of knot k the distribution function of the variable r rk should become the same gaussian distribution for any large value of n and any knot k we also introduce a fitting formula to the distribution gk r of gyration radius r for random polygons under some topological constraint k

sections of convex bodies the generalized busemann petty problem asks if the volume of i dimensional central section of a centrally symmetric convex body in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is smaller than that of another such body is the volume of the body also smaller it is proved that the answer is negative if 2 i n the case of a 2 dimensional section remains open the proof uses techniques in functional analysis and radon transforms on grassmannians it also requires the notion of an i intersection body which generalizes the notion of an intersection body inequalities among the volumes of projection bodies polar projection bodies and their central sections are proved they are related to the maximal slice problem

convolutions transforms and convex bodies the paper studies convex bodies and star bodies in bbb r n by using radon transforms on grassmann manifolds p cosine transforms on the unit sphere and convolutions on the rotation group of bbb r n it presents dual mixed volume characterizations of i intersection bodies and l p balls which are related to certain volume inequalities for cross sections of convex bodies it considers approximations of special convex bodies by analytic bodies and various finite sums of ellipsoids which preserve special geometric properties convolution techniques are used to derive formulas for mixed volumes mixed surface measures and p cosine transforms they are also used to prove characterizations of geometric functionals such as surface area and dual quermassintegrals

ray patterns of matrices and nonsingularity a complex matrix a is ray nonsingular if det x 0 a f 0 for every matrix x with positive entries a sufficient condition for ray nonsingularity is that the origin is not in the relative interior of the convex hull of the signed transversal products of a the concept of an isolated set of transversals is defined and used to obtain a neces

sections of convex bodies with symmetries abstract in this paper we study how certain symmetries of convex bodies affect their geometric properties in particular we consider the impact of symmetries generated by the block diagonal subgroup of orthogonal transformations generalizing complex and quaternionic convex bodies we conduct a systematic study of sections of bodies with symmetries of this type with the emphasis on problems of the busemann petty type and hyperplane inequalities the main role belongs to the class of intersection bodies with symmetries

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

intersection bodies in r4 intersection bodies were introduced in 1988 by lutwak who found a close connection between those bodies and the well known 1956 busemann petty problem which asks whether origin symmetric convex bodies with larger central hyperplane sections also have greater volume the author has recently shown that an origin symmetric star bodykis an intersection body if and only if the function x 1kis a positive definite distribution where x k min a 0 x ak we use this result to prove that the unit balls of the spaces l4q 2 q are intersection bodies forn 5 the unit balls of the spaces lnq 2 q are not intersection bodies the technique of this paper allows to find precise expressions for the generating measures signed measures of these bodies the result that the unit cube in r4is an intersection body shows that the negative solution to the busemann petty problem forn 4 in 20 was incorrect

critical exponents hyperscaling and universal amplitude ratios for two and three dimensional self avoiding walks we make a high precision monte carlo study of two and three dimensional self avoiding walks saws of length up to 80 000 steps using the pivot algorithm and the karp luby algorithm we study the critical exponents v and 2 triangle sub 4 gamma as well as several universal amplitude ratios in particular we make an extremely sensitive test of the hyperscaling relation dv 2 triangle sub 4 gamma in two dimensions we confirm the predicted exponent v 3 4 and the hyperscaling relation we estimate the universal ratios 0 14026 0 00007 0 43961 0 00034 and psi 0 66296 0 00043 68 confidence limits in three dimensions we estimate v 0 5877 0 0006 with a correction to scaling exponent triangle sub 1 0 56 0 03 subjective 68 confidence limits this value for v agrees excellently with the field theoretic renormalization group prediction but there is some discrepancy for triangle 1 earlier monte carlo estimates of v which were approximately 0 592 are now seen to be biased by corrections to scaling we estimate the universal ratios 0 1599 0 0002 and psi 0 2471 0 0003 since psi 0 hyperscaling holds the approach to psi is from above contrary to the prediction of the two parameter renormalization group theory we critically reexamine this theory and explain where the error lies in an appendix we prove rigorously modulo some standard scalingmore assumptions the hyperscaling relation dv 2 triangle sub 4 gamma for two dimensional saws less

on a matrix group constructed from an r s 1 k potent matrix abstract let r c n n be a k involutory matrix that is r k i n for some integer k 2 and let s be a nonnegative integer a matrix a c n n is called an r s 1 k potent matrix if a satisfies r a a s 1 r in this paper a matrix group corresponding to a fixed r s 1 k potent matrix is explicitly constructed and properties of this group are derived and investigated this group is then reconciled with the classical matrix group g a that is associated with a generalized group invertible matrix a

irreducible pattern k potent ray pattern matrices abstract a ray pattern is the class of all complex matrices with a specified zero nonzero pattern such that the argument of each nonzero entry is specified the ray pattern a is called pattern k potent if k is the smallest positive integer for which ak 1 a as ray patterns we characterize the irreducible ray patterns that are pattern k potent and we provide a canonical form for patterns with respect to permutation similarity and signature similarity by diagonal unitary matrices

matrices with multiple symmetry properties applications of centrohermitian and perhermitian matrices abstract twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system we investigate matrices satisfying more than one symmetry pattern we show how a group of operators on gl n c gives rise to distinct types of matrices which satisfy sets of patterns and which give unique decompositions of matrices into components of each type these give a new characterization of normal and unitary matrices we extend symmetry patterns to vectors to study spectral properties of these matrices when a skew symmetric basis of eigenvectors exist we can infer symmetry properties of these matrices

on the critical behaviour of self avoiding walks for pt i see ibid vol 22 p 1839 1989 for the self avoiding walk problem the coefficients of the chain generating function and of the generating function for the sum of square end to end distances have been extended to 20 terms for the triangular lattice to 27 terms for the diamond lattice to 21 terms for the simple cubic lattice and to 16 terms for the bcc lattice precise estimates of the critical points are obtained and for the exponents we find that gamma 1 161 or 0 002 and nu 0 592 or 0 003 encompasses all the three dimensional lattice data

isomorphic properties of intersection bodies we study isomorphic properties of two generalizations of intersection bodies the class ikn of k intersection bodies in rn and the class bpkn of generalized k intersection bodies in rn in particular we show that all convex bodies can be in a certain sense approximated by intersection bodies namely if k is any symmetric convex body in rn and 1 k n 1 then the outer volume ratio distance from k to the class bpkn can be estimated by o v r k bpkn inf c k 1n c bpkn k c cnklogenk where c 0 is an absolute constant next we prove that if k is a symmetric convex body in rn 1 k n 1 and its k intersection body ik k exists and is convex then dbm ik k b2n c k where c k is a constant depending only on k dbm is the banach mazur distance and b2n is the unit euclidean ball in rn this generalizes a well known result of hensley and borell we conclude the paper with volumetric estimates for k intersection bodies

algorithms for solving the inverse problem associated with k a k a s 1 in previous papers the authors introduced and characterized a class of matrices called k s 1 potent also they established a method to construct these matrices the purpose of this paper is to solve the associated inverse problem several algorithms are developed in order to find all involutory matrices k satisfying k a s 1 k a for a given matrix a c n n and a given natural number s the cases s 0 and s 1 are separately studied since they produce different situations in addition some examples are presented showing the numerical performance of the methods

average size of random polygons with fixed knot topology we have evaluated by numerical simulation the average size r k of random polygons of fixed knot topology k ensuremath emptyset 3 1 3 1 ensuremath sharp 4 1 and we have confirmed the scaling law r k 2 ensuremath sim n 2 ensuremath nu k for the number n of polygonal nodes in a wide range n 100 char21 2200 the best fit gives 2 ensuremath nu k ensuremath simeq 1 11 char21 1 16 with good fitting curves in the whole range of n the estimate of 2 ensuremath nu k is consistent with the exponent of self avoiding polygons in a limited range of n n ensuremath gtrsim 600 however we have another fit with 2 ensuremath nu k ensuremath simeq 1 01 char21 1 07 which is close to the exponent of random polygons

interacting self avoiding walks and polygons in three dimensions self interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the point we consider self avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon we prove that these walks and polygons have the same limiting free energy if the interactions between nearest neighbour vertices are repulsive the attractive interaction regime is investigated using monte carlo methods and we find evidence that the limiting free energies are also equal here in particular this means that these models have the same point in the asymptotic limit the dimensions and shapes of walks and polygons are also examined as a function of the interaction strength

spectral study of r s 1 k and r s 1 k potent matrices abstract the r s 1 k and r s 1 k potent matrices have been studied in several recent papers we continue these investigations from a spectral point of view specifically a spectral study of r s 1 k potent matrices is developed using characterizations involving an associated matrix pencil a r the corresponding spectral study for r s 1 k potent matrices involves the pencil a r in order to present some properties the relevance of the projector i a a where a is the group inverse of a is highlighted in addition some applications and numerical examples are given particularly involving pauli matrices and the quaternions

generalized intersection bodies are not equivalent abstract in a koldobsky a functional analytic approach to intersection bodies geom funct anal 10 2000 1507 1526 a koldobsky asked whether two types of generalizations of the notion of an intersection body are in fact equivalent the structures of these two types of generalized intersection bodies have been studied by the author in e milman generalized intersection bodies j funct anal 240 2 2006 530 567 providing substantial evidence for a positive answer to this question the purpose of this note is to construct a counter example which provides a surprising negative answer to this question in a strong sense this implies the existence of non trivial non negative functions in the range of the spherical radon transform and the existence of non trivial spaces which embed in l p for certain negative values of p

intersection bodies and valuations all gl n covariant star body valued valuations on convex polytopes are completely classified it is shown that there is a unique nontrivial such valuation this valuation turns out to be the so called intersection operator an operator that played a critical role in the solution of the busemann petty problem

tridiagonal commuting matrices and fractionalizations of dct and dst matrices of types i iv v and viii in this paper we first establish new relationships in matrix forms among discrete fourier transform dft generalized dft gdft and various types of discrete cosine transform dct and discrete sine transform dst matrices two new independent tridiagonal commuting matrices for each of dct and dst matrices of types i iv v and viii are then derived from the existing commuting matrices of dft and gdft with these new commuting matrices the orthonormal sets of hermite like eigenvectors for dct and dst matrices can be determined and the discrete fractional cosine transform dfrct and the discrete fractional sine transform dfrst are defined the relationships among the discrete fractional fourier transform dfrft fractional gdft and various types of dfrct and dfrst are developed to reduce computations for dfrft and fractional gdft

characterization and properties of matrices with generalized symmetry or skew symmetry let r cn n be a nontrivial involution i e r r 1 i we say that a cn n is r symmetric r skew symmetric if rar a rar a there are positive integers r and s with r s n and matrices p cn r and q cn s such that p p ir q q is rp p and rq q we give an explicit representation of an arbitrary r symmetric matrix a in terms of p and q and show that solving az w and the eigenvalue problem for a reduce to the corresponding problems for matrices app cr r and aqq cs s we also express a 1 in terms of app 1 and aqq 1 under the additional assumption that r r we show that moore penrose inversion and singular value decomposition of a reduce to the corresponding problems for app and aqq we give similar results for r skew symmetric matrices these results are known for the case where r j i n j 1 i j 1n however our proofs are simpler even in this case we say that a cn n is r conjugate if rar a where r rn n and r r 1 i in this case r a is r symmetric and i a is r skew symmetric so our results provide explicit representations for r conjugate matrices in terms of p and q which are now in rn r and rn s respectively we show that solving az w inverting a and the eigenvalue problem for a reduce to the corresponding problems for a related matrix s rn n if rt r this is also true for moore penrose inversion and singular value decomposition of a

on lossless feedback delay networks lossless feedback delay networks fdns are commonly used as a design prototype for artificial reverberation algorithms the lossless property is dependent on the feedback matrix which connects the output of a set of delays to their inputs and the lengths of the delays both unitary and triangular feedback matrices are known to constitute lossless fdns however the most general class of lossless feedback matrices has not been identified in this contribution it is shown that the fdn is lossless for any set of delays if all irreducible components of the feedback matrix are diagonally similar to a unitary matrix the necessity of the generalized class of feedback matrices is demonstrated by examples of fdn designs proposed in literature

characterization and properties of matrices with k involutory symmetries we say that a matrix r cn n is k involutory if its minimal polynomial is xk 1 for some k 2 so rk 1 r 1 and the eigenvalues of r are 1 2 k 1 where e2 i k let 0 1 k 1 if r cm m a cm n s cn n and r and s are k involutory we say that a is r s symmetric if ras a we show that an r s symmetric matrix a can be usefully represented in terms of matrices fl cc l dl 0 l k 1 where cl and dl are respectively the dimensions of the l eigenspaces of r and s this continues a theme initiated in an earlier paper with the same title in which we assumed that 1 we say that a k involution is equidimensional with width d if all of its eigenspaces have dimension d we show that if r and s are equidimensional k involutions with widths d1 and d2 respectively then r s symmetric matrices are closely related to generalized circulants ras r r s 0k 1 where a0 a1 ak 1 cd1 d2 for this case our results are new even if 1 we also give an explicit formula for the moore penrose inverse of a unilevel block circulant as r r s 0k 1 for any 0 1 k 1 generalizing a result previously obtained for the case where gcd k 1

diagonally scaled permutations and circulant matrices abstract if r is an n n matrix over the complex field which is the product of a diagonal matrix d and a permutation matrix p then r is called a diagonally scaled permutation matrix we present the eigenstructure of r by observing that r is permutation similar to the direct sum of diagonally scaled permutation matrices of the form dc where d is a diagonal matrix and c is the circulant permutation 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 the matrix dc is called a scaled circulant permutation matrix we consider two cases for r dc when the scaling matrix d is nonsingular and when d is singular in the singular case r is nilpotent and we are able to obtain upper and lower bounds on the index of nilpotency of r we conclude with information about matrices that commute with a scaled permutation matrix we are also able to represent an arbitrary n n toeplitz matrix as a sum of matrices of the form d k ck for k 1 n where d k is a diagonal matrix

some properties of centrosymmetric matrices in this paper some properties of centrosymmetric matrices which often appear in the construction of orthonormal wavelet basis in wavelet analysis are investigated

the busemann petty problem in hyperbolic and spherical spaces abstract the busemann petty problem asks whether origin symmetric convex bodies in r n with smaller central hyperplane sections necessarily have smaller n dimensional volume it is known that the answer to this problem is affirmative if n 4 and negative if n 5 we study this problem in hyperbolic and spherical spaces

sign k potent sign patterns and ray k potent ray patterns that admit k potence abstract we prove that for each positive integer k all irreducible k potent ray pattern classes allow k potence we construct all irreducible real complex k potent matrices in an irreducible sign ray k potent sign ray pattern class for a given reducible sign ray pattern that is sign ray k potent for some positive integer k we determine necessary conditions for that pattern to allow k potence this generalizes recent work by lee and park 2015 2 who determined necessary conditions for a sign idempotent sign pattern to allow idempotence and who showed that every irreducible sign k potent sign pattern must allow k potence

statistical topology and knotting of fluctuating filaments abstract the aim of these notes is to provide an introduction to the topic of statistical topology with this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments some questions that we are going to address are the following i what is a knot and how can we identify it ii which is the probability of finding a random curve that is knotted iii how complex are these knots iv how big are they to try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some state of the art theoretical and numerical techniques exercises and lists of open problems will be provided too

real eigenvalues of nonnegative matrices which commute with a symmetric matrix involution abstract it is well known that if p is a nonnegative matrix then its spectral radius is an eigenvalue of p perron frobenius theorem in this paper it is shown that if p is an n n nonnegative matrix and it commutes with a nonnegative symmetric involution when n 4 m 3 then 1 p has at least two real eigenvalues if n 4 m or 4 m 2 2 p has at least one real eigenvalue if n 4 m 1 and 3 p has at least three real eigenvalues if n 4 m 3 where m is a nonnegative integer and n 1 examples are given to show that these results are the best possible and nonnegative symmetric involutions are classified

entropic exponents of lattice polygons with specified knot type ring polymers in three dimensions can be knotted and the dependence of their critical behaviour on knot type is an open question we study this problem for polygons on the simple cubic lattice using a novel grand canonical monte carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon we conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

the knot probability in lattice polygons the incidence of knots in lattice polygons in the face centred cubic lattice is investigated numerically the authors generate a sample of polygons using a pivot algorithm and detect knotted polygons by calculating the alexander polynomial if p0n phi is the probability that the polygon with n edges is the unknot then it is known that lim supn to infinity p0n phi 1n e varies as 0 1 they find that varies as phi 7 6 or 0 9 10 6 the effect of the solvent quality on p0n phi is considered the data show that the probability of a polygon being knotted increases rapidly as the quality of the solvent deteriorates

persymmetric and bisymmetric nonnegative inverse eigenvalue problem abstract the nonnegative inverse eigenvalue problem niep is the problem of finding conditions for the existence of an n n entrywise nonnegative matrix a with prescribed spectrum this problem remains open for n 5 if the matrix a is required to be persymmetric bisymmetric the problem will be called persymmetric bisymmetric nonnegative inverse eigenvalue problem pniep bniep persymmetric and bisymmetric matrices are common in physical sciences and engineering they arise for instance in the control of mechanical and electric vibrations a persymmetric version of a perturbation result due to rado and presented by h perfect in 5 is developed and used to give sufficient conditions for the pniep to have a solution our results generate an algorithmic procedure to compute the solution matrix

projection functions area measures and the alesker fourier transform abstract dual to koldobsky s notion of j intersection bodies the class of j projection bodies is introduced generalizing minkowski s classical notion of projection bodies of convex bodies a fourier analytic characterization of j projection bodies in terms of their area measures of order j is obtained in turn this yields an equivalent characterization of j projection bodies involving alesker s fourier type transform on translation invariant smooth spherical valuations as applications of these results several basic properties of j projection bodies are established and new non trivial examples are constructed

irreducible powerful ray pattern matrices a ray pattern is a matrix each of whose entries is either 0 or a ray in the complex plane originating from 0 but not including 0 a ray pattern is a natural generalization of the concept of a sign pattern whose entries are from the set 0 powers of sign patterns and ray patterns especially patterns whose powers are periodic have been studied in several recent papers a ray pattern a is said to be powerful if ak is unambiguously defined for all positive integers k irreducible powerful ray patterns have been characterized recently in this paper reducible powerful ray patterns are investigated in particular for a powerful ray pattern in frobenius normal form it is shown that the existence of a nonzero entry in an off diagonal block implies that the corresponding irreducible components are related in a certain way further the structure of each of the off diagonal blocks is characterized

sign k potent sign patterns and ray k potent ray patterns that allow k potence in this paper we characterize sign k potent sign pattern matrices that allow k potence in particular the structure of a sign idempotent sign pattern matrix that allows idempotence is given thus an open problem posed by eschenbach is affirmatively solved we also extend these results to ray k potent ray pattern matrices providing the structure of a ray k potent ray pattern matrix that allows k potence

entanglement complexity of semiflexible lattice polygons we use monte carlo methods to study knotting in polygons on the simple cubic lattice with a stiffness fugacity we investigate how the knot probability depends on stiffness and how the relative frequency of trefoils and figure eight knots changes as the stiffness changes in addition we examine the effect of stiffness on the writhe of the polygons

characterizations of k s 1 potent matrices and applications abstract recently situations where a matrix coincides with some of its powers have been studied this kind of matrices is related to the generalized inverse matrices on the other hand it is possible to introduce another class of matrices that involve an involutory matrix generalizing the well known idempotent matrix widely useful in many applications in this paper we introduce a new kind of matrices called k s 1 potent as an extension of the aforementioned ones first different properties of k s 1 potent matrices have been developed later the main result developed in this paper is the characterization of this kind of matrices from a spectral point of view in terms of powers of the matrix by means of the group inverse and via a block representation of a matrix of index 1 finally an application of the above results to study linear combinations of k s 1 potent matrices is derived

centrosymmetric nonnegative realization of spectra abstract a list 1 2 n of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix in this paper we intent to characterize those lists of complex numbers which are realizable by a centrosymmetric nonnegative matrix in particular we show that lists of nonnegative real numbers and lists of complex numbers of suleimanova type except in one particular case are always the spectrum of some centrosymmetric nonnegative matrix for the general lists we give sufficient conditions via a perturbation result we also show that for n 4 every realizable list of real numbers is also realizable by a nonnegative centrosymmetric matrix

algorithms for k s 1 potent matrix constructions in this paper we deal with k s 1 potent matrices these matrices generalize all the following classes of matrices k potent matrices periodic matrices idempotent matrices involutory matrices centrosymmetric matrices mirrorsymmetric matrices circulant matrices among others several applications of these classes of matrices can be found in the literature we develop algorithms in order to compute k s 1 potent matrices and k s 1 potent linear combinations of k s 1 potent matrices in addition some examples are presented in order to show the numerical performance of the method

pseudo centrosymmetric matrices with applications to counting perfect matchings we consider square matrices a that commute with a fixed square matrix k both with entries in a field f not of characteristic 2 when k 2 i tao and yasuda defined a to be generalized centrosymmetric with respect to k when k 2 i we define a to be pseudo centrosymmetric with respect to k we show that the determinant of every even order pseudo centrosymmetric matrix is the sum of two squares over f as long as 1 is not a square in f when a pseudo centrosymmetric matrix a contains only integral entries and is pseudo centrosymmetric with respect to a matrix with rational entries the determinant of a is the sum of two integral squares this result when specialized to when k is the even order alternating exchange matrix applies to enumerative combinatorics using solely matrix based methods we reprove a weak form of jockusch s theorem for enumerating perfect matchings of 2 even symmetric graphs as a corollary we reprove that the number of domino tilings of regions known as aztec diamonds and aztec pillows is a sum of two integral squares

periodic irreducible powerful ray pattern matrices abstract a ray pattern is a complex matrix each of whose entries is either 0 or a ray e i where is a real number for a ray pattern a a st we define the ray pattern a a st of a where a st 1 if a st 0 and a st 0 if a st 0 in this paper we first show that an irreducible powerful ray pattern a is ray diagonally similar to a for some ray by using this representation we obtain several results on irreducible powerful ray patterns and irreducible periodic ray patterns then we show that the number of such rays is k a where k a is the index of imprimitivity of a as an application to complex matrices we generalize the perron frobenius theorem to a subclass of complex matrices

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

matrices that commute with a permutation matrix abstract let p be an n n permutation matrix and let p be the corresponding permutation let a be a matrix such that ap pa it is well known that when p is an n cycle a is permutation similar to a circulant matrix we present results for the band patterns in a and for the eigenstructure of a when p consists of several disjoint cycles these results depend on the greatest common divisors of pairs of cycle lengths

intersection bodies and lp spaces abstract we prove that the convex intersection bodies are isomorphically equivalent to unit balls of subspaces of l q for each q 0 1 this is done by extending to negative values of p the factorization theorem of maurey and nikishin which states that for any 0 p q 1 every banach subspace of l p is isomorphic to a subspace of l q

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

a survey on bases of sign pattern matrices abstract an n n sign pattern matrix has entries in 1 1 0 this paper surveys the following problems concerning the bases of sign pattern matrices base and base set of sign pattern matrices the generalizations of the base of sign pattern matrices other indices of sign pattern matrices several open problems are identified

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

generalizations of the busemann petty problem for sections of convex bodies abstract we present generalizations of the busemann petty problem for dual volumes of intermediate central sections of symmetric convex bodies it is proved that the answer is negative when the dimension of the sections is greater than or equal to 4 for two three dimensional sections both negative and positive answers are given depending on the orders of dual volumes involved and certain cases remain open for bodies of revolution a complete solution is obtained in all dimensions

the effect of juxtaposition angle on knot reduction in a lattice polygon model of strand passage a model of local strand passage in self avoiding polygons saps on the simple cubic lattice is investigated numerically the polygons studied called saps contain a specific strand passage structure called at a fixed strand passage site after strand passage knot probabilities are estimated from a monte carlo study of unknotted and trefoil saps and the estimates are used here to investigate how knot reduction depends on the local juxtaposition structure at the strand passage site in particular we observe a correlation between knot reduction and the angle of the crossing at the strand passage site this same angle has been shown experimentally by neuman et al 2009 proc natl acad sci usa 106 6986 91 to be important in explaining topoisomerase action on dna the angle of the crossing is crossing sign dependent and from our observations so is knot reduction this can be used to understand experimentally observed knot type chirality biases

linking probabilities of off lattice self avoiding polygons and the effects of excluded volume we evaluate numerically the probability of linking i e the probability of a given pair of self avoiding polygons saps being entangled and forming a nontrivial link type l in the simulation we generate pairs of saps of n spherical segments of radius rd such that they have no overlaps among the segments and each of the saps has the trivial knot type we evaluate the probability of a self avoiding pair of saps forming a given link type l for various link types with fixed distance r between the centers of mass of the two saps we define normalized distance r by where denotes the square root of the mean square radius of gyration of sap of the trivial knot 01 we introduce formulae expressing the linking probability as a function of normalized distance r which gives good fitting curves with respect to 2 values we also investigate the dependence of linking probabilities on the excluded volume parameter rd and the number of segments n quite interestingly the graph of linking probability versus normalized distance r shows no n dependence at a particular value of the excluded volume parameter rd 0 2

matrices a such that ra as 1r when rk i abstract this paper examines matrices a c n n such that ra a s 1 r where r k i the identity matrix and where s and k are nonnegative integers with k 2 spectral theory is used to characterize these matrices the cases s 0 and s 1 are considered separately since they are analyzed by different techniques

the busemann petty problem for arbitrary measures the busemann petty problem asks whether symmetric convex bodies in n with smaller n 1 dimensional volume of central hyperplane sections necessarily have smaller n dimensional volume the answer to this problem is affirmative for n 4 and negative for n 5 in this paper we generalize the busemann petty problem to essentially arbitrary measure in place of the volume we also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in n

on the dominance of trivial knots among saps on a cubic lattice the knotting probability is defined by the probability with which an n step self avoiding polygon sap with a fixed type of knot appears in the configuration space we evaluate these probabilities for some knot types on a simple cubic lattice for the trivial knot we find that the knotting probability decays much slower for the sap on the cubic lattice than for continuum models of the sap as a function of n in particular the characteristic length of the trivial knot that corresponds to a half life of the knotting probability is estimated to be 2 5 105 on the cubic lattice

irreducible sign k potent sign pattern matrices abstract the sign pattern matrix a is called sign k potent if k is the smallest positive integer such that a k 1 a the structure of irreducible sign k potent pattern matrices was characterized by stuart et al j stuart c eschenbach s kirkland linear algebra appl 294 1999 85 92 we extend those results to the reducible case providing necessary conditions that characterize the structure of each off diagonal block of the frobenius normal form of a reducible sign k potent matrix

on the period and base of a sign pattern matrix abstract a square sign pattern matrix a whose entries are or 0 is said to be powerful if all the powers a 1 a 2 a 3 are unambiguously defined for a powerful pattern a if a l a l p with l and p minimal then l is called the base of a and p is called the period of a we characterize irreducible powerful sign pattern matrices and investigate the period and base of a powerful sign pattern matrix we also consider some connections with real matrices and give some significant classes of powerful patterns

on κ real and κ hermitian matrices abstract a body of theory for real and hermitian matrices is developed some basic results for these matrices their spectral properties and characterizations of linear transformations which preserve them are given

asymptotics of knotted lattice polygons we use monte carlo methods to investigate the asymptotic behaviour of the number and mean square radius of gyration of polygons in the simple cubic lattice with fixed knot type let be the number of n edge polygons of a fixed knot type in the cubic lattice and let be the mean square radius of gyration of all the polygons counted by if we assume that where is the growth constant of polygons of knot type and is the entropic exponent of polygons of knot type then our numerical data are consistent with the relation where is the unknot and is the number of prime factors of the knot if we assume that then our data are consistent with both and being independent of these results support the claims made in janse van rensburg and whittington 1991a 24 3935 and orlandini et al 1996 29 l299 1998 topology and geometry in polymer science ima volumes in mathematics and its applications berlin springer