lower bounds for the perron root of a nonnegative matrix abstract new lower bounds frequently better than previous bounds and readily computable for the perron root r a of a nonnegative matrix a are obtained the theoretical comparison of new bounds with the known ones is supplemented with two numerical examples

on the geometrical structure of symmetric matrices abstract the identity ray i for 0 can be seen as the center ray of the cone of symmetric and positive definite spd matrices in that sense the angle that any spd matrix forms with the identity plays a very important role to understand the geometrical structure of the cone in this work we extend this relationship and analyze the geometrical structure of symmetric matrices including the location of all orthogonal matrices not only the identity matrix this geometrical understanding leads to new results in the subspace of symmetric matrices we also extend some of the geometrical results for the case of general not necessarily symmetric nonsingular matrices

on the best constants in inequalities of the markov and wirtinger types for polynomials on the half line abstract the main topic of the paper is best constants in markov type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves the norm is the l 2 norm with laguerre weight the leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term which is the operator norm of a volterra operator are given for best constants in inequalities of the wirtinger type the limit is computed and an asymptotic formula for the error term is presented

numerical radius and zero pattern of matrices abstract let a be an n n complex matrix and r be the maximum size of its principal submatrices with no off diagonal zero entries suppose a has zero main diagonal and x is a unit n vector then letting a be the frobenius norm of a we show that a x x 2 1 1 2 r 1 2 n a 2 this inequality is tight within an additive term o r n 2 if the matrix a is hermitian then a x x 2 1 1 r a 2 this inequality is sharp moreover it implies the turan theorem for graphs

upper and lower bounds on the renyi dimensions and the uniformity of multifractals abstract we prove that for arbitrary probability measures the generalized renyi dimensions d q satisfy the inequality d q q 1 q d q q 1 q d q q 1 q q q q q 0 1 sign q sign q i n particular 1 2 d 2 d d 2 and 1 2 d d 1 d the upper bound corresponds to uniform measures whereas the lower bound describes non uniform measures such as measures with a power law singularity in order to characterize intermediate cases we introduce a quantity that measures the uniformity of multifractals we discuss the relation to the spectrum and apply the formalism to several examples such as power law measures the feigenbaum attractor and the attractor and mode locking structure of the critical circle map

lower bounds for the spectral radius of a matrix abstract we develop lower bounds for the spectral radius of symmetric skew symmetric and arbitrary real matrices our approach utilizes the well known leverrier faddeev algorithm for calculating the coefficients of the characteristic polynomial of a matrix in conjunction with a theorem by lucas which states that the critical points of a polynomial lie within the convex hull of its roots our results generalize and simplify a proof recently published by tarazaga for a lower bound on the spectral radius of a symmetric positive definite matrix in addition we provide new lower bounds for the spectral radius of skew symmetric matrices we apply these results to a problem involving the stability of fixed points in recurrent neural networks

improving eigenvalue bounds using extra bounds abstract bounds for various functions of the eigenvalues of a hermitian matrix a based on the traces of a and a 2 are improved a technique is presented whereby these bounds can be improved by combining them with other bounds in particular the diagonal of a in conjunction with majorization is used to improve the bounds these bounds all require o n 2 multiplications

geometrical properties of the frobenius condition number for positive definite matrices we study the geometrical properties of the frobenius condition number on the cone of symmetric and positive definite matrices this number related to the cosine of the angle between a given matrix and its inverse is equivalent to the classical 2 norm condition number but has a direct and natural geometrical interpretation in particular we establish bounds for the ratio between the angle that a matrix forms with the identity ray and the angle that the inverse of that matrix forms with the identity ray these bounds allow us to establish new lower bounds for the condition number that only require the trace and the frobenius norm of the matrix

new eigenvalue estimates for complex matrices abstract in a recent work 1 we have proved that all the eigenvalues j of a complex matrix a of order n lie in a disk with center at tr a n and radius n 1 n k 1 n k 2 tr a 2 n 1 2 in this paper we prove that the eigenvalues of a are contained in a rectangle with vertices on the boundary circle of the mentioned disk improving in this way the result in 1 we recall some inequalities which alloww us to bound the radius of the disk and the rectangle the bounds can be computed without knowing the eigenvalues of a

localization of eigenvalues in elliptic regions abstract we prove that the eigenvalues j of an n by n complex matrix a with its characteristic polynomial having real coefficients lie in the elliptic region defined by 2 tr a n 2 2 y 2 2 2 where n 1 n n 1 n re k 2 re te a 2 n 1 2 and n 1 n k 1 n im k 2 1 2 this region is intersected with the strip y 1 2 k 1 u im k 2 1 2 to obtain an improved eigenvalue localization region we also give bounds for the semiaxes which can be computed without knowing the eigenvalues of a when a has r n nonzero eigenvalues we obtain a smaller elliptic region containing such nonzero eigenvalues

how much resolution is enough influence of downscaling the pixel resolution of digital images on the generalised dimensions fractals are very often illustrated by digital images this is a consequence of the widespread availability of electronic and computational equipment these images are always limited by their pixel resolution subsequently the determination of dimensions for such images is restricted practically because for their calculation the range of scales is not infinite downscaling the pixel resolution of images to get images with reduced resolution worsens the achievable absolute computational results but surprisingly the changes in these values and especially the changes of the courses of the calculated generalised dimensions give very promising results this is due to the qualitative nature of these changes with respect to pixel downscaling this opens the possibility of determining the quality of the estimation of the fractal dimensions of any image with any particular pixel resolution furthermore the pattern of changes itself caused by downscaling is different between the individual fractals several geometrical fractals sierpinski gasket a fern a menger gasket and a modified menger gasket with circles as well as an example of a measured fractal the digital images of cancer invasion studies in vitro were investigated to study the influence of downscaling the pixel resolution of digital images on the estimation of fractal dimensions

eigenvalue estimates for symmetric matrices abstract we use the fact that the set of symmetric positive semidefinite matrices of order n form a cone with a special structure in order to find bounds for the eigenvalues of a symmetric matrix

on the cone of positive semidefinite matrices abstract a survey of some general properties of the cone of positive semidefinite matrices its faces two isometric isomorphisms and linear transformations on it is given

algebras with involution with linear codimension growth we study the varieties of associative algebras with involution over a field of characteristic zero which are generated by a finite dimensional algebra in this setting we give a list of algebras classifying all such varieties whose sequence of codimensions is linearly bounded moreover we exhibit a finite list of algebras to be excluded from the varieties with such property as a consequence we find all possible linearly bounded codimension sequences

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

bounds for eigenvalues using the trace and determinant abstract let a be a square matrix with real and positive eigenvalues 1 n 0 and let 1 k l n bounds for kl l and k l involving k l n tr a and det a only are presented

more estimates for eigenvalues and singular values abstract there is an interesting relation between the angle that a matrix forms with the identity and its eigenvalues then we show for symmetric matrices that some information about the matrix can be known if the angle is computed a lower bound for the maximum eigenvalue is obtained for symmetric positive semidefinite matrices this inequality and the upper bound for eigenvalues given in 5 provide us an interval in which must lie the maximum eigenvalue finally some results from 5 and the first part of this paper are generalized for the non symmetric case

the exponent for superalgebras with superinvolution abstract let a be a superalgebra with superinvolution over a field of characteristic zero and let c n a n 1 2 be its sequence of codimensions in 6 it was proved that such a sequence is exponentially bounded in this paper we capture this exponential growth for finitely generated superalgebras with superinvolution a over an algebraically closed field of characteristic zero we shall prove that lim n c n a n exists and it is an integer denoted exp a and called exponent of a moreover we shall characterize finitely generated superalgebras with superinvolution according to their exponent

anomalous multifractal spectrum of aggregating lennard jones particles with brownian dynamics abstract we perform a brownian dynamics simulation of lennard jones particles at low density and temperature from an initially homogeneous but random configuration the system flocculates into a fractal gel we have found that the bicontinous structures formed can be characterized by a fractal dimensionality but the fractal regime is only very small and the scaling depends upon the volume fraction of the particles in this paper we investigate a possibly more universal multifractal spectrum of the aggregates because of the small scaling regime of less than a decade we cannot extrapolate the box counting results for the partition function to zero box size as a consequence we observe a multifractal spectrum dq which is an increasing function of q we show that within a multifractal model a similar anomalous result can be obtained because in practice the scaling regime of any physical system will always be bounded from below it appears necessary to either redefine the notion of a multifractal spectrum or accept the existence of the anomalous spectra and assess their relevance

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

a star variety with almost polynomial growth abstract let f be a field of characteristic zero in this paper we construct a finite dimensional f algebra with involution m and we study its polynomial identities on one hand we determine a generator of the corresponding t ideal of the free algebra with involution and on the other we give a complete description of the multilinear identities through the representation theory of the hyperoctahedral group as an outcome of this study we show that the variety generated by m var m has almost polynomial growth i e the sequence of codimensions of m cannot be bounded by any polynomial function but any proper subvariety of var m has polynomial growth if g 2 is the algebra constructed in giambruno and mishchenko preprint we next prove that m and g 2 are the only two finite dimensional algebras with involution generating varieties with almost polynomial growth

pi algebras with slow codimension growth let cn a n 1 2 be the sequence of codimensions of an algebra a over a field f of characteristic zero we classify the algebras a up to pi equivalence in case this sequence is bounded by a linear function we also show that this property is closely related to the following if ln a n 1 2 denotes the sequence of colengths of a counting the number of sn irreducibles appearing in the nth cocharacter of a then limn ln a exists and is bounded by 2

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

some results concerning the multiplicities of cocharacters of superalgebras with graded involution abstract let a be a finitely generated superalgebra with graded involution over a field f of characteristic zero and let n 1 n 4 a n 1 n 4 n be the n 1 n 4 cocharacter of a in this paper we present some results concerning the multiplicities of cocharacters for this kind of algebras

on the structure of the set of symmetric matrices with the perron frobenius property abstract in this paper we explore the geometry of the set of symmetric perron frobenius matrices whose spectral radius is an eigenvalue with a nonnegative associated eigenvector we prove that the set which is not convex is star convex with the identity matrix as the center we also describe a few aspects of the geometric structure of the set

graded algebras with polynomial growth of their codimensions abstract let a be an algebra over a field of characteristic 0 and assume a is graded by a finite group g we study combinatorial and asymptotic properties of the g graded polynomial identities of a provided a is of polynomial growth of the sequence of its graded codimensions roughly speaking this means that the ideal of graded identities is very large we relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free g graded algebra in the variety generated by a we describe the irreducible modules that can appear in the decomposition we show that their multiplicities are eventually constant depending on the shape obtained by the corresponding multipartition after removing its first row we relate moreover the polynomial growth to the colengths finally we describe in detail the algebras whose graded codimensions are of linear growth

polynomial codimension growth of algebras with involutions and superinvolutions abstract let a be an associative algebra over a field f of characteristic zero endowed with a graded involution or a superinvolution and let c n a be its sequence of codimensions in 4 12 it was proved that if a is finite dimensional such sequence is polynomially bounded if and only if a generates a variety not containing a finite number of algebras the group algebra of z 2 and a 4 dimensional subalgebra of the 4 4 upper triangular matrices with suitable graded involutions or superinvolutions in this paper we focus our attention on such algebras since they are the only finite dimensional algebras up to t 2 equivalence generating varieties of almost polynomial growth i e varieties of exponential growth such that any proper subvariety has polynomial growth we classify the subvarieties of such varieties by giving a complete list of generating finite dimensional algebras along the way we classify all minimal varieties of polynomial growth and surprisingly we show that their number is finite for any given growth finally we describe the algebras whose codimensions are bounded by a linear function

topological indices of the line graph of subdivision graphs and their schur bounds in this paper we investigate topological indices of the line graphs of the tadpole graphs wheel graphs and ladder graphs using the subdivision concept which generalize the results of ranjini et al 2011 in addition we present the schur bound for the general sum connectivity co index

bounds for singular values using traces abstract bounds for the singular values ratios of singular values and rank of a square matrix a involving tr a tr a 2 and tr a h a are presented some of the results are analogous to the well known bounds for the eigenvalues ratios of eigenvalues and rank of a involving tr a and tr a 2

lower bounds for the smallest singular value abstract let a aij be an n n complex matrix suppose that g a the undirected graph of a has no isolated vertex let e be the set of edges of g a we prove that the smallest singular value of a n satisfies n min ij i j e where gij ai aj ai aj 2 ri ci rj cj 1 2 2 with ai aii and ri ci are the ith deleted absolute row sum and column sum of a respectively the result simplifies and improves that of johnson and szulc n mini j ij see 1

a decreasing sequence of eigenvalue localization regions abstract we construct a decreasing sequence of rectangles rp in such a way that all the eigenvalues of a complex matrix a are contained in each rectangle when a is a matrix with real spectrum or a normal matrix each rp can be obtained without knowing the eigenvalues of a

varieties of superalgebras of almost polynomial growth abstract let v gr be a variety of superalgebras and let c n gr v gr n 1 2 be its sequence of graded codimensions such a sequence is polynomially bounded if and only if v gr does not contain a list of five superalgebras consisting of a commutative superalgebra the infinite dimensional grassmann algebra and the algebra of 2 2 upper triangular matrices with trivial and natural z 2 gradings in this paper we completely classify all subvarieties of the varieties generated by these five superalgebras by giving a complete list of finite dimensional generating superalgebras

a decreasing sequence of upper bounds for the perron root abstract we obtain a decreasing sequence of upper bounds for the perron root of a nonnegative matrix these upper bounds are new and readily computable for any nonnegative matrix with zero trace the mentioned sequence converges to the perron root we apply these results to the adjacency matrix of a graph

matricial inner products and pointed cones of hermitian preserving linear transformations abstract let be a conjugate homogeneous reflector on a vector space v over r or c with p a pointed cone contained in spec v a mapping on v v whose range is contained in p which generalizes the usual inner product properties is called a vectorial inner product a certain family of these vectorial inner products on matrices which we call matricial inner products is used to generate a set of pointed cones in the ambient space of hermitian preserving linear transformations some basic results on these cones including psd psd and c p and on the partial orders that they induce are given

characterizations and lower bounds for the spread of a normal matrix abstract the spread of an n n matrix a with eigenvalues 1 n is defined by spr a max j k j k we prove that if a is normal then spr a max x ax y ay x y c n x y 1 max x ay y ax x y c n x y 1 re x y 0 max x ay y ax x y c n x y 1 x y 0 max spr z a z a 2 z c z 1 2 max x a 2 x x ax 2 x a ax x ax 2 1 2 x c n x 1 we also present several lower bounds for spr a given by these characterizations

upper bounds for singular values abstract let a be an n n matrix with singular values 1 n if 1 r n then r min h s r h where sr is the set of n n matrices h such that rank a h r 1 and denotes the spectral norm i e the largest singular value we find upper bounds for r by choosing h suitably

normalized frobenius condition number of the orthogonal projections of the identity abstract this paper deals with the orthogonal projection in the frobenius sense a n of the identity matrix i onto the matrix subspace a s a r n n s being an arbitrary subspace of r n n lower and upper bounds on the normalized frobenius condition number of matrix a n are given furthermore for every matrix subspace s r n n a new index f a s which generalizes the normalized frobenius condition number of matrix a is defined and analyzed

inequalities for the numerical radius the norm and the maximum of the real part of bounded linear operators in hilbert spaces some inequalities for the numerical radius the operator norm and the maximum of the real part of bounded linear operators in hilbert spaces under suitable assumptions for the involved operator are given

graded polynomial identities as identities of universal algebras abstract let a and b be finite dimensional simple algebras with arbitrary signature over an algebraically closed field suppose a and b are graded by a semigroup s so that the graded identical relations of a are the same as those of b then a is isomorphic to b as an s graded algebra

bounds for ratios of eigenvalues using traces abstract let a be an n n matrix with real eigenvalues 1 n and let 1 k l n bounds involving tr a and tr a 2 are introduced for k l k l k l and k k n l 1 l 2 k 2 k n l 1 2 l also included are conditions for l 0 and for k l 0

bounds for the spectral radius and the largest singular value abstract we derive an increasing sequence of lower bounds for the spectral radius of a matrix with real spectrum and progressively improved bounds for the largest singular value of a complex matrix we also find estimates for the rank of normal matrices with real spectrum and for the rank of normal nonnegative matrices including some sufficient condition for such matrices to be invertible

lower bounds for the numerical radius abstract let a be a complex n n matrix we find lower bounds for its numerical radius r a max x ax x c n x x 1 first we choose x satisfying x x 1 and compute x ax we also improve the simple bound so obtained second we apply r a max z a z a 2 z c z 1 where denotes the largest eigenvalue

the origin of an increasing or decreasing multifractal spectrum the multifractal dimensionality dq as a function of q expresses the distribution of measure over space when all the moments scale with resolution in exactly the same way we have a flat spectrum and a single monofractal dimensionality we argue that for multifractal spectra the scaling of the moments of the measure distribution should be considered rather than the usual definition which considers only the limit at high resolution a difference in scaling of the zeroth and second order moment must lead to zero variance at some resolution at that point the scaling must stop as the variance cannot become negative for discrete systems that happens at the particle level with the scaling region above and d2 d0 or in general with an increasing spectrum the variance increases with increasing box size if on the other hand the variance of the distribution decreases with increasing box size the spectrum dq is the standard non increasing function of q then scaling must stop at the level where homogeneity is reached we present a string of bits either unity or zero which has uniform scaling properties over many levels with the number of unity bits in a substring taken as the measure for substrings of size up to 2n the inhomogeneity of the measure increases monotonously while for larger substrings it decreases again thus we have a multifractal set which at small sizes has an increasing spectrum while for large size the scaling is identical to that of a bimodal cantor set with a decreasing spectrum