structure of chevalley groups over rings via universal localization abstract in the current article we study the structure of a chevalley group g r over a commutative ring r we generalize and improve the following results on the standard relative and multi relative commutator formulas the nilpotent structure of relative k 1 the bounded word length of commutators to this end we enlarge the elementary group construct a generic element for the extended elementary group and use localization in the universal ring the key step is a construction of a generic element for the principal congruence subgroup corresponding to a principal ideal

fractional diffusion waiting time distributions and cattaneo type equations we discuss a generalized diffusion equation resulting from an additive two state process in combination with an asymptotically fractal asymptotic power law waiting time distribution the obtained equation is an extension to previously discussed fractional diffusion equations our description leads to a mean squared displacement which describes enhanced subballistic transport for long times the short time behavior however is of a ballistic nature this separation into two domains results from the introduction of a time scale through the asymptotically fractal waiting time distribution this is also mirrored by the observation that for small times our generalized diffusion equation reduces to the standard cattaneo equation the asymptotic probability density is of compressed gaussian type and thus differs from the l evy tail generally found for these kinds of processes

subordination approach to multi term time fractional diffusion wave equations abstract this paper is concerned with the fractional evolution equation with a discrete distribution of caputo time derivatives such that the largest and the smallest orders and m satisfy the conditions 1 2 and m 1 first based on a study of the related propagation function the nonnegativity of the fundamental solutions to the spatially one dimensional cauchy and signaling problems is proven and propagation speed of a disturbance is discussed next we study the equation with a general linear spatial differential operator defined in a banach space and suppose it generates a cosine family a subordination principle is established which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function explicit representation of the probability density function is derived the subordination principle is applied for obtaining regularity results the analytical findings are supported by numerical work

strong euler class of a stably free module of odd rank abstract let r be a commutative noetherian ring of dimension n 3 following a suggestion of fasel we establish a group homomorphism from van der kallen s group um n 1 r e n 1 r to the n th euler class group e n r so that 1 when n is even coincides with the homomorphism given by bhatwadekar and sridharan through euler classes 2 when n is odd is non trivial in general for an important class of rings 3 the sequence um n 1 r e n 1 r e n r e 0 n r 0 is exact where e 0 n r is the n th weak euler class group if x spec r is a smooth affine variety of dimension n over r so that the complex points of x are complete intersections and the canonical module k r is trivial then the sequence is proved to be exact on the left as well more generally let r be a commutative noetherian ring of dimension d and n be an integer such that n d 2 n 3 we also indicate how to extend our arguments to this setup to obtain a group homomorphism from um n 1 r e n 1 r to e n r

time fractional telegraph equations and telegraph processes with brownian time we study the fundamental solutions to time fractional telegraph equations of order 2 we are able to obtain the fourier transform of the solutions for any and to give a representation of their inverse in terms of stable densities for the special case 1 2 we can show that the fundamental solution is the distribution of a telegraph process with brownian time in a special case this becomes the density of the iterated brownian motion which is therefore the fundamental solution to a fractional diffusion equation of order 1 2 with respect to time

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

a diffusion wave equation with two fractional derivatives of different order we analyse a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains thus our model represents a generalized telegraph equation solutions to signalling and cauchy problems in terms of a series and integral representation are given classical wave and heat conduction equations are obtained as limiting cases

on tridiagonalization of matrices abstract we consider the question is every n n complex matrix unitarily similar to a tridiagonal one it is shown that the answer is negative if n 6 and is affirmative if n 3 additionally some positive partial answers and related results are given for example 1 every pair of hermitian projections is simultaneously unitarily similar to a pair of tridiagonal matrices 2 if a a has a rank one then a is unitarily similar to a tridiagonal matrix

homotopy and commutativity principle abstract we give evidence for the principle that a special linear symplectic orthogonal matrix over a commutative ring which is homotopic to the identity will commute up to an elementary matrix with all elements of the respective group

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

canonical forms for symmetric skew symmetric real matrix pairs under strict equivalence and congruence a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the reals one of which is skew symmetric and the other is either symmetric or skew symmetric these reductions are by strict equivalence and by congruence over the reals or over the complex numbers and essentially complete proofs are presented the proofs are based on canonical forms attributed to jordan and kronecker some closely related results which can be derived from the canonical forms of pairs of symmetric skew symmetric real forms are also included they concern simultaneously neutral subspaces hamiltonian and skew hamiltonian matrices and canonical structures of real matrices which are selfadjoint or skew adjoint in a regular skew symmetric indefinite inner product and real matrices which are skew adjoint in a regular symmetric indefinite inner product the paper is largely expository and continues the comprehensive account of the reduction of pairs of matrices started in p lancaster l rodman canonical forms for hermitian matrix pairs under strict equivalence and congruence siam rev in press

generalized cattaneo telegrapher s equations in modeling anomalous diffusion phenomena we study generalized cattaneo telegrapher s equations involving memory effects introduced by smearing the time derivatives consistency conditions where the smearing functions obey restrict freedom in their choice but the proposed scheme goes beyond the approach based on using fractional derivatives we find conditions under which solutions of the equations considered so far can be recognized as probability distributions i e are normalizable and nonnegative on their domains nonnegativity of solutions is demonstrated by methods of positive definite and completely monotonic functions with the bernstein theorem being the cornerstone of the ongoing proofs analysis of exactly solvable examples and relevant mean squared displacements enables us to classify diffusion processes described by such got solutions and to identify them with either ordinary or anomalous diffusion which character may change over time to complete the present research we compare our results with those obtained using the continuous time random walk and the continuous time persistent random walk approaches

band diagonal operators abstract we show that there are operators on a five dimensional hilbert space which are not tridiagonal and that there are compact operators and normal operators on separable infinite dimensional spaces which are not band diagonal

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

a nice group structure on the orbit space of unimodular rows ii abstract we establish an excision type theorem for niceness of group structure on the orbit space of unimodular rows of length n modulo elementary action this permits us to establish niceness for relative versions of results for the cases when n d 1 d being the dimension of the base algebra we then study and establish niceness for the case when n d and also establish a relative version when the base ring is a smooth affine algebra over an algebraically closed field

some applications of persistent random walks and the telegrapher s equation abstract a persistent random walk can be regarded as a multidimensional markov process the bias free telegraphers equation is 2 p t 2 1 t p t v 2 2 p it can be regarded as interpolating between the wave equation t and the diffusion equation t 0 previously it has found application in thermodynamics cf the review in rev mod phys 61 1989 41 62 1990 375 more recent applications are reviewed in the present article

tridiagonal normal forms for orthogonal similarity classes of symmetric matrices abstract let f be an algebraically closed field of characteristic different from 2 define the orthogonal group o n f as the group of n by n matrices x over f such that xx i n where x is the transpose of x and i n the identity matrix we show that every n by n symmetric matrix over f is orthogonally similar to a tridiagonal symmetric matrix if further the characteristic is 0 we construct the tridiagonal normal form for the o n f similarity classes of symmetric matrices we point out that in this case the known normal forms as presented in the well known book by gantmacher are not tridiagonal

on completing unimodular polynomial vectors of length three it is shown that if r is a local ring of dimension three with r then a polynomial three vector vq x vx x v2 x over r x can be completed to an invertible matrix if and only if it is unimodular in particular if 1 3 j then every stably free projective r xx y module is free

change of the congruence canonical form of 2 by 2 and 3 by 3 matrices under perturbations and bundles of matrices under congruence we construct the hasse diagrams g2 and g3 for the closure ordering on the sets of congruence classes of 2 2 and 3 3 complex matrices in other words we construct two directed graphs whose vert

multiple commutators of elementary subgroups end of the line abstract in our previous joint papers with roozbeh hazrat and alexei stepanov we established commutator formulas for relative elementary subgroups in gl n r n 3 and other similar groups such as bak s unitary groups or chevalley groups in particular there it was shown that multiple commutators of elementary subgroups can be reduced to double such commutators however since the proofs of these results depended on the standard commutator formulas it was assumed that the ground ring r is quasi finite here we propose a different approach which allows to lift any such assumptions and establish almost definitive results in particular we prove multiple commutator formulas and other related facts for gl n r over an arbitrary associative ring r

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

tridiagonal forms in low dimensions abstract pati showed that every 4 4 matrix is unitarily similar to a tridiagonal matrix we give a simple proof in addition we show that in an appropriate sense there are generically precisely 12 ways to do this when the real part is diagonal it is shown that the unitary can be chosen with the form u pd where d is diagonal and p is real orthogonal however even if both real and imaginary parts are real symmetric there may be no real orthogonal matrices which tridiagonalize it on the other hand if the matrix belongs to the lie algebra sp 4 c then it can be tridiagonalized by a unitary in the symplectic group sp 2 in dimension 5 or greater there are always rank three matrices which are not tridiagonalizable

on invariance of the euler class groups under a subintegral base change abstract let r be a commutative noetherian ring and r s be a subintegral extension it is proved that the top euler class group of r is isomorphic to the top euler class group of s an example is given to show that similar conclusion does not hold for integral but not subintegral extension of rings

explicit and implicit finite difference schemes for fractional cattaneo equation in this paper the numerical solution of fractional non integer order cattaneo equation for describing anomalous diffusion has been investigated two finite difference schemes namely an explicit predictor corrector and totally implicit schemes have been developed in developing each scheme a separate formulation approach for the governing equations has been considered the explicit predictor corrector scheme is the fractional generalization of well known maccormack scheme and has been called generalized maccormack scheme this scheme solves two coupled low order equations and simultaneously computes the flux term with the main variable fully implicit scheme however solves a single high order undecomposed equation for generalized maccormack scheme stability analysis has been studied through fourier method through a numerical test the experimental order of convergency of both schemes has been found then the domain of applicability and some numerical properties of each scheme have been discussed

solutions of the space time fractional cattaneo diffusion equation abstract the object of this paper is to present the exact solution of the fractional cattaneo equation for describing anomalous diffusion the classical cattaneo model has been generalised to the space time fractional cattaneo model the method of the joint laplace and fourier transform is used in deriving the solution the solutions of the fractional cattaneo equation are obtained under integral and series forms in terms of the h functions finally the fractional order moments are also investigated

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

exact solutions of triple order time fractional differential equations for anomalous relaxation and diffusion i the accelerating case a b s t r a c t in recent years the interest around the study of anomalous relaxation and diffusion processes is increased due to their importance in several natural phenomena moreover a further generalization has been developed by introducing time fractional differentiation of distributed order which ranges between 0 and 1 we refer to accelerating processes when the driving power law has a changing in time exponent whose modulus tends from less than 1 to 1 and to decelerating processes when such an exponent modulus decreases in time moving away from the linear behaviour accelerating processes are modelled by a time fractional derivative in the riemann liouville sense while decelerating processes by a time fractional derivative in the caputo sense here the focus is on the accelerating case while the decelerating one is considered in the companion paper after a short reminder about the derivation of the fundamental solution for a general distribution of time derivative orders we consider in detail the triple order case for both accelerating relaxation and accelerating diffusion processes and the exact results are derived in terms of an infinite series of h functions the method adopted is new and it makes use of certain properties of the generalized mittag leffler function and the h function moreover it provides an elegant generalization of the method introduced by langlands 2006 t a m langlands physica a 367 2006 136 to study the double order case of accelerating diffusion processes

the cattaneo type time fractional heat conduction equation for laser heating in this paper the laser short pulse heating of a solid surface is considered the time fractional cattaneo model is used as the heat conduction model and the corresponding fractional heat conduction equation with a volumetric heat source is built the analytical solution for the temperature distribution is obtained using the laplace transformation method finally the numerical results are presented graphically for various values of model parameters and their effects on the speed of heat conduction propagation are discussed

on the time fractional cattaneo equation of distributed order abstract the present work revisits the time fractional cattaneo equation tfce by considering a generic class of time fractional derivatives of distributed order 0 1 at first we shed light on two main forms of tfce one natural generalization uses the caputo fractional derivative and the other compte metzler generalization is defined in the riemann liouville sense and the equivalence condition between their solutions is provided some special cases of these two forms are addressed hence different versions of distributed order tfce are proposed the fundamental solution of the two general forms of distributed order tfce are derived using integral transform technique the non negativity of solutions of tfce and its distributed order versions is examined using the special type functions technique gorenflo luchko and stojanovic 2013 two classes of order distribution are considered discrete order and continuous order distributions and some random choices of non negative order densities affirm graphically the non negativity of the general forms of distributed order tfce the double order as an example on the discrete order and exponentially exp and sinusoidally sin distributed order as an example on the continuous order are chosen finally connections between some models of distributed order tfce and continuous time random walk theory are established and the corresponding mean squared displacements are discussed

fractional order theory of thermoelasticity in this work a new theory of thermoelasticity is derived using the methodology of fractional calculus the theories of coupled thermoelasticity and of generalized thermoelasticity with one relaxation time follow as limit cases a uniqueness theorem for this model is proved a variational principle and a reciprocity theorem are derived

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

normal forms for orthogonal similarity classes of skew symmetric matrices let f be an algebraically closed field of characteristic different from 2 define the orthogonal group on f as the group of n by n matrices x over f such that xx in where x is the transpose of x and in the identity matrix we show that every nonsingular n by n skew symmetric matrix over f is orthogonally similar to a bidiagonal skew symmetric matrix in the singular case one has to allow some 4 diagonal blocks as well if further the characteristic is 0 we construct the normal form for the on f similarity classes of skew symmetric matrices in this case the known normal forms as presented in the well known book by gantmacher are quite different finally we study some related varieties of matrices we prove that the variety of normalized nilpotent n by n bidiagonal matrices for n 2s 1 is irreducible of dimension s as a consequence the skew symmetric nilpotent n by n bidiagonal matrices are shown to form a variety of pure dimension s

solution of a modified fractional diffusion equation abstract recently a modified fractional diffusion equation has been proposed i sokolov j klafter from diffusion to anomalous diffusion a century after einstein s brownian motion chaos 15 2005 026103 a v chechkin r gorenflo i m sokolov v yu gonchar distributed order time fractional diffusion equation frac calc appl anal 6 3 2003 259279 i m sokolov a v checkin j klafter distributed order fractional kinetics acta phys pol b 35 2004 1323 for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term in this letter we give the solution of the modified equation on an infinite domain in contrast to the solution of the traditional fractional diffusion equation the solution of the modified equation requires an infinite series of fox functions instead of a single fox function

stochastic foundation of normal and anomalous cattaneo type transport abstract we investigate the connection of the cattaneo equation and the stochastic continuous time random walk ctrw theory we show that the velocity model in a ctrw scheme is suited to derive the standard cattaneo equation and allows in principle for a generalisation to anomalous transport as a result for a broad waiting time distribution with diverging mean we find a strong memory to the initial condition of the system the ballistic behaviour subsists also for long times only if a characteristic waiting time exists a non ballistic enhanced motion is found in the limit of long times no transition to subdiffusion can be found

fractional telegrapher s equation from fractional persistent random walks we generalize the telegrapher s equation to allow for anomalous transport we derive the space time fractional telegrapher s equation using the formalism of the persistent random walk in continuous time we also obtain the characteristic function of the space time fractional process and study some particular cases and asymptotic approximations similarly to the ordinary telegrapher s equation the time fractional equation also presents distinct behaviors for different time scales specifically transitions between different subdiffusive regimes or from superdiffusion to subdiffusion are shown by the fractional equation as time progresses

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

canonical forms for complex matrix congruence and congruence abstract canonical forms for congruence and congruence of square complex matrices were given by horn and sergeichuk in linear algebra appl 389 2004 347 353 based on sergeichuk s paper math ussr izvestiya 31 3 1988 481 501 which employed the theory of representations of quivers with involution we use standard methods of matrix analysis to prove directly that these forms are canonical our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms we use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew symmetric matrices as well as canonical forms for simultaneous congruence of pairs of complex hermitian matrices

tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric skew symmetric or hermitian forms abstract tridiagonal canonical forms of square matrices under congruence or congruence pairs of symmetric or skew symmetric matrices under congruence and pairs of hermitian matrices under congruence are given over an algebraically closed field of characteristic different from 2

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

the role of the fox wright functions in fractional sub diffusion of distributed order the fundamental solution of the fractional diffusion equation of distributed order in time usually adopted for modelling sub diffusion processes is obtained based on its mellin barnes integral representation such solution is proved to be related via a laplace type integral to the fox wright functions a series expansion is also provided in order to point out the distribution of time scales related to the distribution of the fractional orders the results of the time fractional diffusion equation of a single order are also recalled and then re obtained from the general theory

on the existence of unimodular elements and cancellation of projective modules over noetherian and non noetherian rings let r be a commutative ring of dimension d s r x or r x 1 x and p a finitely generated projective s module of rank r then p is cancellative if p has a unimodular element and r d 1 moreover if r dim s then p has a unimodular element and therefore p is cancellative as an application we prove that if r is a ring of dimension d of finite type over a prufer domain and p is a projective r x or r x 1 x module of rank at least d 1 then p has a unimodular element and is cancellative