dealing with uncertainties on phase weighting of linear antenna arrays by means of interval based tolerance analysis the analysis of the tolerances on the power patterns radiated by linear antenna arrays whose excitations are affected by phase errors is carried out by means of an approach based on the interval analysis ia the rules and the features of the arithmetic of intervals already effectively applied to analyze the effects of errors on the amplification levels are exploited here to derive a suitable set of analytical relationships predicting the maximum deviations i e upper and lower bounds with respect to the nominal power pattern once the phase errors are defined as belonging to intervals of values representative numerical examples are reported and discussed to assess potentialities and limitations of the ia based approach toward this end different configurations of the beamforming weights affording low sidelobes and steered beam patterns are analyzed

probabilistic interval analysis for the analytic prediction of the pattern tolerance distribution in linear phased arrays with random excitation errors a statistical approach based on the interval analysis ia is proposed for the analysis of the effects on the radiation patterns radiated by phased arrays of random errors and tolerances in the amplitudes and phases of the array elements excitations starting from the efficient reliable and inclusive computation of the bounds of the complex valued interval array pattern function by means of ia an analytic method is presented to yield closed form expressions for the probability of occurrence of a user chosen value of the power pattern or of its features within the corresponding ia derived bounds a set of numerical examples is reported and discussed to assess the reliability of the proposed probabilistic interval analysis pia method with the results from monte carlo simulations as well as to point out its effectiveness and potentialities advantages efficiency in real applications of great industrial interest

approximation of the levy feller advection dispersion process by random walk and finite difference method in this paper we present a random walk model for approximating a levy feller advection dispersion process governed by the levy feller advection dispersion differential equation lfade we show that the random walk model converges to lfade by use of a properly scaled transition to vanishing space and time steps we propose an explicit finite difference approximation efda for lfade resulting from the grunwald letnikov discretization of fractional derivatives as a result of the interpretation of the random walk model the stability and convergence of efda for lfade in a bounded domain are discussed finally some numerical examples are presented to show the application of the present technique

a second order accurate numerical approximation for the fractional diffusion equation fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes in this paper we examine a practical numerical method which is second order accurate in time and in space to solve a class of initial boundary value fractional diffusive equations with variable coefficients on a finite domain an approach based on the classical crank nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates stability consistency and therefore convergence of the method are examined it is shown that the fractional crank nicholson method based on the shifted grunwald formula is unconditionally stable a numerical example is presented and compared with the exact analytical solution for its order of convergence

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

smooth contractible hypersurfaces in cn and exotic algebraic structures on c3 in this paper we study complex affine algebraic varieties that are diffeomorphic to r e but not isomorphic to c according to z1 z3 we shall refer to these varieties as to exotic algebraic structures o n r 2 or on c n it was ramanujam who first tried to find these structures his famous theorem r implies that the only complex algebraic structure o n r 4 is the standard structure of c 2 in the case of higher dimensions zaidenberg zi z3 constructed examples of exotic algebraic structures on c n 3 moreover he proved that these exotic structures on c are not biholomorphically equivalent to c his proof is based on some analytic version of the cancellation theorem for n 3 all varieties in his examples have kodaira logarithmic dimension o c petrie has informed us that he and tom dieck also knew about the existence of exotic algebraic structures in their paper ptd there is a list of all smooth contractible hypersurfaces in c 3 of kodaira logarithmic dimension 1 the authors note in this paper that an analogue of their construction holds for higher dimensions but all contractible hypersurfaces in c with n 4 are diffeomorphic to r 2 1 due to the h cobordism theorem thus they are exotic structures when their kodaira logarithmic dimension is 1 it is worth mentioning that a combination of zaidenberg s approach and the examples from ptd give curious polynomials on c 4 consider a polynomial p x y z z z x 3 3 z x 3 3 x z y 2 2 y in three variables the zero fiber of this polynomial has a singular point and all other fibers are contractible hypersurfaces ptd if we consider p in c 4 with the coordinate system x y z t then by the h cobordism theorem all nonzero fibers are diffeomorphic to r 6 as zaidenberg mentions in z2 these fibers must be exotic algebraic structures on c 3 due to the strong cancellation theorem if i e the polynomial p is not equivalent to a linear polynomial but all its nonzero fibers are diffeomorphic to the euclidean space our main aim is to construct new examples of exotic algebraic structures on c a our examples have kodaira logarithmic dimension 2 and they admit a natural c action with one fixed point compared with the above examples

simple birational extensions of the polynomial algebra c 3 the abhyankar sathaye problem asks whether any biregular embedding c k c n can be rectified that is whether there exists an automorphism a e aut c n such that a o is a linear embedding here we study this problem for the embeddings c 3 c 4 whose image x c 3 is given in c 4 by an equation p f x y u g x y z 0 where f c x y 0 and g e c x y z under certain additional assumptions we show that indeed the polynomial p is a variable of the polynomial ring c 4 c x y z u i e a coordinate of a polynomial automorphism of c 4 this is an analog of a theorem due to sathaye 1976 which concerns the case of embeddings c 2 c 3 besides we generalize a theorem of miyanishi 1984 giving for a polynomial p as above a criterion for when x p 1 0 c 3

numerical methods and analysis for a class of fractional advection dispersion models in this paper a class of fractional advection dispersion models fadms is considered these models include five fractional advection dispersion models i e the time fadm the mobile immobile time fadm with a time caputo fractional derivative 0 c 1 the space fadm with two sides riemann liouville derivatives the time space fadm and the time fractional advection diffusion wave model with damping with index 1 c 2 these equations can be used to simulate the regional scale anomalous dispersion with heavy tails we propose computationally effective implicit numerical methods for these fadms the stability and convergence of the implicit numerical methods are analysed and compared systematically finally some results are given to demonstrate the effectiveness of theoretical analysis

numerical methods for the solution of partial differential equations of fractional order anomalous diffusion is a possible mechanism underlying plasma transport in magnetically confined plasmas to model this transport mechanism fractional order space derivative operators can be used here the numerical properties of partial differential equations of fractional order 1 2 are studied two numerical schemes an explicit and a semi implicit one are used in solving these equations two different discretization methods of the fractional derivative operator have also been used the accuracy and stability of these methods are investigated for several standard types of problems involving partial differential equations of fractional order

preconditioned iterative methods for fractional diffusion equation in this paper we are concerned with numerical methods for the solution of initial boundary value problems of anomalous diffusion equations of order 1 2 the classical crank nicholson method is used to discretize the fractional diffusion equation and then the spatial extrapolation is used to obtain temporally and spatially second order accurate numerical estimates two preconditioned iterative methods namely the preconditioned generalized minimal residual preconditioned gmres method and the preconditioned conjugate gradient for normal residual preconditioned cgnr method are proposed to solve relevant linear systems numerical experiments are given to illustrate the efficiency of the methods

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

tolerance analysis of antenna arrays through interval arithmetic an analytical method based on interval analysis ia is proposed to predict the impact of the manufacturing tolerances of the excitation amplitudes on the radiated array pattern by expressing the array factor according to the rules of the interval arithmetic the radiation features of the linear array are described in terms of intervals whose bounds are analytically determined as functions of the nominal value and the tolerances of the array amplitudes a set of representative numerical experiments dealing with different radiated beams and linear array sizes is reported and discussed to point out the features and potentials of the proposed approach

numerical solutions for fractional reaction diffusion equations fractional diffusion equations are useful for applications in which a cloud of particles spreads faster than predicted by the classical equation in a fractional diffusion equation the second derivative in the spatial variable is replaced by a fractional derivative of order less than two the resulting solutions spread faster than the classical solutions and may exhibit asymmetry depending on the fractional derivative used fractional reaction diffusion equations combine the fractional diffusion with a classical reaction term in this paper we develop a practical method for numerical solution of fractional reaction diffusion equations based on operator splitting then we present results of numerical simulations to illustrate the method and investigate properties of numerical solutions we also discuss applications to biology where the reaction term models species growth and the diffusion term accounts for movements

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

minkowski sum method for planar arrays sensitivity analysis with uncertain but bounded excitation tolerances the sensitivity of the power pattern generated by a planar array to both amplitude and phase excitation errors is addressed under the hypothesis of uncertain but bounded tolerances on the element excitations an innovative approach based on the integration of the interval analysis and the minkowski sum is proposed to predict tight reliable and inclusive pattern bounds representative numerical results are reported and discussed to point out the potentialities and the effectiveness of the proposed approach also in comparison with the state of the art interval based methods

representations of toeplitz plus hankel matrices using trigonometric transformations with application to fast matrix vector multiplication abstract representations of real toeplitz and toeplitz plus hankel matrices are presented that involve real trigonometric transformations dct dst dht and diagonal matrices these representations can be used for fast matrix vector multiplication in particular it is shown that the multiplication of an n n toeplitz plus hankel matrix by a vector requires only 4 transformations of length n plus o n operations

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

trigonometric transform splitting methods for real symmetric toeplitz systems abstract in this paper we study efficient iterative methods for real symmetric toeplitz systems based on the trigonometric transformation splitting tts of the real symmetric toeplitz matrix a theoretical analyses show that if the generating function f of the n n toeplitz matrix a is a real positive even function then the tts iterative methods converge to the unique solution of the linear system of equations for sufficient large n moreover we derive an upper bound of the contraction factor of the tts iteration which is dependent solely on the spectra of the two tts matrices involved different from the cscs iterative method in ng 2003 in which all operations counts concern complex operations when the dfts are employed even if the toeplitz matrix a is real and symmetric our method only involves real arithmetics when the dcts and dsts are used the numerical experiments show that our method works better than cscs iterative method and much better than the positive definite and skew symmetric splitting pss iterative method in bai et al 2005 and the symmetric gauss seidel sgs iterative method

interval arithmetic for pattern tolerance analysis of parabolic reflectors an innovative analytic strategy for the estimation of the average pattern behavior of reflector antennas when surface deformations are present on the parabolic disc is proposed since measuring the surface deviations from a reference ideal shape is a complex task the root mean square rms surface deformations are modeled as intervals and their impact on the power pattern as well as the sensitivity of the beam features is efficiently predicted by exploiting the rules of the interval arithmetic throughout interval analysis ia the result is the definition of closed form relationships between the bounds upper and lower values of the radiated power pattern and the rms interval surface deformations uniformly and non uniformly distributed perturbations of the ideal surface are analyzed to assess the effectiveness of proposed approach comparisons with state of the art solutions are also carried out by varying the deformation thickness as well as the reflector geometry and illumination

test elements retracts and automorphic orbits abstract let a 2 be a free associative or polynomial algebra of rank two over a field k of characteristic zero based on the degree estimate of makar limanov and j t yu we prove 1 an element p a 2 is a test element if p does not belong to any proper retract of a 2 2 every endomorphism preserving the automorphic orbit of a nonconstant element of a 2 is an automorphism

embeddings of hypersurfaces in affine spaces abstract in this paper we address the following two general problems given two algebraic varieties in cn find out whether or not they are 1 isomorphic and 2 equivalent under an automorphism of cn although a complete solution of either of those problems is out of the question at this time we give here some handy and useful invariants of isomorphic as well as of equivalent varieties furthermore and more importantly we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one and use it to construct numerous examples of isomorphic but inequivalent algebraic varieties in cn among other things we establish the following interesting fact for isomorphic hypersurfaces p x1 xn 0 and q x1 xn 0 the number of zeros of grad p might be different from that of grad q this implies in particular that although the fibers p 0 and q 0 are isomorphic there are some other fibers p c and q c which are not we construct examples like this for any n 2

finite element method for the space and time fractional fokker planck equation we develop the finite element method for the numerical resolution of the space and time fractional fokker planck equation which is an effective tool for describing a process with both traps and flights the time fractional derivative of the equation is used to characterize the traps and the flights are depicted by the space fractional derivative the stability and error estimates are rigorously established and we prove that the convergent order is o k 2 alpha h mu where k is the time step size and h the space step size numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims

an o n log2n alternating direction finite difference method for two dimensional fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that cannot be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods for fractional diffusion equations often generate dense or even full coefficient matrices consequently the numerical solution of these methods often require computational work of o n 3 per time step and memory of o n 2 for where n is the number of grid points in this paper we develop a fast alternating direction implicit finite difference method for space fractional diffusion equations in two space dimensions the method only requires computational work of o n log 2n per time step and memory of o n while retaining the same accuracy and approximation property as the regular finite difference method with gaussian elimination our preliminary numerical example runs for two dimensional model problem of intermediate size seem to indicate the observations to achieve the same accuracy the new method has a significant reduction of the cpu time from more than 2 months and 1 week consumed by a traditional finite difference method to 1 5h using less than one thousandth of memory the standard method does this demonstrates the utility of the method

automorphisms fixing a variable of k x y z we study automorphisms of the free associative algebra k x y z over a field k such that x y are linear with respect to x y and z z we establish a sufficient and necessary condition for the tameness of these automorphisms in the class of all automorphisms fixing z which gives an algorithm to recognize the wild ones in particular we prove that the well known anick automorphism is wild in this sense this class of automorphisms induces tame automorphisms of the polynomial algebra k x y z for n 2 the automorphisms of k x1 xn z which fix z and are linear in the xis are tame

optimal synthesis of robust beamformer weights exploiting interval analysis and convex optimization an approach for the optimal design of linear arrays in the presence of tolerances on the amplitude weights is presented starting from the knowledge of the maximum deviations from the nominal values of the amplitude coefficients of the beam forming network bfn an analytical tool based on interval analysis ia and integrating a convex programming cp method is exploited to maximize the minimum peak value of the main beam of the radiated power pattern along the boresight direction while fitting an arbitrary user defined bound on the secondary lobes in a globally optimal fashion in order to analyze the behavior of the proposed approach also assessing its effectiveness a set of representative numerical examples regarding arrays of ideal and realistic radiating elements is reported where different sidelobe constraints array configurations and tolerance errors are considered

generalized fractional diffusion equations for accelerating subdiffusion and truncated levy flights fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time for processes lacking such scaling the corresponding description may be given by diffusion equations with fractional derivatives of distributed order such equations were introduced in a v chechkin r gorenflo and i sokolov phys rev e 66 046129 2002 for the description of the processes getting more anomalous in the course of time decelerating subdiffusion and accelerating superdiffusion here we discuss the properties of diffusion equations with fractional derivatives of the distributed order for the description of anomalous relaxation and diffusion phenomena getting less anomalous in the course of time which we call respectively accelerating subdiffusion and decelerating superdiffusion for the former process by taking a relatively simple particular example with two fixed anomalous diffusion exponents we show that the proposed equation effectively describes the subdiffusion phenomenon with diffusion exponent varying in time for the latter process we demonstrate by a particular example how the power law truncated levy stable distribution evolves in time to the distribution with power law asymptotics and gaussian shape in the central part the special case of two different orders is characteristic for the general situation in which the extreme orders dominate the asymptotics

matrix based interval arithmetic for linear array tolerance analysis with excitation amplitude errors this communication proposes a novel matrix based interval arithmetic mia method for linear antenna array pattern tolerance analysis with excitation amplitude errors it derives the closed form array power pattern bounds in an analytical fashion which can predict the maximum deviations with respect to the referenced value efficiently the array power pattern upper bound is its midpoint value summing two error items the first order derivative and the array error pattern while the lower bound is its midpoint value subtracting the first order derivative the new expression derived by the mia method only involves matrix manipulation and requires no condition judgment which inherits simple coding and easy implementation representative numerical examples are presented to validate and demonstrate the effectiveness of the proposed mia method

poisson brackets and two generated subalgebras of rings of polynomials let a f x1 x2 xn be a ring of polynomials over a field f on the variables x1 x2 xn it is well known see for example 11 that the study of automorphisms of the algebra a is closely related with the description of its subalgebras by the theorem of p m cohn 4 a subalgebra of the algebra f x is free if and only if it is integrally closed the theorem of a zaks 13 says that the dedekind subalgebras of the algebra a are rings of polynomials in a single variable a nowicki and m nagata 8 proved that the kernel of any nontrivial derivation of the algebra f x y char f 0 is also a ring of polynomials in a single generator an original solution of the occurrence problem for the algebra a using the groebner basis was given by d shannon and m sweedler 9 however the method of the groebner basis does not give any information about the structure of concrete subalgebras recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by g noskov 7 the present paper is devoted to the investigation of the structure of twogenerated subalgebras of a in the sequel we always assume that f is an arbitrary field of characteristic 0 let us denote by f the highest homogeneous part of an element f a and by f1 f2 fk the subalgebra of a generated by the elements f1 f2 fk a definition 1 a pair of polynomials f1 f2 a is called reduced if they satisfy the following conditions 1 f1 f2 are algebraically dependent 2 f1 f2 are algebraically independent 3 f1 f2 f2 f1 recall that a pair f1 f2 with condition 3 is usually called reduced condition 1 means that we exclude the trivial case when f1 f2 are algebraically independent we do not consider the case when f1 f2 are algebraically dependent concerning this case recall the well known theorem of s s abhyankar and t t moh 1 which says that if f g f x and f g f x then f or f

embeddings of curves in the plane abstract let k x y be the polynomial algebra in two variables over a field k of characteristic 0 in this paper we contribute toward a classification of two variable polynomials by classifying up to an automorphism of k x y polynomials of the form ax n by m im jn mn c ij x i y j a b c ij k i e polynomials whose newton polygon is either a triangle or a line segment our classification has several applications to the study of embeddings of algebraic curves in the plane in particular we show that for any k 2 there is an irreducible curve with one place at infinity which has at least k equivalent embeddings in c 2 also upon combining our method with a well known theorem of zaidenberg and lin we show that one can decide almost just by inspection whether or not a polynomial fiber p x y 0 is an irreducible simply connected curve

the nagata automorphism is wild it is proved that the well known nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild that is it can not be decomposed into a product of elementary automorphisms

preconditioning techniques for diagonal times toeplitz matrices in fractional diffusion equations the fractional diffusion equation is discretized by an implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable the coefficient matrix of the discretized linear system is equal to the sum of a scaled identity matrix and two diagonal times toeplitz matrices standard circulant preconditioners may not work for such toeplitz like linear systems the main aim of this paper is to propose and develop approximate inverse preconditioners for such toeplitz like matrices an approximate inverse preconditioner is constructed to approximate the inverses of weighted toeplitz matrices by circulant matrices and then combine them together row by row because of toeplitz structure both the discretized coefficient matrix and the preconditioner can be implemented very efficiently by using fast fourier transforms theoretically we show that the spectra of the resulting preconditioned matrices are clustered around one thus krylov subspace methods with the proposed preconditione

a direct o nlog2n finite difference method for fractional diffusion equations fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations because of the nonlocal property of fractional differential operators the numerical methods have full coefficient matrices which require storage of o n 2 and computational cost of o n 3 where n is the number of grid points in this paper we develop a fast finite difference method for fractional diffusion equations which only requires storage of o n and computational cost of o nlog 2n while retaining the same accuracy and approximation property as the regular finite difference method numerical experiments are presented to show the utility of the method

polynomial automorphisms and grobner reductions abstract letpn k x1 xn be the polynomial algebra over a fieldkof characteristic 0 we show that applying an automorphism to a given polynomialp pnis mimicked by grobner transformations of a basis of the ideal ofpngenerated by partial derivatives of this polynomial in the case ofp2 this yields a miraculously simple algorithm for deciding whether or not a given polynomial fromp2is part of a basis another application is an algorithm which given a polynomialp p2that is part of a basis finds a sequence of elementary automorphisms that reducesptox1 we also speculate on how our method may be used for constructing a possible counterexample to the jacobian conjecture in higher dimensions

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

affine varieties with equivalent cylinders abstract a well known cancellation problem asks when for two algebraic varieties v 1 v 2 c n the isomorphism of the cylinders v 1 c and v 2 c implies the isomorphism of v 1 and v 2 in this paper we address a related problem when the equivalence under an automorphism of c n 1 of two cylinders v 1 c and v 2 c implies the equivalence of their bases v 1 and v 2 under an automorphism of c n we concentrate here on hypersurfaces and show that this problem establishes a strong connection between the cancellation conjecture of zariski and the embedding conjecture of abhyankar and sathaye we settle the problem in the affirmative for a large class of polynomials on the other hand we give examples of equivalent cylinders with inequivalent bases those cylinders however are not hypersurfaces another result of interest is that for an arbitrary field k the equivalence of two polynomials in m variables under an automorphism of k x 1 x n n m implies their equivalence under a tame automorphism of k x 1 x 2 n

a circulant preconditioner for fractional diffusion equations the implicit finite difference scheme with the shifted grunwald formula which is unconditionally stable is employed to discretize fractional diffusion equations the resulting systems are toeplitz like and then the fast fourier transform can be used to reduce the computational cost of the matrix vector multiplication the preconditioned conjugate gradient normal residual method with a circulant preconditioner is proposed to solve the discretized linear systems the spectrum of the preconditioned matrix is proven to be clustered around 1 if diffusion coefficients are constant hence the convergence rate of the proposed iterative algorithm is superlinear numerical experiments are carried out to demonstrate that our circulant preconditioner works very well even though for cases of variable diffusion coefficients

the tame and the wild automorphisms of polynomial rings in three variables let c f x1 x2 xn be the polynomial ring in the variables x1 x2 xn over a field f and let autc be the group of automorphisms of c as an algebra over f an automorphism autc is called elementary if it has a form x1 xi 1 xi xi 1 xn 7 x1 xi 1 xi f xi 1 xn where 0 6 f f f x1 xi 1 xi 1 xn the subgroup of autc generated by all the elementary automorphisms is called the tame subgroup and the elements from this subgroup are called tame automorphisms of c non tame automorphisms of the algebra c are called wild it is well known 6 9 10 11 that the automorphisms of polynomial rings and free associative algebras in two variables are tame at present a few new proofs of these results have been found see 5 8 however in the case of three or more variables the similar question was open and known as the generation gap problem 2 3 or tame generators problem 8 the general belief was that the answer is negative and there were several candidate counterexamples see 5 8 12 7 19 the best known of them is the following automorphism aut f x y z constructed by nagata in 1972 see 12 x x x yz z y y 2 x yz x x yz z z z

fractional diffusion in inhomogeneous media starting from the continuous time random walk ctrw scheme with the space dependent waiting time probability density function pdf we obtain the time fractional diffusion equation with varying in space fractional order of time derivative as an example we study the evolution of a composite system consisting of two separate regions with different subdiffusion exponents and demonstrate the effects of non trivial drift and subdiffusion whose laws are changed in the course of time

a generalization of trigonometric transform splitting methods for spatial fractional diffusion equations abstract the spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme with the shifted grunwald formula and the given discretized linear systems have a diagonal plus toeplitz structure in this paper based on trigonometric transformation splitting tts we study efficient iterative method called gtts has two parameters as a focus we give the simple and effective optimal forms of these two parameters respectively we carry out some numerical experiments to illustrate the effectiveness and accuracy of this new algorithm

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

degree estimate for commutators let kh xi be a free associative algebra over a field k of characteristic 0 and let each of the noncommuting polynomials f g kh xi generate its centralizer assume that the leading ho mogeneous components of f and g are algebraically dependent with degrees which do not divide each other we give a counterexample to the recent conjecture of jie tai yu that deg f g deg fg gf min deg f deg g our example satisfies 1 2 deg g deg f g deg g deg f and deg f g can be made as close to deg g 2 as we want we ob tain also a counterexample to another related conjecture of makar limanov and jie tai yu stated in terms of malcev neumann formal power series these counterexamples are found using the description of the free algebra kh xi considered as a bimodule of k u where u is a monomial which is not a power of another mono mial and then solving the equation u m s u n r with unknowns

the strong nagata conjecture it is proved that there exist wild coordinates in the polynomial algebra in three variables over a field of characteristic zero this result implies the famous nagata conjecture

the strong anick conjecture recently umirbaev proved the long standing anick conjecture that is there exist wild automorphisms of the free associative algebra kx y z over a field k of characteristic 0 in particular the well known anick automorphism is wild in this work we obtain a stronger result the strong anick conjecture that implies the anick conjecture namely we prove that there exist wild coordinates of kx y z in particular the two nontrivial coordinates in the anick automorphism are both wild we establish a similar result for several large classes of automorphisms of kx y z we also discover a large previously undescribed class of wild automorphisms of kx y z that is not covered by the results of umirbaev