weighted shift matrices unitary equivalence reducibility and numerical ranges abstract an n by n n 3 weighted shift matrix a is one of the form 0 a 1 0 a n 1 a n 0 where the a j s called the weights of a are complex numbers assume that all a j s are nonzero and b is an n by n weighted shift matrix with weights b 1 b n we show that b is unitarily equivalent to a if and only if b 1 b n a 1 a n and for some fixed k 1 k n b j a k j a n j a j for all j next we show that a is reducible if and only if a j j 1 n is periodic that is for some fixed k 1 k n 2 n is divisible by k and a j a k j for all j 1 j n k finally we prove that a and b have the same numerical range if and only if a 1 a n b 1 b n and s r a 1 2 a n 2 s r b 1 2 b n 2 for all 1 r n 2 where s r s are the circularly symmetric functions

on generalized fractional integral inequalities for twice differentiable convex functions abstract in this article some new generalized fractional integral inequalities of midpoint and trapezoid type for twice differentiable convex functions are obtained in view of this we obtain new integral inequalities of midpoint and trapezoid type for twice differentiable convex functions in a form classical integral and riemann liouville fractional integrals finally we apply our new inequalities to construct inequalities involving moments of a continuous random variable

a note on k theory and triangulated categories we provide an example of two closed model categories having equivalent homotopy categories but different waldhausen k theories we also show that there cannot exist a functor from small triangulated categories to spaces which recovers quillen s k theory for exact categories and which satisfies localization

on the gegenbauer transformation the gegenbauer transformation g k is defined for 1 2 k 0 1 2 by where if being the gegenbauer polynomial of index and degree k and l 0 k is the tchebichef polynomial of degree k the transformation is studied on the spaces l p denned by the norm and its boundedness and range on these spaces is determined and inversion formulae are found

application of subival in solving initial value problems with fractional derivatives abstract the application of a numerical method for the approximation of the fractional derivative in riemann liouville and caputo definitions in initial value problems is discussed the method previously known as the subinterval based method is now referred to by its acronym subival for simpler future references it is dependent on subinterval partitions performed according to a proposed algorithm interpolations using selected time axis nodes i e nodes where solutions have been computed and analytical monomial integrodifferentiation formulae two exemplary circuit problems have been introduced as a test for the method these problems have analytical solutions available in literature the evaluations of these solutions have been compared with results obtained through an adaptive step size predictor corrector scheme where the core computations relied on the proposed numerical method subival has been implemented into an activex dynamic link library dll the paper contains instructions on how the predictor corrector algorithm can be implemented into a computer algebra system where the subival library is applied for the fractional derivative approximation examples of this are given in the form of scripts in matlab and mathematica these scripts generally allow to solve systems of fractional order state equations to which the introduced circuit examples can be brought to

a note on k theory and triangulated derivators abstract in this paper we show an example of two differential graded algebras that have the same derivator k theory but non isomorphic waldhausen k theory we also prove that maltsiniotis s comparison and localization conjectures for derivator k theory cannot be simultaneously true

global convergence of successive approximations for partial hadamard integral equations and inclusions abstract this paper deals with the global convergence of successive approximations as well as the uniqueness of solutions for some classes of partial functional hadamard integral equations and inclusions we prove a theorem on the global convergence of successive approximations to the unique solution of our problems

generalized conformable fractional operators abstract in 2015 abdeljawad 1 has put an open problem which is stated as is it hard to fractionalize the conformable fractional calculus either by iterating the conformable fractional derivative grunwald letnikov approach or by iterating the conformable fractional integral of order 0 1 riemann approach notice that when 0 we obtain hadamard type fractional integrals in this article we claim that yes it is possible to iterate the conformable fractional integral of order 0 1 riemann approach such that when 0 we obtain hadamard fractional integrals first of all we prove cauchy integral formula for repeated conformable fractional integral and proceed to define new generalized conformable fractional integral and derivative operators left and right sided we also prove some basic properties which are satisfied by these operators these operators integral and derivative are the generalizations of katugampola operators riemann liouville fractional operators hadamard fractional operators we apply our results to a simple function also we consider a nonlinear fractional differential equation using this new formulation we show that this equation is equivalent to a volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear problem at the end we give conclusion and point out an open problem

on the regional gradient observability of time fractional diffusion processes this paper for the first time addresses the concepts of regional gradient observability for the riemann liouville time fractional order diffusion system in an interested subregion of the whole domain without the knowledge of the initial vector and its gradient the riemann liouville time fractional order diffusion system which replaces the first order time derivative of normal diffusion system by a riemann liouville time fractional order derivative of order 0 1 is used to well characterize those anomalous sub diffusion processes the characterizations of the strategic sensors when the system under consideration is regional gradient observability are explored we then describe an approach leading to the reconstruction of the initial gradient in the considered subregion with zero residual gradient vector at last to illustrate the effectiveness of our results we present several application examples where the sensors are zone pointwise or filament ones

synchronization of nonlinear fractional order systems by means of pirα reduced order observer this paper deals with the master slave synchronization scheme for partially known nonlinear fractional order systems where the unknown dynamics are considered as the master system and we propose the slave system structure which estimates the unknown state variables besides it is introduced a new fractional model free reduced order observer inspired on the new concept of fractional algebraic observability fao we applied the results to a rossler hyperchaotic fractional order system and lorenz fractional order system and by means of some simulations we show the effectiveness of the suggested approach

fuzzy fractional differential equations under caputo katugampola fractional derivative approach abstract in this work an initial value problem of caputo katugampola ck fractional differential equations in fuzzy setting is considered and an idea of successive approximations under generalized lipschitz condition is used to prove the existence and uniqueness results of the solution to the given problem in order to obtain the above results some necessary comparison theorems in real valued differential equation under ck fractional derivative are established finally a new technique to find analytical solutions of ck fuzzy fractional differential equations by using the solutions of fuzzy integer order differential equations is proposed some examples illustrating the applications of our results are presented

non local fractional calculus from different viewpoint generated by truncated m derivative abstract the core of this paper is to propose more favored novel non local fractional integral and derivative operators with three parameters by using the iteration manner on truncated m derivative involving truncated mittag leffler function this is related closely to the fact that in a certain sense to iterate a convenient local derivative for real phenomena under investigation is one of the best ways for obtaining more comprehensive fractional integration and differentiation in order to gain a deeper grip we address some significant theorems reflected accomplishedly the features of the standard fractional calculus about these aidful operators encompassing some other types of fractional operators the presented new operators in riemann liouville and caputo sense are utilized to solve sturm liouville problem subjected to the initial conditions as a useful method laplace transform of the aforementioned fractional integral and derivative is introduced to solve anomalous dynamics of various complex systems moreover we carry out comparison analysis by plotting governing sturm liouville equation in order to grasp the advantages of these new fractional operators performing much better on account of the additional parameter

existence and stability results for nonlinear fractional order riemann liouville volterra stieltjes quadratic integral equations our aim in this paper is to study the existence and the stability of solutions for riemann liouville volterra stieltjes quadratic integral equations of fractional order our results are obtained by using some fixed point theorems some examples are provided to illustrate the main results

singular points of the algebraic curves of symmetric hyperbolic forms abstract we investigate singular points of the algebraic curve defined by a hyperbolic form satisfying some symmetric properties we show that all singular points of such a curve are real nodes

existence of mild solutions for fractional neutral evolution equations abstract in this paper by using the fractional power of operators and some fixed point theorems we discuss a class of fractional neutral evolution equations with nonlocal conditions and obtain various criteria on the existence and uniqueness of mild solutions in the end we give an example to illustrate the applications of the abstract results

on the range and invertibility of a class of mellin multiplier transforms i abstract we study the range and invertibility in weighted versions of lp 0 of the operators nm and ml whose mellin multipliers are s m and s l respectively we deduce corresponding results for the laplace transform which generalise those of widder the paper is the first of a series concerning operators whose multipliers involve products and quotients of gamma functions and which include as special cases operators with g function kernels studied by rooney

fractional order integral equations of two independent variables in this paper we present some results concerning the existence the uniqueness and the attractivity of solutions for some functional integral equations of riemann liouville fractional order by using some fixed point theorems

systems of diagram categories and k theory ii the additivity theorem for derivateurs associated to complicial biwaldhausen categories is proved also to any exact category in the sense of quillen a k theory space is associated this k theory is shown to satisfy the additivity approximation and resolution theorems

applications of fractional calculus in solving abel type integral equations surface volume reaction problem abstract in this paper we consider a class of partial integro differential equations of fractional order motivated by an equation which arises as a result of modeling surface volume reactions in optical biosensors we solve these equations by employing techniques from fractional calculus several examples are discussed furthermore for the first time we encounter an order of the fractional derivative other than 1 2 in an applied problem hence in this paper we explore the applicability of fractional calculus in real world applications further strengthening the true nature of fractional calculus

properties of the katugampola fractional derivative with potential application in quantum mechanics katugampola e print arxiv 1410 6535 recently introduced a limit based fractional derivative d referred to in this work as the katugampola fractional derivative that maintains many of the familiar properties of standard derivatives such as the product quotient and chain rules typically fractional derivatives are handled using an integral representation and as such are non local in character the current work starts with a key property of the katugampola fractional derivative d y t1 dydt and the associated differential operator d t1 d1 these operators their inverses commutators anti commutators and several important differential equations are studied the anti commutator serves as a basis for the development of a self adjoint operator which could potentially be useful in quantum mechanics a hamiltonian is constructed from this operator and applied to the particle in a box model

mellin transforms of generalized fractional integrals and derivatives we obtain the mellin transforms of the generalized fractional integrals and derivatives that generalize the riemann liouville and the hadamard fractional integrals and derivatives we also obtain interesting results which combine generalized r m operators with generalized stirling numbers and lah numbers for example we show that 1 1 corresponds to the stirling numbers of the 2nd kind and 2 1 corresponds to the unsigned lah numbers further we show that the two operators r m and m r r m n generate the same sequence given by the recurrence relation s n k i 0 r m m r n 2 k i 1 r i r i s n 1 k i 0 k n with s 0 0 1 and s n 0 s n k 0 for n 0 and 1 min r m n 1 k or k 0 finally we define a new class of sequences for r 1 3 1 4 1 5 1 6 and in turn show that 1 2 1 corresponds to the generalized laguerre polynomials

hyperbolic forms associated with cyclic weighted shift matrices abstract the singular points of the curve of a hyperbolic form associated with a cyclic weighted shift matrix are examined it is shown that the singular points of such a curve are real nodes some results related the numerical ranges of cyclic weighted shift matrices are presented in particular the existence of flat portions on the boundary of the numerical range depends on the reducibility of the hyperbolic form further an algebraic method is provided for the decomposition of reducible form which leads to a criterion for the periodicity of the weights

approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique abstract this paper is concerned with the approximate controllability of the semilinear fractional evolution equations with nonlocal and impulsive conditions our main results are obtained by utilizing the technique of approximate solution and the theory of fixed point in addition the impulsive functions in this paper are supposed to be continuous and the nonlocal item is divided into two cases lipschitz continuous and only continuous which generalizes the previous contributions finally two examples are worked out to illustrate our obtained results

additivity for derivator k theory abstract we prove the additivity theorem for the k theory of triangulated derivators this solves one of the conjectures made by maltsiniotis in g maltsiniotis la k theorie d un derivateur triangule in alexei davydov michael batanin michael johnson stephen lack amnon neeman eds categories in algebra geometry and physics conference and workshop in honor of ross street s 60th birthday in contemp math vol 431 amer math soc 2007 pp 341 368 we also review some basic definitions and results in the theory of derivators in the sense of grothendieck

mellin transforms and asymptotics harmonic sums this survey presents a unified and essentially self contained approach to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics discrete probabilistic models and the average case analysis of algorithms it relies on the mellin transform a close relative of the integral transforms of laplace and fourier the method applies to harmonic sums that are superpositions of rather arbitrary harmonics of a common base function its principle is a precise correspondence between individual terms in the asymptotic expansion of an original function and singularities of the transformed function the main applications are in the area of digital data structures probabilistic algorithms and communication theory

regional observability for hadamard caputo time fractional distributed parameter systems abstract in this paper regional gradient exact and approximate observability problems are studied on hadamard caputo time fractional distributed parameter systems without any knowledge of the initial vector and its gradient several equivalent criteria are first provided to achieve the regional observability based on these characterizations for both strategic and gradient strategic zone sensors are developed then by employing the hilbert uniqueness method hum we explicitly reconstruct the initial vector and its gradient respectively a one dimension example is finally included to illustrate our results

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

compositions of hadamard type fractional integration operators and the semigroup property abstract this paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of hadamard in the space xcp of functions f on r 0 such that 0 u c f u p du u ess sup u 0 u c f u for c r in particular in the space lp 0 1 p the semigroup property and its generalizations are established for the generalized hadamard type fractional integration operators under consideration conditions are also given for the boundedness in xcp of these operators they involve kummer confluent hypergeometric functions as kernels

numerical ranges of weighted shift matrices abstract we show that if a is an n by n n 3 matrix of the form 0 a 1 0 a n 1 a n 0 then the boundary of its numerical range contains a line segment if and only if the a j s are nonzero and the numerical ranges of the n 1 by n 1 principal submatrices of a are all equal for n 3 this is the case if and only if a 1 a 2 a 3 0 in which case w a the numerical range of a is the equilateral triangular region with vertices the three cubic roots of a 1 a 2 a 3 for n 4 the condition becomes a 1 a 3 0 and a 2 a 4 0 in which case w a is the convex hull of two degenerate or otherwise ellipses

a modified bessel type integral transform and its compositions with fractional calculus operators on spaces f p m and f p m abstract the paper is devoted to study the integral transform l f x 0 xt f t d t x 0 with the kernel z 1 1 1 t 1 1 t e zt d t for 0 re 1 1 r re z 0 which is a generalization of the modified bessel function of the third kind or macdonald function k z properties of z are investigated and compositions of the operator l with the left and right sided liouville fractional integrals and derivatives are proved

hermite hadamard and hermite hadamard fejer type inequalities for generalized fractional integrals abstract in this paper we obtain the hermite hadamard and hermite hadamard fejer type inequalities for fractional integrals which generalize the two familiar fractional integrals namely the riemann liouville and the hadamard fractional integrals into a single form we prove that in most cases we obtain the riemann liouville and the hadamard equivalence just by taking limits when a parameter 1 and 0 respectively

asymptotic representations for hypergeometric bessel type function and fractional integrals the paper is devoted to the study of asymptotic relations for the function z 1 1 1 t 1 1 t e ztdt generalising tricomi confluent hypergeometric function and modified bessel function of the third kind the full asymptotic representations for z at zero and infinity are established applications are given to obtain full asymptotic expansions near zero and infinity for the liouville fractional integral i f x 1 x f t dt t x 1 x 0 c re 0 and for the erdelyi kober type fractional intergral i f x x x t 1 1 f t dt t x 1 x 0 c re 0 with 0 and c of power exponential function f t and for three other fractional integrals

fractional differential equations of caputo katugampola type and numerical solutions abstract this paper is concerned with a numerical method for solving generalized fractional differential equation of caputo katugampola derivative a corresponding discretization technique is proposed numerical solutions are obtained and convergence of numerical formulae is discussed the convergence speed arrives at o t 1 numerical examples are given to test the accuracy

fractional calculus in the mellin setting and hadamard type fractional integrals abstract the purpose of this paper and some to follow is to present a new approach to fractional integration and differentiation on the half axis r 0 in terms of mellin analysis the natural operator of fractional integration in this setting is not the classical liouville fractional integral i 0 f but j 0 c f x 1 0 x u x c log x u 1 f u du u x 0 for 0 c r the mellin transform of this operator is simply c s m f s for s c it c t r the mellin transform of the associated fractional differentiation operator d 0 c f is similar c s m f s the operator d 0 c f may even be represented as a series in terms of xkf k x k n 0 the coefficients being certain generalized stirling functions sc k of second kind it turns out that the new fractional integral j 0 c f and three further related ones are not the classical fractional integrals of hadamard j mat pure appl ser 4 8 1892 101 186 but far reaching generalizations and modifications of these these four new integral operators are first studied in detail in this paper more specifically conditions will be given for these four operators to be bounded in the space xcp of lebesgue measurable functions f on 0 for c such that 0 u c f u p du u for 1 p ess sup u 0 u c f u for p in particular in the space lp 0 for 1 p connections of these operators with the liouville fractional integration operators are discussed the mellin convolution product in the above spaces plays an important role

numerical ranges of weighted shift matrices with periodic weights abstract let a be an n by n n 2 matrix of the form 0 a 1 0 a n 1 a n 0 we show that if the a j s are nonzero and their moduli are periodic then the boundary of its numerical range contains a line segment we also prove that w a contains a noncircular elliptic arc if and only if the a j s are nonzero n is even a 1 a 3 a n 1 a 2 a 4 a n and a 1 a 2 finally we give a criterion for a to be reducible and completely characterize the numerical ranges of such matrices

mellin transform analysis and integration by parts for hadamard type fractional integrals abstract this paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of hadamard in the space xpc of lebesgue measurable functions f on r 0 such that 0 u c f u p du u ess sup u 0 u c f u for c r in particular in the space lp 0 1 p formulas for the mellin transforms of the four hadamard type fractional integral operators are established as well as relations of fractional integration by parts for them

simplest equation method for some time fractional partial differential equations with conformable derivative abstract the conformable fractional derivative was proposed by r khalil et al in 2014 which is natural and obeys the leibniz rule and chain rule based on the properties a class of time fractional partial differential equations can be reduced into odes using traveling wave transformation then the simplest equation method is applied to find exact solutions of some time fractional partial differential equations the exact solutions solitary wave solutions periodic function solutions rational function solutions of time fractional generalized burgers equation time fractional generalized kdv equation time fractional generalized sharma tasso olver fsto equation and time fractional fifth order kdv equation 3 1 dimensional time fractional kdv zakharov kuznetsov kdv zk equation are constructed this method presents a wide applicability to solve some nonlinear time fractional differential equations with conformable derivative

a note on katugampola fractional calculus and fractal dimensions abstract the goal of this paper is to study the katugampola fractional integral of a continuous function of bounded variation defined on a closed bounded interval we note that the katugampola fractional integral of a function shares some analytical properties such as boundedness continuity and bounded variation of the function defining it consequently we deduce that fractal dimensions minkowski dimension and hausdorff dimension of the graph of the katugampola fractional integral of a continuous function of bounded variation are one a natural question then arises is whether there exists a continuous function which is not of bounded variation with its graph having fractal dimensions one in the last part of the article we construct a continuous function which is not of bounded variation and for which the graph has fractal dimensions one the construction enunciated herein includes previous constructions found in the recent literature as special cases the article also hints at an upper bound for the upper box dimension of the graph of the katugampola fractional derivative of a continuously differentiable function

determinantal representations of hyperbolic forms via weighted shift matrices we characterize hyperbolic ternary forms of degrees 3 4 and 5 that admit determinantal representations via cyclic weighted shift matrices

successive approximations for solutions of functional integral equations in a banach space x where r 0 j o b or icj b and the integral is in the sense of bochner this equation is a unification and extension of a nonlinear volterra integral equation and a functional differential equation of retarded type indeed if in 1 1 the function f satisfies that f f s a for t e j then it is reduced to a functional differential equation in a banach space de balasi and myjak 21 gave a global convergence version of successive approximations to solutions for such reduced equations under the assumption that g is lipschitz continuous and bounded if in addition r 0 then 1 1 is reduced to an ordinary differential equation for this vidossich 161 discussed the global convergence of picard s successive approximations to solutions under some less restrictive conditions on g the main purpose of this paper is to extend vidossich s results to the solutions of eq 1 1 our conditions made on g are similar to what miller 4 and artzstein 1 made for the integral equation 1 1 with r 0 and what vidossich 61 made for the ordinary differential equation referred above

on integral transformations with g function kernels integral transformations t of the formal form where g is a meijer g function are studied on the spaces r of functions f such that results are obtained on the boundedness representation and range of the transformations and on when they have an inverse of the same formal form

a review of definitions of fractional derivatives and other operators abstract given the increasing number of proposals and definitions of operators in the scope of fractional calculus it is important to introduce a systematic classification nonetheless many of the definitions that emerged in the literature can not be considered as fractional derivatives we analyze a list of expressions to have a general overview of the concept of fractional integrals derivatives moreover some formulae that do not involve the term fractional are also included due to their particular interest in the area

on the issue of imposing boundary conditions on quantum fields an interesting example of the deep interrelation between physics and mathematics is obtained when trying to impose mathematical boundary conditions on physical quantum fields this procedure has recently been re examined with care comments on that and previous analysis are provided here together with considerations on the results of the purely mathematical zeta function method in an attempt at clarifying the issue hadamard regularization is invoked in order to fill the gap between the infinities appearing in the qft renormalized results and the finite values obtained in the literature with other procedures

maximizing numerical radii of weighted shifts under weight permutations abstract let w i c 1 i n and l s n the symmetric group of all permutations of 1 2 n suppose a l is the weighted cyclic matrix a l 0 w l 1 0 w l n 1 w l n 0 and w a l denotes its numerical radius we characterize those s n which satisfy w a max l s n w a l the characterizations for unilateral and bilateral weighted backward shifts are also obtained

diagonals and numerical ranges of weighted shift matrices for any n by n matrix a we consider the maximum number k k a forwhichthereisak by kcompressionofawithallitsdiagonal entries in the boundary w a of the numerical range w a of a for any such compression we give a standard model under unitary equivalence fora this is then applied to determine the value ofk a for a of size 3 in terms of the shape of w a w hena is a matrix of the form

condition for the numerical range to contain an elliptic disc for an n by n matrix a and an elliptic disc e in the plane we show that the sum of the number of common supporting lines and the number of common intersection points to e and the numerical range w a of a should be at least 2n 1 in order to guarantee that e be contained in w a this generalizes previous results of anderson and thompson as an application our result is used to verify a special case of the poncelet property conjecture 2003 elsevier science inc all rights reserved

on generalized gruss type inequalities for k fractional integrals the aim of the present paper is to investigate some new integral inequalities of gruss type for k riemann liouville fractional integrals from our results new weighted or classical gruss type inequalities have been established for some special cases moreover special cases of the integral inequalities in this paper have been obtained by dahmani and tabharit 2010 in 5

singular points of the algebraic curves associated to unitary bordering matrices abstract let a be an n n complex matrix a ternary form associated to a is defined as the homogeneous polynomial f a t x y det t i n x a y a we prove for a unitary boarding matrix a the ternary form f a t x y is strongly hyperbolic and the algebraic curve f a t x y 0 has no real singular points as a consequence we obtain that the higher rank numerical range of a unitary boarding matrix is strictly convex

new approach to a generalized fractional integral abstract the paper presents a new fractional integration which generalizes the riemann liouville and hadamard fractional integrals into a single form conditions are given for such a fractional integration operator to be bounded in an extended lebesgue measurable space semigroup property for the above operator is also proved we give a general definition of the fractional derivatives and give some examples

computing hadamard type operators of variable fractional order abstract we consider hadamard fractional derivatives and integrals of variable fractional order a new type of fractional operator which we call the hadamard marchaud fractional derivative is also considered the objective is to represent these operators as series of terms involving integer order derivatives only and then approximate the fractional operators by a finite sum an upper bound formula for the error is provided we exemplify our method by applying the proposed numerical procedure to the solution of a fractional differential equation and a fractional variational problem with dependence on the hadamard marchaud fractional derivative