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

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

discrete aggregation operators with annihilator abstract mas et al have investigated a special kind of commutative and associative aggregation operators with annihilator on the unit interval 0 1 which includes s uninorms s nullnorms t uninorms t nullnorms bi uninorms and bi nullnorms this paper is devoted to extending this kind of operators from the unit interval 0 1 to a finite chain a characterization of this kind of operators on a finite chain is given including some examples this class of operators can be viewed as a generalization of both uninorms and nullnorms on a finite chain

hermite hadamard s type inequalities for operator convex functions some hermite hadamard s type inequalities for operator convex functions of selfadjoint operators in hilbert spaces are given applications for particular cases of interest are also provided

asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials a general doubly adaptive biased coin design is proposed for the allocation of subjects to k treatments in a clinical trial this design follows the same spirit as efron s biased coin design and applies to the cases where the desired allocation proportions are unknown but estimated sequentially strong consistency a law of the iterated logarithm and asymptotic normality of this design are obtained under some widely satisfied conditions for two treatments a new family of designs is proposed and shown to be less variable than both the randomized play the winner rule and the adaptive randomized design also the proposed design tends toward a randomization scheme with a fixed target proportion as the size of the experiment increases

vertex reinforced random walk on z has finite range a stochastic process called vertex reinforced random walk vrrw is defined in pemantle ann probab 16 1229 1241 we consider this process in the case where the underlying graph is an infinite chain i e process in the case where the underlying graph is an infinite chain i e the one dimensional integer lattice we show that the range is almost surely finite that at least five points are visited infinitely often almost surely and that with positive probability the range contains exactly five points there are always points visited infinitely often but at a set of times of zero density and we show that the number of visits to such a point to time n may be asymptotically n for a dense set of values 0 1 the power law analysis relies on analysis of a related urn model

weighted networks as randomly reinforced urn processes we analyze weighted networks as randomly reinforced urn processes in which the edge total weights are determined by a reinforcement mechanism we develop a new statistical test and a new procedure based on it to study the evolution of networks over time detecting the dominance of some edges with respect to the others and then assessing if a given instance of the network is taken at its steady state or not distance from the steady state can be considered as a measure of the relevance of the observed properties of the network our results are quite general in the sense that they are not based on a particular probability distribution or functional form of the random weights moreover the proposed tool can be applied also to dense networks which have received little attention by network community so far since they are often problematic we apply our procedure in the context of the international trade network determining a core of dominant edges

asymptotics in response adaptive designs generated by a two color randomly reinforced urn this paper illustrates asymptotic properties for a response adaptive design generated by a two color randomly reinforced urn model the design considered is optimal in the sense that it assigns patients to the best treatment with probability converging to one an approach to show the joint asymptotic normality of the estimators of the mean responses to the treatments is provided in spite of the fact that allocation proportions converge to zero and one results on the rate of convergence of the number of patients assigned to each treatment are also obtained finally we study the asymptotic behavior of a suitable test statistic

adaptive biased coin designs for skewing the allocation proportion in clinical trials with normal responses response adaptive designs are used in phase iii clinical trial for skewing the allocation pattern towards the better treatments we use optimum design theory to derive adaptive designs when the responses are normally distributed the performance of the designs is studied with respect to the loss and the proportion of allocation to different treatments the adaptive design does not affect inference

characterization of idempotent n uninorms abstract the structure of idempotent n uninorms is studied we show that each idempotent 2 uninorm can be expressed as an ordinal sum of an idempotent uninorm possibly also of a countable number of idempotent semigroups with operations min and max and a 2 uninorm from class 1 possibly restricted to open or half open unit square similar results are shown also for idempotent n uninorms further it is shown that idempotent n uninorms are in one to one correspondence with special lower semi lattices defined on the unit interval the z ordinal sum construction for partially ordered semigroups is also defined

a generalization of hh f divergence abstract the discrimination between two probability distributions is an impotent problem in 1991 lin ieee transactions on information theory 37 1 1991 introduced a novel class of information theoretic divergence measures based on the shannon entropy as a generalization of lin s divergence a new divergence called hermite hadamard hh f divergence based on lin s method of constructing the divergence was introduced by shioya and da te in 1995 in this paper we expand the applicability of hh f divergence by combining the properties of fractional calculus with hh f divergence and then introduce the concept of some fractional hh f divergences which are generalizations of the hh f divergence then some inequalities related to fractional hh f divergence are proposed

on the structure of 2 uninorms abstract in this paper we describe the structure of 2 uninorms defined on the unit interval we divide the class of 2 uninorms into five mutually exclusive subclasses based on their boundary behaviour as determined by the absorbing element and the conjunctive disjunctive nature of the underlying uninorms which every 2 uninorm is known to have for each of these subclasses we fully characterize the structure of its members albeit for two subclasses only under an additional continuity assumption a common practice in the study of associative operations

convexity according to a pair of quasi arithmetic means and inequalities abstract we consider a class of generalized convex functions which are defined according to a pair of quasi arithmetic means and called m m convex functions and establish various fejer type inequalities for such a function class these inequalities not merely provide a natural and intrinsic characterization of the m m convex functions but actually offer a generalization and refinement of the most part of the concrete hermite hadamard and fejer type inequalities obtained in earlier studies for different kinds of convexity and fractional integrals applications to inequalities involving the gamma function and special means are also included

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

weak uninorm aggregation operators abstract by weakening the identity condition of uninorm a general class of associative aggregation operators called weak uninorm is introduced it is proved that continuous weak uninorms are the combination of continuous t norm and continuous t conorm through a median operator the representation theorem for archimedean weak uninorm is also given finally the counting number of the identity elements of weak uninorm is also investigated

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

van melle s combining function in mycin is a representable uninorm an alternative proof abstract recently tsadiras and margaritis have shown that the function implemented in the expert system mycin for combining certainty factors is a uninorm in this paper we present a compact alternative to the given direct and lengthy proof based on the construction of representable uninorms by means of an additive generator consequently we show that van melle s combining function is a representable uninorm in fact it is a rescaled version of the representable uninorm on the unit interval constructed from the algebraic product and the probabilistic sum

extensions of the hermite hadamard inequality for harmonically convex functions via fractional integrals the main aim of this paper is to give extensions of the hermite hadamard inequality for harmonically convex functions via riemann liouville fractional integrals we show how to relax the harmonically convexity property of the function f obtained results in this work involve a larger class of functions our results are the refinements of the existing results for harmonically convex functions

hermite hadamard type inequalities via α m convexity in this paper we establish several new inequalities for functions whose second derivative in absolute value aroused to the qth q 1 power are a m convex some applications to special means of positive real numbers are also given

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

hermite hadamard type inequalities for harmonically convex functions via fractional integrals abstract in this paper the authors established hermite hadamard s inequalities for harmonically convex functions via fractional integrals and obtained some hermite hadamard type inequalities of these classes of functions

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

an advanced interpretable fuzzy neural network model based on uni nullneuron constructed from n uninorms abstract this paper formulates a fuzzy logic neuron that uses n uninorms to construct uni nullneurons a fuzzy neural network fnn composed of these neurons is easy to operate with nullnorms and uninorms at different times which results in high accuracy of the model outputs and increases the flexibility in connecting the rule antecedents enabling and and or connections within one rule this in turn may allow experts operators to extract more knowledge from data the fnn uses a one versus rest classifier learning scheme for multi class classification problems where neuron activation levels construct the indicator regression matrix this results in a non linear regression by indicator which can resolve the inherent class masking problem apparent in the linear case we propose a specific neuron selection strategy in the learning stage that applies lasso to bootstrap replications in order to ensure that the rule base is as compact as possible and induced by a low number of neurons to evaluate the new neuron acting in fnns we performed pattern classification and regression tests compared with traditional fnn in the literature our variant showed improved model accuracies for several high dimensional real world datasets in binary and multi class classification and regression problems combined with the ability to generate human readable rules this offers the ability to generate parsimonious responses with a high degree of confidence

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

compound optimal allocation for individual and collective ethics in binary clinical trials in recent years several authors have investigated response adaptive allocation rules for comparative clinical trials in order to favour at each stage of the trial the treatment that appears to be best in this paper we define admissible allocations namely treatment assignments that cannot be simultaneously improved upon with respect to both a specific design criterion reflecting the inferential properties of the experiment and the proportion of patients assigned to the best treatment or treatments we survey existing designs from this viewpoint we also suggest combining information and ethical considerations by taking a suitable weighted mean of two corresponding standardized criteria with weights that depend on the actual treatment effects this compound criterion leads to a locally optimal allocation that can be targeted by some response adaptive randomization rule the paper mainly deals with the case of two treatments but the suggested methodology is shown to extend to more than two

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

what is a fractional derivative this paper discusses the concepts underlying the formulation of operators capable of being interpreted as fractional derivatives or fractional integrals two criteria for required by a fractional operator are formulated the grunwald letnikov riemann liouville and caputo fractional derivatives and the riesz potential are accessed in the light of the proposed criteria a leibniz rule is also obtained for the riesz potential

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

distributivity and conditional distributivity of a uninorm and a continuous t conorm the open problem recalled by klement in the linz2000 closing session related to distributivity and conditional distributivity of a uninorm and a continuous t conorm is solved for the most usual known classes of uninorms from the obtained results it is deduced that distributivity and conditional distributivity are equivalent for these cases it is remarkable that solutions appear involving not only strict t conorms but also ordinal sums of the maximum with a strict t conorm conversely the distributivity of a t conorm over a uninorm is also studied leading only to already known solutions moreover the dual case of distributivity and conditional distributivity involving uninorms and continuous t norms is also solved proving again the equivalence of both kinds of distributivities

hermite hadamard hermite hadamard fejer dragomir agarwal and pachpatte type inequalities for convex functions via new fractional integrals abstract the aim of this paper is to establish hermite hadamard hermite hadamard fejer dragomir agarwal and pachpatte type inequalities for new fractional integral operators with exponential kernel these results allow us to obtain a new class of functional inequalities which generalizes known inequalities involving convex functions furthermore the obtained results may act as a useful source of inspiration for future research in convex analysis and related optimization fields

nullnorms on bounded lattices derived from t norms and t conorms abstract nullnorms with an annihilator a in any point of a bounded lattice are generalizations and unifications of t norms and t conorms this study continues to investigate the construction of nullnorms on bounded lattices we propose some methods to construct nullnorms derived from t norms and t conorms on bounded lattices where some sufficient and necessary conditions on theirs annihilator are required as a by product of these constructions idempotent nullnorms on bounded lattices are obtained further we provide some illustrative examples of the new classes of nullnorms idempotent nullnorms on bounded lattices

structure of n uninorms in this paper we study binary operators on 0 1 which are associative monotone non decreasing in both variables and commutative amc with neutral element in this work we generalize the concept of neutral element and this generalization gives rise to a new class of amc binary operators on 0 1 called n uninorms n uninorms are denoted as u n where n comes from the generalization of the neutral element we study the structure of n uninorms the structure resembles an ordinal sum structure made up of n uninorms we characterize some special cases of them based on some continuity considerations and show that t norms t conorms uninorms and nullnorms t operators are special cases of n uninorms we also show that given n there are n 1 classes of operators in u n and each of them has many subclasses we also study the frank equation involving n uninorms and show that we need to consider only n uninorms for the study finally we show that the total number of subclasses of operators in u n follows the famous series called catalan numbers

correction to what is a fractional derivative by ortigueira and machado journal of computational physics volume 293 15 july 2015 pages 4 13 special issue on fractional pdes there is a debate among contemporary mathematicians about what it really means by a fractional derivative the question arose as a consequence of introducing a new definition of a fractional derivative in 1 in a reply ortigueira and machado 2 came up with several very important criteria to determine whether a given derivative is a fractional derivative according to their criterion the new fractional derivative called conformable fractional derivative introduced by khalil et al 1 turns out not to be a fractional derivative but rather a controlled or conformable derivative in proving the claim the authors in 2 use an example 2 p 6 it turns out that the explanation given there needs some corrections and it is the sole purpose of this note

optimality variability power we provide a theoretical template for the comparison of response adaptive randomization procedures for clinical trials using a taylor expansion of the noncentrality parameter of the usual chi squared test for binary responses we show explicitly the relationship among the target allocation proportion the bias of the randomization procedure from that target and the variability induced by the randomization procedure we also generalize this relationship for more than two treatments under various multivariate alternatives this formulation allows us to directly evaluate and compare different response adaptive randomization procedures and different target allocations in terms of power and expected treatment failure rate without relying on simulation for k 2 treatments we compare four response adaptive randomization procedures and three target allocations based on multiple objective optimality criteria we conclude that the drop the loser rule and the doubly adaptive biased coin design are clearly super

a single point characterization of representable uninorms we study uninorms of which both the underlying t norm and underlying t conorm are strict such uninorms are the only candidates for being representable by an additive generator we prove that the representability of such a uninorm depends solely on its value at a single arbitrary point in the remaining open part of the unit square more explicitly such a uninorm turns out to be representable if and only if this single value is located strictly between the minimum and the maximum of the corresponding arguments if this single value coincides with one of these bounds then the value of the uninorm at any point in the remaining open part is determined by the same bound

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

on new inequalities of hermite hadamard fejer type for convex functions via fractional integrals in this paper we establish some weighted fractional inequalities for differentiable mappings whose derivatives in absolute value are convex these results are connected with the celebrated hermite hadamard fejer type integral inequality the results presented here would provide extensions of those given in earlier works

uninorms on bounded lattices with the underlying t norms and t conorms abstract uninorms are a generalization of the concepts of triangular norms t norms and triangular conorms t conorms that admit a neutral element as any element in a bounded lattice in this study we discuss the characterization of uninorms on bounded lattices we introduce some new constructions for uninorms on bounded lattices which are derived from both t norms and t conorms in particular two idempotent uninorms on bounded lattices are obtained we also provide some corresponding examples to help understand the structure of these new classes of uninorms in addition we provide an answer to the open problem proposed by cayli 2019 14

the hermite hadamard inequality for log convex functions abstract we discuss the existence of a strengthening of hermite hadamard inequality in the case of log convex functions unlike the classical case which belongs to the field of linear functional analysis this analogue involves nonlinear means such as the geometric mean and the logarithmic mean