variational formulations of differential equations and asymmetric fractional embedding abstract variational formulations for classical dissipative equations namely friction and diffusion equations are given by means of fractional derivatives in this way the solutions of those equations are exactly the extremal of some fractional lagrangian actions the formalism used is a generalization of the fractional embedding developed by cresson fractional embedding of differential operators and lagrangian systems j math phys 48 2007 033504 where the functional space has been split in two in order to take into account the asymmetry between left and right fractional derivatives moreover this asymmetric fractional embedding is compatible with the least action principle and respects the physical causality principle

hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

including the size of regions in image segmentation by region based graph applying a fast over segmentation algorithm to image and working on a region based graph instead of the pixel based graph is an efficient approach to reduce the computational complexity of graph based image segmentation methods nevertheless some undesirable effects may arise if the conventional cost functions such as ncut averagecut and mincut are employed for partitioning the region based graph this is because these cost functions are generally tailored to pixel based graphs in order to resolve this problem we first introduce a new class of cost functions containing ncut and averagecut for graph partitioning whose corresponding suboptimal solution can be efficiently computed by solving a generalized eigenvalue problem then among these cost functions we propose one that considers the size of regions in the partitioning procedure by simulation the performance of the proposed cost function is quantitatively compared with that of the ncut and averagecut

fractional variational problems depending on indefinite integrals abstract we obtain necessary optimality conditions for variational problems with a lagrangian depending on a caputo fractional derivative a fractional and an indefinite integral main results give fractional euler lagrange type equations and natural boundary conditions which provide a generalization of the previous results found in the literature isoperimetric problems problems with holonomic constraints and depending on higher order caputo derivatives as well as fractional lagrange problems are considered

the global analysis on the spectral collocation method for time fractional schrodinger equation abstract in this paper a spectral collocation method is proposed and analyzed for solving the time fractional schrodinger equation the space derivative is discretized using the collocation method and the time fractional derivative using grunwald letnikov formulation the stability and convergence of the full discretization scheme are analyzed based on the z transform the global behavior of the finite difference spectral collocation method is derived numerical examples show a good agreement with the theoretical analysis

a survey on fuzzy fractional variational problems abstract this paper focuses on introducing and studying generalized fuzzy euler lagrange equation and fuzzy isoperimetric problem according to the concept of caputo type fuzzy fractional derivative in the sense of the generalized fuzzy differentiability we extend and establish some definitions on fuzzy fractional calculus of variation and provide some necessary conditions to obtain the fuzzy fractional euler lagrange equation for both constrained and unconstrained fuzzy fractional variational problems the fuzzy isoperimetric problem is also investigated in relation to the generalized fuzzy euler lagrange finally two examples are given to describe the proposed approach

a new method of dynamical stability i e fractional generalized hamiltonian method and its applications fractional generalized hamiltonian method of dynamical stability is presented six criterions for the new method of dynamical stability are given the new method s applications are studied five kinds of actual fractional dynamical models are constructed dynamical stability of the five actual fractional dynamical models is explored in the paper we present fractional generalized hamiltonian method of dynamical stability in terms of riesz riemann liouville derivative and study its applications for an actual dynamical system the fractional generalized hamiltonian method of constructing a fractional dynamical model is given and then the six criterions for fractional generalized hamiltonian method of dynamical stability are presented as applications by using the fractional generalized hamiltonian method we construct five kinds of actual fractional dynamical models which include a fractional euler poinsot model of rigid body that rotates with respect to a fixed point a fractional hojman urrutia model a fractional lorentz dirac model a fractional whittaker model and a fractional robbins lorenz model and we explore the dynamical stability of these models respectively this work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering

mixed convolved action for classical and fractional derivative dissipative dynamical systems the principle of mixed convolved action provides a new rigorous weak variational formalism for a broad range of initial value problems in mathematical physics and mechanics here the focus is initially on classical single degree of freedom oscillators incorporating either kelvin voigt or maxwell dissipative elements and then subsequently on systems that utilize fractional derivative constitutive models in each case an appropriate mixed convolved action is formulated and a corresponding weak form is discretized in time using temporal shape functions to produce an algorithm suitable for numerical solution several examples are considered to validate the mixed convolved action principles and to investigate the performance of the numerical algorithms for undamped systems the algorithm is found to be symplectic and unconditionally stable with respect to the time step in the case of dissipative systems the approach is shown to be robust and to be accurate with good convergence characteristics for both classical and fractional derivative based models as part of the derivations some interesting results in the calculus of caputo fractional derivatives also are presented

variational problems with fractional derivatives invariance conditions and nother s theorem abstract a variational principle for lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases necessary and sufficient conditions for an infinitesimal transformation group basic nother s identity are obtained these conditions extend the classical results valid for integer order derivatives a generalization of nother s theorem leading to conservation laws for fractional euler lagrangian equation is obtained as well results are illustrated by several concrete examples finally an approximation of a fractional euler lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws

a polynomial algorithm for the k cut problem for fixed k the k cut problem is to find a partition of an edge weighted graph into k nonempty components such that the total edge weight between components is minimum this problem is np complete for an arbitrary k and its version involving fixing a vertex in each component is np hard even for k 3 we present a polynomial algorithm for k fixed that runs in onki 2 3k 2 4tn m steps where tn m is the running time required to find the minimum s t cut on a graph with n vertices and m edges

new fractional operators and application to fractional variational problem abstract motivated by hilfer fractional derivative a class of generalized fractional integral and derivative operators is discussed the generalized fractional integrals are the counterparts of generalized fractional derivatives we study their semigroup property and fractional integration by parts formulae which play an important role in deducing the euler lagrange equation of extremal problem involving generalized fractional operators as applications we discuss a minimizing functional problem and an isoperimetric problem containing these new operators numerical simulations based on polynomial approximation and galerkin approach are presented our study shows that the polynomial approximation and galerkin approach can be easily applied to fractional variational problem depending on both left and right generalized fractional integrals and derivatives

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

improved algorithms for bipartite network flow in this paper network flow algorithms for bipartite networks are studied a network g v e is called bipartite if its vertex set v can be partitioned into two subsets v 1 and v 2 such that all edges have one endpoint in v 1 and the other in v 2 let n v n 1 v 1 n 2 v 2 m e and assume without loss of generality that n 1 leq n 2 a bipartite network is called unbalanced if n 1 ll n 2 and balanced otherwise this notion is necessarily imprecise it is shown that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks the basic idea in these improvements is a two edge push rule that allows one to charge most computation to vertices in v 1 and hence develop algorithms whose running times depend on n 1 rather than n for example it is shown that the two edge push version of goldberg and tarjan s fifo preflow push algorithm runs in o n 1 m n 1 3 time and that the analogous version of ahuja and orlin s excess scaling algorithm runs in o n 1 m n 1 2 log u time where u is the largest edge capacity these ideas are also extended to dynamic tree implementations parametric maximum flows and minimum cost flows

the pseudoflow algorithm a new algorithm for the maximum flow problem we introduce the pseudoflow algorithm for the maximum flow problem that employs only pseudoflows and does not generate flows explicitly the algorithm solves directly a problem equivalent to the minimum cut problem the maximum blocking cut problem once the maximum blocking cut solution is available the additional complexity required to find the respective maximum flow is o m log n a variant of the algorithm is a new parametric maximum flow algorithm generating all breakpoints in the same complexity required to solve the constant capacities maximum flow problem the pseudoflow algorithm has also a simplex variant pseudoflow simplex that can be implemented to solve the maximum flow problem one feature of the pseudoflow algorithm is that it can initialize with any pseudoflow this feature allows it to reach an optimal solution quickly when the initial pseudoflow is close to an optimal solution the complexities of the pseudoflow algorithm the pseudoflow simplex and the parametric variants of pseudoflow and pseudoflow simplex algorithms are all o mn log n on a graph with n nodes and m arcs therefore the pseudoflow simplex algorithm is the fastest simplex algorithm known for the parametric maximum flow problem the pseudoflow algorithm is also shown to solve the maximum flow problem on s t tree networks in linear time where s t tree networks are formed by joining a forest of capacitated arcs with nodes s and t adjacent to any subset of the nodes

the generalized fractional trace variational identity and fractional integrable couplings of kaup newell hierarchy starting from fractional isospectral problems and general bilinear forms the generalized fractional trace identity is presented then a new explicit lie algebra is introduced for which the fractional integrable couplings of kaup newell hierarchy are derived furthermore we obtained the fractional hamiltonian structures of the fractional integrable couplings of kaup newell hierarchy the methods derived by us can be generalized to other fractional integrable couplings

fractional noether s theorem in the riesz caputo sense abstract we prove a noether s theorem for fractional variational problems with riesz caputo derivatives both lagrangian and hamiltonian formulations are obtained illustrative examples in the fractional context of the calculus of variations and optimal control are given

on analogues of exponential functions for antisymmetric fractional derivatives an equation with the antisymmetric fractional derivative of order email protected 1 2 containing the t b potential is solved using the mellin transform method the solutions are analogues of exponential functions of a new type they are represented as meijer g function series in a finite time interval in the classical limit email protected 1 the eigenfunction equation for a derivative of the first order and its solution an exponential function are recovered then an analogy between the derivation of euler lagrange equations in fractional mechanics and in classical mechanics is discussed the results are applied to a simple fractional euler lagrange equation containing an antisymmetric fractional derivative and its general solution is obtained

blurred derivatives and meshless methods in this work we first introduce and describe the concept of blurred derivatives it is shown how they can be used both to approximate differential equations and to re express them in alternative ways in particular formulations in terms of functional integrals can be obtained using blurred derivatives and extended to non linear problems blurred derivatives are shown to provide higher flexibility for selection of approximation functions than strong and weak formulations some computational implementations of one dimensional problems are discussed and the relationship between several well known numerical methods is analysed finally a meshless numerical scheme for the poisson equation is described in detail its performance is compared with linear finite elements and generalized finite differences on unstructured meshes of points copyright 2002 john wiley sons ltd

a fractional dirac equation and its solution this paper presents a fractional dirac equation and its solution the fractional dirac equation may be obtained using a fractional variational principle and a fractional klein gordon equation both methods are considered here we extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of caputo derivatives by applying the variational principle to a fractional action s we obtain the fractional euler lagrange equations of motion we present a lagrangian and a hamiltonian for the fractional dirac equation of order we also use a fractional klein gordon equation to obtain the fractional dirac equation which is the same as that obtained using the fractional variational principle eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard dirac equation we also provide expressions for the path integral quantization for the fractional dirac field which in the limit 1 approaches to the path integral for the regular dirac field it is hoped that the fractional dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics

dynamic graph cuts for efficient inference in markov random fields in this paper we present a fast new fully dynamic algorithm for the st mincut max flow problem we show how this algorithm can be used to efficiently compute map solutions for certain dynamically changing mrf models in computer vision such as image segmentation specifically given the solution of the max flow problem on a graph the dynamic algorithm efficiently computes the maximum flow in a modified version of the graph the time taken by it is roughly proportional to the total amount of change in the edge weights of the graph our experiments show that when the number of changes in the graph is small the dynamic algorithm is significantly faster than the best known static graph cut algorithm we test the performance of our algorithm on one particular problem the object background segmentation problem for video it should be noted that the application of our algorithm is not limited to the above problem the algorithm is generic and can be used to yield similar improvements in many other cases that involve dynamic change

isoperimetric graph partitioning for image segmentation spectral graph partitioning provides a powerful approach to image segmentation we introduce an alternate idea that finds partitions with a small isoperimetric constant requiring solution to a linear system rather than an eigenvector problem this approach produces the high quality segmentations of spectral methods but with improved speed and stability

hamilton jacobi fractional mechanics abstract as a continuation of rabei et al work eqab m rabei khaled i nawafleh raed s hijjawi sami i muslih dumitru baleanu the hamilton formalism with fractional derivatives j math anal appl 327 2007 891 897 the hamilton jacobi partial differential equation is generalized to be applicable for systems containing fractional derivatives the hamilton jacobi function in configuration space is obtained in a similar manner to the usual mechanics two problems are considered to demonstrate the application of the formalism the result is found to be in exact agreement with agrawal s formalism

the fractional quadratic form identity and hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy starting from a general isospectral problem of fractional order we propose a fractional quadratic form identity from which the hamiltonian structure of an integrable coupling of the fractional ablowitz kaup newell segur hierarchy is derived the method can be generalized to other fractional integrable couplings

generalized hamilton s principle with fractional derivatives we generalize hamilton s principle with fractional derivatives in the lagrangian l t y t 0d ty t so that the function y and the order of the fractional derivative are varied in the minimization procedure we derive stationarity conditions and discuss them through several examples

discrete direct methods in the fractional calculus of variations finite differences as a subclass of direct methods in the calculus of variations consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem this article generalizes the same idea for fractional variational problems we consider a minimization problem with a lagrangian that depends on the left riemann liouville fractional derivative using the grunwald letnikov definition we approximate the objective functional in an equispaced grid as a multi variable function of the values of the unknown function on mesh points the problem is then transformed to an ordinary static optimization problem the solution to the latter problem gives an approximation to the original fractional problem on mesh points

mixed convolved action a stationary principle is developed for dynamical systems by formulating the concept of mixed convolved action which is written in terms of displacement and force variables using temporal convolutions and fractional derivatives the classical canonical single degree of freedom dynamical system is considered as an initial application with this new formulation a single real scalar functional provides the governing differential equations along with all the pertinent initial conditions as the euler lagrange equations emanating from the stationarity of the mixed convolved action both conservative and non conservative processes can be considered within a common framework thus resolving a long standing limitation of variational approaches for dynamical systems several results in fractional calculus also are developed

an efficient algorithm for image segmentation markov random fields and related problems problems of statistical inference involve the adjustment of sample observations so they fit some a priori rank requirements or order constraints in such problems the objective is to minimize the deviation cost function that depends on the distance between the observed value and the modify value in markov random field problems there is also a pairwise relationship between the objects the objective in markov random field problem is to minimize the sum of the deviation cost function and a penalty function that grows with the distance between the values of related pairs separation function we discuss markov random fields problems in the context of a representative application the image segmentation problem in this problem the goal is to modify color shades assigned to pixels of an image so that the penalty function consisting of one term due to the deviation from the initial color shade and a second term that penalizes differences in assigned values to neighboring pixels is minimized we present here an algorithm that solves the problem in polynomial time when the deviation function is convex and separation function is linear and in strongly polynomial time when the deviation cost function is linear quadratic or piecewise linear convex with few pieces where few means a number exponential in a polynomial function of the number of variables and constraints the complexity of the algorithm for a problem on n pixels or variables m adjacency relations or constraints and range of variable values colors u is o t n m n log u where t n m is the complexity of solving the minimum s t cut problem on a graph with n nodes and m arcs furthermore other algorithms are shown to solve the problem with convex deviation and convex separation in running time o mn log n log nu and the problem with nonconvex deviation and convex separation in running time o t nu mu the nonconvex separation problem is np hard even for fixed value of u for the family of problems with convex deviation functions and linear separation function the algorithm described here runs in polynomial time which is demonstrated to be fastest possible

the elastic ratio introducing curvature into ratio based image segmentation we present the first ratio based image segmentation method that allows imposing curvature regularity of the region boundary our approach is a generalization of the ratio framework pioneered by jermyn and ishikawa so as to allow penalty functions that take into account the local curvature of the curve the key idea is to cast the segmentation problem as one of finding cyclic paths of minimal ratio in a graph where each graph node represents a line segment among ratios whose discrete counterparts can be globally minimized with our approach we focus in particular on the elastic ratio 0l c i c s c s ds vl c 0l c c s qds that depends given an image i on the oriented boundary c of the segmented region candidate minimizing this ratio amounts to finding a curve neither small nor too curvy through which the brightness flux is maximal we prove the existence of minimizers for this criterion among continuous curves with mild regularity assumptions we also prove that the discrete minimizers provided by our graph based algorithm converge as the resolution increases to continuous minimizers in contrast to most existing segmentation methods with computable and meaningful i e nondegenerate global optima the proposed approach is fully unsupervised in the sense that it does not require any kind of user input such as seed nodes numerical experiments demonstrate that curvature regularity allows substantial improvement of the quality of segmentations furthermore our results allow drawing conclusions about global optima of a parameterization independent version of the snakes functional the proposed algorithm allows determining parameter values where the functional has a meaningful solution and simultaneously provides the corresponding global solution

the fractional boltzmann transport equation the fractional boltzmann transport equation is derived making use of the fractional hamilton s equations based on the fractional actionlike variational approach by simply defining a distribution function and inspecting its time derivative many important results in statistical physics can be derived

polynomial time algorithms for ratio regions and a variant of normalized cut in partitioning clustering and grouping problems a typical goal is to group together similar objects or pixels in the case of image processing at the same time another goal is to have each group distinctly dissimilar from the rest and possibly to have the group size fairly large these goals are often combined as a ratio optimization problem one example of such a problem is a variant of the normalized cut problem another is the ratio regions problem we devise here the first polynomial time algorithms solving optimally the ratio region problem and the variant of normalized cut as well as a few other ratio problems the algorithms are efficient and combinatorial in contrast with nonlinear continuous approaches used in the image segmentation literature which often employ spectral techniques such techniques deliver solutions in real numbers which are not feasible to the discrete partitioning problem furthermore these continuous approaches are computationally expensive compared to the algorithms proposed here the algorithms presented here use as a subroutine a minimum s t cut procedure on a related graph which is of polynomial size the output consists of the optimal solution to the respective ratio problem as well as a sequence of nested solutions with respect to any relative weighting of the objectives of the numerator and denominator

multiobjective fractional variational calculus in terms of a combined caputo derivative abstract the study of fractional variational problems in terms of a combined fractional caputo derivative is introduced necessary optimality conditions of euler lagrange type for the basic isoperimetric and lagrange variational problems are proved as well as transversality and sufficient optimality conditions this allows to obtain necessary and sufficient pareto optimality conditions for multiobjective fractional variational problems

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

fractional variational principles with delay the fractional variational principles within riemann liouville fractional derivatives in the presence of delay are analyzed the corresponding euler lagrange equations are obtained and one example is analyzed in detail

asymptotic integration of 1 α order fractional differential equations we establish the long time asymptotic formula of solutions to the 1 a order fractional differential equation 0 io t 1 ax a t x 0 t 0 under some simple restrictions on the functional coefficient a t where 0 io t 1 a is one of the fractional differential operators 0d t a x 0d t ax 0d t 1 ax and 0d t a tx x here 0d t a designates the riemann liouville derivative of order a 0 1 the asymptotic formula reads as b o 1 x s m a l l c x l a r g e as t for given b c r where x s m a l l and x l a r g e represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation 0 io t 1 ax 0 t 0

extending bauer s corollary to fractional derivatives we comment on the method of dreisigmeyer and young j phys a math gen 368297 to model nonconservative systems with fractional derivatives it was previously hoped that using fractional derivatives in an action would allow us to derive a single retarded equation of motion using a variational principle it is proven that under certain reasonable assumptions the method of dreisigmeyer and young fails

fractional hamiltonian analysis of higher order derivatives systems the fractional hamiltonian analysis of 1 1 dimensional field theory is investigated and the fractional ostrogradski s formulation is obtained the fractional path integral of both simple harmonic oscillator with an acceleration squares part and a damped oscillator are analyzed the classical results are obtained when fractional derivatives are replaced with the integer order derivatives

fractional newton mechanics with conformable fractional derivative in this paper we use the conformable fractional derivative and integral to discuss the fractional newtonian mechanics the fractional version of the calculus of variations is introduced and the fractional euler lagrange equation is constructed some mechanical problems such as fractional harmonic oscillator problem the fractional damped oscillator problem and the forced oscillator problem are discussed in the one dimensional fractional dynamics

generalized euler lagrange equations for fractional variational problems with free boundary conditions in this article we are going to present necessary conditions which must be satisfied to make the fractional variational problems fvps with completely free boundary conditions have an extremum the fractional derivatives are defined in the caputo sense first we present the necessary conditions for the problem with only one dependent variable and then we generalize them to problems with multiple dependent variables we also find the transversality conditions for when each end point lies on a given arbitrary curve in the case of a single variable or a surface in the case of multiple variables it is also shown that in special cases such as those with specified and unspecified boundary conditions and problems with integer order derivatives the new results reduce to the known necessary conditions some examples are presented to demonstrate the applicability of the new formulations

space time fractional schrodinger equation with time independent potentials abstract we develop a space time fractional schrodinger equation containing caputo fractional derivative and the quantum riesz fractional operator from a space fractional schrodinger equation in this paper by use of the new equation we study the time evolution behaviors of the space time fractional quantum system in the time independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively the space time fractional schrodinger equation with time independent potentials is divided into a space equation and a time one a general solution which is composed of oscillatory terms and decay ones is obtained we investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative but also on the sign positive or negative of the eigenvalues of the space equation we also find that the limit value of the total probability can be greater or less than one which means the space time fractional schrodinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials additionally the non markovian time evolution laws of the space time fractional quantum system are discussed the formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space time fractional quantum system we also get a mittag leffler type of time evolution operator of wave functions and then establish a heisenberg equation containing fractional operators

optimal attack and reinforcement of a network in a nonnegative edge weighted network the weight of an edge represents the effort required by an attacker to destroy the edge and the attacker derives a benefit for each new component created by destroying edges the attacker may want to minimize over subsets of edges the difference between or the ratio of the effort incurred and the benefit received this idea leads to the definition of the strength of the network a measure of the resistance of the network to such attacks efficient algorithms for the optimal attack problem the problem of computing the strength and the problem of finding a minimum cost reinforcement to achieve a desired strength are given these problems are also solved for a different model in which the attacker wants to separate vertices from a fixed central vertex

fractional variational calculus with classical and combined caputo derivatives abstract we give a proper fractional extension of the classical calculus of variations by considering variational functionals with a lagrangian depending on a combined caputo fractional derivative and the classical derivative euler lagrange equations to the basic and isoperimetric problems as well as transversality conditions are proved

regularized tree partitioning and its application to unsupervised image segmentation in this paper we propose regularized tree partitioning approaches we study normalized cut ncut and average cut acut criteria over a tree forming two approaches 1 normalized tree partitioning ntp and 2 average tree partitioning atp we give the properties that result in an efficient algorithm for ntp and atp in addition we present the relations between the solutions of ntp and atp over the maximum weight spanning tree of a graph and ncut and acut over this graph to demonstrate the effectiveness of the proposed approaches we show its application to image segmentation over the berkeley image segmentation data set and present qualitative and quantitative comparisons with state of the art methods

a direct approach to the construction of standard and non standard lagrangians for dissipative like dynamical systems with variable coefficients we present a direct approach to the construction of lagrangians for a large class of one dimensional dynamical systems with a simple dependence monomial or polynomial on the velocity we rederive and generalize some recent results and find lagrangian formulations which seem to be new some of the considered systems e g motions with the friction proportional to the velocity and to the square of the velocity admit infinite families of different explicit lagrangian formulations

solving integer programs over monotone inequalities in three variables a framework for half integrality and good approximations abstract we define a class of monotone integer programs with constraints that involve up to three variables each a generic constraint in such integer program is of the form ax by z c where a and b are nonnegative and the variable z appears only in that constraint we devise an algorithm solving such problems in time polynomial in the length of the input and the range of variables u the solution is obtained from a minimum cut on a graph with o nu nodes and o mu arcs where n is the number of variables of the types x and y and m is the number of constraints our algorithm is also valid for nonlinear objective functions nonmonotone integer programs are optimization problems with constraints of the type ax by z c without restriction on the signs of a and b such problems are in general np hard we devise here an algorithm relying on a transformation to the monotone case that delivers half integral superoptimal solutions in polynomial time such solutions provide bounds on the optimum value that can only be superior to bounds provided by linear programming relaxation when the half integral solution can be rounded to an integer feasible solution this is a 2 approximate solution in that the technique is a unified 2 approximation technique for a large class of problems the results apply also for general integer programming problems with worse approximation factors that depend on a quantifier measuring how far the problem is from the class of problems we describe the algorithm described here has a wide array of problem applications an additional important consequence of our results is that nonmonotone problems in the framework are max snp hard and at least as hard to approximate as vertex cover problems that are amenable to the analysis provided here are easily recognized the analysis itself is entirely technical and involves manipulating the constraints and transforming them to a totally unimodular system while losing no more than a factor of 2 in the integrality

standard lagrangians for wave laplace and tricomi like equations with variable coefficients general methods to derive lagrangians for first and second order linear partial differential equations with variable coefficients are developed the methods are used to obtain standard lagrangians for different physical problems that are described by wave laplace and tricomi like equations

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

image segmentation with ratio cut this paper proposes a new cost function cut ratio for segmenting images using graph based methods the cut ratio is defined as the ratio of the corresponding sums of two different weights of edges along the cut boundary and models the mean affinity between the segments separated by the boundary per unit boundary length this new cost function allows the image perimeter to be segmented guarantees that the segments produced by bipartitioning are connected and does not introduce a size shape smoothness or boundary length bias the latter allows it to produce segmentations where boundaries are aligned with image edges furthermore the cut ratio cost function allows efficient iterated region based segmentation as well as pixel based segmentation these properties may be useful for some image segmentation applications while the problem of finding a minimum ratio cut in an arbitrary graph is np hard one can find a minimum ratio cut in the connected planar graphs that arise during image segmentation in polynomial time while the cut ratio alone is not sufficient as a baseline method for image segmentation it forms a good basis for an extended method of image segmentation when combined with a small number of standard techniques we present an implemented algorithm for finding a minimum ratio cut prove its correctness discuss its application to image segmentation and present the results of segmenting a number of medical and natural images using our techniques

a general finite element formulation for fractional variational problems abstract this paper presents a general finite element formulation for a class of fractional variational problems fvps the fractional derivative is defined in the riemann liouville sense for fvps the euler lagrange and the transversality conditions are developed in the fractional finite element formulation ffef presented here the domain of the equations is divided into several elements and the functional is approximated in terms of nodal variables minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme three examples are considered to show the performance of the algorithm results show that as the number of discretization is increased the numerical solutions approach the analytical solutions and as the order of the derivative approaches an integer value the solution for the integer order system is recovered for unspecified boundary conditions the numerical solutions satisfy the transversality conditions this indicates that for the class of problems considered the numerical solutions can be obtained directly from the functional and there is no need to solve the fractional euler lagrange equations thus the formulation extends the traditional finite element approach to fvps

the hamilton formalism with fractional derivatives recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives in this paper the passage from the lagrangian containing fractional derivatives to the hamiltonian is achieved the hamilton s equations of motion are obtained in a similar manner to the usual mechanics in addition the classical fields with fractional derivatives are investigated using hamiltonian formalism two discrete problems and one continuous are considered to demonstrate the application of the formalism the results are obtained to be in exact agreement with agrawal s formalism

central limit theorem for the lorentz process via perturbation theory the markov partition of the sinai billiard allows the following heuristic interpretation for the lorentz process with a 2 periodic configuration of scatterers while executing a non markovian random walk on 2 and particle changes its internal state according to the symbolic dynamics defined by the markov partition this picture can be formalized and then the lorentz process appears as the limit of a sequence of markovian random walks with a finite but increasing number of internal states and the central limit theorem can be proved for it by perturbational expansions with uniformly bounded in a sence related to the perron frobenius theorem coefficients and uniform remainder terms

dynamics of a rod made of generalized kelvin voigt visco elastic material abstract in this paper we develop a new mathematical model for the lateral vibration of an axially compressed visco elastic rod as the basis for this model we use a fractional derivative type of stress strain relation we show that the dynamics of the lateral vibration is governed by two coupled linear differential equations with fractional derivatives for a special case of the generalized kelvin voigt body this system is reduced to a single fractional derivative differential equation eq 19 for a class of problems to which 19 belongs the questions of the existence of a solution and its regularity are analyzed both continuous and impulsive loading are treated

on a connection between a class of q deformed algebras and the hausdorff derivative in a medium with fractal metric abstract over the recent decades diverse formalisms have emerged that are adopted to approach complex systems amongst those we may quote the q calculus in tsallis version of non extensive statistics with its undeniable success whenever applied to a wide class of different systems kaniadakis approach based on the compatibility between relativity and thermodynamics fractional calculus fc that deals with the dynamics of anomalous transport and other natural phenomena and also some local versions of fc that claim to be able to study fractal and multifractal spaces and to describe dynamics in these spaces by means of fractional differential equations the question we might ask is whether or not there are common aspects that connect these alternative approaches in this short communication we discuss a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely tsallis framework and kaniadakis scenario with local form of fractional derivative operators defined in fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure this connection opens up new perspectives for theories that satisfactorily describe the dynamics for the transport in media with fractal metrics such as porous or granular media possible connections with other alternative definitions of fc are also contemplated insights on complexity connected to concepts like coarse grained space time and physics in general are pointed out

a computational study of the pseudoflow and push relabel algorithms for the maximum flow problem we present the results of a computational investigation of the pseudoflow and push relabel algorithms for the maximum flow and minimum s t cut problems the two algorithms were tested on several problem instances from the literature our results show that our implementation of the pseudoflow algorithm is faster than the best known implementation of push relabel on most of the problem instances within our computational study

discussing the extension and applications of a variational approach with deformed derivatives one has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives both in the lagrangian of systems of particles and in the lagrangian density of field theoretic models classical euler lagrange equations and the hamiltonian formalism have been reassessed in this approach whenever applied to a number of physical systems the resulting dynamical equations come out to be the correct ones found in the literature especially with mass dependent and with nonlinear equations for classical and quantum mechanical systems in the present contribution one extends the variational approach including a piecewise form of deformed derivatives to study higher order dissipative systems and to obtain as an option deformed equations as well applications to concrete situations are contemplated such as an accelerated point charge this is the problem of the abraham lorentz dirac force stochastic dynamics like the langevin the advection convection reaction and fokker planck equations the korteweg de vries equation the landau lifshitz gilbert problem and the caldirola kanai hamiltonian and heat transfer equation of the fourier and non fourier types by considering these different examples it is shown that the formulation proposed in this paper may be a simple but promising path for dealing for example with dissipative nonlinear stochastic systems and the anomalous heat transfer problem by adopting a variational approach one has recently presented an extension of the standard variational calculus to include the presence of deformed derivatives both in the lagrangian of systems of particles and in the lagrangian density of field theoretic models classical euler lagrange equations and the hamiltonian formalism have been reassessed in this approach whenever applied to a number of physical systems the resulting dynamical equations come out to be the correct ones found in the literature especially with mass dependent and with nonlinear equations for classical and quantum mechanical systems in the present contribution one extends the variational approach including a piecewise form of deformed derivatives to study higher order dissipative systems and to obtain as an option deformed equations as well applications to concrete situations are contemplated such as an accelerated point charge this is the problem of the abraham lorentz dirac force stochastic dynamics like the langevin the advection convection reaction and

variational approach and deformed derivatives recently we have demonstrated that there exists a possible relationship between q deformed algebras in two different contexts of statistical mechanics namely the tsallis framework and the kaniadakis scenario with a local form of fractional derivative operators for fractal media the so called hausdorff derivatives mapped into a continuous medium with a fractal measure here in this paper we present an extension of the traditional calculus of variations for systems containing deformed derivatives embedded into the lagrangian and the lagrangian densities for classical and field systems the results extend the classical euler lagrange equations and the hamiltonian formalism the resulting dynamical equations seem to be compatible with those found in the literature specially with mass dependent and with nonlinear equations for systems in classical and quantum mechanics examples are presented to illustrate applications of the formulation also the conserved noether current is worked out

fractional variational principle of herglotz for a new class of problems with dependence on the boundaries and a real parameter the fractional variational problem of herglotz type for the case where the lagrangian depends on generalized fractional derivatives the free endpoints conditions and a real parameter is studied this type of problem generalizes several problems recently studied in the literature moreover it allows us to unify conservative and non conservative dynamical processes in the same model the dependence of the lagrangian with respect to the boundaries and a free parameter is effective and transforms the standard herglotz s variational problem into a problem of a different nature

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

the problem of recurrence for lorentz processes we prove that for the planar lorentz process with a periodic configuration of scatterers the quasi local clt of the gaussian log n type holds for any 1 consequently for arbitrary 3 2 the probabilities that at the moment of thenth reflection this process lies in a square of size log n are asymptotically gaussian this implies that these events occur for infinitely many values ofn i e a weaker form of recurrence

a polynomial time algorithm for rayleigh ratio on discrete variables replacing spectral techniques for expander ratio normalized cut and cheeger constant a general form of minimizing the rayleigh ratio on discrete variables is shown here for the first time to be polynomial time solvable this is significant because major problems in clustering partitioning and imaging can be presented as the rayleigh ratio minimization on discrete variables and an orthogonality constraint these challenging problems are modeled as the normalized cut problem the graph expander ratio problem the cheeger constant problem or the conductance problem all of which are np hard these problems have traditionally been solved heuristically using the spectral technique a unified framework is provided here whereby all these problems are formulated as a constrained minimization form of a quadratic ratio referred to here as the rayleigh ratio the quadratic ratio is to be minimized on discrete variables and a single sum constraint that we call the balance or orthogonality constraint when the discreteness constraints on the variables are disregarded the resulting continuous relaxation is solved by the spectral method it is shown here that the rayleigh ratio minimization subject to the discreteness constraints requiring each variable to assume one of two values in b 1 is solvable in strongly polynomial time equivalent to a single minimum s t cut algorithm on a graph of same size as the input graph for any nonnegative value of b this discrete form for the rayleigh ratio problem was often assumed to be np hard not only is it shown here that the discrete rayleigh ratio problem is polynomial time solvable but also the algorithm is more efficient than the spectral algorithm furthermore an experimental study demonstrates that the new algorithm provides in practice an improvement often dramatic on the quality of the results of the spectral method both in terms of approximating the true optimum of the rayleigh ratio problem on both the discrete variables and the balance constraint and in terms of the subjective partition quality a further contribution here is the introduction of a problem the quantity normalized cut generalizing all the rayleigh ratio problems the discrete version of that problem is also solved with the efficient algorithm presented this problem is shown in a companion paper to enable the modeling of features essential to clustering that are valuable in practical applications

fractional actionlike variational problems fractional actionlike variational problems have recently gained importance in studying dynamics of nonconservative systems in this note we address multidimensional fractional actionlike problems of the calculus of variations

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

random walks with internal degrees of freedom between two absorbing barriers consider a random walk with a finite number of internal degrees of freedom and with zero drift by using a functional analytic approach based on the spectral theory of matrix polynomials the asymptotics of the first hitting probabilities is obtained when the distance of the barriers tends to infinity

numerical solution of fractional oscillator equation abstract we focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval this type of equation includes a complex form of left and right sided fractional derivatives its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations here we elaborated two numerical schemes being dependent on a fractional order of the equation the results of numerical calculations are compared with analytical solutions then we illustrate convergence and stability of our schemes

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

a generalized fractional kn equation hierarchy and its fractional hamiltonian structure a generalized hamiltonian structure of the fractional soliton equation hierarchy is presented by using differential forms and exterior derivatives of fractional orders we construct the generalized fractional trace identity through the riemann liouville fractional derivative an example of the fractional kn soliton equation hierarchy and hamiltonian structure is presented which is a new integrable hierarchy and possesses hamiltonian structure