gravity capillary waves in finite depth on flows of constant vorticity this paper considers two dimensional periodic gravity capillary waves propagating steadily in finite depth on a linear shear current constant vorticity a perturbation series solution for steady periodic waves accurate up to the third order is derived using a classical stokes expansion procedure which allows us to include surface tension effects in the analysis of wave current interactions in the presence of constant vorticity the analytical results are then compared with numerical computations with the full equations the main results are i the phase velocity is strongly dependent on the value of the vorticity ii the singularities wilton singularities in the stokes expansion in powers of wave amplitude that correspond to a bond number of 1 2 and 1 3 which are the consequences of the non uniformity in the ordering of the fourier coefficients are found to be influenced by vorticity iii different surface profiles of capillary gravity waves are computed and the effect of vorticity on those profiles is shown to be important in particular that the solutions exhibit type 2 like wave features characterized by a secondary maximum on the surface profile with a trough between the two maxima

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

global bifurcation of steady gravity water waves with critical layers we construct families of two dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed in particular establishing the existence of waves of large amplitude a riemann hilbert problem approach is used to recast the governing equations as a one dimensional elliptic pseudodifferential equation with a scalar constraint the structural properties of this formulation which arises as the euler lagrange equation of an energy functional enable us to develop a theory of analytic global bifurcation

analyticity of periodic traveling free surface water waves with vorticity we prove that the prole of a periodic traveling wave propagating at the surface of water above a at bed in a ow with a real analytic vorticity must be real analytic provided the wave speed exceeds the horizontal uid velocity throughout the ow the real analyticity of each streamline be

noether s theorem of fractional birkhoffian systems abstract in this paper we study noether type symmetry theorem to fractional birkhoffian systems with riemann liouville derivatives this theorem provides an explicit algorithmic way to compute a constant for any birkhoffian systems admitting a symmetry finally we extend our noether s theorem to fractional birkhoffian systems base on caputo or riesz derivatives

fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

a bliss type multiplier rule for constrained variational problems with time delay abstract this paper gives a bliss type multiplier rule for general constrained variational problems with time delay our formulation allows the time delay terms and their derivatives to be present in both the objective functional and the equality and inequality constraint equations our major results include a lagrange multiplier rule involving euler lagrange equations for delay differential equations and transversality conditions of special note is that these results will be used in later work by the authors to obtain new analytical techniques to solve general constrained problems involving delay differential equations for the calculus of variations and for optimal control theory they will also lead to new general accurate and efficient numerical methods to solve these important problems where no such methods exist at this time thus for the first time we will be able to solve this important class of problems obtaining analytic solutions for simpler problems and accurate numerical results for more complex problems when such a solution exists

steady periodic water waves with constant vorticity regularity and local bifurcation this paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed using conformal mappings the free boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable the new formulation leads to a regularity result and by use of bifurcation theory to the existence of waves of small amplitude even in the presence of stagnation points in the flow

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

fractional integral and differential equations for a class of levy type probability densities an application of the fractional calculus to a class of levy distribution functions leads to the conclusion that the levy index fractal dimension mu is identical to the order of the fractional liouville riemann integral operator the corresponding fractional integral and differential equations are presented and solutions of levy type one sided probability densities are given and discussed

fractional radiative transfer equation within chebyshev spectral approach in this work we report the convergence of the chebyshev polynomials combined with the s n method for the steady state transport equation using the fractional derivative the procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a system of fractional differential equations the convergence of this approach is studied in the context of the multidimensional discrete ordinates equations

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

range characterization of the cosine transform on higher grassmannians abstract we characterize the range of the cosine transform on real grassmannians in terms of the decomposition under the action of the special orthogonal group so n we also give a geometric interpretation of this image in terms of valuations in addition we discuss the non archimedean analogues

on some degenerate principal series representations of u n n we study the infinitesimal action of u n n on the degenerate principal series representations of u n n associated with a parabolic subgroup which stabilizes a maximal isotropic subspace of c2n relative to the indefinite hermitian form with signature n n this allows us in particular to determine the reducibility socle series and unitarity of these representations

bessel operators on jordan pairs and small representations of semisimple lie groups abstract we provide a uniform construction of l 2 models for all small unitary representations in degenerate principal series of semisimple lie groups which are induced from maximal parabolic subgroups with abelian nilradical this generalizes previous constructions to the case of a maximal parabolic subgroup which is not necessarily conjugate to its opposite and hence the previously used jordan algebra methods have to be generalized to jordan pairs the crucial ingredients for the construction of the l 2 models are the lie algebra action and the spherical vector working in the so called fourier transformed picture of the degenerate principal series the lie algebra action is given in terms of bessel operators on jordan pairs we prove that precisely for those parameters of the principal series where small quotients occur the bessel operators are tangential to certain submanifolds further we show that the small quotients are unitarizable if and only if these submanifolds carry equivariant measures in this case we can express the spherical vectors in terms of multivariable k bessel functions and some delicate estimates for these bessel functions imply the existence of the l 2 models

intertwining operators for the generalized principal series on a symmetric r space three questions about the intertwining operators for the generalized principal series on a symmetric r space are solved description of the functional kernel both in the noncompact and in the compact picture domain of convergence meromorphic continuation a large use is made of the theory of positive jordan triple systems the meromorphic continuation of the intertwining integral is achieved via a bernstein sato identity and a precise description of the poles is obtained 0 2000 mathematics subject classification 22e45 43a80

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

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

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

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

explicit hilbert spaces for certain unipotent representations iii abstract in this paper we construct a family of small unitary representations for real semisimple lie groups associated with jordan algebras these representations are realized on l2 spaces of certain orbits in the jordan algebra the representations are spherical and one of our key results is a precise l2 estimate for the fourier transform of the spherical vector we also consider the tensor products of these representations and describe their decomposition

eigenvalue problems of the model from nonlocal continuum mechanics this article studies the eigenvalue problem of a fractional differential equation which is a foundation model of a bar of finite length with long range interactions arising from non local continuum mechanics we show that this problem has countable simple real eigenvalues and the corresponding eigenfunctions form a complete orthogonal system in the hilbert space l2 furthermore the asymptotic behavior of eigenvalues and the numbers of zeros of eigenfunctions are studied by using the analytic perturbation theory

symmetry of steady periodic gravity water waves with vorticity we prove that steady periodic two dimensional rotational gravity water waves with a monotone surface profile between troughs and crests have to be symmetric about the crest irrespective of the vorticity distribution within the fluid

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

recovering the water wave profile from pressure measurements a new method is proposed to recover the water wave surface elevation from pressure data obtained at the bottom of the fluid the new method requires the numerical solution of a nonlocal nonlinear equation relating the pressure and the surface elevation which is obtained from the euler formulation of the water wave problem without approximation from this new equation a variety of different asymptotic formulas are derived the nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments the solvability properties of the nonlocal equation are rigorously analyzed using the implicit function theorem

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

inversion of fractional integrals related to the spherical radon transform explicit inversion formulas are obtained for the analytic family of fractional integrals t f x n sn xy 1 f y dyon the unit sphere in rn 1 arbitrary complex andn 2 are considered in the ease 0 the integralt fcoincides with the spherical radon transform for 1 1 3 5 such integrals are known as the blaschke levy representations and arise in convex geometry probability and the banach space theory for 1 3 5 the integralt fis defined by continuity as the spherical convolution with the power logarithmic kernel different inversion methods are discussed

explicit hilbert spaces for certain unipotent representations ii letg be the universal cover of the group of automorphisms of a symmetric tube domain and letp ln be its shilov boundary parabolic subgroup this paper attaches an irreducible unitary representation ofg to each of the finitely many l orbits onn the hilbert space of the representation consists of functions on the orbit which are square integrable with respect to a certainl equivariant measure the representation remains irreducible when restricted top and descends to a quotient ofg which is at worst thedouble cover of a linear group if thel orbit isnot open inn the construction gives a unipotent representation ofg

on the degenerate principal series of complex symplectic groups we apply techniques introduced by clerc kobayashi orsted and pevzner to study the degenerate principal series of sp n c an explicit description of the k types is provided and knapp stein normalised operators are realised as symplectic fourier transforms and their k spectrum explicitly computed reducibility phenomena are analysed in terms of k types and eigenvalues of intertwining operators we also construct a new model for these representations in which knapp stein intertwiners take an algebraic form

noether symmetries and conserved quantities for fractional forced birkhoffian systems abstract in this paper noether symmetries and conserved quantities for fractional forced birkhoffian systems are studied firstly a new fractional pfaff birkhoff variational principle with riemann liouville derivatives and generalized force is established and from which the fractional forced birkhoff equations are derived secondly the definitions and the criteria of the invariance of a functional under an e parameter group of infinitesimal transformations are given finally the fractional noether theorems for forced birkhoffian systems are obtained and an illustrative example is presented

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

on the meromorphic extension of the spherical functions on noncompactly causal symmetric spaces abstract we determine integral formulas for the meromorphic extension in the parameter of the spherical functions on a noncompactly causal symmetric space the main tool is bernstein s theorem on the meromorphic extension of complex powers of polynomials the regularity properties of are deduced in particular the possible poles of are located among the translates of the zeros of the bernstein polynomial the translation parameter depends only on the structure of the symmetric space the expression of the bernstein polynomial is conjectured the relation between the bernstein polynomial and the product formula of the c function is analyzed the conjecture is verified in the rank one case the explicit formulas obtained in this case yield a detailed description of singularities of in the general higher rank case the integral formulas are applied to find asymptotic estimates for the spherical functions in the appendix the spherical functions on noncompactly causal symmetric spaces are regarded as a special instance of harish chandra type expansions associated with roots systems with arbitrary multiplicities we study expansions obtained by taking averages over arbitrary parabolic subgroups of the weyl group of the root system the possible singularities are located in this general context

analysis on the minimal representation of o p q abstract for the group o p q we give a new construction of its minimal unitary representation via euclidean fourier analysis this is an extension of the q 2 case where the representation is the mass zero spin zero representation realized in a hilbert space of solutions to the wave equation the group o p q acts as the mobius group of conformal transformations on r p 1 q 1 and preserves a space of solutions of the ultrahyperbolic laplace equation on r p 1 q 1 we construct in an intrinsic and natural way a hilbert space of solutions so that o p q becomes a continuous irreducible unitary representation in this hilbert space we also prove that this representation is unitarily equivalent to the representation on l2 c where c is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the lie algebra of o p q

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

processes of flats induced by higher dimensional processes stationary processes of k flats in mathbb e d can be thought of as point processes on the grassmannian mathcal l kd of k dimensional subspaces of mathbb e d if such a process is sampled by a d k j dimensional space f it induces a process of j flats in f in this work we will investigate the possibility of determining the original k process from knowledge of the intensity measures of the induced j processes we will see that this is impossible precisely when 1 k d 1 and j 0 2 r 2 1 where r is the rank of the manifold mathcal l kd we will show how the problem is equivalent to the study of the kernel of various integral transforms these will then be investigated using harmonic analysis on grassmannian manifolds

a constructive approach to tensor product decompositions soit g r un groupe de lie semi simple non compact a centre fini et on suppose k un sous groupe compact maximal de g r soient g r et k les algebres de lie respectives de g r et k avec g r x p la decomposition de cartan de g r soit la representation induite de k sur p et soit k gl v une representation ineductible on etudie la decomposition produit tensoriel de on construit les operateurs differentiels g invariants on obtient des formules de clebsch gordan pour les differents groupes classiques

covariants of spn c and degenerate principal series of gln h let m n and let p m2n m slm be the algebra of all slm c invariant polynomials on the space m2n m of all 2n m complex matrices we determine the free generators of its subalgebra of spn c covariants these generators are given as pfaffians they are used to study the degenerate principal series representations of gln h induced from characters of a maximal parabolic subgroup we determine the composition series and unitarity of these representations

the complex busemann petty problem on sections of convex bodies abstract the complex busemann petty problem asks whether origin symmetric convex bodies in c n with smaller central hyperplane sections necessarily have smaller volume we prove that the answer is affirmative if n 3 and negative if n 4

existence and uniqueness theorem for a class of delay differential equations with left and right caputo fractional derivatives the existence and uniqueness theorems for functional right left delay and left right advanced fractional functional differential equations with bounded delay and advance respectively are proved the continuity with respect to the initial function for these equations is also proved under some lipschitz kind conditions the q operator is used to transform the delay type equation to an advanced one and vice versa an example is given to clarify the results

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

eigenvalue problems for fractional differential equations with right and left fractional derivatives this paper studies the eigenvalue problem of a class of fractional differential equations with right and left fractional derivatives with the aid of the spectral theory of compact self adjoint operators in hilbert spaces we show that the spectrum of this problem consists of only countable real eigenvalues with finite multiplicity and the corresponding eigenfunctions form a complete orthogonal system furthermore the lower bound of the eigenvalues is obtained

fractional curve flows and solitonic hierarchies in gravity and geometric mechanics methods from the geometry of nonholonomic manifolds and lagrange finsler spaces are applied in fractional calculus with caputo derivatives and for elaborating models of fractional gravity and fractional lagrange mechanics the geometric data for such models are encoded into fractional bi hamiltonian structures and associated solitonic hierarchies the constructions yield horizontal vertical pairs of fractional vector sine gordon equations and fractional vector mkdv equations when the hierarchies for corresponding curve fractional flows are described in explicit forms by fractional wave maps and analogs of schrodinger maps

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

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

fourier and poisson transformation associated to a semisimple symmetric space soit g un groupe de lie semisimple connexe contenu dans un groupe de lie semisimple simplement connexe dont l algebre de lie est la complexification de celle de g soit une involution non triviale de g et une involution de cartan commutant avec soit h un groupe point fixe de on etudie les representations h spheriques de dimension finie de g contenant un vecteur k fixe ou k est un sous groupe compact maximal stable de g sous representations de dimension finie de le noyau de poisson la distribution u i la transformation de fourier et poisson formules integrales

the trajectories of particles in stokes waves analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed within a period each particle experiences a backward forward motion with a slight forward drift

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

generalizations of the busemann petty problem for sections of convex bodies abstract we present generalizations of the busemann petty problem for dual volumes of intermediate central sections of symmetric convex bodies it is proved that the answer is negative when the dimension of the sections is greater than or equal to 4 for two three dimensional sections both negative and positive answers are given depending on the orders of dual volumes involved and certain cases remain open for bodies of revolution a complete solution is obtained in all dimensions

spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup abstract we present a new method of calculating intertwining operators between principal series representations of semisimple lie groups g working in the compact realization we find the eigenvalues of the operators on the k types and give several examples among the advantages of our method is its applicability to bundle valued cases

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

canonical representations related to hyperbolic spaces abstract we extend the notion of canonical representation introduced by a m vershik i m gel fand and m i graev 1982 in representation theory cambridge univ press cambridge u k to all classical rank one semisimple lie groups we determine the spectral decomposition of the canonical representations in detail

structure of the degenerate principal series on symmetric r spaces and small representations abstract let g be a simple real lie group with maximal parabolic subgroup p whose nilradical is abelian then x g p is called a symmetric r space we study the degenerate principal series representations of g on c x in the case where p is not conjugate to its opposite parabolic we find the points of reducibility the composition series and all unitarizable constituents among the unitarizable constituents we identify some small representations having as associated variety the minimal nilpotent k c orbit in p c where k c is the complexification of a maximal compact subgroup k g and g k p the corresponding cartan decomposition

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

a penalization method for calculating the flow beneath traveling water waves of large amplitude we suggest an algorithm for the calculation of large amplitude traveling water waves in irrotational flows and in flows with constant and nonconstant vorticity respectively the algorithm is based on a quite general partial differential equation constrained optimization formulation maximizing the wave amplitude subject to the pde constraint that the euler equations hold numerical examples illustrate the performance of the algorithm for irrotational flows and in flows with constant vorticity where the results are compared with the literature as a proof of principle we also consider a test case with a nonconstant vorticity

exact steady periodic water waves with vorticity we consider the classical water wave problem described by the euler equations with a free surface under the influence of gravity over a flat bottom we construct two dimensional inviscid periodic traveling waves with vorticity they are symmetric waves whose profiles are monotone between each crest and trough we use bifurcation and degree theory to construct a global connected set of such solutions c 2004 wiley periodicals inc

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

on a nonlinear distributed order fractional differential equation abstract we study the existence and the uniqueness of mild and classical solutions for a class of equations of the form y 2 t 0 2 y t d f y t such equations arise in distributed derivatives models of viscoelasticity and system identification theory we also formulate a variational principle for a more general equation based on a method of doubling of variables for such equations

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

on the asymptotic integration of a class of sublinear fractional differential equations we estimate the growth in time of the solutions to a class of nonlinear fractional differential equations d0 x x0 f t x which includes d0 x x0 h t x with 0 1 for the case of slowly decaying coefficients h the proof is based on the triple interpolation inequality on the real line and the growth estimate reads as x t o ta when t for 1 1 a 0 our result can be thought of as a noninteger counterpart of the classical bihari asymptotic integration result for nonlinear ordinary differential equations by a carefully designed example we show that in some circumstances such an estimate is optimal

degenerate principal series representations of gln c and gln r let f be the field c or r for 1 k n 2 let pk be the maximal parabolic subgroup of gln f with its levi factor isomorphic to gln k f glk f in this paper we explicitly calculate the action of the enveloping algebra of gln f on the degenerate principal series representations induced from characters of pk we address composition series k types and unitarity