global regularity for a class of 2d generalized tropical climate models abstract this paper establishes the global existence and regularity of solutions to a two dimensional 2d tropical climate model tcm with fractional dissipation the inviscid counterpart of this model was derived by frierson majda and pauluis 8 as a model for tropical geophysical flows this model reflects the interaction and coupling among the barotropic mode u the first baroclinic mode v of the velocity and the temperature the systems with fractional dissipation studied here may arise in the modeling of geophysical circumstances mathematically these systems allow simultaneous examination of a family of systems with various levels of regularization the aim here is the global regularity with the least dissipation we prove two main results first the global regularity of the system with v and for 1 and 3 2 and second the global regularity of the system with v for 3 2 the proofs of these results are not trivial and the requirements on the fractional indices appear to be optimal the key tools employed here include the maximal regularity for general fractional heat operators the littlewood paley decomposition and besov space techniques lower bounds involving fractional laplacian and simultaneous estimates of several coupled quantities

stancu schurer kantorovich operators based on q integers the goal of this paper is to introduce and study q analogue of stancu schurer kantorovich operators a convergence theorem using the well known bohman korovkin criterion is proven and the rate of convergence involving the modulus of continuity is established the estimate of the rate of convergence by means of the lipshitz function is considered furthermore we obtained a voronovskaja type result for these operators also we investigate the statistical approximation properties of these operators using korovkin type statistical approximation theorem

initial boundary value problem for two dimensional viscous boussinesq equations we study the initial boundary value problem of two dimensional viscous boussinesq equations over a bounded domain with smooth boundary we show that the equations have a unique classical solution for h3 initial data and the no slip boundary condition in addition we show that the kinetic energy is uniformly bounded in time

stancu kantorovich operators based on inverse polya eggenberger distribution the purpose of this paper is to investigate approximation properties of stancu kantorovich operators based on inverse polya eggenberger distribution for these new operators we establish some approximation properties including uniform convergence asymptotic formula and degree of approximation

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

global well posedness for a boussinesq navier stokes system with critical dissipation abstract in this paper we study a fractional diffusion boussinesq model which couples a navier stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature we establish global well posedness results with rough initial data

on dual pairs in dirac geometry in this note we discuss weak dual pairs in dirac geometry we show that this notion appears naturally when studying the problem of pushing forward a dirac structure along a surjective submersion and we prove a dirac theoretic version of libermann s theorem from poisson geometry our main result is an explicit construction of self dual pairs for dirac structures this theorem not only recovers the global construction of symplectic realizations from crainic and m rcu j symplectic geom 9 4 435 444 2011 but allows for a more conceptual understanding of it yielding a simpler and more natural proof as an application of the main theorem we present a different approach to the recent normal form theorem around dirac transversals from bursztyn et al j fur die reine und angewandte mathematik crelles j doi 10 1515 crelle 2017 0014 2017

the 2d boussinesq equations with logarithmically supercritical velocities abstract this paper investigates the global in time regularity of solutions to a system of equations that generalize the vorticity formulation of the 2d boussinesq navier stokes equations the velocity u in this system is related to the vorticity through the relations u and log i which reduces to the standard velocity vorticity relation when 0 when either 0 or 0 the velocity u is more singular the quasi velocity v determined by v satisfies an equation of very special structure this paper establishes the global regularity and uniqueness of solutions for the case when 0 and 0 in addition the vorticity is shown to be globally bounded in several functional settings such as l 2 for 0 in a suitable range

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

remarks on global regularity of the 2d boussinesq equations with fractional dissipation abstract in this paper we are interested in the study of the cauchy problem to the two dimensional 2d incompressible boussinesq equations with fractional dissipation by making use of the nonlinear lower bounds for the fractional laplacian established in constantin and vicol 2012 we establish the global regularity of the smooth solutions of the 2d boussinesq equations with a new range of fractional powers of the laplacian this result significantly improves the recent works of constantin and vicol 2012 and yang et al 2014

global well posedness issues for the inviscid boussinesq system with yudovich s type data the present paper is dedicated to the study of the global existence for the inviscid two dimensional boussinesq system we focus on finite energy data with bounded vorticity and we find out that under quite a natural additional assumption on the initial temperature there exists a global unique solution no smallness conditions are imposed on the data the global existence issues for infinite energy initial velocity and for the benard system are also discussed

global well posedness for the 2d boussinesq equations with partial temperature dependent dissipative terms abstract in this paper we obtain the global existence and uniqueness results for the 2d boussinesq equations with mixed partial temperature dependent viscosity and thermal diffusivity because of the deficiency of the dissipation in some directions there is not enough smooth effect in those directions which makes the problem of regularity become more difficult in order to overcome these difficulties we construct some crucial inequalities and the global estimates for the higher fractional derivatives of temperature in the necessary directions

global well posedness for the 2d inviscid benard system with fractional diffusivity and yudovichʼs type data abstract in this paper we consider the cauchy problem of the two dimensional inviscid benard system with fractional diffusivity we show that there is a global unique solution to this system with yudovich s type data

generalized 2d euler boussinesq equations with a singular velocity abstract this paper studies the global in time regularity problem concerning a system of equations generalizing the two dimensional incompressible boussinesq equations the velocity here is determined by the vorticity through a more singular relation than the standard biot savart law and involves a fourier multiplier operator the temperature equation has a dissipative term given by the fractional laplacian operator we establish the global existence and uniqueness of solutions to the initial value problem of this generalized boussinesq equations when the velocity is double logarithmically more singular than the one given by the biot savart law this global regularity result goes beyond the critical case in addition we recover a result of chae constantin and wu 8 when the initial temperature is set to zero

global regularity of solutions of 2d boussinesq equations with fractional diffusion abstract the goal of this work is to study the boussinesq equations for an incompressible fluid in r 2 with diffusion modeled by fractional laplacian the existence the uniqueness and the regularity of solution has been proved

global regularity results for the 2d boussinesq equations and micropolar equations with partial dissipation abstract this paper establishes the global regularity of the two dimensional 2d boussinesq equations and micropolar equations with partial dissipation our first result is the global regularity of the 2d boussinesq equations with fractional vertical dissipation in the horizontal velocity horizontal dissipation in the vertical velocity and zero thermal diffusion which is shown by taking advantage of the nice structure of the 2d boussinesq equations and several refined commutator estimates the second goal of this paper is to consider a system of the 2d incompressible micropolar equations with vertical dissipation in the horizontal velocity equation horizontal dissipation in the vertical velocity equation and the fractional dissipation in the micro rotation velocity in order to overcome the difficulty caused by the lack of full laplacian diffusion in the velocity equations we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small 0 always possesses a unique global classical solution

banach lie poisson spaces and reduction the category of banach lie poisson spaces is introduced and studied it is shown that the category of w algebras can be considered as one of its subcategories examples and applications of banach lie poisson spaces to quantization and integration of hamiltonian systems are given the relationship between classical and quantum reduction is discussed

small global solutions to the damped two dimensional boussinesq equations abstract the two dimensional 2d incompressible euler equations have been thoroughly investigated and the resolution of the global in time existence and uniqueness issue is currently in a satisfactory status in contrast the global regularity problem concerning the 2d inviscid boussinesq equations remains widely open in an attempt to understand this problem we examine the damped 2d boussinesq equations and study how damping affects the regularity of solutions since the damping effect is insufficient in overcoming the difficulty due to the vortex stretching we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption by positioning the solutions in a suitable functional setting more precisely the homogeneous besov space b 1 1 we are able to obtain a unique global solution under a minimal smallness assumption

on the profiles of locally self similar solutions for the 2d inviscid boussinesq equations abstract in this paper we investigate the locally self similar solutions of the 2d inviscid boussinesq equations and mainly focus on the possible velocity and temperature blowup profiles we derive the formula of the pressure profiles in terms of the velocity and temperature profiles in the case that the possible nontrivial profiles have non decaying spatial asymptotics then using the local energy inequality of the profiles and iterative method we prove some nonexistence results and obtain the energy behavior concerning these possible profiles

szasz durrmeyer type operators based on charlier polynomials in the present paper we introduce a durrmeyer type modification of szasz operators based on charlier polynomials and study a voronovskaja type asymptotic theorem local approximation theorem weighted approximation statistical convergence and approximation of functions with derivatives of bounded variation for these operators

statistical approximation of certain positive linear operators constructed by means of the chan chyan srivastava polynomials in this study by obtaining some korovkin type approximation results in statistical sense for certain positive linear operators constructed by means of the chan chyan srivastava multivariable polynomials w c c chan c j chyan h m srivastava the lagrange polynomials in several variables integral transform spec funct 12 2001 139 148 we show that our approximation method is stronger than the corresponding classical aspects in the approximation theory settings furthermore we investigate their statistical rates by means of the modulus of continuity and the elements of the lipschitz class

global regularity results for the 2d boussinesq equations with vertical dissipation abstract the two dimensional 2d incompressible boussinesq equations model geophysical fluids and play an important role in the study of the raleigh bernard convection mathematically this 2d system retains some key features of the 3d navier stokes and euler equations such as the vortex stretching mechanism the issue of whether the 2d boussinesq equations always possess global in time classical solutions can be difficult when there is only partial dissipation or no dissipation at all this paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases these results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2d boussinesq equations

global well posedness of the 2d boussinesq equations with vertical dissipation we prove the global well posedness of the two dimensional boussinesq equations with only vertical dissipation the initial data u 0 theta 0 are required to be only in the space x f in l 2 mathbb r 2 partial x f in l 2 mathbb r 2 and thus our result generalizes that of cao and wu arch rational mech anal 208 985 1004 2013 where the initial data are assumed to be in h 2 mathbb r 2 the assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions a logarithmic type limiting sobolev embedding inequality for the l infty mathbb r 2 norm in terms of anisotropic sobolev norms and a logarithmic type gronwall inequality are established to obtain the global in time a priori estimates which guarantee the local solution to be a global one

the 2d regularized incompressible boussinesq equations with general critical dissipations abstract considering the 2d regularized boussinesq equations with fractional dissipations u and convection terms u u u where and 0 we prove the global existence and uniqueness of the solution in two critical cases the first case has fractional dissipations u where 1 0 and the second one has particular dissipation 1 u 0 in particular for the case 0 we give some decay estimates for u and the uniform estimate for g independent of time where g 1 u 2 2 u 1 1

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

a kantorovich variant of a new type bernstein stancu polynomials abstract a new generalization of bernstein stancu kantorovich type polynomials are constructed and the theorems on convergence and the degree of convergence are established in addition r th generalization is given

input output decoupling of hamiltonian systems the linear case in this note we give necessary and sufficient conditions for a linear hamiltonian system to be input output decouplable by hamiltonian feedback i e feedback that preserves the hamiltonian structure in a second paper we treat the same problem for nonlinear hamiltonian systems

poisson integrators for lie poisson structures on mathbb r 3 this paper is concerned with the study of poisson integrators we are interested in lie poisson systems on first we focus on poisson integrators for constant poisson systems and the transformations used for transforming lie poisson structures to constant poisson structures then we construct local poisson integrators for lie poisson systems on finally we present the results of numerical experiments for two lie poisson systems and compare our poisson integrators with other known methods

blow up criterion of smooth solutions for the boussinesq equations abstract this paper is dedicated to the study the boussinesq equations with fractional dissipation in n dimensions n 2 we obtain a blow up criterion for the local in time classical solution to the boussinesq equations with arbitrarily small fractional powers of the laplacian therefore our results significantly improve the previous works and are more closer to the resolution of the well known regularity criterion issue on the inviscid boussinesq equations

global regularity for the two dimensional anisotropic boussinesq equations with vertical dissipation this paper establishes the global in time existence of classical solutions to the two dimensional anisotropic boussinesq equations with vertical dissipation when only vertical dissipation is present there is no direct control on the horizontal derivatives and the global regularity problem is very challenging to solve this problem we bound the derivatives in terms of the l infty norm of the vertical velocity v and prove that v l r with 2 leqq r infty does not grow faster than sqrt r log r at any time as r increases a delicate interpolation inequality connecting v l infty and v l r then yields the desired global regularity

well posedness for density dependent boussinesq equations without dissipation terms in besov spaces abstract in this paper we consider the n dimensional incompressible density dependent boussinesq equations without dissipation terms n 2 we establish the local well posedness for the incompressible boussinesq system under the framework of the besov spaces in addition we also obtain a beale kato majda type regularity criterion

hamiltonian formalism for general relativistic adiabatic fluids abstract we derive the hamiltonian structures of three theories non relativistic special relativistic and general relativistic adiabatic fluids each in the eulerian representation in riemannian space or lorentzian spacetime all by the same procedure using standard variational principles the evolution in each case is generated by a hamiltonian that is equivalent to that obtained from a canonical analysis for the gravitational variables the poisson bracket has the usual canonical symplectic structure however for the fluid variables the three theories all share the same lie poisson bracket when expressed in the appropriate spaces of physical variables constructed here this shared lie poisson bracket is associated to the dual of the semidirect product lie algebra of vector fields acting on differential forms an immediate consequence of this shared structure is that each of these theories possesses an infinite family of conservation laws the so called casimirs that belong to the kernel of the lie poisson bracket the role of these casimirs in the study of lyapunov stability or dynamic stability for fluid equilibria is discussed the relationship of this approach to other approaches in the literature is also discussed

lie algebroid foliations and script e1 m dirac structures we prove some general results about the relation between the 1 cocycles of an arbitrary lie algebroid a over m and the leaves of the lie algebroid foliation on m associated with a using these results we show that a 1 m dirac structure l induces on every leaf f of its characteristic foliation a 1 f dirac structure lf which comes from a precontact structure or from a locally conformal presymplectic structure on f in addition we prove that a dirac structure on m can be obtained from l and we discuss the relation between the leaves of the characteristic foliations of l and

a generalization of bernstein kantorovic operators abstract in this paper we use a probabilistic setting to introduce a double sequence l k n of linear polynomial operators which includes as particular cases the classical bernstein operators the kantorovic operators and the operators recently introduced by cao for these operators we discuss several approximation properties in particular we deal with the convergence properties according to the way in which the different parameters vary and the preservation of global smoothness and classes of functions determined by concave moduli of continuity a remarkable feature of our approach is that if f is differentiable the approximation properties of both l k nf and its derivatives can be discussed simultaneously throughout the paper probabilistic methods play an important role

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

poisson structures and integrable systems a poisson coalgebra is used to construct integrable hamiltonian systems we consider a poisson structure given by the bivector which does not form a lie algebra with respect to the poisson bracket xa xb p pab x we prove that this coalgebra may be used to generate integrable hamiltonian systems as an example we give the poisson tensor and we show that it is linked with the calogero system

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

higher order regularity and blow up criterion for semi dissipative and ideal boussinesq equations in this paper we establish local in time existence and uniqueness of strong solutions in hs for s n2 to the viscous zero thermal diffusive boussinesq equations in rn n 2 3 beale kato majda type blow up criterion has been established in three dimensions with respect to the bmo norm of the vorticity we further prove the local in time existence for nonviscous and fully ideal boussinesq systems in rn n 2 3 moreover we establish blow up criterion for nonviscous boussinesq system in three dimensions and for fully ideal boussinesq system in both two and three dimensions commutator estimates from the work of kato and ponce comm pure appl math 41 891 1988 and fefferman et al j funct anal 267 1035 2014 play important roles in the calculations

two families of bernstein durrmeyer type operators this paper deals with two families of bernstein durrmeyer type operators that arise as integral modifications respectively from the classical bernstein operators and from bernstein type operators based on the polya distribution an important role is played by the so called genuine elements of each class the only ones that reproduce linear functions asymptotic formulae and direct convergence results are stated finally some modifications of the sequences of the families are introduced in such a way that the resulting operators reproduce linear functions and are comparable with the genuine sequences

yudovich type solution for the 2d inviscid boussinesq system with critical and supercritical dissipation abstract in this paper we consider the yudovich type solution of the 2d inviscid boussinesq system with critical and supercritical dissipation for the critical case we show that the system admits a global and unique yudovich type solution for the supercritical case we prove the local and unique existence of yudovich type solution and the global result under a smallness condition of 0 we also give a refined blowup criterion in the supercritical case

global well posedness for the 2d boussinesq system with anisotropic viscosity and without heat diffusion abstract we establish global existence and uniqueness theorems for the two dimensional non diffusive boussinesq system with anisotropic viscosity acting only in the horizontal direction which arises in ocean dynamics models global well posedness for this system was proven by danchin and paicu however an additional smoothness assumption on the initial density was needed to prove uniqueness they stated that it is not clear whether uniqueness holds without this additional assumption the present work resolves this question and we establish uniqueness without this additional assumption furthermore the proof provided here is more elementary we use only tools available in the standard theory of sobolev spaces and without resorting to para product calculus we use a new approach by defining an auxiliary stream function associated with the density analogous to the stream function associated with the vorticity in 2d incompressible euler equations then we adapt some of the ideas of yudovich for proving uniqueness for 2d euler equations

regularity criterion of the 2d benard equations with critical and supercritical dissipation abstract in this paper we investigate the cauchy problem for the two dimensional 2d incompressible benard equations on the one hand we prove global in time existence of smooth solutions to the 2d benard equations with critical dissipation on the other hand we establish several regularity criteria involving temperature for 2d benard equations with supercritical dissipation

on the regularity criteria of the 2d boussinesq equations with partial dissipation abstract in this paper we consider the two dimensional 2d boussinesq equations with partial dissipation the issue of whether the 2d boussinesq equations always possess global in time classical solutions can be difficult when there is only partial dissipation therefore the goal of this paper is to establish some regularity criteria of the corresponding systems

approximation of baskakov type polya durrmeyer operators in the present paper we propose the durrmeyer type modification of baskakov operators based on inverse polya eggenberger distribution first we estimate a recurrence relation by using hypergeometric series we give a global approximation theorem in terms of second order modulus of continuity a direct approximation theorem by means of the ditzian totik modulus of smoothness and a voronovskaja type theorem some approximation results in weighted space are obtained also we show the rate of convergence of these operators to certain functions by illustrative graphics using the maple algorithms

on the geometry of lie algebras and poisson tensors a geometric programme to analyse the structure of lie algebras is presented with special emphasis on the geometry of linear poisson tensors the notion of decomposable poisson tensors is introduced and an algorithm to construct all solvable lie algebras is presented poisson liouville structures are also introduced to discuss a new class of lie algebras which include as a subclass semi simple lie algebras a decomposition theorem for poisson tensors is proved for a class of poisson manifolds including linear ones simple lie algebras are also discussed from this viewpoint and lower dimensional real lie algebras are analysed

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

global well posedness of the 2d boussinesq equations with fractional laplacian dissipation abstract as a continuation of the previous work 48 in this paper we focus on the cauchy problem of the two dimensional 2d incompressible boussinesq equations with fractional laplacian dissipation we give an elementary proof of the global regularity of the smooth solutions of the 2d boussinesq equations with a new range of fractional powers of the laplacian the argument is based on the nonlinear lower bounds for the fractional laplacian established in 13 consequently this result significantly improves the recent works 13 45 48

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

the kantorovich variant of an operator defined by d d stancu abstract in this note we introduce kantorovich variant of the operators considered by stancu 1998 based on two nonnegative parameters here we prove an approximation theorem with the help of bohman korovkin s principle and find the estimate of the rate of convergence by means of modulus of smoothness and lipschitz type function for these operators in the last section of the paper we show the convergence of the operators by illustrative graphics in mathematica to certain functions

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

the durrmeyer variant of an operator defined by d d stancu in 1984 d d stancu defined a sequence of positive linear operators generalizing the bernstein polynomials in this paper a durrmeyer variant of these operators is considered we derive some basic approximation properties and present a complete asymptotic expansion for the sequence of these operators all coefficients are calculated in a concise form

global well posedness for a class of 2d boussinesq systems with fractional dissipation abstract the incompressible boussinesq equations not only have many applications in modeling fluids and geophysical fluids but also are mathematically important the well posedness and related problem on the boussinesq equations have recently attracted considerable interest this paper examines the global regularity issue on the 2d boussinesq equations with fractional laplacian dissipation and thermal diffusion attention is focused on the case when the thermal diffusion dominates we establish the global well posedness for the 2d boussinesq equations with a new range of fractional powers of the laplacian

particle like structure of lie algebras if a lie algebra structure x1d524 on a vector space is the sum of a family of mutually compatible lie algebra structures x1d524 i s we say that x1d524 is simply assembled from the x1d524 i s repeating this procedure with a number of lie algebras themselves simply assembled from the x1d524 i s one obtains a lie algebra assembled in two steps from x1d524 i s and so on we describe the process of modular disassembling of a lie algebra into a unimodular and a non unimodular part we then study two inverse questions which lie algebras can be assembled from a given family of lie algebras and from which lie algebras can a given lie algebra be assembled we develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of lie algebras the main result of our theory is that any finite dimensional lie algebra over an algebraically closed field of characteristic zero or over r can be assembled in a finite number of steps from two elementary constituents which we call dyons and