the gould hopper polynomials in the novikov veselov equation we use the gould hopper gh polynomials to investigate the novikov veselov nv equation the root dynamics of the flow in the nv equation is studied using the gh polynomials and then the lax pair is found in particular when n 3 4 5 one can get the gold fish model the smooth rational solutions of the nv equation are also constructed via the extended moutard transformation and the gh polynomials the asymptotic behavior is discussed and then the smooth rational solution of the liouville equation is obtained

explicit solution of a stochastic irreversible investment problem and its moving threshold we consider a firm producing a single consumption good that makes irreversible investments to expand its production capacity the firm aims to maximize its expected total discounted real profit net of investment on a finite horizont the capacity is modeled as a controlled ita process where the control is the real investment which is not necessarily a rate but more generally a monotone process the result is a singular stochastic control problem we introduce the associated optimal stopping problem that is the optimal cost of not investing its variational formulation turns out to be a parabolic obstacle problem which we explicitly solve in the case of constant relative risk aversion crra production functions the moving free boundary is the threshold at which the shadow value of installed capital exceeds the capital s replacement cost then we use the equation of the free boundary to evaluate the optimal investment policy and its corresponding optimal profits

on stable constant mean curvature surfaces with free boundary ros and vergasta geom dedicata 56 19 33 1995 proved among others interesting results a theorem which states that an immersed orientable compact stable constant mean curvature surface sigma with free boundary in a closed ball b subset mathbb r 3 must be a planar equator a spherical cap or a surface of genus 1 with at most two boundary components in this article by using a modified hersch type balancing argument we complete their work by proving that sigma cannot have genus 1

free boundary minimal annuli in convex three manifolds we prove the existence of free boundary minimal annuli inside suitably convex subsets of three dimensional riemannian manifolds with nonnegative ricci curvature including strictly convex domains of the euclidean space r

optimal boundary surface for irreversible investment with stochastic costs this paper examines a markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production we model market uncertainty and the cost of investment per unit of production capacity as two independent one dimensional regular diffusions and we consider a general convex running cost function the optimization problem is set as a three dimensional degenerate singular stochastic control problem we provide the optimal control as the solution of a reflected diffusion at a suitable boundary surface such boundary arises from the analysis of a family of two dimensional parameter dependent optimal stopping problems and it is characterized in terms of the family of unique continuous solutions to parameter dependent nonlinear integral equations of fredholm type

novel isochronous n body problems featuring n arbitrary rational coupling constants a novel class of n body problems is identified with n an arbitrary positive integer n 2 these models are characterized by newtonian accelerations equal forces equations of motion describing n equal point particles moving in the complex z plane these highly nonlinear equations feature n arbitrary coupling constants yet they can be solved by algebraic operations and if all the n coupling constants are real and rational the corresponding n body problem is isochronous its generic solutions are all completely periodic with an overall period t independent of the initial data but many solutions feature subperiods t p with p integer it is moreover shown that these models are hamiltonian

identifying the free boundary of a stochastic irreversible investment problem via the bank el karoui representation theorem we study a stochastic continuous time model on a finite horizon for a firm that produces a single good we model the production capacity as an ito diffusion controlled by a nondecreasing process representing the cumulative investment the firm aims to maximize its expected total net profit by choosing the optimal investment process that is a singular stochastic control problem we derive some first order conditions for optimality and we characterize the optimal solution in terms of the base capacity process l t i e the unique solution of a representation problem in the spirit of bank and el karoui p bank and n el karoui ann probab 32 2004 pp 1030 1067 we show that the base capacity is deterministic and it is identified with the free boundary hat y t of the associated optimal stopping problem when the coefficients of the controlled diffusion are deterministic functions of time this is a novelty in the literature on finite horizon singular stochastic control problems as a sub

on a free boundary problem for minimal surfaces forc 4 embedded manifoldss 3 which are differmorphic to the standard sphere in 3 the existence of non constant minimal surfaces bounded bys and intersectings orthogonally along their boundaries is deduced

generations of solvable discrete time dynamical systems a technique is introduced which allows to generate starting from any solvable discrete time dynamical system involving n time dependent variables new generally nonlinear generations of discrete time dynamical systems also involving n time dependent variables and being as well solvable by algebraic operations essentially by finding the n zeros of explicitly known polynomials of degree n the dynamical systems constructed using this technique may also feature large numbers of arbitrary constants and they need not be autonomous the solvable character of these models allows to identify special cases with remarkable time evolutions for instance isochronous or asymptotically isochronous discrete time dynamical systems the technique is illustrated by a few examples

new solvable discrete time many body problem featuring several arbitrary parameters ii a new solvable discrete time many body problem is identified it extends a model treated in a previous paper by introducing in its equations of motion an additional free parameter hence it features 6 parameters 2 of which can be eliminated say replaced by unity by appropriate rescalings assignments of these parameters are identified which entail that the many body model is asymptotically isochronous namely that its generic solution when the discrete time variable l diverges l becomes completely periodic up to exponentially vanishing corrections with a fixed period independent of the initial data

on a stochastic irreversible investment problem the productive sector of the economy represented by a single firm employing labor to produce the consumption good is studied in a stochastic continuous time model on a finite time interval the firm must choose the optimal level of employment and capital investment in order to maximize its expected total profits in this stochastic control problem the firm s capacity is modeled as an ito process controlled by a monotone process possibly singular that represents the cumulative real investment it is optimal to invest when the shadow value of installed capital exceeds the capital s replacement cost this threshold is the free boundary of a related optimal stopping problem which we recast as a stopping problem without integral cost similar to the american option problem then under a regularity condition we characterize the free boundary as the unique solution of a nonlinear integral equation

properties of the zeros of generalized basic hypergeometric polynomials we define the generalized basic hypergeometric polynomial of degree n in terms of the generalized basic hypergeometric function by choosing one of its parameters to allow the termination of the series after a finite number of summands in this paper we obtain a set of nonlinear algebraic equations satisfied by the n zeros of the polynomial moreover we obtain an n n matrix m defined in terms of the zeros of the polynomial which in turn depend on the parameters of the polynomial the eigenvalues of this remarkable matrix m are given by neat expressions that depend only on some of the parameters of the polynomial that is the matrix m is isospectral moreover in case the parameters that appear in the expressions for the eigenvalues of m are rational the matrix m has rational eigenvalues a diophantine property

the neatest many body problem amenable to exact treatments a goldfish various formulations findings and conjectures are reviewed which relate to a many body problem that is arguably the neatest nontrivial such model amenable to exact treatments

some isoperimetric inequalities and eigenvalue estimates in weighted manifolds abstract in this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds as a consequence we obtain a relative linear isoperimetric inequality for such submanifolds we also prove an extrinsic upper bound to the first non zero eigenvalue of the drift laplacian on closed submanifolds of weighted manifolds

a nonautonomous yet solvable discrete time n body problem a new discrete time n body problem is introduced its equations of motion which become newtonian equations of motion accelerations equal forces in the continuous time limit are nonautonomous featuring an arbitrary function f of the discrete time variable 0 1 2 3 they are nevertheless solvable by algebraic operations the solution of their initial value problem are the zeros of a polynomial of degree n in z pn z explicitly known for all time via an appropriate discrete time quadrature in terms of the initial data this model generalizes a previously known model in which the arbitrary function f is an arbitrary constant

time dependent polynomials with one multiple root and new solvable dynamical systems a time dependent monic polynomial in the z variable with n distinct roots such that exactly one root has multiplicity m 2 is considered for k 1 2 the kth derivatives of the n roots are expressed in terms of the derivatives of order j k of the first n coefficients of the polynomial and of the derivatives of order j k 1 of the roots themselves these relations are utilized to construct new classes of algebraically solvable first order systems of ordinary differential equations as well as n body problems multiple examples of solvable isochronous all solutions are periodic with the same period 2 and 3 body problems are provided a time dependent monic polynomial in the z variable with n distinct roots such that exactly one root has multiplicity m 2 is considered for k 1 2 the kth derivatives of the n roots are expressed in terms of the derivatives of order j k of the first n coefficients of the polynomial and of the derivatives of order j k 1 of the roots themselves these relations are utilized to construct new classes of algebraically solvable first order systems of ordinary differential equations as well as n body problems multiple examples of solvable isochronous all solutions are periodic with the same period 2 and 3 body problems are provided

characterization of the optimal boundaries in reversible investment problems this paper studies a reversible investment problem where a social planner aims to control its capacity production in order to fit optimally the random demand of a good our model allows for general diffusion dynamics on the demand as well as general cost functional the resulting optimization problem leads to a degenerate two dimensional bounded variation singular stochastic control problem for which explicit solution is not available in general and the standard verification approach cannot be applied a priori we use a direct viscosity solutions approach for deriving some features of the optimal free boundary function and for displaying the structure of the solution in the quadratic cost case we are able to prove a smooth fit c 2 property which gives rise to a full characterization of the optimal boundaries and value function

a complex deformation of the classical gravitational many body problem that features many completely periodic motions a complex deformation of the newtonian equations of motion of the classical gravitational many body problem is introduced namely a many body problem that features a parameter and that reduces when this parameter vanishes to the standard equations of motion of newtonian gravitation for an arbitrary number of pointlike bodies with arbitrary masses and it is shown that when this parameter is instead positive 0 there is an open set of complex initial data such that all the complex motions originating from it are completely periodic with period t 2 and that the infinite measure of this set is a finite nonvanishing fraction of the measure of the entire set of initial data

a class of integrable hamiltonian systems whose solutions are perhaps all completely periodic we show that the dynamical system characterized by the complex equations of motion qj i qj k 1 k jnqjqkf qj qk j 1 n with f x x x is hamiltonian and integrable and we conjecture that all its solutions qj t j 1 n are completely periodic with a period that is a finite integral multiple of t 2 here n is an arbitrary positive integer is an arbitrary nonvanishing real constant y y is the weierstrass function with arbitrary semiperiods and are two arbitrary constants special cases are f x 2 coth x 1 r2 sinh2 x f x 2 coth x f x 2 sinh x f x 2 x 1 2x2 and of course f x 2 x these findings as well as the conjecture which is shown to be true in some of these special cases are based on the possibility to recast these equations of motion in the modified lax form l i l l m with l and m appropriate n n matrix functions of the n dynamical variables qj and of their time derivatives qj

characterization of f extremal disks abstract we show uniqueness for overdetermined elliptic problems defined on topological disks with c 2 boundary i e positive solutions u to u f u 0 in m 2 g so that u 0 and u c t e along the unit outward normal along under the assumption of the existence of a candidate family to do so we adapt the galvez mira generalized hopf type theorem 19 to the realm of overdetermined elliptic problem when m 2 g is the standard sphere s 2 and f is a c 1 function so that f x 0 and f x x f x for any x r we construct such candidate family considering rotationally symmetric solutions this proves the berestycki caffarelli nirenberg conjecture in s 2 for this choice of f more precisely this shows that if u is a positive solution to u f u 0 on a topological disk s 2 with c 2 boundary so that u 0 and u c t e along then must be a geodesic disk and u is rotationally symmetric in particular this gives a positive answer to the schiffer conjecture d cf 33 35 for the first dirichlet eigenvalue and classifies simply connected harmonic domains cf 28 also called serrin problem in s 2

overdetermined elliptic problems in topological disks abstract we introduce a method based on the poincare hopf index theorem to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk applications to some well known nonlinear elliptic pdes are provided our result can be seen as the analogue of hopf s uniqueness theorem for constant mean curvature spheres but for the general analytic context of overdetermined elliptic problems

a further solvable three body problem in the plane the solution is provided of a three body problem in the plane which is the third of a trio recently identified as likely to display a particularly simple time evolution hence to be amenable to exact treatment this conjecture already validated by providing the solution of the first two of these three models is now completely proven by exhibiting the solution of the third this finding also demonstrates the conjectured super painleve character of certain nonlinear ordinary differential equations namely the fact that their general solution is an entire function of the independent variable

an approach for obtaining integrable hamiltonians from poisson commuting polynomial families we discuss a general approach permitting the identification of a broad class of sets of poisson commuting hamiltonians which are integrable in the sense of liouville it is shown that all such hamiltonians can be solved explicitly by a separation of variables ansatz the method leads in particular to a proof that the so called goldfish hamiltonian is maximally superintegrable and leads to an elementary identification of a full set of integrals of motion the hamiltonians in involution with the goldfish hamiltonian are also explicitly integrated new integrable hamiltonians are identified among which some have the property of being isochronous that is all their orbits have the same period finally a peculiar structure is identified in the poisson brackets between the elementary symmetric functions and the set of hamiltonians commuting with the goldfish hamiltonian these can be expressed as products between elementary symmetric functions and hamiltonians the structure displays an invariance pro

a stochastic representation theorem with applications to optimization and obstacle problems we study a new type of representation problem for optional processes with connections to singular control optimal stopping and dynamic allocation problems as an application we show how to solve a variant of skorohod s obstacle problem in the context of backward stochastic differential equations

a technique to identify solvable dynamical systems and a solvable generalization of the goldfish many body problem a simple approach is discussed which associates to solvable matrix equations solvable dynamical systems generally interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles we then focus on cases in which the auxiliary variables can be altogether eliminated reobtaining thereby via this unified approach well known solvable many body problems and moreover a solvable extension of the goldfish model

a note on the continuity of free boundaries in finite horizon optimal stopping problems for one dimensional diffusions we provide sufficient conditions for the continuity of the free boundary in a general class of finite horizon optimal stopping problems arising for instance in finance and economics the underlying process is a strong solution of a one dimensional time homogeneous stochastic differential equation sde the proof relies on both analytic and probabilistic arguments and is based on a contradiction scheme inspired by the maximum principle in partial differential equations theory mild local regularity of the coefficients of the sde and smoothness of the gain function locally at the boundary are required

capillary surfaces with free boundary in a wedge abstract we study constant mean curvature compact surfaces with free boundary in a wedge under the hypothesis of stability or embeddedness we prove that the surface is part of a sphere centered at the vertex of the wedge

a large class of solvable discrete time many body problems a class of n body problems is identified characterized by second order discrete time evolution equations determining the motion in the complex z plane of an arbitrary number n of points zn zn l where l 0 1 2 is the discrete time independent variable both these equations of motion and the solution of their initial value problem only involve algebraic operations finding the zeros of explicitly known polynomials of degree n in z finding the eigenvectors and eigenvalues of explicitly known n n matrices these models feature an arbitrarily large number of arbitrary parameters coupling constants

two novel classes of solvable many body problems of goldfish type with constraints two novel classes of many body models with nonlinear interactions of goldfish type are introduced they are solvable provided the initial data satisfy a single constraint in one case in the other two constraints i e for such initial data the solution of their initial value problem can be achieved via algebraic operations such as finding the eigenvalues of given matrices or equivalently the zeros of known polynomials entirely isochronous versions of some of these models are also exhibited i e versions of these models whose nonsingular solutions are all completely periodic with the same period

multivariable utility functions utility functions of several variables are ubiquitous in economics their maximization requires inversion of the gradient map using convex analysis tools we provide a representation of an extension of this inverse that accounts for possible constraints to solve economic equilibrium problems the utility functions of the agents are frequently aggregated into a representative agent s utility function we establish regularity and inversion properties of such representative utility functions

hamiltonian character of the motion of the zeros of a polynomial whose coefficients oscillate over time a hamiltonian is explicitly exhibited whose equations of motion yield the time evolution of the n zeros of a polynomial of degree n in z when its coefficients oscillate or evolve in some other hamiltonian manner

additive control of stochastic linear systems with finite horizon we consider a dynamic system whose state is governed by a linear stochastic differential equation with time dependent coefficients the control acts additively on the state of the system our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process it is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space time domain by using some decomposition arguments the problems of a two sided control i e optimal corrections and the case with constraints on the resources i e finite fuel can be reduced to a simpler case of only one sided control i e a monotone follower these results are applied to solving some examples by the so called method of similarity solutions

a general existence theorem for embedded minimal surfaces with free boundary in this paper we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact riemannian 3 manifold m with boundary m these minimal surfaces are either disjoint from m or meet m orthogonally the main feature of our result is that there is no assumptions on the curvature of m or convexity of m we prove the boundary regularity of the minimal surfaces at their free boundaries furthermore we define a topological invariant the filling genus for compact 3 manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold m our proof employs a variant of the min max construction used by colding and de lellis on closed embedded minimal surfaces which were first developed by almgren and pitts 2014 wiley periodicals inc

generalized kuhn tucker conditions for n firm stochastic irreversible investment under limited resources in this paper we study a continuous time optimal stochastic investment problem under limited resources in a market with n firms the investment processes are subject to a time dependent stochastic constraint rather than using a dynamic programming approach we exploit the concavity of the profit functional to derive some necessary and sufficient first order conditions for the corresponding social planner optimal policy our conditions are a stochastic infinite dimensional generalization of the kuhn tucker theorem the lagrange multiplier takes the form of a nonnegative optional random measure on 0 t which is flat off the set of times for which the constraint is binding i e when all the fuel is spent as a subproduct we obtain an enlightening interpretation of the first order conditions for a single firm in bank siam j control optim 44 2005 pp 1529 1541 in the infinite horizon case with operating profit functions of cobb douglas type our method allows the explicit calculation of t

a solvable n body problem in the plane i we introduce and discuss an n body problem in the plane characterized by equations of motion of newtonian type rj k 1nfjk j 1 n with given forces fjk having the following characteristics fjk depends only on rj rk rj rk i e only one body and two body forces are present fjk behaves as a two dimensional vector under rotations in the plane i e the model is rotation invariant for j k fjk is linear in rj and rj for j k fjk rj rk 2fjk with fjk a homogeneous polynomial of third degree in rj rk rj rk hence fjk is homogeneous of degree one in rj rk rj rk fjk contains linearly 8 arbitrary coupling constants the n body problem is solvable for arbitrary n and for arbitrary values of the 8 coupling constants its solutions display a rich phenomenology if the 8 coupling constants are suitably restricted the model is translation invariant and or hamiltonian of course when it is hamiltonian it is integrable indeed in some ca

a model for reversible investment capacity expansion we consider the problem of determining the optimal investment level that a firm should maintain in the presence of random price and or demand fluctuations we model market uncertainty by means of a geometric brownian motion and we consider general running payoff functions our model allows for capacity expansion as well as for capacity reduction with each of these actions being associated with proportional costs the resulting optimization problem takes the form of a singular stochastic control problem that we solve explicitly we illustrate our results by means of the so called cobb douglas production function the problem that we study presents a model in which the associated hamilton jacobi bellman equation admits a classical solution that conforms with the underlying economic intuition but does not necessarily identify with the corresponding value function which may be identically equal to infty thus our model provides a situation that highlights the need for rigorous mathematical analysis when addressing stochastic optimization applications in finance and economics as well as in other fields

an irreversible investment problem with maintenance expenditure on a finite horizon free boundary analysis abstract this paper concerns a continuous time finite horizon optimal irreversible investment problem with maintenance expenditure of a firm under uncertainty we assume that the firm can make irreversible investments to expand its production capacity and spend maintenance expenditure to achieve better performance of the productivity the objective of the firm is to construct optimal investment and maintenance policies to maximize its expected total profit over a finite horizon mathematically it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraint the problem gives rise to two free boundaries which stand for the optimal investment and maintenance strategies respectively we investigate behaviors of free boundaries study regularities of the value function and give optimal investment and maintenance policies as we know this is a first integral result for an investment maintenance problem with a finite time horizon due to use of partial differential equation pde technique

uniqueness of stable capillary hypersurfaces in a ball in this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical our result also provides a proof of a conjecture proposed by sternberg and zumbrun j reine angew math 503 63 85 1998 we also prove a heintze karcher ros type inequality for hypersurfaces with free boundary in a ball which together with the new minkowski formula yields a new proof of alexandrov s theorem for embedded cmc hypersurfaces in a ball with free boundary

a solvable two dimensional degenerate singular stochastic control problem with non convex costs in this paper we provide a complete theoretical analysis of a two dimensional degenerate nonconvex singular stochastic control problem the optimisation is motivated by a storage consumption model

optimal control under a dynamic fuel constraint we present a new approach to solve optimal control problems of the monotone follower type the key feature of our approach is that it allows us to include an arbitrary dynamic fuel constraint instead of dynamic programming we use the convexity of our cost functional to derive a first order characterization of optimal policies based on the snell envelope of the objective functional s gradient at the optimum the optimal control policy is constructed explicitly in terms of the solution to a representation theorem for stochastic processes obtained in bank and el karoui 2004 ann probab 32 pp 1030 1067 as an illustration we show how our methodology allows us to extend the scope of the explicit solutions obtained for the classical monotone follower problem and for an irreversible investment problem arising in economics

goldfishing by gauge theory a new solvable many body problem of goldfish type is identified and used to revisit the connection between two different approaches to solvable dynamical systems an isochronous variant of this model is identified and investigated alternative versions of these models are presented the behavior of the alternative isochronous model near its equilibrium configurations is investigated and a remarkable diophantine result as well as related diophantine conjectures are thereby obtained

on the starting and stopping problem application in reversible investments in this work we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process we use backward stochastic differential equations bsdes and snell envelopes finally we give some numerical results

calogero s goldfish is indeed a school of free particles a many body system of n nonlinear ordinary differential equations of second order which is amenable to exact treatments a goldfish calogero 2001 the neatest many body problem amenable to exact treatments a goldfish physica d 152 153 78 84 is shown to be equivalent through an exact transformation to the equations of one dimensional motion of n 1 free particles a school of free particles indeed the transformation is obtained by applying the reduction method and lie group analysis as introduced in nucci 1996 the complete kepler group can be derived by lie group analysis j math phys 37 1772 5

the lax representation for an integrable class of relativistic dynamical systems we exhibit the lax pair for the class of relativistic dynamical systems recently introduced by ruijsenaars and schneider whose equations of motion read whereb is an arbitrary constant and the weierstrass elliptic function

the painleve property for quasihomogenous systems and a many body problem in the plane we investigate a many body problem in the plane introduced by calogero and intensively studied by calogero francoise and sommacal an ad hoc complexification transforms the many body problem to a system of second order autonomous complex equations depending on some complex constants that describe the two body interactions we investigate the sets of two body interaction constants that make the complexified equation have the painleve property this is its solutions are given by single valued meromorphic functions in this case the original system has only periodic isochronous solutions we exhibit a family of settings where the system displays this property and show that it is not present in the three and four body problems that do not fall within our class for this we introduce a necessary condition for the presence of the painleve property in some quasihomogenous systems

properties of the zeros of generalized hypergeometric polynomials abstract we define the generalized hypergeometric polynomial of degree n as follows p n 1 p 1 q z m 0 n n m 1 m p m z n m m 1 m q m z n f q p 1 n 1 p 1 q 1 z here n is an arbitrary positive integer p and q are arbitrary nonnegative integers the p q parameters j and k are arbitrary generic possibly complex numbers m is the pochhammer symbol and f q p 1 0 1 p 1 q z is the generalized hypergeometric function in this paper we obtain a set of n nonlinear algebraic equations satisfied by the n zeros n of this polynomial we moreover manufacture an n n matrix l in terms of the 1 p q parameters n j k characterizing this polynomial and of its n zeros n and we show that it features the n eigenvalues m m k 1 q k 1 m m 1 n these n eigenvalues depend only on the q parameters k implying that the n n matrix l is isospectral for variations of the p parameters j and they clearly are integer or rational numbers if the q parameters k are themselves integer or rational numbers a nontrivial diophantine property

lax pairs galore a formula that yields an apparently but only apparently nontrivial lax pair for any nonlinear evolution pde in 1 1 dimensions possessing a local conservation law is presented several examples are exhibited

general solution of a three body problem in the plane we provide and discuss the general solution of a hamiltonian three body problem in the plane characterized by newtonian equations of motion with rotation and translation invariant velocity dependent one body and two body forces the model features a nonnegative real parameter when it does not vanish all solutions are completely periodic with period t 2 when it vanishes both unbounded and confined motions are possible with a rather rich phenomenology of possible behaviour in the latter case including completely periodic motions and limit cycles

integrable and solvable many body problems in the plane via complexification a simple prescription allows us to transform by appropriate complexification any one dimensional many body problem describing motions on a line with newtonian acceleration equals forces equations of motion featuring forces which depend analytically on the positions and possibly on the velocities of the particles and which are moreover scaling or translation invariant into a two dimensional many body problem describing motions in the plane with rotation invariant equations of motion if the original one dimensional model is hamiltonian the two dimensional model is also hamiltonian in fact bi hamiltonian in this manner starting from known integrable or solvable one dimensional many body problems integrable or solvable many body problems in the plane generally featuring much richer dynamics are obtained several examples are exhibited finally another more direct prescription is outlined to transform by complexification almost any one dimensional many body problem into a rota

stationary partitioning of convex bodies on considere une version particuliere du probleme de partage soit 7b b un corps convexe dans r 3 de volume m 3 7b b une surface de type disque s est a l interieur de 7b b divisant cet interieur en deux cellules simplement connexes 7b b et 7b b de volumes m 3 7b b m 3 7b b m 3 7b b 1 m 3 7b b 0 1 on considere comme fixee et on etudie les cas ou s a l aire la plus petite possible

sufficient conditions for constant mean curvature surfaces to be round we study under what condition a constant mean curvature surface can be round i if the boundary of a compact immersed disk type constant mean curvature surface in vec r 3 consists of lines of curvature and has less than 4 vertices with angle pi then the surface is spherical ii a compact immersed disk type capillary surface with less than 4 vertices in a domain of vec r 3 bounded by spheres or planes is spherical iii the mean curvature vector of a compact embedded capillary hypersurface of vec r n with smooth boundary in an unbounded polyhedral domain with unbalanced boundary should point inward iv if the kth order 2 leq k leq n 1 mean curvature of a compact immersed constant mean curvature hypersurface of vec r n without boundary is constant then the hypersurface is a sphere

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

capillary surfaces in a convex cone we show that a compact embedded hypersurface s subset mathbb r n 1 with constant higher order mean curvature in a convex piecewise smooth cone c which is perpendicular to c is part of a hypersphere also we prove that an embedded disk type constant mean curvature surface s subset mathbb r 3 in a convex polyhedral cone c which makes constant contact angles with c is a spherical cap if c has at most five faces this condition on the number of faces can be dropped if c is a right cone over a regular n gon and the contact angles are the same on s

a technique to identify solvable dynamical systems and another solvable extension of the goldfish many body problem we take advantage of the simple approach recently discussed which associates to solvable matrix equations solvable dynamical systems interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles starting from a solvable matrix evolution equation we obtain the corresponding many body model and note that in one case the auxiliary variables can be altogether eliminated obtaining thereby an also hamiltonian extension of the goldfish model the solvability of this novel model and of its isochronous variant is exhibited a related as well solvable model is also introduced as well as its isochronous variant finally the small oscillations of the isochronous models around their equilibrium configurations are investigated and from their isochronicity certain diophantine relations are evinced

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

novel solvable variants of the goldfish many body model a recent technique to identify solvable many body problems in two dimensional space yields via a new twist new many body problems of goldfish type some of these models are isochronous namely their generic solutions are completely periodic with a fixed period independent of the initial data the investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations

constant mean curvature surfaces with boundary on a sphere in this article we study the shape of a compact surface of constant mean curvature of euclidean space whose boundary is contained in a round sphere we consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle we study under what geometric conditions the surface must be spherical our results apply in many scenarios in physics where in absence of gravity a liquid drop is deposited on a round solid ball and the air liquid interface is a critical point for area under all variations that preserve the enclosed volume

goldfishing a new solvable many body problem a recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical many body problems being characterized by equations of motion of newtonian type generally in two dimensional space in this paper we tersely review results previously obtained in this manner and present novel findings of this kind mainly solvable variants of the goldfish many body model including models that feature isochronous classes of completely periodic solutions different formulations of these models are presented the behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated and in this manner some remarkable diophantine findings are obtained