domain contraction in nonlinear programming minimizing a quadratic concave objective over a polyhedron using linear underestimating approximations and their dual solutions we develop new results for the nonconvex problem of minimizing a quadratic concave function under linear constraints for solving this problem which has such important management applications as economies of scale fixed charge and quadratic assignment problems some results provide and tighten bounds on the global minimum value while others identify the domains of the variables which may be excluded from further consideration this leads to successive reduction of domains containing a global solution called domain contraction and ideally to a solution with the objective value within some acceptable error level for most results specific problem instances are provided to establish their use in theory in practice experience with solving the problems presented here indicates that a combination of the proposed domain contraction via linear underestimation and its dual information with the branch and bound approach offers a computa

invariant conservation law preserving discretizations of linear and nonlinear wave equations symmetry and conservation law preserving finite difference discretizations are obtained for linear and nonlinear one dimensional wave equations on five and nine point stencils using the theory of lie point symmetries of difference equations and the discrete direct multiplier method of conservation law construction in particular for the linear wave equation an explicit five point scheme is presented that preserves the discrete analogs of its basic geometric point symmetries and six of the corresponding conservation laws for a class of nonlinear wave equations arising in hyperelasticity a nine point implicit scheme is constructed preserving four point symmetries and three local conservation laws other discretizations of the nonlinear wave equations preserving different subsets of conservation laws are discussed

potential conservation laws we prove that potential conservation laws have characteristics depending only on local variables if and only if they are induced by local conservation laws therefore characteristics of pure potential conservation laws have to essentially depend on potential variables this statement provides a significant generalization of results of the recent paper by bluman et al j math phys 47 113505 2006 moreover we present extensions to gauged potential systems abelian and general coverings and general foliated systems of differential equations an example illustrating possible applications of these results is given a special version of the hadamard lemma for fiber bundles and the notions of weighted jet spaces are proposed as new tools for the investigation of potential conservation laws

the capacity region of the gaussian multiple input multiple output broadcast channel the gaussian multiple input multiple output mimo broadcast channel bc is considered the dirty paper coding dpc rate region is shown to coincide with the capacity region to that end a new notion of an enhanced broadcast channel is introduced and is used jointly with the entropy power inequality to show that a superposition of gaussian codes is optimal for the degraded vector broadcast channel and that dpc is optimal for the nondegraded case furthermore the capacity region is characterized under a wide range of input constraints accounting as special cases for the total power and the per antenna power constraints

a recursion formula for the construction of local conservation laws of differential equations abstract a simple formula is presented that for any given local divergence type conservation law of a system of partial or ordinary differential equations pde ode generates a divergence expression involving an arbitrary function of all independent variables in the cases when the new flux vector is a local expression inequivalent to the initial local conservation law flux vector a new local conservation law is obtained for odes this can yield additional integrated factors examples of systems of differential equations are presented for which the proposed new relationship yields important local conservation laws starting from basic ones examples include a nonlinear ode and several fundamental physical pde models in particular general classes of nonlinear wave and diffusion equations vorticity type equations and a shear wave propagation model in hyper viscoelastic fiber reinforced solids

restoration of blurred star field images by maximally sparse optimization the problem of removing blur from or sharpening astronomical star field intensity images is discussed an approach to image restoration that recovers image detail using a constrained optimization theoretic approach is introduced ideal star images may be modeled as a few point sources in a uniform background it is argued that a direct measure of image sparseness is the appropriate optimization criterion for deconvolving the image blurring function a sparseness criterion based on the l sub p is presented and candidate algorithms for solving the ensuing nonlinear constrained optimization problem are presented and reviewed synthetic and actual star image reconstruction examples are presented to demonstrate the method s superior performance as compared with several image deconvolution methods

fair multiuser channel allocation for ofdma networks using nash bargaining solutions and coalitions in this paper a fair scheme to allocate subcarrier rate and power for multiuser orthogonal frequency division multiple access systems is proposed the problem is to maximize the overall system rate under each user s maximal power and minimal rate constraints while considering the fairness among users the approach considers a new fairness criterion which is a generalized proportional fairness based on nash bargaining solutions and coalitions first a two user algorithm is developed to bargain subcarrier usage between two users then a multiuser bargaining algorithm is developed based on optimal coalition pairs among users the simulation results show that the proposed algorithms not only provide fair resource allocation among users but also have a comparable overall system rate with the scheme maximizing the total rate without considering fairness they also have much higher rates than that of the scheme with max min fairness moreover the proposed iterative fast implementation has the complexity for each iteration of only o k sup 2 nlog sub 2 n k sup 4 where n is the number of subcarriers and k is the number of users

duality achievable rates and sum rate capacity of gaussian mimo broadcast channels we consider a multiuser multiple input multiple output mimo gaussian broadcast channel bc where the transmitter and receivers have multiple antennas since the mimo bc is in general a nondegraded bc its capacity region remains an unsolved problem we establish a duality between what is termed the dirty paper achievable region the caire shamai see proc ieee int symp information theory washington dc june 2001 p 322 achievable region for the mimo bc and the capacity region of the mimo multiple access channel mac which is easy to compute using this duality we greatly reduce the computational complexity required for obtaining the dirty paper achievable region for the mimo bc we also show that the dirty paper achievable region achieves the sum rate capacity of the mimo bc by establishing that the maximum sum rate of this region equals an upper bound on the sum rate of the mimo bc

a comparison of conservation law construction approaches for the two dimensional incompressible mooney rivlin hyperelasticity model an extended kovalevskaya form is derived for the two dimensional incompressible mooney rivlin nonlinear hyperelasticity equations and is used to compute a complete set of local conservation laws of the model through the direct method conserved densities and fluxes of the conservation laws are derived and their physical interpretation is discussed since the model admits a variational formulation the equations are rewritten in the self adjoint form computation of local conservation laws through the direct method applied to the self adjoint form as well as a conservation law computation through the local symmetry analysis and the noether s first theorem is performed a correspondence between local variational symmetries and conservation law multipliers is illustrated it is argued that even though it leads to more complicated forms of multipliers the direct conservation law construction method applied to the kovalevskaya form of the equations is a preferred systematic way of conservation law computat

new conservation laws obtained directly from symmetry action on a known conservation law two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries the first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation when the contact transformation is a symmetry of the given system then the corresponding conservation law is a conservation law of the given system the second formula checks a priori whether or not the action of a symmetry continuous or discrete on a conservation law can yield one or more new conservation laws of the given system several examples are considered including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws

an optimal design method for magnetic resonance imaging gradient waveforms a method of using nonlinear constrained optimization to design gradient waveforms for magnetic resonance imaging is described formulation and solution of the waveform optimization problem are described and example waveforms are presented for a variety of design objectives and constraint sets most design objectives can be expressed as linear or quadratic functions of the discrete parameter set and most constraint functions are linear thus linear and quadratic programming techniques can be utilized to solve the optimization problem among the objectives considered are minimize rms current minimize waveform slewing minimize waveform moments to reduce motion induced dephasing minimize echo time te for given imaging and motion refocusing conditions maximize the gradient amplitude during rf application and sampling and the area of the phase encoding waveform to maximize resolution and minimize or maximize the gradient b factor or diffusion sensitivity this optimal design procedure produces physically realizable waveforms which optimally achieve specific imaging and motion artifact reduction goals and it is likely to reduce waveform design time by making it more scientifically rather than heuristically based

fairness in multiuser systems with polymatroid capacity region for a wide class of multiuser systems a subset of capacity region which includes the corner points and the sum capacity facet has a special structure known as polymatroid multiple access channels with fixed input distributions and multiple antenna broadcast channels are examples of such systems any interior point of the sum capacity facet can be achieved by time sharing among corner points or by an alternative method known as rate splitting the main purpose of this paper is to find a point on the sum capacity facet which satisfies a notion of fairness among the active users this problem is addressed in two cases i where the complexity of achieving interior points is not feasible and ii where the complexity of achieving interior points is feasible for the first case the corner point for which the minimum rate of the active users is maximized is desired a simple greedy algorithm is introduced to find such an optimum corner point in addition it is shown for single antenna gaussian multiple access channels the resulting corner point is leximin maximal with respect to the set of the corner points for the second case the properties of the unique leximin maximal rate vector with respect to the polymatroid are reviewed it is shown that the problems of deriving the time sharing coefficients or rate splitting scheme to attain the leximin maximal vector can be solved by decomposing the problem into some lower dimensional subproblems in addition a fast algorithm to compute the time sharing coefficients to attain a general point on the sum capacity facet is presented

nonlocally related systems linearization and nonlocal symmetries for the nonlinear wave equation abstract the nonlinear wave equation u t t c 2 u u x x arises in various physical applications ames et al w f ames r j lohner e adams group properties of u t t f u u x x int j nonlin mech 16 1981 439 447 did the complete group classification for its admitted point symmetries with respect to the wave speed function c u and as a consequence constructed explicit invariant solutions for some specific cases by considering conservation laws for arbitrary c u we find a tree of nonlocally related systems and subsystems which include related linear systems through hodograph transformations we use existing work on such related linear systems to extend the known symmetry classification in w f ames r j lohner e adams group properties of u t t f u u x x int j nonlin mech 16 1981 439 447 to include nonlocal symmetries moreover we find sets of c u for which such nonlinear wave equations admit further nonlocal symmetries and hence significantly further extend the group classification of the nonlinear wave equation

global minimization of a linearly constrained concave function by partition of feasible domain the problem of minimizing a concave function subject to linear inequality constraints may have many local solutions therefore finding the global constrained minimum is a computationally difficult problem a computational method is described which finds the global minimum of a smooth concave function over a polyhedron in rn the feasible domain is partitioned into a rectangular domain which can be excluded from further consideration and r 2n subdomains at least one of which contains the global minimum a known algorithm can be applied sequentially or in parallel to each of these r subdomains to compute the global minimum a method is also presented appendix b for the construction of nontrivial test problems for which the global minimum point is known given an arbitrary polyhedron and a selected vertex it is shown how to determine a concave quadratic function generally with many local minima with its global minimum at the selected vertex

solution of the multisource weber and conditional weberproblems by d c programming d c programming is a recent technique of global optimization that allows the solution of problems whose objective function and constraints can be expressed as differences of convex i e d c functions many such problems arise in continuous location theory the problem first considered is to locate a known number of source facilities to minimize the sum of weighted euclidean distances between a user s fixed location and the source facility closest to the location of each user we also apply d c programming to the solution of the conditional weber problem an extension of the multisource weber problem in which some facilities are assumed to be already established in addition we consider a generalization of weber s problem the facility location problem with limited distances where the effective service distance becomes a constant when the actual distance attains a given value computational results are reported for problems with up to 10 000 users and two new facilities 50 users and three new facilities 1 000 users 20 existing facilities and one new facility or 200 users 10 existing and two new facilities

on the design of maximally sparse beamforming arrays the problem of designing nonuniformly spaced arrays is formulated as one of constrained optimization in which the cost function is chosen to select the array with the minimum number of elements the response of the array is controlled by a set of inequality point response constraints it is shown that a suitable cost function for this problem is the l sub p quasi norm for 0

hierarchy of conservation laws of diffusion convection equations we introduce notions of equivalence of conservation laws with respect to lie symmetry groups for fixed systems of differential equations and with respect to equivalence groups or sets of admissible transformations for classes of such systems we also revise the notion of linear dependence of conservation laws and define the notion of local dependence of potentials to construct conservation laws we develop and apply the most direct method which is effective to use in the case of two independent variables admitting possibility of dependence of conserved vectors on a number of potentials we generalize the iteration procedure proposed by bluman and doran wu for finding nonlocal potential conservation laws as an example we completely classify potential conservation laws including arbitrary order local ones of diffusion convection equations with respect to the equivalence group and construct an exhaustive list of locally inequivalent potential systems corresponding to these equations

framework for nonlocally related partial differential equation systems and nonlocal symmetries extension simplification and examples any partial differential equation pde system can be effectively analyzed through consideration of its tree of nonlocally related systems if a given pde system has n local conservation laws then each conservation law yields potential equations and a corresponding nonlocally related potential system moreover from these n conservation laws one can directly construct 2n 1 independent nonlocally related systems by considering these potential systems individually n singlets in pairs n n 1 2couplets taken all together one n plet in turn any one of these 2n 1 systems could lead to the discovery of new nonlocal symmetries and or nonlocal conservation laws of the given pde system moreover such nonlocal conservation laws could yield further nonlocally related systems a theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems the planar gas dynamics equations and nonlinear telegraph equations are used as illustrative examples many new local

comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics abstract the conservation laws for second order scalar partial differential equations and systems of partial differential equations which occur in fluid mechanics are constructed using different approaches the direct method noether s theorem the characteristic method the variational approach multiplier approach for arbitrary functions as well as on the solution space symmetry conditions on the conserved quantities the direct construction formula approach the partial noether approach and the noether approach for the equation and its adjoint are discussed and explained with the help of an illustrative example on a non linear field equation describing the relaxation to a maxwellian distribution the conservation laws for the non linear diffusion equation for the spreading of an axisymmetric thin liquid drop the system of two partial differential equations governing flow in a laminar two dimensional jet and the system of two partial differential equations governing flow in a laminar radial jet are discussed via these approaches

conservation properties and potential systems of vorticity type equations partial differential equations of the form divn 0 nt curlm 0 involving two vector functions in r3 depending on t x y z appear in different physical contexts including the vorticity formulation of fluid dynamics magnetohydrodynamics mhd equations and maxwell s equations it is shown that these equations possess an infinite family of local divergence type conservation laws involving arbitrary functions of space and time moreover it is demonstrated that the equations of interest have a rather special structure of a lower degree degree two conservation law in r4 t x y z the corresponding potential system has a clear physical meaning for the maxwell s equations it gives rise to the scalar electric and the vector magnetic potentials for the vorticity equations of fluid dynamics the potentialization inverts the curl operator to yield the fluid dynamics equations in primitive variables for mhd equations the potential equations yield a generalization of the galas bogoyavlenskij potential that

framework for potential systems and nonlocal symmetries algorithmic approach an algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations des each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree useful conservation laws play an essential role in the construction of an extended tree a useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system nonlocal symmetries for a given system of des can arise from any system in its extended tree we construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations and in both cases obtain new nonlocal symmetries more importantly due to the equivalence of solution sets any coordinate independent method of analysis qualitative numerical perturbation etc can be applied to any system within the tree and may yield simpler computations and or results that cannot be obtained when the method is directly applied to the given system

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

concave minimization via collapsing polytopes we present a procedure for globally minimizing a concave function over a bounded polytope by successively minimizing the function over polytopes containing the feasible region and collapsing to the feasible region the initial containing polytope is a simplex and at the kth iteration the procedure chooses the most promising vertex of the current containing polytope to refine the approximation the method generates a tree whose ultimate terminal nodes coincide with the vertices of the feasible region and accounts for the vertices of the containing polytopes

local conservation laws of second order evolution equations generalizing results by bryant and griffiths 1995 duke math j 78 531 we completely describe local conservation laws of second order 1 1 dimensional evolution equations up to contact equivalence the possible dimensions of spaces of conservation laws prove to be 0 1 2 and infinity the canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws