regularity of the hamiltonian along optimal trajectories this paper concerns state constrained optimal control problems in which the dynamic constraint takes the form of a differential inclusion if the differential inclusion does not depend on time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is independent of time if the differential inclusion is lipschitz continuous then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is lipschitz continuous these two well known results are examples of the following principle the hamiltonian evaluated along the optimal state trajectory and the co state trajectory inherits the regularity properties of the differential inclusion regarding its time dependence we show that this principle also applies to another kind of regularity if the differential inclusion has bounded variation w r t time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory has bounded variation two application

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

a one dimensional variational problem with continuous lagrangian and singular minimizer we construct a continuous lagrangian strictly convex and superlinear in the third variable such that the associated variational problem has a lipschitz minimizer which is non differentiable on a dense set more precisely the upper and lower dini derivatives of the minimizer differ by a constant on a dense hence second category set in particular we show that mere continuity is an insufficient smoothness assumption for tonelli s partial regularity theorem

universal singular sets in the calculus of variations for regular one dimensional variational problems ball and nadirashvilli introduced the notion of the universal singular set of a lagrangian l and established its topological negligibility this set is defined to be the set of all points in the plane through which the graph of some absolutely continuous l minimizer passes with infinite derivative motivated by tonelli s partial regularity results the question of the size of the universal singular set in measure naturally arises here we show that universal singular sets are characterized by being essentially purely unrectifiable that is they intersect most lipschitz curves in sets of zero length and any compact purely unrectifiable set is contained within the universal singular set of some smooth lagrangian with given superlinear growth this gives examples of universal singular sets of hausdorff dimension two filling the gap between previously known one dimensional examples and sychev s result that universal singular sets are lebesgue null we show that some smoothness of the lagrangian is necessary for the topological size estimate and investigate the relationship between growth of the lagrangian and the existence of pathological rectifiable pieces in the universal singular set we also show that tonelli s partial regularity result is stable in that the energy of a near minimizer u over the set where it has large derivative is controlled by how far u is from being a minimizer

the semigroup property of value functions in lagrange problems the lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form the binary operation of inf composition applied to the value functions of a lagrange problem equates the principle of optimality with a semigroup property this paper finds the infinitesimal generator of the semigroup by differentiating at t 0 the type of limit is epigraphical convergence in a uniform sense moreover the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed

singular sets and the lavrentiev phenomenon we show that non occurrence of the lavrentiev phenomenon does not imply that the singular set is small precisely given a compact lebesgue null subset e and an arbitrary superlinearity there exists a smooth strictly convex lagrangian with this superlinear growth such that all minimizers of the associated variational problem have singular set exactly e but still admit approximation in energy by smooth functions

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

coupled points in the calculus of variations and applications to periodic problems the aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions and to develop second order necessary conditions for optimality when one of the end points is fixed our necessary conditions reduce to the known ones involving conjugate points we also apply our results to the periodic problems of the calculus of variations

egalitarian solutions of large games i a continuum of players this is the first of two papers devoted to the study of egalitarian solutions for nontransferable utility ntu games with a large number of players this paper is concerned with the construction of egalitarian solutions for the limit situation with a continuum of players we build on the concept of the potential introduced in hart and mas colell hart s a mas colell 1989 potential value and consistency econometrica57 589 614 and exploit partial differential equations techniques in particular we associate to the egalitarian solutions a variational problem which may be viewed as a generalization to the nontransferable utility context of the diagonal principle of aumann and shapley aumann r j l s shapley 1974 values of non atomic games princeton university press princeton new jersey in the transferable utility setting

low complexity user selection algorithms for multiuser mimo systems with block diagonalization block diagonalization bd is a precoding technique that eliminates interuser interference in downlink multiuser multiple input multiple output mimo systems with the assumptions that all users have the same number of receive antennas and utilize all receive antennas when scheduled for transmission the number of simultaneously supportable users with bd is limited by the ratio of the number of base station transmit antennas to the number of user receive antennas in a downlink mimo system with a large number of users the base station may select a subset of users to serve in order to maximize the total throughput the brute force search for the optimal user set however is computationally prohibitive we propose two low complexity suboptimal user selection algorithms for multiuser mimo systems with bd both algorithms aim to select a subset of users such that the total throughput is nearly maximized the first user selection algorithm greedily maximizes the total throughput whereas the criterion of the second algorithm is based on the channel energy we show that both algorithms have linear complexity in the total number of users and achieve around 95 of the total throughput of the complete search method in simulations

element removal method for singular minimizers in variational problems involving lavrentiev phenomenon a numerical method called element removal method is designed to calculate singular minimizers which cannot be approximated by simple applications of standard numerical methods because of the so called lavrentiev phenomenon the convergence of the method is proved the results of numerical experiments show that the method is effective

a regularity theory for variational problems with higher order derivatives we consider problems in the calculus of variations in one independent variable and where the lagrangian involves derivatives up to order n n 1 existence theory supplies mild hypotheses under which there are minimizers for such problems but they need to be strengthened for standard necessary conditions to apply for problems with n 1 this paper initiates investigation of regularity properties and associated necessary conditions which obtain strictly under the hypotheses of existence theory it is shown that the nth derivative of a minimizer is locally essentially bounded off a closed set of zero measure the set of points of bad behaviour additional hypotheses are shown to exclude occurrence of points of bad behaviour finally a counter example suggests respects in which problems with n 1 exhibit pathologies not present in the n i case

precoding for full duplex multiuser mimo systems spectral and energy efficiency maximization we consider data transmissions in a full duplex fd multiuser multiple input multiple output mu mimo system where a base station bs bidirectionally communicates with multiple users in the downlink dl and uplink ul channels on the same system resources the system model of consideration has been thought to be impractical due to the self interference si between transmit and receive antennas at the bs interestingly recent advanced techniques in hardware design have demonstrated that the si can be suppressed to a degree that possibly allows for fd transmission this paper goes one step further in exploring the potential gains in terms of the spectral efficiency se and energy efficiency ee that can be brought by the fd mu mimo model toward this end we propose low complexity designs for maximizing the se and ee and evaluate their performance numerically for the se maximization problem we present an iterative design that obtains a locally optimal solution based on a sequential convex approximation method in this way the nonconvex precoder design problem is approximated by a convex program at each iteration then we propose a numerical algorithm to solve the resulting convex program based on the alternating and dual decomposition approaches where analytical expressions for precoders are derived for the ee maximization problem using the same method we first transform it into a concave convex fractional program which then can be reformulated as a convex program using the parametric approach we will show that the resulting problem can be solved similarly to the se maximization problem numerical results demonstrate that compared to a half duplex system the fd system of interest with the proposed designs achieves a better se and a slightly smaller ee when the si is small

differential and numerically invariant signature curves applied to object recognition we introduce a new paradigm the differential invariant signature curve or manifold for the invariant recognition of visual objects a general theorem of e cartan implies that two curves are related by a group transformation if and only if their signature curves are identical the important examples of the euclidean and equi affine groups are discussed in detail secondly we show how a new approach to the numerical approximation of differential invariants based on suitable combination of joint invariants of the underlying group action allows one to numerically compute differential invariant signatures in a fully group invariant manner applications to a variety of fundamental issues in vision including detection of symmetries visual tracking and reconstruction of occlusions are discussed

energy efficient zero forcing precoding design for small cell networks we consider small cell networks with multiple antenna transceivers and base stations bss cooperating to jointly design linear precoders to maximize the network energy efficiency subject to a sum power and per antenna power constraints at individual bss as well as user specific quality of service qos requirements assuming zero forcing precoding we formulate the problem of interest as a concave convex fractional program to which we proposed a centralized optimal solution based on the prevailing dinkelbach algorithm to facilitate distributed implementations we transform the design problem into an equivalent convex program using charnes cooper s transformation then based on the framework of alternative direction method of multipliers admm we develop a decentralized algorithm which is numerically shown to achieve fast convergence since bss are generally power hungry it may be more energy efficient if some bss can be shut down while still satisfying the qos constraints toward this end we investigate the problem of joint precoder design and bs selection which is a mixed boolean nonlinear program and then provide an optimal solution by customizing the branch and bound method for real time applications we propose a greedy algorithm which achieves near optimal performance in polynomial time numerical results are provided to demonstrate the effectiveness of the proposed algorithms

structure from motion theoretical foundations of a novel approach using custom built invariants we present a novel simple formulation of the problem of 3d object reconstruction from images in this formulation the object is seen as lying at the intersection of the projection of orbits of custom built lie group actions the group parameters correspond to unknown irrelevant quantities such as the camera orientation the depth parameters of the object with respect to the camera and the focal length we then use an algorithmic method based on moving frames a la fels olver to obtain a fundamental set of invariants of these group actions the invariants are used to define a set of equations determining the 3d object thus providing a mathematical formulation of the problem where the irrelevant parameters do not appear

globally optimal energy efficient power control and receiver design in wireless networks the characterization of the global maximum of energy efficiency ee problems in wireless networks is a challenging problem due to their nonconvex nature in interference channels the aim of this paper is to develop a new and general framework to achieve globally optimal solutions first the hidden monotonic structure of the most common ee maximization problems is exploited jointly with fractional programming theory to obtain globally optimal solutions with exponential complexity in the number of network links to overcome the high complexity we also propose a framework to compute suboptimal power control strategies with affordable complexity this is achieved by merging fractional programming and sequential optimization the proposed monotonic framework is used to shed light on the ultimate performance of wireless networks in terms of ee and also to benchmark the performance of the lower complexity framework based on sequential programming numerical evidence is provided to show that the sequential fractional programming framework achieves global optimality in several practical communication scenarios

sum capacity computation for the gaussian vector broadcast channel via dual decomposition a numerical algorithm for the computation of sum capacity for the gaussian vector broadcast channel is proposed the sum capacity computation relies on a duality between the gaussian vector broadcast channel and the sum power constrained gaussian multiple access channel the numerical algorithm is based on a lagrangian dual decomposition technique and it uses a modified iterative water filling approach for the gaussian multiple access channel the algorithm converges to the sum capacity globally and efficiently

relaxation of multiple integrals in the space bv ω rp a characterisation of the surface energy density for the relaxation in v r p of the functional is obtained a lemma of de giorgi is used to modify a sequence near the boundary without increasing its total energy

maurer cartan forms for fields on surfaces application to heart fiber geometry we study the space of first order models of smooth frame fields using the method of moving frames by exploiting the maurer cartan matrix of connection forms we develop geometrical embeddings for frame fields which lie on spherical ellipsoidal and generalized helicoid surfaces we design methods for optimizing connection forms in local neighborhoods and apply these to a statistical analysis of heart fiber geometry using diffusion magnetic resonance imaging this application of moving frames corroborates and extends recent characterizations of muscle fiber orientation in the heart wall but also provides for a rich geometrical interpretation in particular we can now obtain direct local measurements of the variation of the helix and transverse angles of fiber fanning and twisting and of the curvatures of the heart wall in which these fibers lie

regularity properties of solutions to the basic problem in the calculus of variations this paper concerns the basic problem in the calculus of variations minimize a functional j defined by j x j l t x t x t dt a over a class of arcs x whose values at a and b have been specified existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs conditions which must be strengthened in order that the standard necessary conditions apply the question arises what necessary conditions hold merely under hypotheses of existence theory say the classical tonelli conditions it is shown that given a solution x there exists a relatively open subset q of a b of full measure on which x is locally lipschitz and satisfies a form of the euler lagrange equation the main theorem of which this is a corollary can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions three such classes are identified and results of bernstein tonelli and morrey are extended one of these classes is of a novel nature and its study implies the new result that when l is independent of t the solution has essentially bounded derivative

another theorem of classical solvability in small for one dimensional variational problems in this paper we suggest a direct method for studying local minimizers of one dimensional variational problems which naturally complements the classical local theory this method allows us both to recover facts of the classical local theory and to resolve a number of problems which were previously unreachable the basis of these results is a regularity theory a priori estimates and compactness in c1 for solutions of obstacle problems with sufficiently close obstacles in these problems we establish that solutions exist and inherit regularity of the obstacles even under assumptions on integrands that are much weaker than those required in the classical local theory

bolza problems with discontinuous lagrangians and lipschitz continuity of the value function we study the local lipschitz continuity of the value function v associated with a bolza problem in the presence of a lagrangian l x q convex and uniformly superlinear in q but only borel measurable in x under these assumptions the associated integral functional is not lower semicontinuous with respect to the suitable topology which ensures the existence of minimizers so all results known in the literature fail to apply yet the lipschitz regularity of v does not depend on the existence of minimizers in fact it is enough to control the derivatives of quasi minimal curves but the problem is nontrivial due to the general growth conditions assumed here on l x cdot we propose a new approach based on suitable reparameterization arguments to obtain suitable a priori estimates on the lipschitz constants of quasi minimizers as a consequence of our analysis we derive the lipschitz continuity of v and a compactness result for value functions associated with sequences of locally equibounded discontinuous lagrangians

a condition on the value function both necessary and sufficient for full regularity of minimizers of one dimensional variational problems we study two point lagrange problems for integrands l l t u v p b f u y l t u t u t dt inf a u c v c w1l a b in lv a avv b 3 under very weak regularity hypotheses l is holder continuous and locally el liptic on each compact subset of ilt x iltn x iltn we obtain when l is of superlinear growth in v a characterization of problems in which the minimizers of p are c1 regular for all boundary data this characterization involves the behavior of the value function s ilt x ilt x r x in ilt defined by s a a b b inf a f namely all minimizers for p are c1 regular in neighborhoods of a and b if and only if s is lipschitz continuous at a a b b consequently problems p possessing no singular minimizers are characterized in cases where not even a weak form of the euler lagrange equations is available for guidance full reg ularity results for problems where l is nearly autonomous nearly independent of u or jointly convex in u v are presented

zero forcing methods for downlink spatial multiplexing in multiuser mimo channels the use of space division multiple access sdma in the downlink of a multiuser multiple input multiple output mimo wireless communications network can provide a substantial gain in system throughput the challenge in such multiuser systems is designing transmit vectors while considering the co channel interference of other users typical optimization problems of interest include the capacity problem maximizing the sum information rate subject to a power constraint or the power control problem minimizing transmitted power such that a certain quality of service metric for each user is met neither of these problems possess closed form solutions for the general multiuser mimo channel but the imposition of certain constraints can lead to closed form solutions this paper presents two such constrained solutions the first referred to as block diagonalization is a generalization of channel inversion when there are multiple antennas at each receiver it is easily adapted to optimize for either maximum transmission rate or minimum power and approaches the optimal solution at high snr the second known as successive optimization is an alternative method for solving the power minimization problem one user at a time and it yields superior results in some e g low snr situations both of these algorithms are limited to cases where the transmitter has more antennas than all receive antennas combined in order to accommodate more general scenarios we also propose a framework for coordinated transmitter receiver processing that generalizes the two algorithms to cases involving more receive than transmit antennas while the proposed algorithms are suboptimal they lead to simpler transmitter and receiver structures and allow for a reasonable tradeoff between performance and complexity

relaxation and regularity in the calculus of variations in this work we prove that if l t u is a continuous function in t and u borel measurable in with bounded non convex pieces in then any absolutely continuous solution u to the variational problem min abl t u t u t dt u w01 1 a b is quasi regular in the sense of tonelli i e u is locally lipschitz on an open set of full measure of a b under the further assumption that either l is lipschitz continuous in u locally uniformly in but not necessarily in t or l is invariant under a group of c1 transformations as in the noether s theorem without one of those further assumptions the solution could be not regular as shown by a recent example in gratwick and preiss 2010 13 our result is then optimal in this sense moreover we improve the standard hypothesis used so far in buttazzo et al 1998 1 clarke and vinter 1985 5 6 csornyei et al 2008 7 tonelli 1915 15 which have been the lipschitz continuity of l in u locally uniform in and t and some growth condition in we also show that the relaxed and the original problem have the same solutions without assuming any of the two further assumptions above this extends a result in mariconda and treu 2004 14 to the non autonomous case

reparametrizations and approximate values of integrals of the calculus of variations abstract we prove an approximation result that implies the non occurrence of the lavrentiev phenomenon

sum rate maximization in two way af mimo relaying polynomial time solutions to a class of dc programming problems sum rate maximization in two way amplify and forward af multiple input multiple output mimo relaying belongs to the class of difference of convex functions dc programming problems dc programming problems occur also in other signal processing applications and are typically solved using different modifications of the branch and bound method which however does not have any polynomial time complexity guarantees in this paper we develop two efficient polynomial time algorithms for the sum rate maximization in two way af mimo relaying the first algorithm guarantees to find at least a karush kuhn tucker kkt solution there is a strong evidence however that such a solution is actually globally optimal the second algorithm that is based on the generalized eigenvectors shows the same performance as the first one with reduced computational complexity the objective function of the problem is represented as a product of quadratic fractional ratios and parameterized so that its convex part versus the concave part contains only one or two optimization variables one of the algorithms is called polynomial time dc potdc and is based on semi definite programming sdp relaxation linearization and an iterative newton type search over a single parameter the other algorithm is called rate maximization via generalized eigenvectors rages and is based on the generalized eigenvectors method and an iterative search over two or one in its approximate version optimization variables we derive an upper bound for the optimal value of the corresponding optimization problem and show by simulations that this upper bound is achieved by both algorithms it provides an evidence that the algorithms find a global optimum the proposed methods are also superior to other state of the art algorithms

interpretation of lavrentiev phenomenon by relaxation the higher order case we consider integral functionals of the calculus of variations of the form f u fj f x u u u n dx defined for u e wn 0 1 and we show that the relaxed functional f with respect to the weak woc 1 0 1 convergence can be written as i f u j xu u n dx l u where the additional term l u the lavrentiev gap is explicitly identified in terms of f

interpretation of the lavrentiev phenomenon by relaxation we consider functionals of the calculus of variations of the form f u 01 f x u u dx defined for u w1 0 1 and we show that the relaxed functional f with respect to weak w1 1 0 1 convergence can be written as f u 01 f x u u dx l u where the additional term l u called the lavrentiev term is explicitly identified in terms of f

cyclic interference alignment for full duplex multi antenna cellular networks this paper studies full duplex fd cellular networks in which a base station bs operated in fd mode with multiple antennas supports multiple uplink and downlink users simultaneously in the same wireless channel two typical fd cellular scenarios are considered one with half duplex hd users and the other with fd users along with the fd bs for both the cases a novel constructive method is developed for finding a closed form interference alignment ia solution named cyclic ia the core idea behind this approach is to construct a set of loop equations enabling ia in a cyclic manner so that beamforming vectors are sequentially determined by solving an eigenvalue problem it is shown analytically that the proposed cyclic ia can achieve the optimal sum degrees of freedom dof when the number of user antennas is large enough to meet the derived conditions in particular it is shown that the proposed scheme achieves a twofold dof gain compared with conventional hd cellular networks even in the presence of inter link interference provided the number of users becomes large enough compared with the ratio of the number of bss and user antennas simulation results demonstrate that not only are the analytical dof results valid but under a practical multi cell scenario the proposed cyclic ia offers significant throughput gains depending on the cell radius

the approximation of higher order integrals of the calculus of variations and the lavrentiev phenomenon we prove the following approximation theorem given a function x a b to mathbb r n in the sobolev space bf w nu 1 1 nu geq 1 and epsilon 0 there exists a function x epsilon in bf w nu 1 infty such that begin gather int a b sum i 1 m l i x epsilon nu x epsilon nu 1 psi i t x epsilon x epsilon dotsc x epsilon nu int a b sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu epsilon begin aligned x epsilon a x a quad x epsilon b x b x epsilon a x a quad x epsilon b x b vdots x epsilon nu a x nu a quad x epsilon nu b x nu b end aligned end gather provided that for every i in 1 dotsc m l i psi i is continuous in a neighborhood of x l i is convex in its second variable and psi i evaluated along x has positive sign we discuss the optimality of our assumptions comparing them with an example of sarychev j dynam control systems 3 1997 pp 565 588 as a consequence we obtain the nonoccurrence of the lavrentiev phenomenon in particular the integral functional int a b l x nu x nu 1 does not exhibit the lavrentiev phenomenon for any given boundary values x a a x b b x a a x b b dotsc x nu a a nu x nu b b nu furthermore we prove the following necessary condition an action functional with lagrangian of the form sum i 1 m l i x nu x nu 1 psi i t x x dotsc x nu with nu geq 0 exhibiting the lavrentiev phenomenon takes the value infty in any neighborhood of a minimizer

iterative precoder design and user scheduling for block diagonalized systems the block diagonalization bd scheme is a low complexity suboptimal precoding technique for multiuser multiple input multiple output mimo downlink channels which completely precancels the multiuser interference accordingly the precoder of each user lies in the null space of other users channel matrices in this paper we propose an iterative algorithm using qr decompositions qrds to compute the precoders specifically to avoid dealing with a large concatenated matrix we apply the qrd to a sequence of matrices of lower dimensions one problem of bd schemes is that the number of users that can be simultaneously supported is limited due to zero interference constraints when the number of users is large a set of users must be selected and selection algorithms should be designed to exploit the multiuser diversity gain finding the optimal set of users requires an exhaustive search which has too high computational complexity to be practically useful based on the iterative precoder design this paper proposes a low complexity user selection algorithm using a greedy method in which the precoders of selected users are recursively updated after each selection step the selection metric of the proposed scheduling algorithm relies on the product of the squared row norms of the effective channel matrices which is related to the eigenvalues by the hadamard and schur inequalities an asymptotic analysis is provided to show that the proposed algorithm can achieve the optimal sum rate scaling of the mimo broadcast channel the numerical results show that the proposed algorithm achieves a good trade off between sum rate performance and computational complexity when users suffer different channel conditions providing fairness among users is of critical importance to address this problem we also propose two fair scheduling fs algorithms one imposing fairness in the approximation of the data rate and another directly imposing fairness in the product of the squared row norms of the effective channel matrices

weighted sum rate maximization in full duplex multi user multi cell mimo networks in this paper we focus on a multi user multi cell scenario with full duplex fd base stations and half duplex hd downlink dl and uplink ul users where all nodes are equipped with multiple antennas our goal is to design filters for weighted sum rate wsr maximization whilst taking into consideration the effect of transmitter and receiver distortion since wsr problems are non convex we exploit the relationship between rate and mean squared error in order to propose low complexity alternating optimization algorithms which are guaranteed to converge while the initial design assumes perfect channel state information csi we also move beyond this assumption and consider wsr problems under imperfect csi this is done using two types of error models the first is a norm bounded error model suitable for cases where the csi error is dominated by quantization issues and the second is a stochastic error model suitable for errors that occur during the channel estimation process itself results show that rates achieved in fd mode are higher than those achieved by the baseline hd schemes and demonstrate the robust performance of the proposed imperfect csi designs in addition we also extend our original wsr problem to one which maximizes the total dl rate subject to each ul user achieving a desired target rate this latter design can be used to overcome potential unfairness issues and ensure that all ul users are equally served in every time slot

existence and regularity in the small in the calculus of variations abstract a local existence theorem is proved for the basic problem in the calculus of variations that of minimizing l t x x dt over a class of functions x assuming given boundary conditions the lagrangian l is only assumed to be locally lipschitz and strictly convex in its x variable

generalized gradients of lipschitz functionals abstract the purpose of this article is threefold i to present in a unified fashion the theory of generalized gradients whose elements are at present scattered in various sources ii to give an account of the ways in which the theory has been applied iii to prove new results concerning generalized gradients of summation functionals pointwise maxima and integral functionals on subspaces of l these last mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by rockafellar ioffe and levin valadier and others

multifunctions of bounded variation abstract consider control systems described by a differential equation with a control term or more generally by a differential inclusion with velocity set f t x certain properties of state trajectories can be derived when it is assumed that f t x is merely measurable w r t the time variable t but sometimes a refined analysis requires the imposition of stronger hypotheses regarding the time dependence stronger forms of necessary conditions for minimizing state trajectories can be derived for example when f t x is lipschitz continuous w r t time it has recently become apparent that significant addition properties of state trajectories can still be derived when the lipschitz continuity hypothesis is replaced by the weaker requirement that f t x has bounded variation w r t time this paper introduces a new concept of multifunctions f t x that have bounded variation w r t time near a given state trajectory of special relevance to control we provide an application to sensitivity analysis

singular minimizers for regular one dimensional problems in the calculus of variations i i f x u x u x dx ja where a b is a finite interval denotes d dx f f x u p is c 0 and fpp 0 regularity we consider the problem of minimizing in the set a of absolutely continuous functions u a b r satisfying u a a u b 3 where a and 5 are given constants as is well known if a minimizer u of i in a is lipschitz continuous then u is smooth and satisfies the euler lagrange equation

one dimensional variational problems whose minimizers do not satisfy the euler lagrange equation in this paper we consider the problem of minimizing i u text int a b text f x u x u x dx 1 1 in the set a of absolutely continuous function u a b satisfying the end conditions u left a right text alpha text quad u left b right text beta text 1 2 where and are given constant in 1 1 a b is a finite interval denotes frac d dx and the integrand f f x u p is assumed to be smooth nonnegative and to satisfy the regularity condition f pp text 0 text 1 3

generalized one sided estimates for solutions of hamilton jacobi equations and applications on etudie les proprietes locales des solutions de l equation de hamilton jacobi u t h t x u 0 et plus generalement de l equation u t h t x u u 0 ou t o t et x r n

full duplex radio for uplink downlink wireless access with spatially random nodes a full duplex fd multiple antenna access point ap communicating with single antenna half duplex hd spatially random users to support simultaneous uplink ul downlink dl transmissions is investigated since fd nodes are inherently constrained by the loopback interference li we study precoding schemes for the ap based on maximum ratio combining mrc maximal ratio transmission mrt zero forcing and the optimal scheme for ul and dl sum rate maximization using tools from stochastic geometry in order to shed insights into the systems performance simple expressions for single antenna perfect li cancellation negligible internode interference cases are also presented we show that fd precoding at ap improves the ul dl sum rate and hence a doubling of the performance of the hd mode is achievable in particular our results show that these impressive performance gains remain substantially intact even if the li cancellation is imperfect furthermore relative performance gap between fd and hd modes increases as the number of transmit receive antennas becomes large while with the mrc mrt scheme increasing the receive antenna number at fd ap is more beneficial in terms of sum rate than increasing the transmit antenna number

a new variational inequality in the calculus of variations and lipschitz regularity of minimizers abstract we consider a problem of the calculus of variations of the form p minimize i x a b t x t x t d t x a x b subject to x w 1 m a b r n x t c a e x t t a b where a b r n r n r is borel measurable c is a cone is a nonempty subset of r n and is an arbitrary extended valued function we prove that if t t x satisfies a nonsmooth version of cesari s condition s then any local minimizer of p in the ac norm is lipschitz whenever is coercive the proof is obtained via a new variational inequality formulated here that holds under the extended condition s just borel measurability if is autonomous for a e t in a b w t x t x t v v t x t x t p t v 1 v 0 where p t is an absolutely continuous arc the derivative of which belongs to a suitable subdifferential of x t x t a e on a b the proof of the directional weierstrass type condition w is based on clarke s nonsmooth recent versions of the maximum principle

the perceptual organization of texture flow a contextual inference approach locally parallel dense patterns sometimes called texture flows define a perceptually coherent structure of particular significance to perceptual organization we argue that with applications ranging from image segmentation and edge classification to shading analysis and shape interpretation texture flows deserve attention equal to edge segment grouping and curve completion this paper develops the notion of texture flow from a geometrical point of view to argue that local measurements of such structures must incorporate two curvatures we show how basic theoretical considerations lead to a unique model for the local behavior of the flow and to a notion of texture flow good continuation this in turn translates to a specification of consistency constraints between nearby flow measurements which we use for the computation of globally piecewise coherent structure through the contextual framework of relaxation labeling we demonstrate the results on synthetic and natural images

local white matter geometry from diffusion tensor gradients we introduce a mathematical framework for computing geometrical properties of white matter fibers directly from diffusion tensor fields the key idea is to isolate the portion of the gradient of the tensor field corresponding to local variation in tensor orientation and to project it onto a coordinate frame of tensor eigenvectors the resulting eigenframe centered representation then makes it possible to define scalar indices or measures that describe the local white matter geometry directly from the diffusion tensor field and its gradient without requiring prior tractography we derive new scalar indices of 1 fiber dispersion and 2 fiber curving and we demonstrate them on synthetic and in vivo data finally we illustrate their applicability to a group study on schizophrenia

max min weighted downlink sinr with uplink sinr constraints for full duplex mimo systems in this paper we investigate a max min weighted signal to interference plus noise ratio sinr problem in a full duplex multiuser multiple input multiple output system where a full duplex capable base station bs equipped with multiple antennas communicates with multiple half duplex downlink and uplink users under the same system resources specifically we consider a practical scenario where the downlink minimum weighted sinr is maximized under specific sinr constraints for uplink users moreover the optimization is conducted by jointly considering the bs transmit power the transmit power of uplink users and bs transmit and receive beamforming this optimization problem is therefore subject to multiple uplink sinr constraints and multiple transmit power constraints due to the sinr constraints negative matrix components arise and hence the optimization problem cannot be directly solved by the standard approach i e perron frobenius theory we have provided an explicit iterative scheme to solve this joint optimization problem with a strict proof the proposed algorithm is proved to converge to karush kuhn tucker points simulation results show that our proposed algorithm has a fast convergence rate and leads to a better performance compared with other optimization techniques that do not jointly consider all parameters