gaussian interference channel capacity to within one bit the capacity of the two user gaussian interference channel has been open for 30 years the understanding on this problem has been limited the best known achievable region is due to han and kobayashi but its characterization is very complicated it is also not known how tight the existing outer bounds are in this work we show that the existing outer bounds can in fact be arbitrarily loose in some parameter ranges and by deriving new outer bounds we show that a very simple and explicit han kobayashi type scheme can achieve to within a single bit per second per hertz bit s hz of the capacity for all values of the channel parameters we also show that the scheme is asymptotically optimal at certain high signal to noise ratio snr regimes using our results we provide a natural generalization of the point to point classical notion of degrees of freedom to interference limited scenarios

on the minimum time function and the minimum energy problem a nonlinear case abstract for a semilinear control system in general banach spaces we prove results on exact and approximate controllability regularity of the minimum time function connections between the minimum time function and the minimum energy

the two user causal cognitive interference channel novel outer bounds and constant gap result for the symmetric gaussian noise channel in weak interference this paper studies the two user causal cognitive interference channel ccic where two transmitters aim to communicate independent messages to two different receivers via a common channel one source referred to as the cognitive is capable of overhearing the other source referred to as the primary through a noisy in band link and thus can assist in sending the primary s data the authors of this paper recently characterized to within a constant gap the capacity of the symmetric gaussian ccic in 1 the strong interference regime and 2 for a subset of the weak interference regime when the cooperation link is larger than a given threshold this paper characterizes to within a constant gap the capacity for the symmetric gaussian ccic in the regime that was still open to this end two novel outer bounds of the types 2 r mathsf p r mathsf c and r mathsf p 2 r mathsf c are derived for the class of injective semideterministic ccics where the noises at the different source destination pairs are independent these outer bounds as well as an achievable rate region based on gelfand pinsker binning superposition coding and simultaneous decoding at the receivers are then specialized to the gaussian noise case it is shown that the novel outer bounds are necessary to characterize the capacity within a constant gap when the cooperation link is weaker than the direct links that is in this regime unilateral cooperation leaves some system resources underutilized

some compact classes of open sets under hausdorff distance and application to shape optimization in this paper we introduce three new classes of open sets in a general euclidean space mathbb r n it is shown that every class of open sets is compact under the hausdorff distance the result is then applied to a shape optimization problem of elliptic equation the existence of the optimal solution is presented

new inner and outer bounds for the memoryless cognitive interference channel and some new capacity results the cognitive interference channel is a two user interference channel in which one transmitter is non causally provided with the message of the other transmitter this channel model has been extensively studied in the past years and capacity results have been proved for certain classes of channels this paper presents new inner and outer bounds for the capacity region of the cognitive interference channel as well as new capacity results previously proposed outer bounds are expressed in terms of auxiliary random variables for which no cardinality constraint of their alphabet is known consequently it is not possible to evaluate such outer bounds explicitly for a given channel the outer bound derived in this work is based on an idea originally devised by sato for channels without receiver cooperation and results in an outer bound that does not contain auxiliary random variables thus allowing it to be more easily evaluated the inner bound presented in this work which includes rate splitting superposition coding a broadcast channel like binning scheme and gel fand pinsker coding is the largest known to date and is explicitly shown to include all previously proposed achievable rate regions the novel inner and outer bounds are shown to coincide in certain cases in particular capacity is proved for a class of channels in the so called better cognitive decoding regime which includes the regimes in which capacity was known finally the capacity region of the semi deterministic cognitive interference channel in which the signal at the cognitive receiver is an arbitrary deterministic function of the channel inputs is established

optimal controls of 3 dimensional navier stokes equations with state constraints this work is concerned with the maximum principles for optimal control problems governed by 3 dimensional navier stokes equations some types of state constraints time variables are considered

continuous dependence of optimal control to controlled domain of actuator for heat equation abstract in this paper we consider continuous dependence of the optimal control with respect to the actuator domain which is varying as open subset in the spatial domain for a multi dimensional heat equation both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to target set in minimal time interval is actually the minimal energy of driving the system to target set in this minimal time interval and visa versa it is shown that both optimal control and optimal cost are continuous with respect to open controlled actuator domain under the lebesgue measure

local langlands correspondence for classical groups and affine hecke algebras using the results of colette moeglin on the representations of p adic classical groups based on methods of james arthur and its relation with representations of affine hecke algebras established by the author we show that the category of smooth complex representations of a quasi split p adic classical group and its pure inner forms is naturally decomposed into subcategories which are equivalent to a tensor product of categories of unipotent representations of classical groups a statement of this kind had been conjectured by g lusztig all classical groups general linear orthogonal symplectic and unitary groups appear in this context we get also parameterizations of representations of affine hecke algebras which seem not all to be in the literature yet all this sheds some light on what is known as the stable bernstein center

feedback capacity of the gaussian interference channel to within 2 bits we characterize the capacity region to within 2 bits s hz and the symmetric capacity to within 1 bit s hz for the two user gaussian interference channel ic with feedback we develop achievable schemes and derive a new outer bound to arrive at this conclusion one consequence of the result is that feedback provides multiplicative gain at high signal to noise ratio the gain becomes arbitrarily large for certain channel parameters this finding is in contrast to point to point and multiple access channels where feedback provides no gain and only bounded additive gain respectively the result makes use of a linear deterministic model to provide insights into the gaussian channel this deterministic model is a special case of the el gamal costa deterministic model and as a side generalization we establish the exact feedback capacity region of this general class of deterministic ics

optimal distribution of the internal null control for the one dimensional heat equation abstract this paper addresses the problem of optimizing the distribution of the support of the internal null control of minimal l 2 norm for the 1 d heat equation a measure constraint is imposed on the support but no topological assumption such as the number of connected components therefore the problem typically lacks of solution in the class of characteristic functions and needs of relaxation we show that the relaxed formulation is obtained by replacing the set of characteristic functions by its convex envelope the proof requires that the observability constant related to the control problem be uniform with respect to the support property which is obtained by the control transmutation method the optimality conditions of the relaxed problem as well as the case where the number of connected components is fixed a priori are also discussed several numerical experiments complete the study and suggest the ill posedness of the problem in contrast to the wave situation

a new achievable rate region for the interference channel a new achievable rate region for the general interference channel which extends previous results is presented and evaluated the technique used is a generalization of superposition coding to the multivariable case a detailed computation for the gaussian channel case clarifies to what extent the new region improves previous ones the capacity of a class of gaussian interference channels is also established

outer bounds on the capacity of gaussian interference channels two outer bounds on the capacity region of the two user gaussian interference channel ifc are derived the idea of the first bound is to let a genie give each receiver just enough information to decode both messages this bound unifies and improves the best known outer bounds of sato and carleial furthermore the bound extends to discrete memoryless ifcs and is shown to be equivalent to another bound of carleial the second bound follows directly from existing results of costa and sato and possesses certain optimality properties for weak interference

optimal control of parabolic differential equations with two point boundary state constraints this paper deals with the necessary conditions of optimality for some optimal control problems governed by some parabolic differential equations involving monotone graphs the two point boundary time variable state constraints are discussed some analyses on the state equations are given

dependence balance bounds for single output two way channels if in a transmission the inputs of a single output two way channel exhibit some interdependence this dependence must have been created during earlier transmissions the idea that no more dependence can be consumed than is produced is used to obtain new upper bounds to the capacity region of the discrete memoryless single output two way channel with these upper bounds it is shown that c e shannon 1961 inner bound region is the capacity region for channels in a certain class and the zhang berger schalkwijk upper bound 1986 for blackwell s multiplying channel is improved upon

regularity of the minimum time function and minimum energy problems the linear case we prove new results on the continuity properties and on reachable sets of the minimum time function associated with a linear dynamical system in a separable hilbert space h part of the results are new also in the finite dimensional case the regularity results are stated thanks to the result we obtain on the connection between the minimum time function and the minimum energy

inner and outer bounds for the gaussian cognitive interference channel and new capacity results the capacity of the gaussian cognitive interference channel a variation of the classical two user interference channel where one of the transmitters referred to as cognitive has knowledge of both messages is known in several parameter regimes but remains unknown in general this paper provides a comparative overview of this channel model as it proceeds through the following contributions first several outer bounds are presented a a new outer bound based on the idea of a broadcast channel with degraded message sets and b an outer bound obtained by transforming the channel into channels with known capacity next a compact fourier motzkin eliminated version of the largest known inner bound derived for the discrete memoryless cognitive interference channel is presented and specialized to the gaussian noise case where several simplified schemes with jointly gaussian input are evaluated in closed form and later used to prove a number of results these include a new set of capacity results for a the primary decodes cognitive regime a subset of the strong interference regime that is not included in the very strong interference regime for which capacity was known and b the s channel in strong interference in which the primary transmitter does not interfere with the cognitive receiver and the primary receiver experiences strong interference next for a general gaussian channel the capacity is determined to within one bit s hz and to within a factor two regardless of the channel parameters thus establishing rate performance guarantees at high and low snr respectively the paper concludes with numerical evaluations and comparisons of the various simplified achievable rate regions and outer bounds in parameter regimes where capacity is unknown leading to further insight on the capacity region

on the han kobayashi region for theinterference channel in this correspondence we derive a simplified description of the han kobayashi rate region for the general interference channel using this result we establish that the recently discovered chong motani garg rate region is a new representation of the han kobayashi region moreover a tighter bound for the cardinality of the time sharing auxiliary random variable emerges from our simplified description

an approach to the optimal time for a time optimal control problem of an internally controlled heat equation in this paper we study the optimal time for a time optimal control problem mathcal p where the state system is an internally controlled heat equation by projecting the original problem via the finite element method we obtain another time optimal control problem mathcal p hl governed by a linear system of ordinary differential equations here h and l are the mesh sizes of the finite element spaces of the state space and the control space respectively the purpose of this study is to approach the optimal time for the problem mathcal p through the optimal time for the problem mathcal p hl we prove that the optimal time for the problem mathcal p hl converges to the optimal time for the problem mathcal p when h and l are approaching zero more significantly we obtain error estimates between the optimal times in terms of h and l for certain cases

on convergence of boundary hausdorff measure and application to a boundary shape optimization problem this is the second part of our recent work siam j control optim 50 2012 pp 222 242 on new open sets class and related shape optimization in this paper we are concerned with a boundary shape optimization problem it is shown that the convergence of the open sets class under the hausdorff distance implies the convergence of the hausdorff measure on the boundary the existence of the boundary shape optimization is concluded

approximation of time optimal controls for heat equations with perturbations in the system potential in this paper we study a certain approximation property for a time optimal control problem of the heat equation with l infty potential we prove that the optimal time and the optimal control to the same time optimal control problem for the heat equation where the potential has a small perturbation are close to those for the original problem we also verify that for the heat equation with a small perturbation in the potential one can construct a new time optimal control problem which has the same target as that of the original problem but has a different control constraint bound from that of the original problem such that the new and the original problems share the same optimal time and meanwhile the optimal control of the new problem is close to that of the original one the main idea to approach such approximation is an appropriate use of an equivalence theorem of minimal norm and minimal time control problems for the heat equations under consideration this theorem was first established by wa

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

on the equivalence of minimal time and minimal norm controls for internally controlled heat equations we prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the euclidean space with homogeneous dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves we consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time as a consequence of this equivalence using the well known variational characterization of minimal norm controls we establish necessary and sufficient conditions for the minimal time and the corresponding control

l infty null controllability for the heat equation and its consequences for the time optimal control problem in this paper we establish a certain l infty null controllability for the internally controlled heat equation in omega times 0 t with the control restricted to a product set of an open nonempty subset in omega and a subset of positive measure in the interval 0 t based on this we obtain a bang bang principle for the time optimal control of the heat equation with controls taken from the set mathcal u ad u cdot t 0 infty rightarrow l 2 omega measurable u cdot t in u a e in t where u is a closed and bounded subset of l 2 omega namely each optimal control u cdot t of the problem satisfies necessarily the bang bang property u cdot t in partial u for almost all t in 0 t where partial u denotes the boundary of the set u and t is the optimal time we also get the uniqueness of the optimal control when the target set s is convex and the control set u is a closed ball

approximate capacity region for the symmetric gaussian interference channel with noisy feedback recent results have shown that feedback can significantly increase the capacity of interference networks this paper considers the impact of noise on such gains due to feedback in particular this paper considers the two user linear deterministic interference channel with noisy feedback as a stepping stone to characterize the approximate capacity region for the two user gaussian interference channel with noisy feedback first the capacity region for the symmetric linear deterministic interference channel with noisy feedback is obtained it is shown that noisy feedback enlarges the capacity region if and only if the number of feedback bits l is greater than a certain threshold l it is found that excluding the regime 1 2 leq alpha leq 2 where alpha is the normalized interference level in which even full feedback does not increase symmetric capacity this threshold l is equal to the per user symmetric capacity without feedback one of the key ideas is a novel converse outer bounding technique for the weighted sum rates 2r 1 r 2 and r 1 2r 2 these results and the techniques developed for the linear deterministic model are then applied to characterize inner bounds and outer bounds for the symmetric gaussian interference channel with noisy feedback the outer bounds are shown to be at most 4 7 b s hz away from the achievable rate region as a corollary the generalized degrees of freedom region which approximates the capacity region of the symmetric gaussian interference channel at high snr is found