a new outer bound via interference localization and the degrees of freedom regions of mimo interference networks with no csit the two user multiple input multiple output mimo interference and cognitive radio channels are considered where the ith transmitter and the ith receiver have mi and ni antennas respectively in particular the degrees of freedom dof regions of these channels are studied under the assumption of having no channel state information at the transmitters making certain assumptions about the distributions of channel matrices of increasing generality respectively huang zhu and guo and the authors of this paper characterized the dof region of the mimo interference channel for all values of the four tuple m1 m2 n1 n2 except when min m1 n1 n2 m2 or min m2 n2 n1 m1 more recently for isotropic fading distributions zhu and guo have solved this latter case by providing a tight outer bound to the dof region here a simpler and more widely applicable proof of that outer bound is given based on the idea of interference localization using the same idea under rayleigh fading the dof region is then also established for the mimo cognitive radio channel when the second transmitter is cognitive with min m1 m2 n1 n2 m2 the only class for which the inner and outer bounds previously reported by the authors were not tight thereby completing the dof region characterization of this channel under rayleigh fading for all values of m1 m2 n1 n2

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

variational mass lumping in the partition of unity method this paper is concerned with the construction of a variational mass lumping scheme for the partition of unity methods the presented approach is applicable to arbitrary local approximation spaces and any nonnegative partition of unity we give improved error bounds for the partition of unity method using a nonnegative partition of unity and show that our lumped mass matrix is conservative at least for any f in v rm pu such that f omega cap omega i in v i omega i for all patches omega i we present numerical results using smooth higher order discontinuous and singular local approximation spaces confirming our theoretical results

the degrees of freedom region of the mimo interference channel with shannon feedback the two user multiple input multiple output mimo fast fading interference channel ic with an arbitrary number of antennas at each of the four terminals is studied under the settings of shannon feedback and output feedback wherein channel matrices and outputs or just the channel outputs respectively are available to the transmitters with a delay while for most numbers of antennas at the four terminals the degree of freedom dof regions with shannon feedback are equal to that with just delayed channel state information csit it is shown that for the rest of the mimo ics the dof region with shannon feedback is strictly larger than the dof region with just delayed csit to realize these dof gains with shannon feedback a new interference alignment scheme is obtained wherein transmitter cooperation made possible by output feedback in addition to delayed csit is employed to effect a more efficient form of interference alignment than is feasible with previously known schemes that use just delayed csit the dof region for output only feedback is also obtained for all but a class of mimo ics that satisfy one of two inequalities involving the numbers of antennas moreover the dof region for shannon feedback is shown to be applicable to two limited shannon feedback settings where the transmitters have knowledge only of certain channel matrices and certain outputs with delay

stable periodic solution of a discrete periodic lotka volterra competition system abstract this paper discusses a discrete lotka volterra competition system we first obtain the persistence of the system assuming that the coefficients in the system are periodic we obtain the existence of a periodic solution moreover under some additional conditions this periodic solution is globally stable our results not only reduce to those for the scalar equation when there is no coupling but also improve and complement some in the literature

harmless delays for uniform persistence abstract this paper studies a two dimensional predator prey system with a finite number of discrete delays our purpose is to demonstrate that the time delays are harmless for uniform persistence of the solutions of the system the results are obtained by constructing suitable persistence functionals

a receiver centric approach to interference management fairness and outage optimization effective interference management in the multiuser interference channel necessitates that the users form their transmission and interference management decisions in coordination and adapt them to the state of the channel establishing such coordination often facilitated through information exchange is prohibitive in fast varying channels especially when the network size grows this paper focuses on the multiuser gaussian interference channel and offers a receiver centric approach to interference management in this approach the transmitters deploy rate splitting and superposition coding to generate their messages according to independent gaussian codebooks the receivers can freely decode any arbitrary set of interfering messages along with their designated messages in any desired joint or ordered fashion and treat the rest of the interferers as gaussian noise the proposed receiver centric interference management approach is applied to two class of problems outage optimization and fairness constrained rate allocation and constructive proofs are provided to establish the following properties for the proposed approach 1 the optimal set of codebooks to be decoded by each receiver is a local decision made by each receiver based on its local channel state information csi 2 the globally optimal transmission rates are related to locally optimal rates computed by the receivers based on their local information which implies that the transmitters do not require explicit knowledge of the csi and can determine their rates via limited feedback from the receivers and 3 obtaining the optimal interference management strategy at each receiver has controlled complexity

approximate capacity of fast fading interference channels with no instantaneous csit we develop a characterization of fading models which assigns a number called logarithmic jensen s gap to a given fading model we show that as a consequence of a finite logarithmic jensen s gap an approximate capacity region can be obtained for fast fading interference channels ff ics for several scenarios we illustrate three instances where a constant capacity gap can be obtained as a function of the logarithmic jensen s gap first for an ff ic with neither feedback nor instantaneous channel state information at transmitter csit if the fading distribution has finite logarithmic jensen s gap we show that a rate splitting scheme based on the average interference to noise ratio can achieve its approximate capacity second we show that a similar scheme can achieve the approximate capacity of ff ic with feedback and delayed csit if the fading distribution has finite logarithmic jensen s gap third when this condition holds we show that point to point codes can achieve approximate capacity for a class of ff ics with feedback we prove that the logarithmic jensen s gap is finite for common fading models including rayleigh and nakagami fading thereby obtaining the approximate capacity region of ff ic with these fading models

interference and x networks with noisy cooperation and feedback the gaussian k user interference and m times k x channels are investigated with no instantaneous channel state information at transmitters csit first it is assumed that the csi is fed back to all nodes after a finite delay delayed csit and furthermore the transmitters operate in full duplex mode i e they can transmit and receive simultaneously achievable results on the degrees of freedom dofs of these channels under the above assumption are obtained it is observed that in contrast with no csit and full csit models when csit is delayed the achievable dofs for both channels with the full duplex transmitter cooperation are greater than the available achievable results on their dof without transmitter cooperation then k user interference and k times k x channels are considered with output feedback wherein the channel output of each receiver is causally fed back to its corresponding transmitter our achievable results with output feedback demonstrate strict dof improvements over those with the full duplex delayed csit when k 5 in the k user interference channel and k 2 in the k times k x channel next the combination of delayed csit and output feedback known as shannon feedback is studied and strictly higher dofs compared with the output feedback model are achieved in the k user interference channel when k 5 or k 6 and in the k times k x channel when k 2

capacity bounds for the gaussian interference channel the capacity region of the two user gaussian interference channel ic is studied three classes of channels are considered weak one sided and mixed gaussian ics for the weak gaussian ic a new outer bound on the capacity region is obtained that outperforms previously known outer bounds the sum capacity for a certain range of channel parameters is derived for this range it is proved that using gaussian codebooks and treating interference as noise are optimal it is shown that when gaussian codebooks are used the full han kobayashi achievable rate region can be obtained by using the naive han kobayashi achievable scheme over three frequency bands equivalently three subspaces for the one sided gaussian ic an alternative proof for the sato s outer bound is presented we derive the full han kobayashi achievable rate region when gaussian codebooks are utilized for the mixed gaussian ic a new outer bound is obtained that outperforms previously known outer bounds for this case the sum capacity for the entire range of channel parameters is derived it is proved that the full han kobayashi achievable rate region using gaussian codebooks is equivalent to that of the one sided gaussian ic for a particular range of channel parameters

the degrees of freedom region and interference alignment for the mimo interference channel with delayed csit the degrees of freedom dof region of the two user multiple input multiple output mimo interference channel ic is studied under the assumptions of fast fading and delayed channel state information csi at the transmitters csit under our fast fading assumption the channel matrices vary independently across time so that delayed csit is equivalent to outdated csit in particular the dof region under delayed csit is established for the general mimo ic with an arbitrary numbers of antennas at each of the four terminals toward this end a set of outer bounds to the dof region of the general mimo ic is derived these bounds are then shown to be tight by developing dof region optimal interference alignment schemes a comparison of the dof region of the mimo ic under the delayed csit assumption with those under the two extremes of instantaneous csit and no csit assumptions is made this comparison reveals that there are nonempty classes of mimo ics defined by certain relationships between the numbers of antennas at the four terminals that correspond to each of the following four scenarios the no csit dof region is strictly contained by or is equal to the delayed csit dof region which in turn is strictly contained by or is equal to the instantaneous csit dof region it is notable that within the class of mimo ics for which the delayed csit dof region is strictly larger than the no csit dof region there is a subclass for which the interference alignment scheme which uses just delayed csit achieves the entire dof region previously known to be achievable only with instantaneous csit

orbital enriched flat top partition of unity method for the schrodinger eigenproblem abstract quantum mechanical calculations require the repeated solution of a schrodinger equation for the wavefunctions of the system from which materials properties follow recent work has shown the effectiveness of enriched finite element type galerkin methods at significantly reducing the degrees of freedom required to obtain accurate solutions however time to solution has been adversely affected by the need to solve a generalized rather than standard eigenvalue problem and the ill conditioning of associated system matrices in this work we address both issues by proposing a stable and efficient orbital enriched partition of unity method to solve the schrodinger boundary value problem in a parallelepiped unit cell subject to bloch periodic boundary conditions in the proposed partition of unity method the three dimensional domain is covered by overlapping patches with a compactly supported weight function associated with each patch a key ingredient in our approach is the use of non negative weight functions that possess the flat top property i e each weight function is identically equal to unity over some finite subset of its support this flat top property provides a pathway to devise a stable approximation over the whole domain on each patch we use p th degree orthogonal legendre polynomials that ensure p th order completeness and in addition include eigenfunctions of the radial schrodinger equation furthermore we adopt a variational lumping approach to construct a block diagonal overlap matrix that yields a standard eigenvalue problem for which there exist efficient eigensolvers the accuracy stability and efficiency of the proposed method is demonstrated for the schrodinger equation with a harmonic potential as well as a localized gaussian potential we show that the proposed approach delivers optimal rates of convergence in the energy and the use of orbital enrichment significantly reduces the number of degrees of freedom for a given desired accuracy in the energy eigenvalues while the stability of the enriched approach is fully maintained

the generalized diversity multiplexing tradeoff of the mimo z interference channel the generalized diversity multiplexing tradeoff gdmt of the two user quasi static fading multi input multi output mimo z interference channel z ic is established for the general case with an arbitrary number of antennas at each node under the assumptions of full channel state information at the transmitters csit and a short term average power constraint in the gdmt framework which captures the rate versus reliability tradeoff in the high signal to noise ratio snr regime the direct link snr and cross link interference to noise ratio inr are allowed to be disparate so that their ratios relative to a nominal snr in the decibel scale i e the snr and inr exponents are arbitrary and fixed it is shown that a simple han kobayashi message splitting partial interference decoding scheme that uses only partial csit in which the second transmitter s signal depends only on its cross link channel matrix and the first user s transmit signal does not need any csit can achieve the full csit gdmt of the mimo z ic the gdmt of the mimo z ic under the no csit assumption is also obtained for some range of multiplexing gains the size of this range depends on the numbers of antennas at the four nodes and the snr and inr exponents of the direct and cross links respectively for certain classes of channels including those in which the interfered receiver has more antennas than do the other nodes or when the inr exponent is greater than a certain threshold the gdmt of the mimo z ic under the no csit assumption is also completely characterized

achievable sum dof of the k user mimo interference channel with delayed csit this paper considers a k user multiple input multiple output mimo interference channel ic where 1 the channel state information obtained by the transmitters csit is completely outdated and 2 the number of transmit antennas at each transmitter i e m is greater than the number of receive antennas at each user i e n the usefulness of the delayed csit was first identified in a k phase retrospective interference alignment ria scheme proposed by maddah ali and tse for the multiple input single output broadcast channel but the extension to the mimo ic is a non trivial step as each transmitter only has the message intended for the corresponding user recently abdoli et al focused on a single input single output ic and solved such bottleneck by inventing a k phase ria with distributed overheard interference retransmission in this paper we propose two k phase ria schemes suitable for the mimo ic by generalizing and integrating some key features of both abdoli s and maddah ali s works the two schemes jointly yield the best known sum degrees of freedom dof performance so far for the case m n geq k the achieved sum dof is asymptotically given by 64 15 n when k to infty

ergodic interference alignment this paper develops a new communication strategy ergodic interference alignment for the k user interference channel with time varying fading at any particular time each receiver will see a superposition of the transmitted signals plus noise the standard approach to such a scenario results in each transmitter receiver pair achieving a rate proportional to 1 k its interference free ergodic capacity however given two well chosen time indices the channel coefficients from interfering users can be made to exactly cancel by adding up these two observations each receiver can obtain its desired signal without any interference if the channel gains have independent uniform phases this technique allows each user to achieve at least 1 2 its interference free ergodic capacity at any signal to noise ratio prior interference alignment techniques were only able to attain this performance as the signal to noise ratio tended to infinity extensions are given for the case where each receiver wants a message from more than one transmitter as well as the x channel case with two receivers where each transmitter has an independent message for each receiver finally it is shown how to generalize this strategy beyond gaussian channel models for a class of finite field interference channels this approach yields the ergodic capacity region

on generation of lumped mass matrices in partition of unity based methods summary the partition of unity based methods such as the extended finite element method and the numerical manifold method are able to construct global functions that accurately reflect local behaviors through introducing locally defined basis functions beyond polynomials in the dynamic analysis of cracked bodies using an explicit time integration algorithm as a result huge difficulties arise in deriving lumped mass matrices because of the presence of those physically meaningless degrees of freedom associated with those locally defined functions observing no spatial derivatives of trial or test functions exist in the virtual work of inertia force we approximate the virtual work of inertia force in a coarser manner than the virtual work of stresses where we inversely utilize the from local to global skill the proposed lumped mass matrix is strictly diagonal and can yield the results in agreement with the consistent mass matrix but has more excellent dynamic property than the latter meanwhile the critical time step of the numerical manifold method equipped with an explicit time integration scheme and the proposed mass lumping scheme does not decrease even if the crack in study approaches the mesh nodes a very excellent dynamic property copyright 2017 john wiley sons ltd

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

almost periodic sequence solution of a discrete hassell varley predator prey system with feedback control in this paper we investigate a discrete hassell varley response function predator prey system with feedback control first sufficient conditions are established for the permanence of the system then assuming that the coefficients of the system is almost periodic sequences we obtain conditions for the existence and unique of the almost periodic solution moreover the almost periodic solution we obtained is uniformly asymptotically stable

stable periodic solution of a discrete periodic lotka volterra competition system with a feedback control this paper discusses a discrete lotka volterra competition system with feedback controls we first obtain the persistence of the system assuming that the coefficients in the system are periodic we obtain the existence of a periodic solution moreover under some additional conditions this periodic solution is globally stable

degrees of freedom for the mimo interference channel in this correspondence we show that the exact number of spatial degrees of freedom dof for a two user nondegenerate full rank channel matrices multiple input multiple output mimo gaussian interference channel with m1 m2 antennas at transmitters 1 2 and n1 n2 antennas at the corresponding receivers and perfect channel knowledge at all transmitters and receivers is min m1 m2 n1 m2 max m1 n2 max m2 n1 a constructive achievability proof shows that zero forcing is sufficient to achieve all the available dof on the two user mimo interference channel we also show through an example of a share and transmit scheme how the gains of transmitter cooperation may be entirely offset by the cost of enabling that cooperation so that the available dof are not increased

isogeometric analysis for explicit elastodynamics using a dual basis diagonal mass formulation abstract we propose a method to obtain diagonal mass matrices for nurbs based approximation spaces by a dual lumping method the use of lumped mass matrices is of great importance in elastodynamics problems as they can be employed in explicit time integration schemes which do not require the solution of a linear system in finite elements several well established methods such as row sum diagonal scaling or nodal quadrature methods have been used to obtain lumped mass matrices for different applications however for higher order and higher continuity approximation spaces such as those derived from nurbs these approaches have only limited second order accuracy in this work we derive a dual basis which has optimal approximation and dispersion properties while maintaining local support the dual space has discontinuities at the element boundaries knots and it is used to provide the test functions in the context of a petrov galerkin method this results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high order accuracy numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity

uniform persistence in functional differential equations abstract in this paper we investigate the question of uniform persistence for retarded functional differential equations by utilizing liapunov like functions razumikhin techniques and differential inequalities we are able to establish criteria for uniform persistence analogous to those obtained by others for ordinary differential equations difference equations and reaction diffusion equations we apply these criteria to some well known biological models with delay our results indicate that the conditions which guarantee the existence of an interior equilibrium are enough to ensure uniform persistence moreover these conditions are equivalent to uniform persistence for the cases without delay as well

degrees of freedom region of the mimo x channel we provide achievability as well as converse results for the degrees of freedom region of a multiple input multiple output mimo x channel i e a system with two transmitters two receivers each equipped with multiple antennas where independent messages need to be conveyed over fixed channels from each transmitter to each receiver the inner and outer bounds on the degrees of freedom region are tight whenever integer degrees of freedom are optimal for each message with m 1 antennas at each node we find that the total sum rate degrees of freedom are bounded above and below as 1 les eta x les 4 3 if m 1 and channel matrices are nondegenerate then the precise degrees of freedom eta x 4 3 m thus the mimo x channel has noninteger degrees of freedom when m is not a multiple of 3 simple zero forcing without dirty paper encoding or successive decoding suffices to achieve the 4 3 m degrees of freedom if the channels vary with time frequency then the channel with single antennas m 1 at all nodes has exactly 4 3 degrees of freedom the key idea for the achievability of the degrees of freedom is interference alignment i e signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired we also explore the increase in degrees of freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio

existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls abstract sufficient conditions are obtained for the existence and global stability of a positive periodic solution in some periodic integrodifferential systems with feedback controls by using the technique of coincidence degree and lyapunov functional

the degrees of freedom of isotropic mimo interference channels without state information at the transmitters this paper fully determines the degree of freedom dof region of two user interference channels with arbitrary number of transmit and receive antennas in the case of isotropic and independent or block wise independent fading where the channel state information is available to the receivers but not to the transmitters thus the dof gap between previous upper and lower bounds due to the authors vaze and varanasi and huang jafar shamai and vishwanath is closed the dof result characterizes the capacity region to the first order of the logarithm of the signal to noise ratio snr in the high snr regime the dof region is achievable by using random gaussian codebooks independent of the channel states which implies that it is impossible to increase the dof using beamforming and interference alignment in the absence of channel state information at the transmitters

the degrees of freedom of two unicast layered mimo interference networks with feedback two unicast layered interference networks under the shannon feedback and limited shannon feedback settings are studied from a degrees of freedom dof perspective the main focus is on the archetypal two hop multiple input multiple output mimo m n times m n times m n network denoted here as the m n 3 network which is a layered two unicast network with two transmitters two relays and two receivers with the first hop network between the transmitters and the relays and the second hop network between the relays and the receivers both being mimo gaussian interference channels in particular the first transmitter receiver pair and the first relay have m antennas each and the second transmitter receiver pair and the second relay having n antennas each the full shannon feedback setting for the m n 3 network is one in which the transmitters have delayed knowledge of the first and second hop channel coefficients and the relay and destination outputs and the relays have delayed knowledge of the second hop channel coefficients and destination outputs the dof region under this setting is established a key result in this paper shows that this shannon feedback dof region can in fact be achieved with much less side information under what we refer to as the limited shannon feedback setting wherein the transmitters have no channel state or output feedback whatsoever and only the m antenna relay assuming m geq n has delayed knowledge of the coefficients of the second hop channel and of only the received signal of the n antenna receiver for this limited shannon feedback setting the dofs region of the m n 3 network is established by introducing a retro cooperative interference alignment scheme these dof region results for the m n 3 network with feedback are also extended to more general layered interference networks including the two unicast l hop layered networks as well as layered networks with more general numbers of antennas at the various terminals

positive periodic solutions of a class of delay differential system with feedback control with the help of continuation theorem based on gaines and mawhin s coincidence degree sufficient and realistic conditions are obtained for the global existence of positive periodic solutions for the delay differential system with feedback control when these results are applied to some special delay population models some new results are obtained and some known results are generalized

the capacity of the gaussian interference channel under strong interference corresp the capacity region of a gaussian interference channel with two separate messages is obtained for the case of moderately strong interference it is shown that the region coincides with the one where both messages are required in both receiving terminals

optimal dof of the k user broadcast channel with delayed and imperfect current csit this work identifies the optimal degrees of freedom dof of the k user miso broadcast channel bc with delayed channel state information at the transmitter csit and with additional current noisy csit where the current channel estimation error scales in p alpha for alpha in 0 1 these two settings had in the past been studied separately the setting of imperfect current csit has attracted considerable interest over the last decade while the setting of delayed csit was studied in the seminal work of maddah ali and tse in 2010 where an optimal dof of k sum k 1 k frac 1 k was established since then there have been several efforts to combine the two settings of delayed and imperfect current csit our work establishes for the first time the optimal dof in this joint setting capitalizing on a novel transmission scheme that is presented here which combines a structurally new approach of handling past and current interference to achieve the optimal performance we establish the once elusive optimal dof to be of the form alpha k 1 alpha k left sum k 1 k frac 1 k right this further shows that the two types of dof gains from current and delayed csit can be combined additively

sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control with the help of a continuation theorem based on gaines and mawhin s coincidence degree easily verifiable criteria are established for the global existence of positive periodic solutions of a neutral model with periodic delays and feedback controldndt n t a t email protected t n t email protected i 1nb i t n email protected i t email protected i 1nc i t n email protected i t email protected t u t email protected i 1nd i t u email protected i t dudt e t u t f t n t email protected i 1ng i t n email protected i t our results extend and improve existing results and have further applications in population dynamics

global stability of discrete population models with time delays and fluctuating environment the global stability of a discrete population model of volterra type is studied the model incorporates time delays and allows for a fluctuating environment by linearization of the model at positive solutions and construction of liapunov functionals sufficient conditions are obtained to ensure a positive solution of the model is stable and attracts all positive solutions

topological interference management with transmitter cooperation interference networks with no channel state information at the transmitter except for the knowledge of the connectivity graph have been recently studied under the topological interference management framework in this paper we consider a similar problem with topological knowledge but in a distributed broadcast channel setting i e a network where transmitter cooperation is enabled we show that the topological information can also be exploited in this case to strictly improve the degrees of freedom dof as long as the network is not fully connected which is a reasonable assumption in practice achievability schemes from graph theoretic and interference alignment perspectives are proposed together with outer bounds built upon generator sequence the concept of compound channel settings and the relation to index coding we characterize the symmetric dof for the so called regular networks with constant number of interfering links and identify the sufficient and or necessary conditions for the arbitrary network topologies to achieve a certain amount of symmetric dof

linear degrees of freedom of the x channel with delayed csit we establish the degrees of freedom dof of the two user x channel with delayed channel knowledge at transmitters i e delayed channel state information at the transmitters csit assuming linear coding strategies at the transmitters we derive a new upper bound and characterize the linear dof of this network to be 6 5 the converse builds upon our development of a general lemma that shows that if two distributed transmitters employ linear strategies the ratio of the dimensions of received linear subspaces at the two receivers cannot exceed 3 2 due to delayed csit as a byproduct we also apply this general lemma to the three user interference channel with delayed csit thereby deriving a new upper bound of 9 7 on its linear dof this is the first bound that captures the impact of delayed csit on the dof of this network under the assumption of linear encoding strategies

interference networks with no csit impact of topology we consider partially connected k user interference networks where the transmitters have no knowledge about the channel gain values but they are aware of network topology we introduce several linear algebraic and graph theoretic concepts to derive new topology based outer bounds and inner bounds on the symmetric degrees of freedom of these networks we evaluate our bounds for two classes of networks to demonstrate their tightness for most networks in these classes quantify the gain of our inner bounds over benchmark interference management strategies and illustrate the effect of network topology on these gains

a necessary and sufficient condition for permanence of a lotka volterra discrete system with delays abstract in this paper we consider the permanence of the following lotka volterra discrete competition system with delays k1 k2 l1 and l2 x n 1 x n exp r1 1 x n k1 1y n k2 y n 1 y n exp r2 1 2x n l1 y n l2 we show the system is permanent for all nonnegative integers k1 k2 l1 and l2 if and only if 1