exact multiplicity results for a class of two point boundary value problems abstract this paper is concerned with the exact number of positive solutions for boundary value problems y p 2 y f y 0 and y 1 y 1 0 where p 1 and 0 is a positive parameter we consider the case in which the nonlinearity f is positive on 0 and p 1 f u u f u changes sign from negative to positive

lattice codes for the gaussian relay channel decode and forward and compress and forward lattice codes are known to achieve capacity in the gaussian point to point channel achieving the same rates as i i d random gaussian codebooks lattice codes are also known to outperform random codes for certain channel models that are able to exploit their linearity in this paper we show that lattice codes may be used to achieve the same performance as known i i d gaussian random coding techniques for the gaussian relay channel and show several examples of how this may be combined with the linearity of lattices codes in multisource relay networks in particular we present a nested lattice list decoding technique in which lattice codes are shown to achieve the decode and forward df rate of single source single destination gaussian relay channels with one or more relays we next present two examples of how this df scheme may be combined with the linearity of lattice codes to achieve new rate regions which for some channel conditions outperform analogous known gaussian random coding techniques in multisource relay channels that is we derive a new achievable rate region for the two way relay channel with direct links and compare it to existing schemes and derive a new achievable rate region for the multiple access relay channel we furthermore present a lattice compress and forward cf scheme for the gaussian relay channel which exploits a lattice wyner ziv binning scheme and achieves the same rate as the cover el gamal cf rate evaluated for gaussian random codes these results suggest that structured lattice codes may be used to mimic and sometimes outperform random gaussian codes in general gaussian networks

lattice network codes based on eisenstein integers in this paper we investigate lattice network codes lncs constructed from eisenstein integer based lattices quantization and encoding algorithms over eisenstein integers are first introduced then a union bound estimation ube of the decoding error probability is derived when the shaping region of the lnc is a product of regular hexagons next the gaussian reduction algorithm is generalized to be applicable to complex lattices over eisenstein integers such that an optimal coefficient vector can be found in the two transmitter single relay system based on the ube design criteria for optimal lncs with minimum decoding error probability are formulated and applied to construct both gaussian integer and eisenstein integer based good lncs from rate 1 2 feed forward convolutional codes by complex construction a the constructed codes provide up to 7 65 db nominal coding gains over rayleigh fading channels furthermore we introduce the construction of lncs from linear codes by complex construction b the nominal coding gains and error performance of the lncs thus constructed are explicitly analyzed examples show that the lncs constructed by complex construction b provide a better tradeoff between code rate and nominal coding gain

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

a minimax inequality and its applications to ordinary differential equations the aim of this paper is to investigate the minimax inequality which plays a fundamental role in the critical points theorem of b ricceri below equivalent formulations are shown and characterization is proved in particular for a special class of functionals as an application a multiplicity result for an ordinary dirichlet problem is emphasized

symmetry breaking for a class of semilinear elliptic problems abstract this document considers a nonlinear elliptic problem suppose this problem has a family of positive radial solutions parameterized by r i e ur x this paper studies the possibility of the existence of nonradial solutions of bifurcating from the radial solutions family answering a question posed by smoller and wasserman it is shown this happens if f satisfies suitable assumptions therefore the global structure of the nonradial solution set is investigated

pointwise topology on completely distributive lattices abstract in this paper the author presents a kind of pointwise topology theory on complete completely distributive lattices which is both a generalization of fuzzy topology and a further generalization of the author s papers 14 15

the global solution set for a class of semilinear problems we use bifurcation theory to study positive negative and sign changing solutions for several classes of boundary value problems depending on a real parameter we show the existence of infinitely many points of pitchfork bifurcation and study global properties of the solution curves

lattices over eisenstein integers for compute and forward in this paper we consider the use of lattice codes over eisenstein integers for implementing a compute and forward protocol in wireless networks when channel state information is not available at the transmitter we extend the compute and forward paradigm of nazer and gastpar to decoding eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the poltyrev limit using this result we show that both the outage performance and error correcting performance of the nested lattice codebooks over eisenstein integers surpass those of lattice codebooks over integers considered by nazer and gastpar with no additional computational complexity

construction a of lattices over number fields and block fading wiretap coding we propose a lattice construction from totally real and complex multiplication fields which naturally generalizes construction a of lattices from p ary codes obtained from the cyclotomic field mathbb q zeta p p a prime which in turn contains the so called construction a of lattices from binary codes as a particular case we focus on the maximal totally real subfield mathbb q zeta p r zeta p r of the cyclotomic field mathbb q zeta p r r geq 1 our construction has applications to coset encoding of algebraic lattice codes for block fading channels and in particular for block fading wiretap channels

universal lattice codes for mimo channels we propose a coding scheme that achieves the capacity of the compound mimo channel with algebraic lattices our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill conditioned channel realizations to shape the constellation a discrete gaussian distribution over the lattice points is applied these techniques along with algebraic properties of the proposed lattices are then used to construct a sub optimal de coupled coding scheme that achieves a constant gap to compound capacity by decoding in a lattice that does not dependent on the channel realization the gap is characterized in terms of algebraic invariants of the code and is shown to be significantly smaller than previous schemes in the literature we also exhibit alternative algebraic constructions that achieve the capacity of ergodic siso fading channels

minimal hausdorff l fuzzy topological spaces abstract in this paper the author introduced the concept of minimal hausdorff l fuzzy topological spaces investigated basic properties of it by means of fuzzy ideals and fuzzy nets and compared it with the concepts of compactness and h closedness

compute and forward harnessing interference through structured codes interference is usually viewed as an obstacle to communication in wireless networks this paper proposes a new strategy compute and forward that exploits interference to obtain significantly higher rates between users in a network the key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise after decoding these linear equations the relays simply send them towards the destinations which given enough equations can recover their desired messages the underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field this scheme is applicable even if the transmitters lack channel state information

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

exact multiplicity results for a class of boundary value problems with cubic nonlinearities abstract we give an exact multiplicity result for a class of boundary value problems for cubic nonlinearities with an explicit x dependence moreover we provide a detailed analysis of global solution curves

some results on i l and r l abstract this paper gives a partial solution to problem 1 in wang and lu 1992 and a suggestion about how to construct a better fuzzy unit interval it defines a new kind of fuzzy real line and investigates a series of its properties

exact multiplicity of positive solutions for a class of semilinear problems we establish an exact multiplicity result of positive solutions of u f u 0inbn u 0on bn wherebnis the unit ball fsatisfiesf changes sign only once and asymptotically sublinear or linear the nonlinearities which we are concerned with here includef u u u b c u for 0 2b candf u up uqfor 1 p q precise global bifurcation diagrams are obtained

exact multiplicity results for a p laplacian problem with concave convex concave nonlinearities we study the exact number of positive solutions of a two point dirichlet boundary value problem involving the p laplacian operator we consider the case p 2 as well as the case p 1 when the nonlinearity f satisfies f 0 0 and has two distinct simple positive zeros and such that f changes sign exactly twice on 0 note that we may allow that f changes sign more than twice on 0 some interesting examples of quartic polynomials are given in particular for f u u2 u 1 u 2 we study the evolution of the bifurcation curves of the p laplacian problem as p increases from 1 to infinity and hence are able to determine the exact multiplicity of positive solutions for each p 1

symmetry breaking for a class of semilinear elliptic problems we study positive solutions of the dirichlet problem for aji up a p 1 a 0 on the unit ball 0 we show that there exists a positive solution uo ao of this problem which satisfies in addition duo dn 0 on fi we prove also that at uo ao the symmetry breaks i e asymmetric solutions bifurcate from the positive radial solutions 1 introduction we consider here the symmetry breaking for semilinear el liptic problems of the type a f u x in bu u 0 on dbi where bi is the unit ball on r n 2 and rxr risac2 function we say that the symmetry breaks at a if the radial solution un ao is a nonradial bifurcation point i e if every neighborhood of u0 x0 contains solutions of 1 1 u a with u nonradial symmetry breaking problems for positive solutions of semilinear elliptic equa tions have been studied by smoller and wasserman 4 5 in fact they prove that for the function g c2 satisfying

ring compute and forward over block fading channels the compute and forward c f protocol in quasi static channels normally employs lattice codes based on the rational integers mathbb z the gaussian integers mathbb z i or the eisenstein integers mathbb z omega while its extension to more general channels often assumes channel state information at transmitters csit in this paper we propose a novel scheme for c f in block fading channels without csit which is referred to as ring c f because the fading coefficients are quantized to the canonical embedding of a ring of algebraic integers owing to the multiplicative closure of the algebraic lattices employed a relay is able to decode an algebraic integer linear combination of lattice codewords we analyze its achievable computation rates and show it outperforms conventional c f based on the mathbb z lattices by investigating the effect of the diophantine approximation by algebraic conjugates we prove that the degrees of freedom dofs of the optimized computation rate are n l where n is the number of blocks and l is the number of users

amplify and forward in wireless relay networks rate diversity and network size a wireless network with fading and a single source destination pair is considered the information reaches the destination via multiple hops through a sequence of layers of single antenna relays at high signal to noise ratio snr the simple amplify and forward strategy is shown to be optimal in terms of degrees of freedom because it achieves the degrees of freedom equal to a point to point multiple input multiple output mimo system hence the lack of coordination in relay nodes does not reduce the achievable degrees of freedom the performance of this amplify and forward strategy degrades with increasing network size this phenomenon is analyzed by finding the tradeoffs between network size rate and diversity a lower bound on the diversity multiplexing tradeoff for concatenation of multiple random gaussian matrices is obtained also it is shown that achievable network size in the outage formulation short codes is a lot smaller than the ergodic formulation long codes

a simple proof for the existence of good pairs of nested lattices this paper provides a simplified proof for the existence of nested lattice codebooks allowing to achieve the capacity of the additive white gaussian noise channel as well as the optimal rate distortion tradeoff for a gaussian source the proof is self contained and relies only on basic probabilistic and geometrical arguments an ensemble of nested lattices that is different and more elementary than the one used in the previous proofs is introduced this ensemble is based on lifting different subcodes of a linear code to the euclidean space using construction a in addition to being simpler the analysis is less sensitive to the assumption that the additive noise is gaussian in particular for additive ergodic noise channels it is shown that the achievable rates of the nested lattice coding scheme depend on the noise distribution only via its power similarly the nested lattice source coding scheme attains the same rate distortion tradeoff for all ergodic sources with the same second moment

exact multiplicity results for boundary value problems with nonlinearities generalizing cubic using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type the first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u the second one deals with a class of nonlinearities with explicit x dependence

boosted kz and lll algorithms there exist two issues among popular lattice reduction algorithms that should cause our concern the first one is korkine zolotarev kz and lenstra lenstra lovasz lll algorithms may increase the lengths of basis vectors the other is kz reduction suffers worse performance than minkowski reduction in terms of providing short basis vectors despite its superior theoretical upper bounds to address these limitations we improve the size reduction steps in kz and lll to set up two new efficient algorithms referred to as boosted kz and lll for solving the shortest basis problem with exponential and polynomial complexity respectively both of them offer better actual performance than their classic counterparts and the performance bounds for kz are also improved we apply them to designing integer forcing if linear receivers for multi input multioutput communications our simulations confirm their rate and complexity advantages

exact multiplicity and ordering properties of positive solutions of a p laplacian dirichlet problem and their applications we study the exact multiplicity and ordering properties of positive solutions of the p laplacian dirichlet problem

achieving 1 2 log 1 snr on the awgn channel with lattice encoding and decoding we address an open question regarding whether a lattice code with lattice decoding as opposed to maximum likelihood ml decoding can achieve the additive white gaussian noise awgn channel capacity we first demonstrate how minimum mean square error mmse scaling along with dithering lattice randomization techniques can transform the power constrained awgn channel into a modulo lattice additive noise channel whose effective noise is reduced by a factor of spl radic 1 snr snr for the resulting channel a uniform input maximizes mutual information which in the limit of large lattice dimension becomes 1 2 log 1 snr i e the full capacity of the original power constrained awgn channel we then show that capacity may also be achieved using nested lattice codes the coarse lattice serving for shaping via the modulo lattice transformation the fine lattice for channel coding we show that such pairs exist for any desired nesting ratio i e for any signal to noise ratio snr furthermore for the modulo lattice additive noise channel lattice decoding is optimal finally we show that the error exponent of the proposed scheme is lower bounded by the poltyrev exponent

symmetry breaking for positive solutions of semilinear elliptic equations in a recent interesting paper gidas ni and nirenberg 2 proved that positive solutions of the dirichlet problem for second order semi linear elliptic equations on balls must themselves be spherically symmetric functions here we consider the bifurcation problem for such solutions specifically we investigate the ways in which the symmetric solution can bifurcate into a nonsymmetric solution when this happens we say that the symmetry breaks to carry out this program we rely on certain results in 6 where we studied the kernel of the associated linearized operator this enables us to give some necessary conditions for symmetry to break we also find a class of functionsf where these conditions are also sufficient see equation 18 below the problem is of course to show that when zero comes into the spectrum of the linearized operator and our conditions are fulfilled bifurcation into the non radial direction actually occurs this is done by showing first that the bifurcation curve is a smooth manifold near the bifurcation point and then appealing to a theorem of vanderbauwhede 8 which gives sufficient conditions for bifurcation to occur in the presence of symmetries

order homomorphisms on fuzzes abstract in this paper basic properties of the concept of order homomorphisms on fuzzes 4 which is a proper generalization of the concept of fuzz functions 1 are studied conditions for fuzz functions to be functions of zadeh s type are obtained moreover the concepts of continuous closed and open order homomorphisms are introduced and studied

the approximate sum capacity of the symmetric gaussian k user interference channel interference alignment has emerged as a powerful tool in the analysis of multiuser networks despite considerable recent progress the capacity region of the gaussian k user interference channel is still unknown in general in part due to the challenges associated with alignment on the signal scale using lattice codes this paper develops a new framework for lattice interference alignment based on the compute and forward approach within this framework each receiver decodes by first recovering two or more linear combinations of the transmitted codewords with integer valued coefficients and then solving these linear combinations for its desired codeword for the special case of symmetric channel gains this framework is used to derive the approximate sum capacity of the gaussian interference channel up to an explicitly defined outage set of the channel gains the key contributions are the capacity lower bounds for the weak through strong interference regimes where each receiver should jointly decode its own codeword along with part of the interfering codewords as part of the analysis it is shown that decoding k linear combinations of the codewords can approach the sum capacity of the k user gaussian multiple access channel up to a gap of no more than k 2 log k bits

on some gross misunderstandings about the theory of topological molecular lattices abstract some gross misunderstandings have arisen since the theory of topological molecular lattices briefly tmls was presented wang 1989 1992 certain scholars think that there is a correspondence between tmls and classical spaces and they could obtain the results in the theory of tmls hence fuzzy topology by simply translating the corresponding results in classical topology the aim of this paper is to clear out such misunderstandings