on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

stability and isolation phenomena for yang mills fields in this article a series of results concerning yang mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods one of our main results is to prove that any weakly stable yang mills field overs4 with groupg su2 su3 oru2 is either self dual or anti self dual the analogous statement for so4 bundles is also proved the other main series of results concerns gap phenomena for yang mills fields as a consequence of the non linearity of the yang mills equations we can give explicitc0 neighbourhoods of the minimal yang mills fields which contain no other yang mills fields in this part of the study the nature of the groupg does not matter neither is the dimension of the base manifold constrained to be four

on variational iterative methods for semilinear problems abstract in this article an iterative method suitable for inverting semilinear problems is presented the method employs a two way parametrization that is able to reduce semilinear boundary value problems into simpler and linear systems governed by the laplacian in contrast to newton picard iterations the method prescribes a well defined procedure for constructing initial iterates and thus does not suffer from lack of global convergence with rectangular domains and periodic boundary conditions the technique also allows fast poisson solvers to be engaged theoretical justifications are provided and supported by several experiments numerical results show that the method yields approximations which are comparable to reference solutions

modulation and coding for throughput efficient optical systems optical direct detection systems are currently being considered for some high speed intersatellite links where data rates of a few hundred megabits per second are envisioned under power and pulsewidth constraints the authors investigate the capacity cutoff rate and error probability performance of uncoded and trellis coded systems for various modulation schemes and under throughput and power constraints modulation schemes considered are on off keying pulse position modulation ppm overlapping ppm and multipulse combinatorial ppm

the inverse mean curvature flow and the riemannian penrose inequality let m be an asymptotically flat 3 manifold of nonnegative scalar curvature the riemannian penrose inequality states that the area of an outermost minimal surface n in m is bounded by the adm mass m according to the formula n 16 m 2 we develop a theory of weak solutions of the inverse mean curvature flow and employ it to prove this inequality for each connected component of n using geroch s monotonicity formula for the adm mass our method also proves positivity of bartnik s gravitational capacity by computing a positive lower bound for the mass purely in terms of local geometry

a new analytic algorithm of lane emden type equations an reliable ease to use analytic algorithm is provided for lane emden type equation which models many phenomena in mathematical physics and astrophysics this algorithm logically contains the well known adomian decomposition method different from all other analytic techniques this algorithm itself provides us with a convenient way to adjust convergence regions even without pade technique some applications are given to show its validity

practical codes for photon communication in a recent paper pierce studied the problems of communicating at optical frequencies using photon counting techniques and concluded that at low temperatures we encounter insuperable problems of encoding long before we approach channel capacity in this paper it is shown that even assuming a noiseless model for photon communication for which capacity measured in nats photon is infinite it is unlikely that a signaling efficiency of even 10 nats photon could be achieved practically on the positive side it is shown that pulse position modulation plus reed solomon coding yields practical results in the range of 2 to 3 nats photon

metrics of negative ricci curvature one of the most natural and important topics in riemannian geometry is the relation between curvature and global structure of the underlying manifold for instance complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic furthermore it seemed to be a natural principle that a closed manifold cannot carry two metrics of different signed curvatures as it is a basic fact that this is true for sectional curvature but it turned out to be wrong much later and from a strongly analytic argument for the scalar curvature s since each manifold m n 3 admits a complete metric with s 1 cf aubin a and bland kalka bik hence the situation for ricci curvature ric lying between sectional and scalar curvature seemed to be quite delicate up to now the most general results concerning ric 0 were proved by gao yau gy and brooks br using thurston s theory of hyperbolic threemanifolds viz each closed three manifold admits a metric with ric 0 this is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities gao and yau resp brooks smoothed these singularities to get a regular metric with ric 0 these methods extend to three manifolds of finite type and certain hyperbolic orbifolds in any case the arguments rely on exploiting some extraordinary metric structures whose existence is neither obvious nor conceptually related to the ricci curvature problem indeed the existence depends on the assumption that the manifold is three dimensional and compact moreover this approach does not provide insight into the typical behaviour of metrics with ric 0 since one is led to very special metrics in this article we approach negative ricci curvature using a completely different and new concept which will become even more significant in l2 as we deliberately produce ric 0 actually we will prove the following results in these notes ric g resp r g denotes the ricci tensor resp curvature of a smooth metric g

deformations of the hemisphere that increase scalar curvature consider a compact riemannian manifold m of dimension n whose boundary m is totally geodesic and is isometric to the standard sphere s n 1 a natural conjecture of min oo asserts that if the scalar curvature of m is at least n n 1 then m is isometric to the hemisphere s n equipped with its standard metric this conjecture is inspired by the positive mass theorem in general relativity and has been verified in many special cases in this paper we construct counterexamples to min oo s conjecture in dimension n 3

information rates of photon limited overlapping pulse position modulation channels a direct detection photon limited optical communication channel that uses pulse position modulation ppm under a pulsewidth constraint is considered overlapping ppm oppm allows multiple positions per pulsewidth as well as fractional modulation indices number of pulsewidths per frame requiring more refined timing than that needed for conventional disjoint ppm djppm it is shown that even at moderate values of the expected photon count per pulse q such as needed for high data rates oppm outperforms on off keying ook in both capacity and cutoff rate even though ook is uniformly superior to djppm moreover efficient use of oppm is possible with equiprobable input symbols whereas ook requires inconvenient asymmetrical inputs to achieve capacity and high cutoff rate efficiencies nats photon at lower data rates where capacity efficiency is the prime criterion a significant advantage 20 percent over djppm can be achieved up to efficiencies of about 0 7 nats photon the m ary photon limited oppm channel can be viewed as an ambiguity and erasure channel in the sense that some channel outputs are ambiguous in only some input symbols and only if no photons are counted is there ambiguity in all input symbols for large m ambiguities cause bursts of erasures of data symbols massey s interlaced encoding as well as conventional encoding followed by interleaving are adaptable to this bursty channel and effect an increase in its cutoff rate comparable to the increase obtainable with djppm by the same techniques

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

synchronizable codes for the optical oppm channel random overlapping pulse position modulation oppm sequences result in an unrecoverable error floor on both the probability of erroneous synchronization and the probability of symbol error when only chip synchronization is present it is known however that for a given sequence length m a subset of the set of all possible sequences is synchronizable in the sense that in the absence of noise the receiver can correctly symbol synchronize by observing m or more symbol intervals the authors design finite state machines and codes over a j ary alphabet which produce sequences with the property that every subsequence of length l is synchronizable some of the codes in addition to being synchronizable produce a coding gain for an optical poisson channel the authors introduce joint synchronization and detection algorithms that utilize the memory in the encoded sequences to produce joint estimates of timing and sequences their performance is analyzed through simulations and analytical results

scalar curvature and hammocks scalar curvature is the simplest generalization of gaussian curvature to higher dimensions however there are many questions open with regard to its relation to other geometric quantities and topology here we will prove and illustrate some features of scalar curvature in higher dimensions related to a general hammock effect for scalar curvature namely the one sided affinity for curvature decreasing deformations the first one is concerned with some prescribed decrease of the scalar curvature scal g of some riemannian metric g on a given manifoldmn of dimension 3 we denote the e neighborhood of some set u with respect to g by ue

a new mass for asymptotically flat manifolds abstract in this paper we introduce a mass for asymptotically flat manifolds by using the gauss bonnet curvature we first prove that the mass is well defined and is a geometric invariant if the gauss bonnet curvature is integrable and the decay order satisfies n 4 3 motivated by an elegant idea of lam 35 we then show a positive mass theorem for this new mass for asymptotically flat graphs over r n moreover we obtain various penrose type inequalities in this case in these penrose type inequalities we establish relationship between the new mass and various geometric integrals

frame synchronization for optical overlapping pulse position modulation systems the maximum likelihood rules for locating data frames in direct detection optical communication systems utilizing overlapping pulse position modulation oppm are derived under two distinct assumptions of symbol and chip synchronization respectively various approximations to the optimal rules are identified as well as upper and lower bounds on the probability of correct synchronization further the problem of sequence design based on a maximum peak to sidelobe distance criterion is studied and upper bounds on peak to sidelobe distance for sequences of oppm symbols are derived some good sequences are obtained through a computer search

bounds on the capacity of a spectrally constrained poisson channel an upper bound is derived on the capacity of a poisson channel that has a stationary input process of a given spectrum and is subjected to peak and average power constraints the bound is shown to be asymptotically tight with the relaxation of the spectral constraints its maximization over a given set of admissible spectra is closely related to an analogous problem in the awgn regime the results are used for bounding the capacity of a poisson channel under a second moment bandwidth constraint as well as the capacity under a strict bandwidth constraint asymptotically tight lower bounds on the channel capacity for the above two cases are also presented the approach for lower bounding the capacity for the latter case yields as a by product improved bounds on the bit error probability in uncoded amplitude shift keying and on off modulation as a special case operating over a poisson channel impaired by intersymbol interference

the cutoff rate for on off keying it is well known that on off keying ook i e the transmission of a pulse for a data one mark and transmission of no signal for a data zero space exhibits a 3 db inferiority to antipodal signalling in terms of the signal to noise ratio required to achieve a specified bit error probability when the channel is characterized by the cutoff rate r sub 0 this 3 db inferiority persists in the sense that the required e sub b n sub 0 to operate at a given value of r sub 0 is 3 db greater for the ook channel than for the antipodal channel however it is shown that if the definition of cutoff rate is generalized in an appropriate way to reflect the fact that signal energy is expended only in the transmission of marks part of this penalty is recovered in fact it is shown that in the limit of low r sub 0 there is no penalty in signal to noise ratio even at rates of realistic interest the penalty is significantly less than the expected 3 db

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

some implications of tcm for optical direct detection channels consideration is given to the optical direct detection channel and it is shown how simple trellis coded modulation tcm can be used to improve performance or increase throughput in bits per second without bandwidth expansion or performance loss in fact a modest performance gain can be achieved although the approach can be used with other signal constellations the authors concentrate on signals derived from the pulse position modulation ppm format by allowing overlap theoretical motivation for using this signal set known as overlapping ppm oppm was recently given by i bar david and g kaplan 1984 who showed a capacity gain when overlap is introduced

derivation of the adomian decomposition method using the homotopy analysis method abstract adomian decomposition method has been used intensively to solve nonlinear boundary and initial value problems it has been proved to be very efficient in generating series solutions of the problem under consideration under the assumption that such series solution exits however very little has been done to address the mathematical foundation of the method and its error analysis in this article the mathematical derivation of the method using the homotopy analysis method is presented in addition an error analysis is addressed as well as the convergence criteria

scalar curvature deformation and a gluing construction for the einstein constraint equations on a compact manifold the scalar curvature map at generic metrics is a local surjection f m we show that this result may be localized to compact subdomains in an arbitrary riemannian manifold the method is extended to establish the existence of asymptotically flat scalar flat metrics on n n 3 which are spherically symmetric hence schwarzschild at infinity i e outside a compact set such metrics provide cauchy data for the einstein vacuum equations which evolve into nontrivial vacuum spacetimes which are identically schwarzschild near spatial infinity

about the mass of certain second order elliptic operators abstract let m g be a closed riemannian manifold of dimension n 3 and let f c m such that the operator p f g f is positive if g is flat near some point p and f vanishes around p we can define the mass of p f as the constant term in the expansion of the green function of p f at p in this paper we establish many results on the mass of such operators in particular if f n 2 4 n 1 s g i e if p f is the yamabe operator we show the following result assume that there exists a closed simply connected non spin manifold m such that the mass is non negative for every metric g as above on m then the mass is non negative for every such metric on every closed manifold of the same dimension as m

adomian s method of decomposition critical review and examples of failure abstract adomian s method of decomposition is considered in application to initial boundary value problems for the one space dimensional spatially homogeneous heat conduction equation it is shown that the fundamental equation of the method is well defined only for certain restricted types of boundary conditions within the class of such boundary conditions examples are given such that the fundamental equation fails to have a unique solution and such that the sequence produced by iteration of this equation is divergent the latter is a counterexample to a published assertion of convergence

energy and volume a proof of the positivity of adm energy using the yamabe invariant of three manifolds we give a new proof of the positivity non negativity of adm energy1 using the yamabe invariant of three manifolds from a physical point of view the new proof is motivated by a formula explicitly non negative for the total adm energy of emerging asymptotically flat stationary solutions on maximally expanding compact cosmologies mathematically the proof is an application of the thurston geometrization of three manifolds

a note on the decomposition method for operator equations we show for linear operator equations that adomian s decomposition method is equivalent to the classical method of successive approximations picard iterations hence in contrast to previous claims it is no more rapidly convergent than previous methods and may in fact be divergent

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

performance of pll based optical oppm communication systems the power spectral density psd of a direct detected optical signal with overlapping pulse position modulation oppm scheme is derived it is demonstrated that this spectrum contains a strong periodic component at the signal symbol frequency thereby enabling a simple pll device to provide joint symbol and slot synchronization the variance of the timing error for a linearized pll synchronizer is shown to be mainly dominated by and is inversely proportional to the product of the pll closed loop bandwidth and symbol duration of the oppm signal the performance of a shot noise limited pll based optical oppm communication system is then assessed in terms of the overall system symbol error rate

trellis codes for the optical direct detection channel it is observed that the reduced complexity minimum distance search method of zehavi and wolf 1987 can be extended from codes designed using the squared euclidean distance measure to any measure which is additive over the code branches c n georghiades 1989 has derived such a metric for codes for the direct detection optical channel the search method is applied to find codes for four and eight point overlapping pulse position modulation oppm with large minimum distance and a small number of codewords at that distance

capacity and cutoff rate for optical overlapping pulse position modulation channels upper and lower bounds on the capacity and cutoff rate for direct detection optical overlapping pulse position modulation channels are derived it is shown that these bounds are asymptotically tight in the sense that the difference between the upper and lower bounds converges to zero as the pulse position multiplicity m and or the average photon count per pulse q goes to infinity the tightness of these bounds for finite values of m and and is investigated by providing some numerical examples orders of magnitude of the rate of convergence of the difference between the bounds are also estimated

positive scalar curvature with skeleton singularities we study positive scalar curvature on the regular part of riemannian manifolds with singular uniformly euclidean l infty metrics that consolidate gromov s scalar curvature polyhedral comparison theory and edge metrics that appear in the study of einstein manifolds we show that in all dimensions edge singularities with cone angles le 2 pi along codimension 2 submanifolds do not affect the yamabe type in three dimensions we prove the same for more general singular sets which are allowed to stratify along 1 skeletons exhibiting edge singularities angles le 2 pi and arbitrary l infty isolated point singularities we derive as an application of our techniques positive mass theorems for asymptotically flat manifolds with analogous singularities