hodge integrals and degenerate contributions hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the gromov witten theory of 3 folds in the calabi yau case the contributions are compared to related m theoretic calculations in the fano case the contributions suggest new integrality conditions

gopakumar vafa invariants via vanishing cycles in this paper we propose an ansatz for defining gopakumar vafa invariants of calabi yau threefolds using perverse sheaves of vanishing cycles our proposal is a modification of a recent approach of kiem li which is itself based on earlier ideas of hosono saito takahashi we conjecture that these invariants are equivalent to other curve counting theories such as gromov witten theory and pandharipande thomas theory our main theorem is that for local surfaces our invariants agree with pt invariants for irreducible one cycles we also give a counter example to the kiem li conjectures where our invariants match the predicted answer finally we give examples where our invariant matches the expected answer in cases where the cycle is non reduced non planar or non primitive

tropical refined curve counting from higher genera and lambda classes block and gottsche have defined a q number refinement of counts of tropical curves in mathbb r 2 under the change of variables q e iu we show that the result is a generating series of higher genus log gromov witten invariants with insertion of a lambda class this gives a geometric interpretation of the block gottsche invariants and makes their deformation invariance manifest

performance analysis of coded modulation with generalized selection combining in rayleigh fading we consider coded modulation with generalized selection combining gsc for bandwidth efficient coded modulation over rayleigh fading channels our results show that reception diversity with generalized selection combining can conveniently trade off system complexity versus performance we provide a number of new results by calculating the cutoff rate and by deriving analytical upper bounds on symbol interleaved trellis coded modulation tcm and bit interleaved coded modulation bicm with gsc all are verified by simulation we show that our new bounds on tcm with gsc which includes maximum ratio combining and selection combining as special cases are tighter than the previously derived bounds a new asymptotic analysis on the pairwise error probability which can be used as a guideline for designing coded modulation over gsc channels is also given finally we show that bicm with iterative decoding bicm id can achieve significant coding gain over conventional coded modulation in a multiple receiving antenna channel

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

new closed form bounds on the performance of coding in correlated rayleigh fading new simple bounds are presented for the probability of error in a binary hypothesis test for communications using diversity signaling in correlated rayleigh fading the bounds are developed in the context of pairwise error event probabilities in decoding an error correction code a long standing conjecture regarding the form of worst case error events in exponentially correlated rayleigh fading is also proven the utility of the results is illustrated by their application to transfer function bounds on the probability of bit error for a system using a convolutional code the closed form transfer function bounds are shown to be tighter than previously developed transfer function bounds for communications in exponentially correlated rayleigh fading

exact evaluation of maximal ratio and equal gain diversity receivers for m ary qam on nakagami fading channels exact integral expressions are derived for calculating the symbol error rate ser of multilevel quadrature amplitude modulation mqam in conjunction with l fold antenna diversity on arbitrary nakagami fading channel both maximal ratio combining mrc in independent and correlated fading and equal gain combining egc predetection in independent fading diversity techniques have been considered exact closed form ser expressions for two restricted nakagami fading cases mrc reception are also derived an exact analysis of egc for mqam has not been reported previously despite its practical interest remarkably the exact ser integrals can also be replaced by a finite series approximation formula a useful procedure for computing the confluent hypergeometric series is also presented

diversity order gain for narrow band multiuser communications with pre combining group detection in narrow band multiuser communication systems with fading diversity it is shown that pre combining group detection can bridge the diversity order gap in performance between the optimum and linear detectors for a system with m diversity channels the group detector diversity order is m g where g is the interfering group size a design parameter group detection thus provides a more substantial improvement in performance in narrow band channels over linear detection than in wide band channels in which the diversity orders of the optimal and linear detectors are equal here the complexity of the receiver is a new parameter which in addition to the number of antennas can be used to control the diversity order exact formulas for the pairwise error probabilities and bounds for the bit error rate are obtained and numerical results are shown

exact pairwise error probability of space time codes we describe a simple technique for the numerical calculation within any desired degree of accuracy of the pairwise error probability pep of space time codes over fading channels this method applies also to the calculation of e q spl radic spl xi for any nonnegative random variable spl xi whose moment generating function spl phi sub spl xi s e exp s spl xi is known its application to the multiple antenna independent rayleigh fading channel and to the rayleigh block fading channel is discussed and illustrated by two simple examples

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

a comparison theorem for gromov witten invariants in the symplectic category abstract we exploit the geometric approach to the virtual fundamental class due to fukaya ono and li tian to compare gromov witten invariants of a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order various special cases of the comparison theorem in this paper have long been used in the algebraic category some of them have also appeared in the symplectic setting combined with the inherent flexibility of the symplectic category the main theorem leads to a confirmation of pandharipande s gopakumar vafa prediction for gw invariants of fano classes in 6 dimensional symplectic manifolds the proof of the main theorem uses deformations of the cauchy riemann equation that respect the submanifold and carleman similarity principle for solutions of perturbed cauchy riemann equations in a forthcoming paper we apply a similar approach to relative gromov witten invariants and the absolute relative correspondence in genus 0

analysis of the error performance of trellis coded modulations in rayleigh fading channels this work presents an exact expression for the pairwise error event probability of trellis coded modulation tcm transmitted over rayleigh fading channels it includes phase shift keying psk and multilevel quadrature amplitude modulation qam codes as well as coherent and partially coherent e g differential pilot tone etc detection due to the form of the exact pairwise error event probabilities this calculation technique cannot be used with the transfer function technique to obtain an upper union bound on the overall bit error probability for this reason the authors estimate the bit error probability by considering only a small number of short error events through simulations they found that the estimation is usually very accurate at high signal to noise ratios but not as accurate at lower signal to noise ratios they study several coded modulation schemes this way among the results are the fact that tcm provides significant improvement in the error floor when detected differentially and an asymmetry in the pairwise error event probability for 16 qam

curve counting via stable pairs in the derived category for a nonsingular projective 3 fold x we define integer invariants virtually enumerating pairs c d where c x is an embedded curve and d c is a divisor a virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of x the resulting invariants are conjecturally equivalent after universal transformations to both the gromov witten and dt theories of x for calabi yau 3 folds the latter equivalence should be viewed as a wall crossing formula in the derived category

curves in calabi yau threefolds and topological quantum field theory we continue our study of the local gromov witten invariants of curves in calabi yau threefolds we define relative invariants for local theory which give rise to a 1 1 dimensional topological quantum field theory tqft taking values in the ring mathbb q t the associated frobenius algebra over mathbb q t is semisimple consequently we obtain a structure result for the local invariants as an easy consequence of our structure formula we recover the closed formulas for the local invariants in the case where either the target genus or the degree equals 1

lower bounds for enumerative counts of positive genus real curves abstract we transform the positive genus real gromov witten invariants of many real orientable symplectic threefolds into signed counts of curves these integer invariants provide lower bounds for counts of real curves of a given genus that pass through conjugate pairs of constraints we conclude with some implications and related conjectures for hodge integrals

reduced genus one gromov witten invariants in a previous paper we described a natural closed subset m 0 1 k x a j of the moduli space m1 k x a j of stable genus one j holomorphic maps into a symplectic manifold x in this paper we generalize the definition of the main component to moduli spaces of perturbed in a restricted way j holomorphic maps this generalization implies that m 0 1 k x a j just like m1 k x a j carries a virtual fundamental class and can be used to define symplectic invariants these truly genus one invariants constitute part of the standard genus one gromovwitten invariants which arise from the entire moduli space m1 k x a j the new invariants are more geometric and can be used to compute the genus one gw invariants of complete intersections as shown in a separate paper

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

analysis of predetection diversity tcm mpsk and postdetection diversity tcm mdpsk systems on a rayleigh fading channel in this paper the bit error rate ber performance of predetection diversity trellis coded multilevel phase shift keying tcm mpsk and postdetection diversity trellis coded multilevel differential phase shift keying tcm mdpsk transmitted over a rayleigh fading channel is presented novel analytical upper bounds using the transfer function bounding technique are obtained and illustrated by several numerical examples a new asymptotically tight upper bound for the maximal ratio combining mrc diversity schemes is also derived in order to analyze practical tcm schemes four or more states only trellis codes holding uniform error property uep and uniform distance property udp are considered enabling the encoder transfer function to be obtained from a modified state transition diagram having no more states than the encoder itself monte carlo simulation results which are more indicative of the exact system performance are also given

on finiteness and rigidity of j holomorphic curves in symplectic three folds abstract given a symplectic three fold m we show that for a generic almost complex structure j compatible with there are finitely many j holomorphic curves in m of genus g representing the homology class for every g 0 and every h 2 m z such that c 1 m 0 and the divisibility of is at most 4 i e if n with h 2 m z and n z then n 4 moreover every such curve is embedded and 4 rigid

virasoro constraints for target curves we prove generalized virasoro constraints for the relative gromov witten theories of all nonsingular target curves descendents of the even cohomology classes are studied first by localization degeneration and completed cycle methods descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations as an outcome of our results the relative theories of target curves are completely and explicitly determined

a general method for calculating error probabilities over fading channels signal fading is a ubiquitous problem in mobile and wireless communications in digital systems fading results in bit errors and evaluating the average error rate under fairly general fading models and multichannel reception is often required predominantly to date most researchers perform the averaging using the probability density function method or the moment generating function mgf method this paper presents a third method called the characteristic function chf method for calculating the average error rates and outage performance of a broad class of coherent differentially coherent and noncoherent communication systems with or without diversity reception in a myriad of fading environments unlike the mgf technique the proposed chf method based on parseval s theorem enables us to unify the average error rate analysis of different modulation formats and all commonly used predetection diversity techniques i e maximal ratio combining equal gain combining selection diversity and switched diversity within a single common framework the chf method also lends itself to the averaging of the conditional error probability involving the complementary incomplete gamma function and the confluent hypergeometric function over fading amplitudes which heretofore resisted to a simple form as an aside we show some previous results as special cases of our unified framework

multiple antenna signaling over fading channels with estimated channel state information performance analysis multiple antenna concepts for wireless communication systems promise high spectral efficiency and low error rates by proper exploitation of the randomness in multipath propagation in this paper we investigate the impact of channel uncertainty caused by channel estimation errors on the error rate performance we consider a training based multiple antenna system that reserves a portion of time to sound the channel training symbols are used to estimate the channel by means of an arbitrary linear filter at the receiver no channel state information csi is assumed at the transmitter we present a new framework to analyze training based multiple antenna systems by introducing an equivalent system model that specifies the channel by the estimated and hence known channel coefficients and an uncorrelated data dependent multiplicative noise we derive the maximum likelihood ml detector and highlight its behavior in the limiting cases of perfect csi and no csi and its relation to several mismatched detectors we deduce new exact expressions and chernoff bounds of the pairwise error probability pep used to assess word error and bit error rate bounds for ml and mismatched detection finally we review the code design guidelines in terms of the deleterious effect of channel uncertainty for coherent and noncoherent signaling schemes and present numerical results

on the tautological ring of mg we prove that any product of tautological classes of g of degreed in the chow ring of g with rational coefficients vanishes ford g 2 and is proportional to the class of the hyperelliptic locus in degreeg 2

gopakumar vafa bps invariants hilbert schemes and quasimodular forms i we prove a closed formula for leading gopakumar vafa bps invariants of local calabi yau geometries given by the canonical line bundles of toric fano surfaces it shares some sim ilar features with gottsche yau zaslow formula connection with hilbert schemes connection with quasimodular forms and qua dratic property after suitable transformation in part i of this paper we will present the case of projective plane more general cases will be presented in part ii the problem of counting curves in algebraic varieties dates back to the nineteenth century classical examples include the famous 27 lines on a cubic surface and the 2875 lines on a quintic 3 fold through the interaction with string theory spectacular progresses in this classi cal branch of algebraic geometry have been made over the years since 1990 s first of all gromov witten theory and its various variants have laid the foundation for the modern treatment of many classical enumerative problems and greatly expanded the range of enumerative problems being considered see pt for an introduction to some of the exciting developments the study of the problems of counting curves in surfaces and calabi yau 3 folds have served both as motivations and applications of gromov witten theory we will recall some results in these directions in 2 and 3 to serve as motivations for this work we emphasize on the following three salient features for the curving counting problems for algebraic surfaces connection with hilbert schemes connection with quasimod ular forms and quadratic properties of the node polynomials after suit able transformation our main results will show that these features are also shared in the curve counting problems of some noncompact calabi yau 3 folds arising as the the total space of the canonical line bundles of toric fano surfaces 1

generalised morphisms of k graphs k morphs in a number of recent papers k l graphs have been constructed from k graphs by inserting new edges in the last l dimensions these constructions have been motivated by c algebraic considerations so they have not been treated systematically at the level of higher rank graphs themselves here we introduce k morphs which provide a systematic unifying framework for these various constructions we think of k morphs as the analogue at the level of k graphs of c correspondences between c algebras to make this analogy explicit we introduce a category whose objects are k graphs and whose morphisms are isomorphism classes of k morphs we show how to extend the assignment c to a functor from this category to the category whose objects are c algebras and whose morphisms are isomorphism classes of c correspondences

a compactification of the space of maps from curves we construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points it is shown to be a proper deligne mumford stack equipped with a natural virtual fundamental class

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

evaluation of the exact union bound for trellis coded modulations over fading channels an analytical technique is presented for computing the exact union bound on the average bit error probability of trellis coded modulation schemes over rayleigh rician or shadowed rician fading channels to this end an integral expression is derived for the pairwise error event probability pep existing bounds can be obtained as special cases of this expression it turns out that a gauss chebyshev quadrature rule offers excellent accuracy for this integral by extension the exact union bound i e the weighted sum of an exact peps of a code can readily be evaluated this method has the same complexity as the union chernoff bound and a few examples are given to show its application

the local gromov witten theory of curves the local gromov witten theory of curves is solved by localization and degeneration methods localization is used for the exact evalu ation of basic integrals in the local gromov witten theory of p 1 a tqft formalism is defined via degeneration to capture higher genus curves together the results provide a compete and effective solution the local gromov witten theory of curves is equivalent to the lo cal donaldson thomas theory of curves the quantum cohomology of the hilbert scheme points of c 2 and the orbifold quantum cohomol ogy the symmetric product of c 2 the results of the paper provide the local gromov witten calculations required for the proofs of these equivalences

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

uniform distance and error probability properties of tcm schemes the class of uniform trellis coded modulation tcm techniques is defined and simple explicit conditions for uniformity are derived uniformity is shown to depend on the metric properties of the two subconstellations resulting from the first step in set partitioning as well as on the assignment of binary labels to channel symbols the uniform distance property and uniform error property which are both derived from uniformity but are not equivalent are discussed the derived concepts are extended to encompass transmission over a not necessarily gaussian memoryless channel in which the metric used for detection may not be maximum likelihood an appropriate distance measure is defined that generalizes the euclidean distance it is proved that uniformity of a tcm scheme can also be defined under this new distance the results obtained are shown to hold for channels with phase offset or independent amplitude only fading examples are included to illustrate the applicability of the results

tight analytical and asymptotic upper bound for the ber and fer of linear codes over exponentially correlated generalized fading channels in this paper the upper bound and asymptotical approximation for the bit error rate ber as well as for the frame error rate of linear block codes under maximum likelihood decoding operating over exponentially correlated ec generalized fading channels are proposed particularly our analysis takes the convolutional codes as an example and three different fading scenarios are considered namely rice nakagami m and weibull fading such scenarios have a common property that is spherically invariant random process sirp our analysis relies on the derivation of new lower bounds per each eigenvalue since the ber as well as the fer expressions are written in terms of the pairwise error probability which by its turn depends on the eigenvalues of the channel covariance matrix based on that the upper bounds and asymptotic expressions of the considered metrics subject to the three above fading models are derived which show to be tighter than the previous results published elsewhere in the literature especially for the high signal to noise ratio regions

hodge integrals partition matrices and the λ g conjecture we prove a closed formula for integrals of the cotangent line classes against the top chern class of the hodge bundle on the moduli space of stable pointed curves these integrals are computed via relations obtained from virtual localization in gromov witten theory an analysis of several natural matrices indexed by partitions is required

enumerative geometry of calabi yau 4 folds gromov witten theory is used to define an enumerative geometry of curves in calabi yau 4 folds the main technique is to find exact solutions to moving multiple cover integrals the resulting invariants are analogous to the bps counts of gopakumar and vafa for calabi yau 3 folds we conjecture the 4 fold invariants to be integers and expect a sheaf theoretic explanation

blow up formulae of high genus gromov witten invariants for threefolds using the degeneration formula and absolute relative correspondence we study the change of gromov witten invariants under blow ups for smooth projective threefolds and obtain several closed blow up formulae for high genus gromov witten invariants our formulae also imply some simple relations among generalized bps numbers

the tropical vertex elements of the tropical vertex group are formal families of symplectomorphisms of the 2 dimensional algebraic torus we prove that ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus zero relative gromov witten invariants of toric surfaces the relative invariants which arise have full tangency to a toric divisor at a single unspecified point the method uses scattering diagrams tropical curve counts degeneration formulas and exact multiple cover calculations in orbifold gromov witten theory

exact calculation of the union bound on performance of trellis coded modulation in fading channels this paper presents a method for the exact calculation of the union bound on the error performance of trellis coded modulation tcm in fading channels previous analyses have approximated the error expression to allow the use of the transfer function or restricted their attention to the most common error events or both in contrast we combine the transfer function which counts all error events with exact calculation of event probabilities and produce an exact expression for the union bound although computationally intensive the technique produces the tightest bound yet published in addition it is very general and is applicable to coherent differential and pilot based detection and to phase shift keying psk and quadrature amplitude modulation qam types of constellations

some new twists to problems involving the gaussian probability integral using an alternate form of the gaussian probability integral discovered a number of years ago it is shown that the solution to a number of previously considered communication problems can be simplified and in some cases made more accurate i e exact rather than bounded these problems include the evaluation of 1 the bit error probability of uncoded phase shift keying psk with costas loop tracking 2 word error probability of antipodal modulation in the presence of fading 3 bit error probability of coded m ary psk mpsk over the memoryless fading channel with given channel state information 4 conditional symbol error probability of mpsk in the presence of carrier synchronization error and 5 the average error probability for the binary additive white gaussian noise awgn intersymbol interference channel also obtained is a generalization of this new alternate form to the case of a two dimensional gaussian probability integral with arbitrary correlation which can be used to evaluate the symbol error probability of mpsk with i q unbalance

error bounds for trellis coded mpsk on a fading mobile satellite channel analytical performance bounds are presented for trellis coded mpsk transmitted over a satellite based land mobile channel upper bounds are evaluated using the well known transfer function bounding technique and lower bounds are achieved through knowledge of exact pairwise error probabilities in order to analyze practical trellis codes four or more states the uniform properties displayed by a certain class of trellis codes are exploited enabling the encoder transfer function to be obtained from a modified state transition diagram having no more states than the encoder itself monte carlo simulation results are presented in confirmation of all performance bounds and indicate a general weakness in the transfer function upper bounds a new asymptotically tight upper bound is derived based on a simple modification to the standard transfer function bound and results are presented for the four and eight state trellis codes in rician and rayleigh fading

gromov witten donaldson thomas correspondence for toric 3 folds we prove the equivariant gromov witten theory of a nonsingular toric 3 fold x with primary insertions is equivalent to the equivariant donaldson thomas theory of x as a corollary the topological vertex calculations by agangic klemm marino and vafa of the gromov witten theory of local calabi yau toric 3 folds are proven to be correct in the full 3 leg setting

on twisted higher rank graph c algebras we define the categorical cohomology of a k graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper this leads to an alternative characterisation of the twisted k graph c algebras introduced there we prove a gauge invariant uniqueness theorem and use it to show that every twisted k graph c algebra is isomorphic to a twisted groupoid c algebra we deduce criteria for simplicity prove a cuntz krieger uniqueness theorem and establish that all twisted k graph c algebras are nuclear and belong to the bootstrap class

topological field theory and rational curves we analyze the quantum field theory corresponding to a string propagating on a calabi yau threefold this theory naturally leads to the consideration of witten s topological non linear model and the structure of rational curves on the calabi yau manifold we study in detail the case of the world sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by candelas de la ossa green and parkes

computing top intersections in the tautological ring of mg we derive effective recursion formulae of top intersections in the tautological ring r mathcal m g of the moduli space of curves of genus g 2 as an application we prove a convolution type tautological relation in r g 2 mathcal m g and some interesting bernoulli number identities

bounds on the pairwise error probability of coded ds ssma communication systems in rayleigh fading channels arbitrarily tight upper and lower bounds on the pairwise error probability pep of a trellis coded or convolutional coded direct sequence spread spectrum multiple access ds ssma communication system over a rayleigh fading channel are derived a new set of probability density functions pdfs and cumulative distribution functions cdfs of the multiple access interference mai statistic is derived and a modified bounding technique is proposed to obtain the bounds the upper bounds and lower bounds together specify the accuracy of the resulting estimation of the pep and give an indication of the system error performance several suboptimum decoding schemes are proposed and their performances are compared to that of the optimum decoding scheme by the average pairwise error probability apep values the approach can be used to accurately study the multiple access capability of the coded ds ssma system without numerical integrations

performance analysis of coded communication systems on nakagami fading channels with selection combining diversity in this paper we develop analytical tools for the performance analysis of coded coherent communication systems on independent and identically distributed nakagami m fading channels with selection combining sc diversity first we derive an exact expression for the moment generation function mgf of the signal to noise ratio snr of a code symbol at the output of the selection combiner next based on gauss chebyshev quadrature and gauss laguerre quadrature rules we propose a simple to compute yet accurate numerical solution for the pairwise error probability pep of coded m phase shift keying psk signals using the pep expressions we present the union bound based bit error performance of trellis coded modulation schemes and turbo codes finally we derive an exact expression for the computational cutoff rate of a coded system with m psk signaling and sc diversity and show that the cutoff rate expression is a simple function of the mgf of the snr at the output of the diversity combiner

the design of trellis coded mpsk for fading channels performance criteria it has been well established that the appropriate criterion for optimum trellis coded modulation design on the additive white gaussian noise channel is maximization of the free euclidean distance it is shown that when the trellis coded modulation is used on a rician fading channel with interleaving deinterleaving the design of the code of optimum performance is guided by other factors in particular the length of the shortest error event path and the product of branch distances possibly normalized by the euclidean distance of the path along the path although maximum free distance d sub free is still an important consideration it plays a less significant role the more severe the fading is on the channel these considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over rician fading channels if no interleaving deinterleaving is used then once again the design of the trellis code is guided by maximizing d sub free

stable pair invariants under blow ups we use the degeneration formula and the absolute relative correspondence tostudy the change of stable pair invariants of threefold under blow ups and obtain some closed blow up formulae which imply some interesting relations among bps state counts our results provide positive evidence for the gw dt p correspondence and also give a partial correspondence for varieties not necessarily toric or complete intersections