intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

on the error probability of coded frequency hopped spread spectrum multiple access systems a simple exact calculation is presented of the probability distribution of the number of hits in a block of n symbols in a frequency hopped spread spectrum multiple access communication system while the sequence of hits is not markovian there is an underlying markovian structure that allows the probability distribution of the number of hits to be calculated in a recursive fashion knowing the probability distribution of the number of hits makes it possible to calculate the probability of error for a system employing error correcting codes for several different types of receivers including receivers with both errors and erasures the numerical results show that both the approximation obtained by assuming the actual sequence of hits is markovian and the approximation obtained by assuming the hits are independent are very good when the number of frequency slots is not too small less than five calculations show that assuming the independence of hits gives an error probability accurate to within 1 of the actual error probability assuming the hits are markovian gives error probabilities which are accurate to within 0 001

tadpole graph in covariant closed string field theory a one loop tadpole graph in covariant closed string field theory must generate all inequivalent tori with one puncture in order to determine the region of integration in moduli space we map the tadpole to a two sheeted sphere and use the analyticity of the modular parameter with respect to the complex propagator parameter it is shown that the naive closed string extension of the witten open string vertex fails to give the correct modular region for this one loop amplitude

noncoherent hybrid ds sfh spread spectrum multiple access communications the performance of noncoherent reception in synchronous and asynchronous hybrid direct sequence slow frequency hopped spread spectrum multiple access communication systems operating through additive white gaussian noise channels is investigated systems with binary and m ary frequency shift keying modulation and noncoherent demodulation as well as systems with differential phase shift keying modulation and differentially coherent demodulation are examined and their probability of error is evaluated for random frequencyhopping patterns and signature sequences the multiple access capability of noncoherent hybrid spread spectrum is shown to be superior to that of noncoherent purely frequency hopped spread spectrum and inferior to that of noncoherent purely directsequence spread spectrum for systems with the same bandwidth expansion comparison of hybrid systems with coherent and noncoherent demodulation shows a considerable loss in the performance of the noncoherent systems

how covariant closed string theory solves a minimal area problem for a given genusg riemann surface withn 0 punctures n 3 forg 0 we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2 the minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures these metrics are induced by jenkins strebel quadratic differentials they arise from the string diagrams corresponding to restricted feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra

minimal area problems and quantum open strings we discuss minimal area problems for surfaces with boundaries and both open and closed string punctures we define open closed string diagrams to be surfaces with metrics of minimal area under the condition that any nontrivial jordan open curve be longer or equal to and any nontrivial jordan closed curve be longer or equal to 2 it is proven that the double of an open closed string diagram is a closed string diagram of covariant closed string field theory

minimal area metrics on the swiss cross and punctured torus the closed string theory minimal area problem asks for the conformal metric of least area on a riemann surface with the condition that all non contractible closed curves have length greater than or equal to an arbitrary constant that can be set to one through every point in such a metric there is a geodesic that saturates the length condition and the saturating geodesics in a given homotopy class form a band the extremal metric is unknown when bands of geodesics cross as it happens for surfaces of non zero genus we use recently proposed convex programs to numerically find the minimal area metric on the square torus with a square boundary for various sizes of the boundary for large enough boundary the problem is equivalent to the swiss cross challenge posed by strebel we find that the metric is positively curved in the two band region and flat in the single band regions for small boundary the metric develops a third band of geodesics wrapping around it and has both regions of positive and of negative curvature this surface can be completed to provide the minimal area metric on a once punctured torus representing a closed string tadpole diagram

the lie algebra associated to a unit form to any unit form q x i 1nxi2 i jqijxixj qij z we associate a lie algebra g q an intersection matrix lie algebra in the terminology of slodowy by means of generalized serre relations for a nonnegative unit form the isomorphism type of g q is determined by the equivalence class of q moreover for q nonnegative and connected with radical of rank zero or one respectively the algebras g q turn out to be exactly the simply laced lie algebras which are finite dimensional simple or affine kac moody respectively in case q is connected nonnegative of corank two and not of dynkin type an the algebra g q is elliptic

capacity of frequency hop spread spectrum multiple access communication systems the information theoretic capacity is considered in order to account for independent encoding and decoding and private to the sender and receiver hopping patterns an interference channel model is adopted with k sender receiver pairs with the ith receiver only interested in the message transmitted by the ith sender both synchronous and asynchronous hopping patterns are investigated although the channel exhibits memory in the latter case it is possible to compute the capacity region the asymptotic normalized sum capacity is also computed

multiuser demodulation and iterative decoding for frequency hopped networks demodulation and decoding for frequency hopped spread spectrum multiple access fh ssma systems have been traditionally conducted by conventional single user noncollaborative demodulation and error and erasure correcting decoding techniques in this paper we study the demodulation and decoding aspects of collaborative multiuser reception for fh ssma and propose methods which increase the number of users the system can support in particular we propose and analyze the optimum maximum a priori probability demodulation of multiple symbols or type and the use of iterative multiuser decoding after the demodulation since hits from one or two other users are the most likely hit events in fh ssma the joint demodulation of two or of three users is performed based on likelihood ratio tests m ary frequency shift keying modulation with noncoherent demodulation and reed solomon codes with hard decision minimum distance decoding are used in the fh ssma system results are derived for both synchronous and asynchronous frequency hop systems the performance of the proposed multiuser detector in additive white gaussian noise and flat rayleigh fading channels is evaluated scenarios when all simultaneous users or only a subset of them are collaboratively demodulated and decoded are simulated

a proof that witten s open string theory gives a single cover of moduli space we show that witten s open string diagrams are surfaces with metrics of minimal area under the condition that all nontrivial open jordan curves be longer or equal to the minimal area property is used together with a mini max problem to establish a new existence and uniqueness theorem for quadratic differentials in open riemann surfaces with or without punctures on the boundaries this theorem implies that the feynman rules of open string theory give a single cover of the moduli of open riemann surfaces

multiple access capability of frequency hopped spread spectrum revisited an analysis of the effect of unequal power levels a method for the evaluation of the probability of error of uncoded asynchronous frequency hopped spread spectrum multiple access communications is presented for systems with binary fsk modulation this method provides an accurate approximation and a tight upper bound to the bit error probability for systems with m ary fsk modulation it provides tight upper bounds to the symbol error probability the method enables the computationally efficient averaging of the error probability with respect to the delays phase angles and data streams of the different users it relies on the integration of the product of the characteristic function of the envelope of the branch of the bfsk demodulator which carries the desired signal and of the derivative of the characteristic function of the envelope of the other branch for sufficient frequency separation between the bfsk tones the method can achieve any desirable accuracy moreover the computational effort required for its evaluation grows linearly with the number of interfering users in the m ary case tight upper bounds based on the union bound and the results of the binary case are derived the method allows the effect of unequal power levels on other user interference in fh ssma systems to be quantified accurately for the first time the results indicate that the fh ssma systems suffer from the near far problem although less seriously than direct sequence ssma systems

error performance of binary ncfsk in the presence of multiple tone interference and system noise this paper presents a general solution to the problem of calculating the error performance of binary ncfsk systems in the presence of multiple tone interference and system noise the development uses the concepts of circularly symmetric signals and expresses the results in terms of a bessel integral this integral is then evaluated using a rapidly converging fourier bessel series the error in this approximation to the integral is controlled by two parameters which may be adjusted so that the desired accuracy is attained it is determined that in the useful probability of error range and when the total tone power is constrained the error rate increases for an increase in the number of interfering tones in addition for equal amplitude tones the performance is independent of the distribution of the tones within the channel the technique used is also applicable to other detection problems where the threshold is a random variable

throughput performance of an fhma system with variable rate coding we compare throughput bounds for a frequency hopped multiple access fhma system employing variable rate as well as fixed rate coding frc nonfading as well as rayleigh fading channels are explored the throughput bounds for the variable rate coding vrc schemes are based on the assumption that the number of active users in the system m is known at any time while the bounds for the frc scheme assumes that the fixed code rate is optimized for the user population mean spl lambda where the population is assumed to be poisson distributed we present bounds for theoretical perfect codes which achieve the capacity as well as bounds for reed solomon rs codes of practical block lengths finally we show the dependence of a vrc scheme upon the accuracy of the estimates of m and present a general comparison of the gains of using vrc versus a realistic frc scheme

probability of error in frequency hop spread spectrum multiple access communication systems with noncoherent reception expressions are developed for the probability of error for asynchronous frequency hop spread spectrum multiple access networks using markov hopping patterns and binary frequency shift keying bfsk with one symbol transmitted per hop the expressions are exact when there is one interfering user and orthogonal bfsk is used they provide excellent approximations when there are more than one interfering user it is also shown that the error probability when markov hopping patterns are used is a good approximation to the error probability when memoryless hopping patterns are used by computing the channel capacity and the associated throughput a simple hard decision receiver is shown to perform much better than a receiver using perfect side information to erase the symbols transmitted on hops that were hit when all the users have the same power and one binary symbol is transmitted per hop

a criterion for flatness in minimal area metrics that define string diagrams it has been proposed that the string diagrams of closed string field theory be defined by a minimal area problem that requires that all nontrivial homotopy curves have length greater than or equal to 2 consistency requires that the minimal area metric be flat in a neighbourhood of the punctures the theorem proven in this paper yieds a criterion which if satisfied will ensure this requirement the theorem states roughly that the metric is flat in an open set u if there is a unique closed curve of length 2 through every point inu and all of these closed curves are in the same free homotopy class

root systems and inflations of non negative quasi cartan matrices abstract cartan matrices quasi cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like lie theory representation theory and algebraic graph theory we study quasi cartan matrices by means of the inflation algorithm an idea used in ovsienko s classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results we prove that it is enough to perform linear number of inflations to reduce positive or non negative principal quasi cartan matrix to its canonical form i e to the cartan matrix of a dynkin or euclidean diagram respectively moreover we show that in the positive case the length of every sequence of inflations has quadratic bound and that the only other natural non negative class having such universal bound is the class of so called pos sincerely principal quasi cartan matrices and in this case the bound is linear we obtain such low bounds by applying among others some new observations on the properties of the reduced root systems in the sense of bourbaki associated with positive quasi cartan matrices

serre type relations for complex semisimple lie algebras associated to positive definite quasi cartan matrices abstract cartan matrices play an important role in the classification of complex semi simple lie algebras the well known serre s theorem states that every finite dimensional complex semisimple lie algebra g can be constructed from a cartan matrix a by using generators and relations we generalize serre s theorem by associating to each positive definite quasi cartan matrix a complex semi simple lie algebra and we prove that two positive definite quasi cartan matrices are equivalent if and only if its corresponding lie algebras are isomorphic this work complements the results obtained by barot and rivera in 1