efficient discrete approximations of quantum gates quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set it has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision e is only polynomial in log 1 e here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1 e a result which matches the lower bound from counting volume up to constant factor

positive solutions of second order three point boundary value problems with change of sign in banach spaces abstract in this paper by using the fixed points of strict set contractions we study the existence of at least one or two positive solutions to the second order three point boundary value problem y t a t f y t 0 t 1 y 0 y 1 y in banach space e where is the zero element of e 0 1 0 1 and a t is allowed to change sign on 0 1 as an application we also give one example to demonstrate our results

expanders and time restricted branching programs the replication number of a branching program is the minimum number r such that along every accepting computation at most r variables are tested more than once the sets of variables re tested along different computations may be different for every branching program this number lies between 0 read once programs and the total number n of variables general branching programs the best results so far were exponential lower bounds on the size of branching programs with r o n logn we improve this to email protected email protected for a constant e 0 this also gives an alternative and simpler proof of an exponential lower bound for email protected n time branching programs for a constant e 0 we prove these lower bounds for quadratic functions of ramanujan graphs

positive solutions for three point boundary value problems with dependence on the first order derivative abstract a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for the second order three point boundary value problem x f t x x 0 0 x 0 0 x 1 x where f is a nonnegative continuous function 0 0 1

global behavior of positive solutions of nonlinear three point boundary value problems we investigate the structure of the positive solution set for nonlinear three point boundary value problems of the form u h t f u 0 u 0 0 u 1 lambdau eta where eta epsilon 0 1 is given lambda epsilon 0 1 n is a parameter f epsilon c 0 infinity 0 infinity satisfies f s 0 for s 0 and h epsilon c 0 1 0 infinity is not identically zero on any subinterval of 0 1 our main results demonstrate the existence of continua of positive solutions of the above problem c 2004 elsevier ltd all rights reserved

a general sequential time space tradeoff for finding unique elements an optimal omega n 2 lower bound is shown for the time space product of any r branching program that determines those values which occur exactly once in a list of n integers in the range 1 r where r geqq n this omega n 2 tradeoff also applies to the sorting problem and thus improves the previous time space tradeoffs for sorting because the r way branching program is such a powerful model these time space product tradeoffs also apply to all models of sequential computation that have a fair measure of space such as off line multitape turing machines and off line log cost random access machines rams

time space lower bounds for satisfiability we establish the first polynomial time space lower bounds for satisfiability on general models of computation we show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic random access turing machine can solve satisfiability in time nc and space nd where d approaches 1 when c does on conondeterministic instead of deterministic machines we prove the same for any constant c less than 2radic our lower bounds apply to nondeterministic linear time and almost all natural np complete problems known in fact they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n1 c our proofs follow the paradigm of indirect diagonalization we also use that paradigm to prove time space lower bounds for languages higher up in the polynomial time hierarchy

on the computational power of probabilistic and quantum branching program in this paper we show that one qubit polynomial time computations are as powerful as nc 1 circuits more generally we define syntactic models for quantum and stochastic branching programs of bounded width and prove upper and lower bounds on their power we show that any nc 1 language can be accepted exactly by a width 2 quantum branching program of polynomial length in contrast to the classical case where width 5 is necessary unless nc 1 acc this separates width 2 quantum programs from width 2 doubly stochastic programs as we show the latter cannot compute the middle bit of multiplication finally we show that bounded width quantum and stochastic programs can be simulated by classical programs of larger but bounded width and thus are in nc 1 for read once quantum branching programs qbps we give a symmetric boolean function which is computable by a read once qbp with o logn width but not by a deterministic read once bp with o n width or by a classical randomized read once bp with o n width which is stable in the sense that its transitions depend on the value of the queried variable but do not vary from step to step finally we present a general lower bound on the width of read once qbps showing that our o logn upper bound for this symmetric function is almost tight

quantum branching programs and space bounded nonuniform quantum complexity in this paper the space complexity of non uniform quantum algorithms is investigated using the model of quantum branching programs qbps in order to clarify the relationship between qbps and non uniform quantum turing machines simulations between these two models are presented which allow to transfer upper and lower bound results exploiting additional insights about the connection between the running time and the precision of amplitudes it is shown that non uniform quantum turing machines with algebraic amplitudes and qbps with a suitable analogous set of amplitudes are equivalent in computational power if both models work with bounded or unbounded error furthermore quantum ordered binary decision diagrams qobdds are considered which are restricted qbps that can be regarded as a non uniform analog of one way quantum finite automata upper and lower bounds are proved that allow a classification of the computational power of qobdds in comparison to usual deterministic and randomized variants of the model finally an extension of qbps is proposed where the performed unitary operation may depend on the result of a previous measurement a simulation of randomized bps by this generalized qbp model as well as exponential lower bounds for its ordered variant are presented

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

positive solutions of a nonlinear three point boundary value problem in this paper by using krasnoselskii s fixed point theorem in a cone we study the existence of single and multiple positive solutions to the three point boundary value problem bvp y t a t f y t 0 0

a time space tradeoff for element distinctness in a time space tradeoff for sorting on non oblivious machines borodin et al b 81 proved that to sort n elements requires ts n2 where t time and s space on a comparison based branching problem although element distinctness and sorting are equivalent problems on a computation tree the stated tradeoff result does not immediately follow for element distinctness or indeed for any decision problem in this paper we are able to show that ts n3 2 for deciding element distinctness or the sign of a permutation

positive solutions of the three point boundary value problem for second order differential equations with an advanced argument abstract in this paper by using the fixed point index method we establish the existence of at least one or at least two positive solutions to the three point boundary value problem for a second order differential equation with an advanced argument u t a t f u h t 0 t 0 1 u 0 0 u u 1 where 0 1 0 and 1 0 as an application we also give an example to demonstrate our result

a global bifurcation phenomenon for second order singular boundary value problems we establish an existence result of unbounded continuum of solutions from the trivial branch for a second order singular boundary value problem as an application we prove several existence and multiplicity results of positive solutions for certain singular boundary value problems

on sign changing solution for some three point boundary value problems abstract in this paper an existence result of sign changing solution for some three point boundary value problems is obtained by using the fixed point index method

analysis of switched diversity systems on generalized fading channels the performances of switched diversity systems operating on generalized nakagami fading channels are analyzed using a discrete time model the average bit error rate ber of binary noncoherent frequency shift keying ncfsk on slow nonselective nakagami fading channels is derived closed form expressions that can be used to determine optimum switching thresholds in a minimum error rate sense are also derived in addition the use of optimum fixed thresholds is considered it is found that a considerable amount of diversity gain can be obtained using an optimum fixed rather than adaptive switching threshold results are obtained for both independent and correlated nakagami fading branch signals the effects of fading severity and the correlation coefficient on both the ber and on the optimum switching threshold are investigated it is shown that useful diversity gain can be obtained with power correlation coefficients as high as 0 9 when the fading is strong the results for a rayleigh channel are obtained and presented as a special case of generalized fading model

unified analysis of switched diversity systems in independent and correlated fading channels the moment generating function mgf of the signal power at the output of dual branch switch and stay selection diversity ssd combiners is derived the first order derivative of the mgf with respect to the switching threshold is also derived these expressions are obtained for the general case of correlated fading and nonidentical diversity branches and hold for any common fading distributions e g rayleigh nakagami m rician nakagami q the mgf yields the performance bit or symbol error probability of a broad class of coherent differentially coherent and noncoherent digital modulation formats with ssd reception the optimum switching threshold in a minimum error rate sense is obtained by solving a nonlinear equation which is formed by using the first order derivative of the mgf this nonlinear equation can be simplified for several special cases for independent and identically distributed diversity branches the optimal switching threshold in closed form is derived for three generic forms of the conditional error probability for correlated rayleigh or nakagami m fading with identical branches the optimal switching threshold in closed form is derived for the noncoherent binary modulation formats we show previously published results as special cases of our unified expression selected numerical examples are presented and discussed

multiplicity results for positive solutions of some semi positone three point boundary value problems in this paper some multiplicity results for positive solutions of some singular semi positone three point boundary value problem be obtained by using the fixed point index method

on multi partition communication complexity we study k partition communication protocols an extension of the standard two party best partition model to k input partitions the main results are as follows 1 a strong explicit hierarchy on the degree of non obliviousness is established by proving that using k 1 partitions instead of k may decrease the communication complexity from n to log k 2 certain linear codes are hard for k partition protocols even when k may be exponentially large in the input size on the other hand one can show that all characteristic functions of linear codes are easy for randomized obdds 3 it is proved that there are subfunctions of the triangle freeness function and the function clique 3 n that are hard for multi partition protocols as an application strongly exponential lower bounds on the size of nondeterministic read once branching programs for these functions are obtained solving an open problem of razborov proceedings of the eighth fct lncs 529 springer 1991 pp 47 60

positive solutions for third order boundary value problems with change of signs abstract in this paper we study the third order triple point nonlinear boundary value problems with sign changing nonlinearities by employing krasnoselkii guo fixed point theorem and avery henderson fixed point theorem and establish results on the existence of at least one or two positive solutions to the boundary value problems

parity graph driven read once branching programs and an exponential lower bound for integer multiplication branching programs are a well established computation model for boolean functions especially read once branching programs have been studied intensively exponential lower bounds for read once branching programs are known for a long time on the other hand the problem of proving superpolynomial lower bounds for parity read once branching programs is still open in this paper restricted parity read once branching programs are considered and an exponential lower bound on the size of the so called well structured parity graph driven read once branching programs for integer multiplication is proven this is the first strongly exponential lower bound on the size of a parity nonoblivious read once branching program model for an explicitly defined boolean function in addition more insight into the structure of integer multiplication is yielded

performance of two dual branch postdetection switch and stay combining schemes in correlated rayleigh and rician fading the performance of binary frequency shift keying bfsk and m ary frequency shift keying mfsk with dual branch postdetection switch and stay combining ssc in correlated rayleigh and rician fading is studied two postdetection ssc receivers are considered and the performances of noncoherent bfsk and mfsk are analyzed closed form expressions are derived for the average bit error rate ber of bfsk and mfsk with postdetection ssc in correlated rayleigh and rician fading optimum switching thresholds that minimize the average ber are obtained extensive monte carlo simulations are presented to test the validity of the analytical results the performance of dual branch postdetection ssc is compared with the performance of dual branch predetection ssc the effects of correlation fading parameter average fading power imbalance and switching threshold on the performances of postdetection ssc receivers are examined

three solutions for three point boundary value problems abstract in this paper we study the existence of at least three solutions for the three point nonlinear boundary value problems u t a t f u 0 0 t 1 u 0 0 u 1 u where 0 1 0 1 a c 0 1 0 and f c r r without any monotonicity assumptions on the nonlinear term f by using the increasing operator theory and approximation process we prove that the three point boundary value problems has at least three solutions

positive solution for m point boundary value problems in this paper we investigate an m point boundary value problem with sign changing nonlinearity the existence of an interval of parameters which ensures the problem has at least one positive solution is determined by constructing available operator and combining the method of lower solution with the method of topology degree moreover the associated green s function for the above problem is also given

separating the eraser turing machine classes l e nl e co nl e and p e abstract by means of exponential lower and polynomial upper bounds for read once only branching programs we separate the logarithmic space bounded complexity classes l e nl e co nl e and p e for eraser turing machines

the computational complexity of universal hashing any implementation of carter wegman universal hashing from n bit strings to m bit strings requires a time space tradeoff of ts nm the bound holds in the general boolean branching program model and thus in essentially any model of computation as a corollary computing a b c in any field f requires a quadratic time space tradeoff and the bound holds for any representation of the elements of the field other lower bounds on the complexity of any implementation of universal hashing are given as well quadratic at2 bound for vlsi implementation logn parallel time bound on a crew pram and exponential size for constant depth circuits

solutions and green s functions for some linear second order three point boundary value problems in this paper we consider the green s functions for a second order linear ordinary differential equation with some three point boundary conditions we give exact expressions of the unique solutions for the linear three point boundary problems by the green s functions as applications we study the iterative solutions for some nonlinear singular second order three point boundary value problems

quantum walk algorithm for element distinctness we use quantum walks to construct a new quantum algorithm for element distinctness and its generalization for element distinctness the problem of finding two equal items among n given items we get an o n 2 3 query quantum algorithm this improves the previous o n 3 4 quantum algorithm of buhrman et al siam j comput 34 2005 pp 1324 1330 and matches the lower bound of aaronson and shi j acm 51 2004 pp 595 605 we also give an o n k k 1 query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among n items

time space trade off lower bounds for randomized computation of decision problems we prove the first time space lower bound trade offs for randomized computation of decision problems the bounds hold even in the case that the computation is allowed to have arbitrary probability of error on a small fraction of inputs our techniques are extension of those used by ajtai and by beame jayram and saks that applied to deterministic branching programs our results also give a quantitative improvement over the previous results previous time space trade off results for decision problems can be divided naturally into results for functions with boolean domain that is each input variable is 0 1 valued and the case of large domain where each input variable takes on values from a set whose size grows with the number of variables in the case of boolean domain ajtai exhibited an explicit class of functions and proved that any deterministic boolean branching program or ram using space s o n requires superlinear time t to compute them the functional form of the superlinear bound is not given in his paper but optimizing the parameters in his arguments gives t n log log n log log log n for s o n1 e for the same functions considered by ajtai we prove a time space trade off for randomized branching programs with error of the form t n s log n s log log n s in particular for space o n1 e this improves the lower bound on time to ns log n log log n in the large domain case we prove lower bounds of the form t ns log n s log log n s for randomized computation of the element distinctness function and lower bounds of the form t n log n s for randomized computation of ajtai s hamming closeness problem and of certain functions associated with quadratic forms over large fields

positive solutions of a nonlinear four point boundary value problems in this paper by using the krasnoselskii s theorem in a cone we study the existence of at least one or two positive solutions to the four point boundary value problemy t a t f y t 0 00 as an application we also give some examples to demonstrate our results

a time space tradeoff for sorting on a general sequential model of computation in a general sequential model of computation no restrictions are placed on the way in which the computation may proceed except that parallel operations are not allowed we show that in such an unrestricted environment text time cdot text space omega n 2 log n in order to sort n integers each in the range 1 n 2

a second order singular boundary value problem abstract we study the second order boundary value problem t t f u t o satisfying u 0 u 0 0 u 1 u 1 0 where a t i 1n ai t and 0 0 we assume that each ai t lpi 0 1 for pi 1 and that each ai t has a singularity in 0 1 to show the existence of countably many positive solutions we apply holder s inequality and krasnosel ski i s fixed point theorem for operators on a cone

approximate solution of three point boundary value problems for second order ordinary differential equations with variable coefficients three point boundary value problem is investigated and solved numerically a novel simple and effective numerical method is proposed the convergence and error estimate of approximate solution are given numerical results are carried out to verify the proposed methods some three point boundary value problems for a second order ordinary differential equation with variable coefficients are investigated in the present paper by using the integration method the second order three point boundary value problems are transformed into a fredholm integral equation of the second kind the solutions and green s functions for some special cases of the second order three point boundary value problems can be determined easily the existence and uniqueness of the solutions of the given fredholm integral equations are considered by using the fixed point theorem in banach spaces a new numerical method is further proposed to solve the second kind fredholm integral equation and an approximate solution is made the convergence and error estimate of the obtained approximate solution are further analyzed numerical results are carried out to verify the feasibility and novelty of the proposed solution procedures

exact solutions of a class of second order nonlocal boundary value problems and applications in this paper solutions of a class of second order differential equations with some multi point boundary conditions are studied we give exact expressions of the solutions for the linear m point boundary problems by the green s functions as applications we study uniqueness and iteration of the positive solutions for a nonlinear singular second order m point boundary value problem

positive solutions of second order three point boundary value problems with change of sign we study the existence of positive solutions of the second order three point boundary value problems t a t f y t 0 0 t 1 y 0 0 y 1 y n where 0 1 0 1 we show the existence of at least one positive solution by applying the krasnoselskii fixed point theorem in a cone here a t is allowed to changes sign on 0 1 applications of our results are provided to yield positive radial solutions of some elliptic boundary value problems on an annulus

fast learning requires good memory a time space lower bound for parity learning we prove that any algorithm for learning parities requires either a memory of quadratic size or an exponential number of samples this proves a recent conjecture of steinhardt et al 2016 and shows that for some learning problems a large storage space is crucial more formally in the problem of parity learning an unknown string x 0 1 n was chosen uniformly at random a learner tries to learn x from a stream of samples a1 b1 a2 b2 where each at is uniformly distributed over 0 1 n and bt is the inner product of at and x modulo 2 we show that any algorithm for parity learning that uses less than n2 25 bits of memory requires an exponential number of samples previously there was no non trivial lower bound on the number of samples needed for any learning problem even if the allowed memory size is o n where n is the space needed to store one sample we also give an application of our result in the field of bounded storage cryptography we show an encryption scheme that requires a private key of length n as well as time complexity of n per encryption decryption of each bit and is provably and unconditionally secure as long as the attacker uses less than n2 25 memory bits and the scheme is used at most an exponential number of times previous works on bounded storage cryptography assumed that the memory size used by the attacker is at most linear in the time needed for encryption decryption

existence of three positive solutions for some second order boundary value problems in this paper a new fixed point theorem of functional type in a cone is established with using the new fixed point theorem and imposing growth conditions on the nonlinearity the existence of three positive solutions for the boundary value problemx t f t x t x t 0 0 0 is continuous finally an example is given to illustrate the importance of results obtained

amplifying lower bounds by means of self reducibility we observe that many important computational problems in nc1 share a simple self reducibility property we then show that for any problem a having this self reducibility property a has polynomial size tc0 circuits if and only if it has tc0 circuits of size n1 for every 0 counting the number of wires in a circuit as the size of the circuit as an example of what this observation yields consider the boolean formula evaluation problem bfe which is complete for nc1 and has the self reducibility property it follows from a lower bound of impagliazzo paturi and saks that bfe requires depth d tc0 circuits of size n1 d if one were able to improve this lower bound to show that there is some constant 0 independent of the depth d such that every tc0 circuit family recognizing bfe has size at least n1 then it would follow that tc0 nc1 we show that proving lower bounds of the form n1 is not ruled out by the natural proof framework of razborov and rudich and hence there is currently no known barrier for separating classes such as acc0 tc0 and nc1 via existing natural approaches to proving circuit lower bounds we also show that problems with small uniform constant depth circuits have algorithms that simultaneously have small space and time bounds we then make use of known time space tradeoff lower bounds to show that sat requires uniform depth d tc0 and ac0 6 circuits of size n1 c for some constant c depending on d