On Isomorphisms of Finite Cayley Graphs

A Cayley graph Cay(G,S) of a groupGis called a CI-graph if wheneverTis another subset ofGfor which Cay(G,S) Cay(G,T), there exists an automorphism σ ofGsuch thatS^σ = T. For a positive integerm, the groupGis said to have them-CI property if all Cayley graphs ofGof valencymare CI-graphs; further, ifGhas thek-CI property for allk ≤ m, thenGis called anm-CI-group, and a |G|-CI-groupGis called a CI-group. In this paper, we prove that Ais not a 5-CI-group, that SL(2,5) is not a 6-CI-group, and that all finite 6-CI-groups are soluble. Then we show that a nonabelian simple group has the 4-CI property if and only if it is A₅, and that no nonabelian simple group has the 5-CI property. Also we give nine new examples of CI-groups of small order, which were found to be CI-groups with the assistance of a computer.     

FFT-based algorithms for the string matching with mismatches problem

The string matching with mismatches problem requires finding the Hamming distance between a pattern P of length m and every length m substring of text T with length n. Fischer and Paterson's FFT-based algorithm solves the problem without error in O(σnlogm), where σ is the size of the alphabet Σ [SIAM–AMS Proc. 7 (1973) 113–125]. However, this in the worst case reduces to O(nmlogm). Atallah, Chyzak and Dumas used the idea of randomly mapping the letters of the alphabet to complex roots of unity to estimate the score vector in time O(nlogm) [Algorithmica 29 (2001) 468–486]. We show that the algorithm's score variance can be substantially lowered by using a bijective mapping, and specifically to zero in the case of binary and ternary alphabets. This result is extended via alphabet remappings to deterministically solve the string matching with mismatches problem with a constant factor of 2 improvement over Fischer–Paterson's method.     

Ovoids and spreads of finite classical polar spaces

LetP be a finite classical polar space of rankr, r2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceW _n (q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q^–(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q ²) arising from a non-singular Hermitian variety inPG(n, q ²)n even andn > 2, have no ovoids.LetS be a generalized hexagon of ordern (1). IfV is a pointset of order n³ + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG ₂ (q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=3^2h+1, has an ovoid, and thatH(q), q even, has no ovoid.A regular system of orderm onH(3,q ²) is a subsetK of the lineset ofH(3,q ²), such that through every point ofH(3,q ²) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3).     

Inplace 2D matching in compressed images

The compressed matching problem is the problem of finding all occurrences of a pattern in a compressed text. In this paper we discuss the 2-dimensional compressed matching problem in Lempel–Ziv compressed images. Given a pattern P of (uncompressed) size m×m, and a text T of (uncompressed) size n×n, both in 2D-LZ compressed form, our algorithm finds all occurrences of P in T. The algorithm is strongly inplace, that is, the amount of extra space used is proportional to the best possible compression of a pattern of size m². The best compression that the 2D-LZ technique can obtain for a file of size m² is O(m). The time for performing the search is O(n²) and the preprocessing time is O(m³). Our algorithm is general in the sense that it can be used for any 2D compression which can be sequentially decompressed in small space.     

Pattern Matching with Swaps

Let a text string T of n symbols and a pattern string P of m symbols from alphabet Σ be given. A swapped version T′ of T is a length n string derived from T by a series of local swaps (i.e., t′_ℓ ← t_ℓ + 1 and t′_ℓ + 1 ← t_ℓ), where each element can participate in no more than one swap. The pattern matching with swaps problem is that of finding all locations i for which there exists a swapped version T′ of T with an exact matching of P in location i of T′. It has been an open problem whether swapped matching can be done in less than O(nm) time. In this paper we show the first algorithm that solves the pattern matching with swaps problem in time o(nm). We present an algorithm whose time complexity is O(nm^1/3 log m log σ) for a general alphabet Σ, where σ = min(m,Σ).     

A direct method for the general solution of a system of linear equations

A computationally stable method for the general solution of a system of linear equations is given. The system isA ^Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA ^T and the augmented matrix [A ^T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx _m and a symmetricn ×n matrixH _m of rankn–m, so thatx=x _m+H _my satisfies the system for ally, ann-vector. Whenm=n, the matrixH _m reduces to zero andx _m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158.     

Bit-parallel -matching and suffix automata

(δ,γ)-matching is a string matching problem with applications to music retrieval. The goal is, given a pattern P_1…m and a text T_1…n on an alphabet of integers, find the occurrences P^′ of the pattern in the text such that (i) , and (ii) . The problem makes sense for δγδm. Several techniques for (δ,γ)-matching have been proposed, based on bit-parallelism or on skipping characters. We first present an O(mnlog(γ)/w) worst-case time and O(n) average-case time bit-parallel algorithm (being w the number of bits in the computer word). It improves the previous O(mnlog(δm)/w) worst-case time algorithm of the same type. Second, we combine our bit-parallel algorithm with suffix automata to obtain the first algorithm that skips characters using both δ and γ. This algorithm examines less characters than any previous approach, as the others do just δ-matching and check the γ-condition on the candidates. We implemented our algorithms and drew experimental results on real music, showing that our algorithms are superior to current alternatives with high values of δ.     

A Fast and Simple Algorithm for Identifying 2-Monotonic Positive Boolean Functions

Consider the problem of identifying min T(f) and max F(f) of a positive (i.e., monotone) Boolean functionf, by using membership queries only, where min T(f) (max F(f)) denotes the set of minimal true vectors (maximum false vectors) off. Moreover, as the existence of a polynomial total time algorithm (i.e., polynomial time in the length of input and output) for this problem is still open, we consider here a restricted problem: given an unknown positive functionfofnvariables, decide whetherfis 2-monotonic or not, and iffis 2-monotonic, output both min T(f) and max F(f). For this problem, we propose a simple algorithm, which is based on the concept of maximum latency, and we show that it usesO(n²m) time andO(n²m) queries, wherem = |min T(f)| + |max F(f)|. This answers affirmatively the conjecture raised in Boroset al.[Lecture Notes in Comput. Sci.557(1991), 104–115], Boroset al.[SIAM J. Comput.26(1997), 93–109], and is an improvement over the two algorithms discussed therein: one usesO(n³m) time andO(n³m) queries, and the other usesO(nm² + n²m) time andO(nm) queries.     

Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

In a sequence ofn independent random variables the pdf changes fromf(x, 0) tof(x, 0 + δv_n⁻¹) after the firstnλ variables. The problem is to estimateλ (0, 1 ), where 0 and δ are unknownd-dim parameters andv_n → ∞ slower thann^1/2. Let_n denote the maximum likelihood estimator (mle) ofλ. Analyzing the local behavior of the likelihood function near the true parameter values it is shown under regularity conditions that ifn_n²(− λ) is bounded in probability asn → ∞, then it converges in law to the timeT_(δjδ)^1/2 at which a two-sided Brownian motion (B.M.) with drift1/2(δ′Jδ)^1/2ton(−∞, ∞) attains its a.s. unique minimum, whereJ denotes the Fisher-information matrix. This generalizes the result for small change in mean of univariate normal random variables obtained by Bhattacharya and Brockwell (1976,Z. Warsch. Verw. Gebiete37, 51–75) who also derived the distribution ofT_μ forμ > 0. For the general case an alternative estimator is constructed by a three-step procedure which is shown to have the above asymptotic distribution. In the important case of multiparameter exponential families, the construction of this estimator is considerably simplified.     

Polynomial time approximation schemes for general multiprocessor job shop scheduling

We study preemptive and non-preemptive versions of the general multiprocessor job shop scheduling problem: Given a set of n tasks each consisting of at most μ ordered operations that can be processed on different (possibly all) subsets of m machines with different processing times, compute a schedule (preemptive or non-preemptive, depending on the model) with minimum makespan where operations belonging to the same task have to be scheduled according to the specified order. We propose algorithms for both preemptive and non-preemptive variants of this problem that compute approximate solutions of any positive ε accuracy and run in O(n) time for any fixed values of m, μ, and ε. These results include (as special cases) many recent developments on polynomial time approximation schemes for scheduling jobs on unrelated machines, multiprocessor tasks, and classical open, flow and job shops.     

Symmetric designs with bruck subdesigns

IfP is a finite projective plane of ordern with a proper subplaneQ of orderm which is not a Baer subplane, then a theorem of Bruck [Trans. AMS 78(1955), 464–481] asserts thatn≧m ²+m. If the equalityn=m ²+m were to occur thenP would be of composite order andQ should be called a Bruck subplane. It can be shown that if a projective planeP contains a Bruck subplaneQ, then in factP contains a designQ′ which has the parameters of the lines in a three dimensional projective geometry of orderm. A well known scheme of Bruck suggests using such aQ′ to constructP. Bruck’s theorem readily extends to symmetric designs [Kantor, Trans. AMS 146 (1969), 1–28], hence the concept of a Bruck subdesign. This paper develops the analoque ofQ′ and shows (by example) that the analogous construction scheme can be used to find symmetric designs.     

Nonparametric Approach for Non-Gaussian Vector Stationary Processes

Suppose that {z(t)} is a non-Gaussian vector stationary process with spectral density matrixf(λ). In this paper we consider the testing problemH: ∫^π_−π K{f(λ)} dλ=cagainstA: ∫^π_−π K{f(λ)} dλ≠c, whereK{·} is an appropriate function andcis a given constant. For this problem we propose a testT_nbased on ∫^π_−π K{f(λ)} dλ=c, wheref(λ) is a nonparametric spectral estimator off(λ), and we define an efficacy ofT_nunder a sequence of nonparametric contiguous alternatives. The efficacy usually depnds on the fourth-order cumulant spectraf₄^Zofz(t). If it does not depend onf₄^Z, we say thatT_nis non-Gaussian robust. We will give sufficient conditions forT_nto be non-Gaussian robust. Since our test setting is very wide we can apply the result to many problems in time series. We discuss interrelation analysis of the components of {z(t)} and eigenvalue analysis off(λ). The essential point of our approach is that we do not assume the parametric form off(λ). Also some numerical studies are given and they confirm the theoretical results.     

A note on injectiveC(T)-spaces and the Amir boundary

We sovle in the negative a problem of Wolfe ifC(T _A ) is an injective Banach space wheneverC(T) is injective,T compact, andT _A is the Amir boundary ofT (i.e., the complement of the maximal open extremally disconnected subset ofT). In particular, we findT such thatC(T) is aP ₃-space andT _A ∼βN\N. The author’s research was partially supported by a grant of MEN, Poland.     

Theories of linear order

LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2^ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2^ω. (iii) IfS ₁(T) is countable then so isS _n(T) for everyn. (iv) In caseS ₁(T) is countable we find a relation between the Cantor Bendixon rank ofS ₁(T) and the Cantor Bendixon rank ofS _n(T). (v) We define a class of modelsL, and show thatS ₁(T) is finite iff the models ofT belong toL. We conclude that ifS ₁(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models. Most of the results in this paper appeared in the author’s Master of Science thesis which was prepared at the Hebrew University under the supervision of Professor H. Gaifman.     

On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) H(T ₂(n)), whereT ₂(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n max(m, r – 1), then there exists a complete (r – 1)-partite graphG* withn vertices such thatK(G) K(G*) holds for everyK _r -free graphG withn vertices. In particular, in the class of allK _r -free graphs withn vertices the complete balanced (r – 1)-partite graphT _r–1(n) has the largest number of subgraphs isomorphic toK _t (t < r),C ₄,K _2,3. These generalize some theorems of Turán, Erdös and Sauer.Dedicated to Paul Turán on his 80th Birthday     

Generalized Sets of Lengths

LetRbe a Dedekind domain and (R) the set of irreducible elements ofR. In this paper, we study the sets _R(n) = {m | α₁,…,α_n, β₁,…,β_m (R) such that α₁,…,α_n = β₁,…,β_m}, wherenis a positive integer. We show, in constrast to indications in some earlier work, that the sets _R(n) are not completely determined by the Davenport constant of the class group ofR. We offer some specific constructions for Dedekind domains with small class groups, and show how these sets are generalizations of the sets studied earlier by Geroldinger [[9], [10]].     

Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm

In this paper we analyze the average number of steps performed by the self-dual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem ofn variables withm constraints. Smale established that for every number of constraintsm, there is a constantc(m) such that the number of pivot steps of the self-dual algorithm,(m, n), is less thanc(m)(lnn) ^m(m+1) . We improve upon this estimate by showing that(m, n) is bounded by a function ofm only. The symmetry of the function inm andn implies that(m, n) is in fact bounded by a function of the smaller ofm andn. Parts of this research were done while the author was visiting Stanford University, XEROX- PARC, Carnegie-Mellon University and Northwestern University and was supported in part by the National Science Foundation under Grants MCS-8300984, ECS-8218181 and ECS-8121741.     

Optimally Computing the Shortest Weakly Visible Subedge of a Simple Polygon

Given ann-vertex simple polygonP, the problem of computing the shortest weakly visible subedge ofPis that of finding a shortest line segmentson the boundary ofPsuch thatPis weakly visible froms(ifsexists). In this paper, we present new geometric observations that are useful for solving this problem. Based on these geometric observations, we obtain optimal sequential and parallel algorithms for solving this problem. Our sequential algorithm runs inO(n) time, and our parallel algorithm runs inO(log n) time usingO(n/log n) processors in the CREW PRAM computational model. Using the previously best known sequential algorithms to solve this problem would takeO(n²) time. We also give geometric observations that lead to extremely simple and optimal algorithms for solving, both sequentially and in parallel, the case of this problem where the polygons are rectilinear.     

Asymptotic behavior of contractions in hilbert space

LetT be a contraction on a closed convex subsetC of a Hilbert spaceH. It is proved thatT ⁿ x/n tends to a limity asn tends to infinity for everyxєC and that the limity is independent ofx.     

Existence and Regularity of a Class of Weak Solutions to the Navier–Stokes Equations

We construct a class of weak solutions to the Navier–Stokes equations, which have second order spatial derivatives and one order time derivatives, ofppower summability for 1 < p ≤ 5/4. Meanwhile, we show thatu L^s(0, T; W^{2, r}(Ω)) with 1/s + 3/2r = 2 for 1 < r ≤ 5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if u L^s(0, T; L^r(Ω)) with 1/s + 3/2r = 1 for Ω satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain inR³and 1 < s ≤ 2. For Ω = Rⁿwithn ≥ 3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412).