On sums of subsets of a set of integers

Forr2 letp(n, r) denote the maximum cardinality of a subsetA ofN={1, 2,...,n} such that there are noBA and an integery with b=y ^r. It is shown that for any>0 andn>n(), (1+o(1))2^1/(r+1) n ^{(r–1)/(r+1)}p(n, r)n ^+2/3 for allr5, and that for every fixedr6,p(n, r)=(1+o(1))·2^1/(r+1) n ^{(r–1)/(r+1)} asn. Letf(n, m) denote the maximum cardinality of a subsetA ofN such that there is noBA the sum of whose elements ism. It is proved that for 3n ^6/3+mn ²/20 log² n andn>n(), f(n, m)=[n/s]+s–2, wheres is the smallest integer that does not dividem. A special case of this result establishes a conjecture of Erds and Graham.Research supported in part by Allon Fellowship, by a Bat-Sheva de Rothschild Grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.     

Sharp bounds for the partition function of integer sequences

By means of a new criterion for higher multiplicities of additive representations of integers as sums of distinct terms of a fixed integer sequence sharp bounds are determined for the partition function of many sequences with the aid of a computer.     

A non-analytic proof of the newman—znám result for disjoint covering systems

A direct combinatorial proof is given to a generalization of the fact that the largest modulusN of a disjoint covering system appears at leastp times in the system, wherep is the smallest prime dividingN. The method is based on geometric properties of lattice parallelotopes. This research was supported by grant 85-00368 from the United States-Is rael Binational Science Foundation, Jerusalem, Israel.     

On Lovász’ lattice reduction and the nearest lattice point problem

Answering a question of Vera Sós, we show how Lovász’ lattice reduction can be used to find a point of a given lattice, nearest within a factor ofc ^d (c = const.) to a given point in R ^d . We prove that each of two straightforward fast heuristic procedures achieves this goal when applied to a lattice given by a Lovász-reduced basis. The verification of one of them requires proving a geometric feature of Lovász-reduced bases: ac ₁ ^d lower bound on the angle between any member of the basis and the hyperplane generated by the other members, wherec ₁ = √2/3. As an application, we obtain a solution to the nonhomogeneous simultaneous diophantine approximation problem, optimal within a factor ofC ^d . In another application, we improve the Grötschel-Lovász-Schrijver version of H. W. Lenstra’s integer linear programming algorithm. The algorithms, when applied to rational input vectors, run in polynomial time.     

A fixed point theorem for discontinuous functions

Any function from a non-empty polytope into itself that is locally gross direction preserving is shown to have the fixed point property. Brouwer's fixed point theorem for continuous functions is a special case. We discuss the application of the result in the area of non-cooperative game theory.     

On Taylor series and Kapteyn series of the first and second type

We study the relation between the coefficients of Taylor series and Kapteyn series representing the same function. We compute explicit formulas for expressing one in terms of the other and give examples to illustrate our method.     

Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X=(X₁,X₂,…,X_n) with positive components has a Liouville distribution with parameter θ=(θ₁,θ₂,…,θ_n) if its joint probability density function is proportional to , θ_i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.     

Interval solution of nonlinear equations using linear programming

A new computational test is proposed for nonexistence of a solution to a system of nonlinear equations in a convex polyhedral regionX. The basic idea proposed here is to formulate a linear programming problem whose feasible region contains all solutions inX. Therefore, if the feasible region is empty (which can be easily checked by Phase I of the simplex method), then the system of nonlinear equations has no solution inX. The linear programming problem is formulated by surrounding the component nonlinear functions by rectangles using interval extensions. This test is much more powerful than the conventional test if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms. By introducing the proposed test to interval analysis, all solutions of nonlinear equations can be found very efficently. This work was partially supported by the Japanese Ministry of Education.     

Planar acyclic oriented graphs

A plane Hasse representation of an acyclic oriented graph is a drawing of the graph in the Euclidean plane such that all arcs are straight-line segments directed upwards and such that no two arcs cross. We characterize completely those oriented graphs which have a plane Hasse representation such that all faces are bounded by convex polygons. From this we derive the Hasse representation analogue, due to Kelly and Rival of Fary's theorem on straight-line representations of planar graphs and the Kuratowski type theorem of Platt for acyclic oriented graphs with only one source and one sink. Finally, we describe completely those acyclic oriented graphs which have a vertex dominating all other vertices and which have no plane Hasse representation, a problem posed by Trotter.     

Interval iteration for zeros of systems of equations

Easily verifiable existence and convergence conditions are given for a class of interval iteration algorithms for the enclosure of a zero of a system of nonlinear equations. In particular, a quadratically convergent method is obtained which throughout the iteration uses the same interval enclosure of the derivative.     

Projective plane and Möbius band obstructions

LetS be a compact surface with possibly non-empty boundary S and letG be a graph. LetK be a subgraph ofG embedded inS such that SK. Anembedding extension ofK toG is an embedding ofG inS which coincides onK with the given embedding ofK. Minimal obstructions for the existence of embedding extensions are classified in cases whenS is the projective plane or the Möbius band (for several canonical choices ofK). Linear time algorithms are presented that either find an embedding extension, or return a nice obstruction for the existence of extensions.Supported in part by the Ministry of Science and Technology of Slovenia, Research Project P1-0210-101-94.     

Rang des courbes elliptiques et sommes d'exponentielles

Lety ²=x ³+a(t)x+b(t) be the equation of an elliptic curve over , wherea(t) andb(t) are polynomials over. We give an upper bound on average of the rank of such curves whent varies over under the assumption that theL-function attached to these curves satisfies classical assumptions. The proof is based on estimates of exponential sums, over varieties, which are treated by Deligne's theorem.

     

Multi-scale anomaly detection algorithm based on infrequent pattern of time series

In this paper, we propose two anomaly detection algorithms PAV and MPAV on time series. The first basic idea of this paper defines that the anomaly pattern is the most infrequent time series pattern, which is the lowest support pattern. The second basic idea of this paper is that PAV detects directly anomalies in the original time series, and MPAV algorithm extraction anomaly in the wavelet approximation coefficient of the time series. For complexity analyses, as the wavelet transform have the functions to compress data, filter noise, and maintain the basic form of time series, the MPAV algorithm, while maintaining the accuracy of the algorithm improves the efficiency. As PAV and MPAV algorithms are simple and easy to realize without training, this proposed multi-scale anomaly detection algorithm based on infrequent pattern of time series can therefore be proved to be very useful for computer science applications.     

Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case

We consider the Sobolev embeddings