The computational complexity of recognizing permutation functions

Let be a finite field withq elements and a rational function over . No polynomial-time deterministic algorithm is known for the problem of deciding whetherf induces a permutation on . The problem has been shown to be in co-R co-NP, and in this paper we prove that it is inR NP and hence inZPP, and it is deterministic polynomial-time reducible to the problem of factoring univariate polynomials over . Besides the problem of recognizing prime numbers, it seems to be the only natural decision problem inZPP unknown to be inP. A deterministic test and a simple probabilistic test for permutation functions are also presented.     

Fast exponentiation using the truncation operation

Given an integerk, and anarbitrary integer greater than , we prove a tight bound of on the time required to compute with operations {+, –, *, /, ·, }, and constants {0, 1}. In contrast, when the floor operation is not available this computation requires (k) time. Using the upper bound, we give an time algorithm for computing log loga, for alln-bit integersa. This upper bound matches the lower bound for computing this function given by Mansour, Schieber, and Tiwari. To the best of our knowledge these are the first non-constant tight bounds for computations involving the floor operation.     

Staircase visibility and computation of kernels

Let be some set of orientations, that is, . We consider the consequences of defining visibility based on curves that are monotone with respect to the orientations in . We call such curves -staircases. Two points p andq in a polygonP are said to -see each other if an -staircase fromp toq exists that is completely contained inP. The -kernel of a polygonP is then the set of all points which -see all other points. The -kernel of a simple polygon can be obtained as the intersection of all {}-kernels, with . With the help of this observation we are able to develop an algorithm to compute the -kernel of a simple polygon, for finite . We also show how to compute theexternal -kernel of a polygon in optimal time . The two algorithms are combined to compute the ( -kernel of a polygon with holes in time .This work was supported by the Deutsche Forschungsgemeinschaft under Grant No. Ot 64/5-4 and the Natural Sciences and Engineering Research Council of Canada and Information Technology Research Centre of Ontario.     

On the rank of certain finite fields

In the present paper we shall show that the rank of the finite field regarded as an -algebra has one of the two values 2n or 2n+1 ifn satisfies 1/2q+1<n<1/2(m(q)–2). Herem(q) denotes the maximum number of -rational points of an algebraic curve of genus 2 over . Using results of Davenport-Hasse, Honda and Rück we shall give lower bounds form(q) which are close to the Hasse-Weil bound . For specialq we shall further show thatm(q) is equal to the Hasse-Weil bound.     

Congruences et Automorphismes des Automates Finis

Summary We study a class of congruences of strongly connected finite automata, called the group congruences, which may be defined in this way: every element fixing any class of the congruence induces a permutation on this class. These congruences form an ideal of the lattice of all congruences of the automaton and we study the group associated with the maximal group congruence (maximal induced group) with respect to the Suschkevitch group of the transition monoid of . The transitivity equivalence of the subgroups of the automorphism group of are found to be the group congruences associated with regular groups, which form also in ideal of the lattice of congruences of . We then characterize the automorphism group of with respect to the maximal induced group. As an application, we show that, given a group G and an automaton , there exists an automaton whose automorphism group is isomorphic to G and such that the quotient by the automorphism congruence is .     

An Algebraic Approach to Propositional Fuzzy Logic

We investigate the variety corresponding to a logic (introduced in Esteva and Godo, 1998, and called there), which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative (and slightly simpler) axiomatization of such variety. We also investigate the variety, called the variety of algebras, corresponding to the logic obtained from by the adding of a constant and of a defining axiom for one half. We also connect algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and prove that every algebra can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield , and whose operations are the truncations of the operations of to [0, 1]. We prove that such a structure is uniquely determined by up to isomorphism, and we establish an equivalence between the category of algebras and that of f-semifields.     

Parallel computation of disease transforms

Distance transforms are an important computational tool for the processing of binary images. For ann ×n image, distance transforms can be computed in time (n) on a mesh-connected computer and in polylogarithmic time on hypercube related structures. We investigate the possibilities of computing distance transforms in polylogarithmic time on the pyramid computer and the mesh of trees. For the pyramid, we obtain a polynomial lower bound using a result by Miller and Stout, so we turn our attention to the mesh of trees. We give a very simple (logn) algorithm for the distance transform with respect to theL ₁-metric, an (log² n) algorithm for the transform with respect to theL -metric, and find that the Euclidean metric is much more difficult. Based on evidence from number theory, we conjecture the impossibility of computing the Euclidean distance transform in polylogarithmic time on a mesh of trees. Instead, we approximate the distance transform up to a given error. This works for anyL _k -metric and takes time (log³ n).This research was supported by the Deutsche Forschungsgemeinschaft under Grant Al 253/1-1, Schwerpunktprogramm Datenstrukturen und effiziente Algorithmen.     

Determination of the roots and of the global extremum of a lipschitz function

Conclusion The obvious deficiency of the method (1.3), (1.9) is the possible difficulty of the operation . In connection with this one can note that all the above given statements remain valid if the number is replaced by some positive lower bound of |f(t _k ,x)| on .In computational methods, the presence of the Lipschitz constant is considered as a deficiency. In connection with this we can note that the Lipschitz constant L can be replaced by any of its upper estimates. For example, for a differentiable function f(z) one can take .Translated from Kibernetika, No. 2, pp. 71–74, March–April, 1987.     

Optimization of rotor blades for combined structural,dynamic, and aerodynamic properties

A helicopter is intrinsically interdisciplinary due to the close coupling among aerodynamics, dynamics, and the blade structural details. Therefore a design optimization with proper interactions among appropriate disciplines (such as structure, dynamics, and aerodynamics) can offer significant benefit to improve rotor performance. This paper studies the integration of structure, dynamics, and aerodynamics in design optimization of helicopter rotor blades. The optimization is performed to minimize the rotor power required and to satisfy design requirements from structure (minimum blade weight and safe stress margin and fatigue life) and dynamics (proper placement of blade natural frequencies and free of flutter). An effort is made to formulate an effective strategy for combining these various requirements in the optimization process. The paper also presents a way for an intelligent phasing of this interdisciplinary optimization to overcome the hurdles due to conflicting demands on the design variables which arise from different disciplines.Notation nondimensional leading edge mass size, = a/R - C _T rotor thrust coefficient - C _P rotor power coefficient - nondimensional chord, =c/R - nondimensional lumped mass size, =d/R - F(x) objective function - G _j (x) j-th inequality constraint function - H _j (x) j-th equality constraint function - R blade radius, meters - nondimensional blade radial coordinate, =r/R - nondimensional web thickness, =s ₁/R - nondimensional web thickness, =s ₂/R - t nondimensional flange thickness, =t/R - x vector of design variables - x _i i-th component of vector of design variables - blade pitch angle     

A Parallel Logarithmic Order Algorithm for General Multibody System Dynamics

A novel optimal order optimal resource parallel multibody algorithm with general system applicability is derived directly from the sequential recursive methods and the most recent developments in recursive constraint treatments. This new Recursive Coordinate Reduction Parallelism (RCRP) is the first optimal order parallel direct method with a sequential implementation that is exactly the efficient algorithm. Consequently, the RCRP sets new benchmarks for performance over a wide range of problem size and parallel resources. Comparisons to existing methods also demonstrate that the RCRP is presently the best general parallel method.     

Stability of Linear Difference and Differential Inclusions

Any given n×n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N×N matrix B of spectral radius (B) arbitrarily close to (A). A difference inclusion , where is a compact set of matrices, is asymptotically stable if and only if can be extended to a set of nonnegative matrices B with or . Similar results are derived for differential inclusions.     

Complexity of optimal scheduling problems with three jobs

We consider the time-optimal scheduling problemn/m/J of n jobs with fixed routes on m machines. The problem3/m/J/ with identical routes and the problem3/5/J/ are shown to be NP-hard. Similar results are obtained for the problem of minimizing the mean processing time of three jobs on m machines.Translated from Kibernetika, No. 5, pp. 50–54, September–October, 1990.     

Solving A\underline x =\underline b Using a Modified Conjugate Gradient Method Based on Roots of A

We consider the modified conjugate gradient procedure for solving A = in which the approximation space is based upon the Krylov space associated with A ^1/p and , for any integer p. For the square-root MCG (p=2) we establish a sharpened bound for the error at each iteration via Chebyshev polynomials in . We discuss the implications of the quickly accumulating effect of an error in in the initial stage, and find an error bound even in the presence of such accumulating errors. Although this accumulation of errors may limit the usefulness of this method when is unknown, it may still be successfully applied to a variety of small, almost-SPD problems, and can be used to jump-start the conjugate gradient method. Finally, we verify these theoretical results with numerical tests.     

A short note on simultaneous splitting

We show that for any alphabet there is a setL ^* such that ifC is any infinite co-infinite context-free language over , thenL splitsC (i.e., each ofL C,L , C, and is infinite).Preparation of this paper was supported in part by the National Science Foundation under Grant No. MCS77-11360.     

A Modular Algorithm for Computing Cohomologies of Lie Algebras and Superalgebras

The paper describes an improved algorithm for computing cohomologies of Lie (super)algebras. The original algorithm developed earlier by the author of this paper is based on the decomposition of the entire cochain complex into minimal subcomplexes. The suggested improvement consists in the replacement of the arithmetic of rational or integer numbers by a more efficient arithmetic of modular fields and the use of the relationship dim H ^k( _p) dimH ^k() between the dimensions of cohomologies over an arbitrary modular field _p = /p and the filed of rational numbers . This inequality allows us to rapidly find subcomplexes for which dimH ^k( _p) > 0 (the number of such subcomplexes is usually not great) using computations over an arbitrary _p and, then, carry out all required computations over in these subcomplexes.     

Biologically Inspired Autonomous Rover Control

Robotic missions beyond 2013 will likely be precursors to a manned habitat deployment on Mars. Such missions require robust control systems for long duration activities. Current single rover missions will evolve into deployment of multiple, heterogeneous cooperating robotic colonies. This paper describes the map-making memory and action selection mechanism of BISMARC ( iologically nspired ystem for ap-based utonomous over ontrol) that is currently under development at the Jet Propulsion Laboratory in Pasadena, CA (Huntsberger and Rose, Neutral Networks, 11(7/8):1497–1510). BISMARC is an integrated control system for long duration missions involving robots performing cooperative tasks.     

Algorithms and optimality of scheduling soft aperiodic requests in fixed-priority preemptive systems

In this paper, we investigate the problem of scheduling soft aperiodic requests in systems where periodic tasks are scheduled on a fixed-priority, preemptive basis. First, we show that given any queueing discipline for the aperiodic requests, no scheduling algorithm can minimize the response time of every aperiodic request and guarantee that the deadlines of the periodic tasks are met when the periodic tasks are scheduled on a fixed-priority, preemptive basis. We then develop two algorithms: Algorithm is locally optimal in that it minimizes the response time of the aperiodic request at the head of the aperiodic service queue. Algorithm is globally optimal in that it completes the current backlog of work in the aperiodic service queue as early as possible.     

Rangierkomplexität von permutationen

Summary We define a measure of complexity for (the group of permutations ofn elements). We thus obtain non trivial lower bounds for the number of Turing steps necessary for applaying a permutation to a tape withn entries. Transposing annxn matrix, for example, stored linearly needs at leastCn ² Ign steps.     

Lower Bound Methods and Separation Results for On-Line Learning Models

We consider the complexity of concept learning in various common models for on-line learning, focusing on methods for proving lower bounds to the learning complexity of a concept class. Among others, we consider the model for learning with equivalence and membership queries. For this model we give lower bounds on the number of queries that are needed to learn a concept class in terms of the Vapnik-Chervonenkis dimension of , and in terms of the complexity of learning with arbitrary equivalence queries. Furthermore, we survey other known lower bound methods and we exhibit all known relationships between learning complexities in the models considered and some relevant combinatorial parameters. As it turns out, the picture is almost complete. This paper has been written so that it can be read without previous knowledge of Computational Learning Theory.