A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

Given a nonempty set of functions

On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups

In this paper, we discuss the minimal number of observables Q ₁, ..., Q , where expectation values at some time instants t ₁, ..., t _r determine the trajectory of a d-level quantum system (qudit) governed by the Gaussian semigroup . We assume that the macroscopic information about the system in question is given by the mean values E _j(Q _i) = tr(Q _i(t _j)) of n selfadjoint operators Q ₁, ..., Q _n at some time instants t ₁ < t ₂ < ... < t _r, where n < d ²– 1 and r deg (, ). Here (, ) stands for the minimal polynomial of the generator of the Gaussian flow (t).     

Spigot Algorithm and Reliable Computation of Natural Logarithm

The spigot approach used in the previous paper (Reliable Computing 7 (3) (2001), pp. 247–273) for root computation is now applied to natural logarithms. The logarithm ln Q with Q , Q > 1 is decomposed into a sum of two addends k ₁× ln Q ₁+k ₂× ln Q ₂ with k ₁, k ₂ , then each of them is computed by the spigot algorithm and summation is carried out using integer arithmetic. The whole procedure is not literally a spigot algorithm, but advantages are the same: only integer arithmetic is needed whereas arbitrary accuracy is achieved and absolute reliability is guaranteed. The concrete procedure based on the decomposition with p, q ( – {0}), p < q is simple and ready for implementation. In addition to the mentioned paper, means for determining an upper bound for the biggest integer occurring in the process of spigot computing are now provided, which is essential for the reliability of machine computation.     

An Estimation Problem for Quasilinear Stochastic Partial Differential Equations

A problem of estimating a functional parameter (x) and functionals () based on observation of a solution u (t, x) of the stochastic partial differential equation is considered. The asymptotic problem setting, as the noise intensity 0, is investigated.     

The network complexity and the Turing machine complexity of finite functions

Summary Let L(f) be the network complexity of a Boolean function L(f). For any n-ary Boolean function L(f) let . Hereby p ranges over all relative Turing programs and ranges over all oracles such that given the oracle , the restriction of p to inputs of length n is a program for L(f). p is the number of instructions of p. T _p (n) is the time bound and S _p of the program p relative to the oracle on inputs of length n. Our main results are (1) L(f) O(TC(L(f))), (2) TC(f) O(L(f) ^{2 2+}) for every O.The results of this paper have been reported in a main lecture at the 1975 annual meeting of GAMM, April 2–5, Göttingen     

Ramsey numbers and an approximation algorithm for the vertex cover problem

Summary We show two results. First we derive an upper bound for the special Ramsey numbers r _k(q) where r _k(q) is the largest number of nodes a graph without odd cycles of length bounded by 2k+1 and without an independent set of size q+1 can have. We prove The proof is constructive and yields an algorithm for computing an independent set of that size. Using this algorithm we secondly describe an O(¦V¦·¦E¦) time bounded approximation algorithm for the vertex cover problem, whose worst case ratio is , for all graphs with at most (2k+3) ^k (2k+2) nodes (e.g. 1.8, if ¦V¦146000).     

Extremal Relations between Additive Loss Functions and the Kolmogorov Complexity

Conditions are presented under which the maximum of the Kolmogorov complexity (algorithmic entropy) K(₁... _N ) is attained, given the cost f( _i ) of a message ₁... _N . Various extremal relations between the message cost and the Kolmogorov complexity are also considered; in particular, the minimization problem for the function f( _i ) – K(₁... _N ) is studied. Here, is a parameter, called the temperature by analogy with thermodynamics. We also study domains of small variation of this function.     

Computing Population Variance and Entropy under Interval Uncertainty: Linear-Time Algorithms

In statistical analysis of measurement results it is often necessary to compute the range of the population variance when we only know the intervals of possible values of the x _i . While can be computed efficiently, the problem of computing is, in general, NP-hard. In our previous paper “Population Variance under Interval Uncertainty: A New Algorithm” (Reliable Computing 12 (4) (2006), pp. 273–280) we showed that in a practically important case we can use constraints techniques to compute in time O(n · log(n)). In this paper we provide new algorithms that compute (in all cases) and (for the above case) in linear time O(n). Similar linear-time algorithms are described for computing the range of the entropy when we only know the intervals of possible values of probabilities p _i . In general, a statistical characteristic ƒ can be more complex so that even computing ƒ can take much longer than linear time. For such ƒ, the question is how to compute the range in as few calls to ƒ as possible. We show that for convex symmetric functions ƒ, we can compute in n calls to ƒ.     

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.     

Exact optimal grillage layouts — Part I: Combinations of free and simply supported edges

In the first instalment of this three-part study, a comprehensive treatment of analytically derived, exact optimal grillage layouts for combinations of simply supported and free edges is given. In part two, grillages with combinations of simply supported, clamped and free edges will be considered.Notation k constant in specific cost function - M beam bending moment - r radius of circular edge - R ⁺,R ^–,S ⁺,S ^–,T optimal regions - x, x _j coordinate along a beam (j) - slope of the adjoint deflection at pointD in directionDA - t, v coordinates along the free edge - adjoint deflection - angle between long beams and free edge - angle between free and simply supported edges - curvature of the adjoint deflection - , angles for layouts with circular edge - total weight (cost) of grillage - coordinate along a beam in anR ⁺ region - distance defined in Fig. 3     

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.     

Population Variance under Interval Uncertainty: A New Algorithm

In statistical analysis of measurement results, it is often beneficial to compute the range V of the population variance when we only know the intervals of possible values of the x_i. In general, this problem is NP-hard; a polynomialtime algorithm is known for the case when the measurements are sufficiently accurate, i.e., when for all In this paper, we show that we can efficiently compute V under a weaker (and more general) condition .     

Enclosing Solutions of Singular Interval Systems Iteratively

Richardson splitting applied to a consistent system of linear equations Cx = b with a singular matrix C yields to an iterative method x^k+1 = Ax^k + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x^* = x^* (x⁰) if and only if A is semi-convergent. In order to enclose such vectors we consider the corresponding interval iteration with (|[A]|) = 1 where |[A]| denotes the absolute value of the interval matrix [A]. If |[A]| is irreducible we derive a necessary and sufficient criterion for the existence of a limit of each sequence of interval iterates. We describe the shape of and give a connection between the convergence of ( ) and the convergence of the powers of [A].Dedicated to Professor G. Mae on the occasion of his 65^th birthday     

Decision procedures for elementary sublanguages of set theory

In this paper we prove the decidability of the class of unquantified formulae of set theory involving the operators , , , \, {·}, pred _< and the predicates =, , , Finite, where pred _<(s) denotes the collection of all sets having rank strictly less than the rank of s.This work generalizes and combines earlier results published in the same series of papers.This work has been partially supported by ENI and ENIDATA within the AXL project.     

One-way multihead finite automata and 2-bounded languages

LanguagesL _n ={1 ^x 2 ^ix :i, x , 1in} were used to show that, for eachk, one-way non-sensing deterministic finite automata (1-MFA) withk+1 heads are more powerful than such automata withk heads, even if we consider only 2-bounded languages (Chrobak). Fork letf(k) be the maximal numbern such that languageL _n can be recognized by a 1-MFA withk heads. We present a precise inductive formula forf(k). It may be shown that, fork3,

Numerical Experiences with a New Generalized Subinterval Selection Criterion for Interval Global Optimization

The convergence properties are studied for interval global optimization algorithms that select the next subinterval to be subdivided with the largest value of the indicator pf(f _k, X) = . In contrast to previous work, here the more general case is investigated, when the global minimum value is unknown, and thus its estimation f _k in the iteration k has an important role.Extensive numerical tests on 40 problems confirm that substantial improvements can be achieved both on simple and sophisticated algorithms by the new method (not utilizing the minimum value).     

Solution of Systems of Linear Equations by the p-Adic Method

A new algorithm for solving systems of linear equations Ax = b in an Euclidean domain is suggested. In the case of the ring of integers, the complexity of this algorithm is O (n ³ mlog² ||A||), where n)$$ " align="middle" border="0"> is a matrix of rank n and , if standard algorithms for the multiplication of integers and matrices are used. Under the same conditions, the best algorithm of this kind among those published earlier, which was suggested by Labahn and Storjohann in [1], has complexity O (n ⁴ mlog² ||A||). True, when using fast algorithms for the multiplication of numbers and matrices, the theoretical complexity estimate for the latter algorithm is O (n mlog² ||A||), which is better than the similar estimate O (n ³ mlog||A||) for the new algorithm.     

Modelling Asynchrony with a Synchronous Model

The I/O automaton paradigm of Lynch and Tuttle models asynchrony through an interleaving parallel composition. The recognition that such interleaving models in fact can be viewed as special cases of synchronous parallel composition has been very limited. Let be any set of finite-state I/O automata drawing actions from a fixed finite set containing a subset . In this article we establish a translation T : to a class of -automata closed under a synchronous parallel composition, for which T is monotonic with respect to implementation relative to , and linear with respect to composition. Thus, for A¹, ..., A, B¹, ..., B and A = A¹ ... A, B = B¹ ... B, if is the set of actions common to both A and B, then A implements B (in the sense of I/O automata) if and only if the -automaton language containment (T(A¹) ... T(A)) (T(B¹) ... T(B)) obtains, where denotes the interleaving parallel composition on and denotes the synchronous parallel composition on . For the class , we use the L-process model of -automata. This result enables one to verify systems specified by I/O automata through model-checkers such as COSPAN or SMV, that operate on models with synchronous parallel composition. The translation technique generalizes to other interleaving models, although in each case, the translation map must match the specific model.     

Tight bounds for oblivious routing in the hypercube

We prove that in anyN-node communication network with maximum degreed, any deterministic oblivious algorithm for routing an arbitrary permutation requires (N/d) parallel communication steps in the worst case. This is an improvement upon the (N/d ^3/2) bound obtained by Borodin and Hopcroft. For theN-node hypercube, in particular, we show a matching upper bound by exhibiting a deterministic oblivious algorithm that routes any permutation in (N/logN) steps. The best previously known upper bound was (N). Our algorithm may be practical for smallN (up to about 2¹⁴ nodes).C. Kaklamanis was supported in part by NSF Grant NSF-CCR-87-04513. T. Tsantilas was supported in part by NSF Grants NSF-DCR-86-00379 and NSF-CCR-89-02500.     

Generalized lowness and highness and probabilistic complexity classes

We introduce generalized notions of low and high complexity classes and study their relation to structural questions concerning bounded probabilistic polynomial-time complexity classes. We show, for example, that for a bounded probabilistic polynomial-time complexity class =BP _k ^P ,L =H implies that the polynomial hierarchy collapses to . This extends Schöning's result for = _k ^P (L andH are the low and high sets defined by ). We also show, with one exception, that containment relations between the bounded probabilistic classes and the polynomial hierarchy are preserved by their low and high counterparts.LBPP andLBPNP are characterized asNP BPP andNP co-BPNP, respectively. These characterizations are then used to recover Boppana, Hastad, and Zachos's result that if co-NP BPNP, then the polynomial hierarchy collapses toBPNP, and Ko's result that ifNP BPP, then the polynomial hierarchy collapses toBPP.