Showing papers on "Bounded function published in 1982"

Protocols for secure computations

Abstract: Two millionaires wish to know who is richer; however, they do not want to find out inadvertently any additional information about each other’s wealth. How can they carry out such a conversation? This is a special case of the following general problem. Suppose m people wish to compute the value of a function f(x1, x2, x3, . . . , xm), which is an integer-valued function of m integer variables xi of bounded range. Assume initially person Pi knows the value of xi and no other x’s. Is it possible for them to compute the value of f , by communicating among themselves, without unduly giving away any information about the values of their own variables? The millionaires’ problem corresponds to the case when m = 2 and f(x1, x2) = 1 if x1 < x2, and 0 otherwise. In this paper, we will give precise formulation of this general problem and describe three ways of solving it by use of one-way functions (i.e., functions which are easy to evaluate but hard to invert). These results have applications to secret voting, private querying of database, oblivious negotiation, playing mental poker, etc. We will also discuss the complexity question “How many bits need to be exchanged for the computation”, and describe methods to prevent participants from cheating. Finally, we study the question “What cannot be accomplished with one-way functions”. Before describing these results, we would like to put this work in perspective by first considering a unified view of secure computation in the next section.

On the computational complexity of Ising spin glass models

Abstract: In a spin glass with Ising spins, the problems of computing the magnetic partition function and finding a ground state are studied. In a finite two-dimensional lattice these problems can be solved by algorithms that require a number of steps bounded by a polynomial function of the size of the lattice. In contrast to this fact, the same problems are shown to belong to the class of NP-hard problems, both in the two-dimensional case within a magnetic field, and in the three-dimensional case. NP-hardness of a problem suggests that it is very unlikely that a polynomial algorithm could exist to solve it.

Connections with l**p bounds on curvature

Abstract: We show by means of the implicit function theorem that Coulomb gauges exist for fields over a ball inR n when the integralL n/2 field norm is sufficiently small. We then are able to prove a weak compactness theorem for fields on compact manifolds withL p integral norms bounded,p>n/2.

The Green function for uniformly elliptic equations

Abstract: The authors discuss a generalization of the usual Green function to equations with only measurable and bounded coefficients The existence and uniqueness as well as several other important properties are shown Such a Green function proves useful in connection with quasilinear elliptic systems of “diagonal type”

Odd Minimum Cut-Sets and b-Matchings

Abstract: We show that the determination of a minimum cut-set of odd cardinality in a graph with even and odd vertices can be dealt with by a minor modification of the polynomially bounded algorithm of Gomory and Hu for multi-terminal networks. We connect this problem to the problem of identifying a matching or blossom constraint that chops off a point which is not contained in the convex hull of matchings or proving that no such inequality exists. Both the b-matching problems without and with upper bounds are considered. We discuss how the results of this paper can be used in conjunction with commercial LP packages lo solve b-matching problems.

Removable singularities in Yang-Mills fields

Abstract: We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR 4 with bounded functional (L 2 norm) may be obtained from a field onS 4=R 4∪{∞}. Hodge (or Coulomb) gauges are constructed for general small fields in arbitrary dimensions including 4.

Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations

Abstract: It is shown that the Ritz projection onto spaces of piecewise linear finite elements is bounded in the Sobolev space, Wl, for 2 - p < oc. This implies that for functions in wf n W2 the error in approximation behaves like 0(h) in Wp, for 2 ? p < oo, and like 0(h2) in Lp, for 2 - p < oo. In all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur.

Invariant Measures for the Zero Range Process

Abstract: On a countable set of sites $S$, the zero range process is constructed when the stochastic matrix $p(x, y)$ determining the one particle motion satisfies a mild assumption. The set of invariant measures for this process is described in the following two cases: a) The system is attractive and $p(x, y)$ is recurrent. b) The system is attractive, $p(x, y)$ corresponds to a simple random walk on the integers and the rate at which particles leave any site is bounded.

On critical point theory for indefinite functionals in the presence of symmetries

Abstract: We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group. We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments. The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited. Introduction. Many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above. However some of these functionals, defined in an appropriate function space, are "semidefinite" in the sense that they are bounded from below (or from above) if perturbed with a weakly continuous functional. This paper deals with functionals which are not semidefinite. Usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space [7]. In [7] some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals. In many physical situations there are problems which have symmetries with respect to the action of some Lie group. In this case we expect the existence of many critical points; this has been established for semidefinite functionals (cf. e.g. [8 and 3] for even functionals, [9] for a Zp-action with p prime numbers [15, 16, 10 and 6] for an S'-action). In this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries. More precisely Received by the editors July 16, 1980 and, in revised form, June 26, 1981. 1980 Mathematics Subject Classification. Primary 58E05, 70H05; Secondary 34C25.

Stable model reference adaptive control in the presence of bounded disturbances

Abstract: The adaptive control of a linear time-invariant plant in the presence of bounded disturbances is considered. In addition to the usual assumptions made regarding the plant transfer function, it is also assumed that the high-frequency gain k p of the plant and an upper bound on the magnitude of the controller parameters are known. Under these conditions the adaptive controller suggested assures the boundedness of all signals in the overall system.

Isomorphism of graphs with bounded eigenvalue multiplicity

Abstract: We investigate the connection between the spectrum of a graph, i.e. the eigenvalues of the adjacency matrix, and the complexity of testing isomorphism. In particular we describe two polynomial time algorithms which test isomorphism of undirected graphs whose eigenvalues have bounded multiplicity. If X and Y are graphs of eigenvalue multiplicity m, then the isomorphism of X and Y can be tested by an O(n4m+c) deterministic and by an O(n2m+c) Las Vegas algorithm, where n is the number of vertices of X and Y.

Robust multivariable PI-controller for infinite dimensional systems

Abstract: A robust multivariable controller is introduced for a class of distributed parameter systems. The system to be controlled is given as \dot{x} = Ax + Bu, y = Cx in a Banach space. The purpose of the control, which is based on the measurement y , is to stabilize and regulate the system so that y(t) \rightarrow y_{r}, as t \rightarrow \infty , where y r is a constant reference vector. Under the assumptions that operator A generates a holomorphic stable semigroup, B is linear and bounded, C is linear and A -bounded, and the input and output spaces are of the same dimension; a necessary and sufficient condition is found for the existence of a robust multivariable controller. This controller appears to be a multivariable PI-controller. Also, a simple necessary criterion for the existence of a decentralized controller is derived. The tuning of the controller is discussed and it is shown that the I-part of the controller can be tuned on the basis of step responses, without exact knowledge of the system's parameters. The presented theory is then used as an example to control the temperature profile of a bar, with the Dirichlet boundary conditions.

HANKEL OPERATORS OF CLASS $ \mathfrak{S}_p$ AND THEIR APPLICATIONS (RATIONAL APPROXIMATION, GAUSSIAN PROCESSES, THE PROBLEM OF MAJORIZING OPERATORS)

Abstract: A criterion is given for a Hankel operator , where is the orthogonal projection of onto ) to belong to the Schatten-von Neumann class in terms of its symbol . Various applications are considered: a precise description is obtained for classes of functions definable in terms of rational approximation in the (bounded mean oscillation) norm; it is proved that the averaging projection onto the set of Hankel operators is bounded in the norm of , ; a counterexample is given to a conjecture of Simon on the majorization property in ; a problem of Ibragimov and Solev on stationary Gaussian processes is solved; and a criterion is obtained for functions of an operator in the Sz.-Nagy-Foia? model to belong to the class .Bibliography: 47 titles.

The complexity of finding maximum disjoint paths with length constraints

Abstract: The following problem is considered: Given an integer K, a graph G with two distinct vertices s and t, find the maximum number of disjoint paths of length K from s to t. The problem has several variants: the paths may be vertex-disjoint or edge-disjoint, the lengths of the paths may be equal to K or bounded by K, the graph may be undirected or directed. It is shown that except for small values of K all the problems are NP-complete. Assuming P ≠ NP, for each problem, the largest value of K for which the problem is not NP-complete is found. Whenever a polynomial algorithm exists, an efficient algorithm is described.

Reasoning with time and chance

Abstract: The temporal propositional logic of linear time is generalized to an uncertain world, in which random events may occur. The formulas do not mention probabilities explicitly, i.e., the only probability appearing explicitly in formulas is probability one. This logic is claimed to be useful for stating and proving properties of probabilistic programs. It is convenient for proving those properties that do not depend on the specific distribution of probabilities used in the program's random draws. The formulas describe properties of execution sequences. The models are stochastic systems, with state transition probabilities. Three different axiomatic systems are proposed and shown complete for general models, finite models, and models with bounded transition probabilities, respectively. All three systems are decidable, by the results of Rabin ( Trans. Amer. Math. Soc. 141 (1969), 1–35).

Complexity of Matroid Property Algorithms

Abstract: A general theorem is proved which can be used to show that for a large number of matroid properties there is no good algorithm of a certain type for determining whether these properties hold for general matroids. Specifically, there exists no algorithm in which the matroid is represented by an independence test oracle (or an oracle polynomially related to an independence test oracle) and which solves the problem in question after a number of calls on the oracle which is bounded by a polynomial in the number of elements of the ground set of the matroid.

Wiener's criterion for the heat equation

Abstract: This paper establishes a necessary and sufficient condition for a boundary point of an arbitrary bounded open subset of R "+1 to be regular for the heat equation. Our criterion is, in the sense described below, the exact analogue of WINNER'S test [W] for boundary regularity of harmonic functions. A careful statement of our result requires that we first of all introduce a great deal of terminology. Let .(2 Q R "+1 (n ~ 1) denote any bounded open subset and 8/2 its topological boundary. We will often write a typical point z E R "+1 as z = (x, t), xER~, tE R. If uE C2(-Q;R) we may define the heat operator

A note on degree theory for gradient mappings

Abstract: In this note we give a simple proof for the essentially known fact, that the Leray-Schauder degree of the gradient of a coercive functional on a large ball of a Hilbert space is one. As a simple application we show that the local index of an isolated local minimum of a Cl-functional on a Hilbert space equals one. 1. Let U be an open subset of some real Hilbert space H and suppose that the gradient Vf: U -H of a given function f E C1 (U, R) is a compact vector field, that is, Vf = id F, where F E C(U, H) maps bounded sets into compact sets. In this note we give a simple proof for the following useful theorem, where B(xo, r) denotes the open ball in H, with center x0 and radius r, and a bar denotes the closure in H. THEOREM. Suppose that, for some / E R, the set V: f 1(-oo, 3) is bounded and V C U. Moreover, suppose that there are numbers a 0, and a point xo E U such that f1(-oo,a] C B(xo,r) C V and Vf(x) $0 Vx E f-1 [a, 3]. Then deg(Vf, V, 0) = 1. As easy consequences of this theorem we obtain the following corollaries: COROLLARY 1. Suppose that U = H and f(x) -oo as |lx|| _ oo. Moreover, suppose that Vf(x) $4 0 for llxii > rO and some rO > 0. Then there is a number r, > rO such that deg(Vf, B(O, r), 0) = 1 Vr > rl. PROOF. Observe that f(x) = (iixi12/2) 4?(x), where Vl) = F. Since F is compact, 4) is weakly sequentially continuous (on convex sets) by Vainberg's theorem [10, Theorem 8.2]. Hence f maps bounded sets into bounded sets. Thus, let a:= supf(B(O,ro)) and ri := sup{llxll Ix E f 1(-oo,a]}. Moreover, given r > r1, fix /3 > sup f(B(O, r)). Then the assertion follows from the theorem (with xo = 0) and the excision property of the degree. El In the following corollary we denote by i(Vf, xo) the index of Vf at an isolated zero x0. Received by the editors November 16, 1981. 1980 Mathematics Subject Classification. Primary 58E05; Secondary 47H15, 47H10, 34G20.

Classical Coulomb systems near a plane wall. II

Abstract: The equilibrium structure of classical Coulomb systems bounded by a plane hard wall is studied near that wall. A general sum rule is derived for the asymptotic form of the charge-charge correlation function along the wall. The exact results which can be obtained for the two-dimensional one-component plasma provide a test for this new sum rule, as well as for other already known sum rules or their generalizations.

The Bergman kernel function and proper holomorphic mappings

Abstract: It is proved that a proper holomorphic mapping f between bounded complete Reinhardt domains extends holomorphically past the boundary and that if, in addition, f'1(0) = (01, then f is a polynomial mapping. The proof is accomplished via a transformation rule for the Bergman kernel function under proper holomorphic mappings.

Homogeneous polynomial identities

Abstract: PI-algebras are studied by attaching invariants to the homogeneous identities analogous to the invariants of the multilinear identities studied by Regev. Also, it is shown that every finitely generated PI-algebra is polynomially bounded.