Showing papers on "Upper and lower bounds published in 1982"

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.

Bent-function sequences

Abstract: In this paper we construct a new family of nonlinear binary signal sets which achieve Welch's lower bound on simultaneous cross correlation and autocorrelation magnitudes. Given a parameter n with n=0 \pmod{4} , the period of the sequences is 2^{n}-1 , the number of sequences in the set is 2^{n/2} , and the cross/auto correlation function has three values with magnitudes \leq 2^{n/2}+1 . The equivalent linear span of the codes is bound above by \sum_{i=1}^{n/4}\left(\stackrel{n}{i} \right) . These new signal sets have the same size and correlation properties as the small set of Kasami codes, but they have important advantages for use in spread spectrum multiple access communications systems. First, the sequences are "balances," which represents only a slight advantage. Second, the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly "hopped" from sequence to sequence for code division multiple access operation. Most importantly, the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it. This high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing.

Las Vegas is better than determinism in VLSI and distributed computing (Extended Abstract)

Abstract: In this paper we describe a new method for proving lower bounds on the complexity of VLSI - computations and more generally distributed computations. Lipton and Sedgewick observed that the crossing sequence arguments used to prove lower bounds in VLSI (or TM or distributed computing) apply to (accepting) nondeterministic computations as well as to deterministic computations. Hence whenever a boolean function f is such that f and f (the complement of f, f = 1 - f) have efficient nondeterministic chips then the known techniques are of no help for proving lower bounds on the complexity of deterministic chips. In this paper we describe a lower bound technique (Thm 1) which only applies to deterministic computations

An improved Bonferroni inequality and applications

Abstract: SUMMARY We present an improved Bonferroni inequality which gives an upper bound for the probability of the union of an arbitrary sequence of events. The bound is constructed in terms of the joint probability of pairs of events, which are represented by edges on a graph. Examples of applications to periodicity, location shift detection, KolmogorovSmirnov tests and outlier detection are given.

Circuit-size lower bounds and non-reducibility to sparse sets

Abstract: As remarked in Cook (“Towards a Complexity Theory of Synchronous Parallel Computation,≓ Univ. of Toronto, 1980), a nonlinear lower bound on the circuit-size of a language in P or even in NP is not known. The best known published lower bound seems to be due to Paul (“Proceedings, 7th ACM Symposium on Theory of Computing,≓ 1975). In this paper it is shown first that for each nonnegative integer k there is a language L k in σ 2 ⌢ π 2 (of the Meyer and Stockmeyer (“Proceedings, 13th IEEE Symposium on Switching and Automata Theory,≓ 1972) hierarchy) which does not have O( n k )-size circuits. Using the same techniques, one is able to prove several similar results. For example, it is shown that for each nonnegative integer k , there is a language L k in NP that does not have O( n k )-size uniform circuits. This follows as a corollary of a stronger result shown in the paper. This result like the others to follow is not provable by direct diagonalization. It thus points to the most interesting feature of the techniques used hereby using the polynomial-time hierarchy, they are able to prove results about NP that cannot seem to proved by direct diagonalization. Finally, it is noted that existence of “small circuits≓ is in suitable contexts equivalent to being reducible to sparse sets. Using this, one is able to prove, for example, that for any time-constructible superpolynomial function f ( n ), NTIME( f ( n )) contains a language which is not many-to-one p -time reducible to any sparse set.

Solution of Large-Scale Optimal Unit Commitment Problems

Abstract: This paper is concerned with the solution of large-scale unit commitment problems. An optimization model has been developed for these problems that incorporates minimum up and down time constraints, demand and reserve constraints, cooling-time dependent startup-costs, and time varying shutdown costs, as well as other practical considerations. A solution methodology has been developed for the optimization model that has two unique features. First, computational requirements grow only linearly with the number of units. Second, performance of the algorithm can be shown (rigorously) to actually improve as the number of units increases. With a preliminary computer implementation of the algorithm, we have been able to reliably solve problems with 250 units over 12 (2-hour) time periods, and we expect to be able to easily double these numbers.

Error Probability for Direct-Sequence Spread-Spectrum Multiple-Access Communications--Part I: Upper and Lower Bounds

Abstract: Upper and lower bounds on the average probability of error are obtained for direct-sequence spread-spectrum multipleaccess communications systems with additive white Gaussian noise channels. The bounds, which are developed from convexity properties of the error probability function, are valid for systems in which the maximum multiple-access interference does not exceed the desired signal and the signature sequence period is equal to the duration of the data pulse. The tightness of the bounds is examined for systems with a small number of simultaneously active transmitters. This is accomplished by comparisons of the upper and lower bounds for several values of the system parameters. The bounds are also compared with an approximation based on the signal-to-noise ratio and with the Chernoff upper bound.

On eigenstructure assignment in linear multivariable systems

Abstract: The eigenvalue-assignment approach of Brogan [1], [2] is generalized and extended to the assignment of the entire closed-loop eigenstructure of linear multivariable systems The set of assignable eigenvectors and generalized eigenvectors emerges naturally in the solution and is given, moreover, in an explicit parametric form Two numerical examples are worked out to demonstrate the application of the procedure

A Lower Bound for Heilbronn'S Problem

Abstract: We disprove Heilbronn's conjecture—that N points lying in the unit disc necessarily contain a triangle of area less than c/N. Introduction We think that the best account of the problem can be achieved by simply copying the corresponding paragraphs from Roth's paper [6]. \"Let Pv, P2,..., Pn (where n ^ 3) be a distribution of n points in a (closed) disc of unit area, such that the minimum of the areas of the triangles PiPjPk (taken over 1 < i < j < k ^ n) assumes its maximum possible value A = A(n). \"Heilbronn conjectured that A(n) c2(n/t)(\\ogt) . Remark 1. For any 3-graph one has the Turan-type estimate a > n/(3t), since a random (spanned) subgraph of size n/(2t) expectedly contains n/(24t) edges; delete all vertices in edges (see Spencer [8]). This is sharp up to constant multiple, for the Turan 3-graph (n/t disjoint cliques of size t) has t ~ t/2 and a = 2n/t. Thus the condition that G is uncrowded improves the bound on a by a factor (log t). This is again sharp, as is shown by random 3-graphs (choose nt triples at random and delete the few short cycles). Remark 2. Lemma 1 is an analogue of Lemma 1 in [1] or Theorem 2 in [2], which state that for a 2-graph with average valency t = 2e/n the Turan bound a ^ n(t+\\) can be improved to a > (i/\\00)(n/t)\\ogt if only the graph is trianglefree. The proof will also be analogous to the (complicated) one in [1], which uses random methods, rather than to the simple inductive one in [2], which was thoroughly rewritten and simplified by Joel Spencer. The reader is challenged to give a simple, non-probabilistic proof for Lemma 1. 2. The proof of the theorem If we drop N points to the unit disc at random, then (as will be seen shortly) we can select half of them with smallest triangle c/N (an alternative proof for Erdos' lower bound). We shall improve on this method by dropping N points and then selecting an appropriate subset of N points. Define the numbers t and n by the implicit equations t = n / 1 0 0 , N = c2(n/t)(\\ogt) 112 . Set A = (1/200) t/n; then n = -Nt(\\ogty' 2 and A = c3(log0/iV 2 = Cl(\\ogN)/N 2 . Let us drop n points to the unit disc at random, independently of each other, each with a uniform distribution. We define a 3-graph G on these n points (as vertices) by {a, b, c} € G if the points a, b, c form a triangle of area less than A. Now the probability that three random points form a triangle of area less than A is less than 2 2 — d{rn) = — Inrdr = 32TIA < t/n r J r o 16 JANOS KOML6S, JANOS PINTZ AND ENDRE SZEMERED1 (fix two points at a distance r, and then average over r). Hence the expected number of triangles of area less than A is less than nt/6. Thus the expected value of f is less than t/2. Hence, by Markov's inequality (see §4), with probability greater than 1/2, weget a 3-graph with t < t. Now we show that, with large probability, only o(n) short cycles occur in this 3-graph. All calculations will be based on the simple remark (already used above) that once two vertices have been chosen at a distance r, in order to get a triangle of area less than A, the third point has to belong to a strip of area less than 8A/r. The number of pairs of points at a distance less than d = n~° 6 is, with large probability, less than We discard these points. The number of 2-cycles is, with large probability, less than 2 f A c 4 n 4 —Inrdr < c5t \\ogn < n . The number of simple 3-cycles is, with large probability, less than

On Upper and Lower Bounds for the Variance of a Function of a Random Variable

Abstract: Chernoff (1981) obtained an upper bound for the variance of a function of a standard normal random variable, using Hermite polynomials. Chen (1980) gave a different proof, using the Cauchy-Schwarz inequality, and extended the inequality to the case of a multivariate normal. Here it is shown how similar upper bounds can be obtained for other distributions, including discrete ones. Moreover, by using a variation of the Cramer-Rao inequality, analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions. Matrix inequalities are also obtained. All these bounds involve the first two moments of derivatives or differences of the function.

On the Problem of Partitioning Planar Graphs

Abstract: The results in this paper are closely related to the effective use of the divide-and-conquer strategy for solving problems on planar graphs. It is shown that every planar graph can be partitioned into two or more components of roughly equal size by deleting only $O ( \sqrt{n} )$ vertices, and such a partitioning can be found in $O ( n )$ time. Some of the theorems proved in the paper are improvements on the previously known theorems while others are of more general form. An upper bound for the minimum size of the partitioning set is found.

Rank-Reducibility of a Symmetric Matrix and Sampling Theory of Minimum Trace Factor Analysis.

Abstract: One of the intriguing questions of factor analysis is the extent to which one can reduce the rank of a symmetric matrix by only changing its diagonal entries. We show in this paper that the set of matrices, which can be reduced to rankr, has positive (Lebesgue) measure if and only ifr is greater or equal to the Ledermann bound. In other words the Ledermann bound is shown to bealmost surely the greatest lower bound to a reduced rank of the sample covariance matrix. Afterwards an asymptotic sampling theory of so-called minimum trace factor analysis (MTFA) is proposed. The theory is based on continuous and differential properties of functions involved in the MTFA. Convex analysis techniques are utilized to obtain conditions for differentiability of these functions.

Growth rates of bending KdV solitons

Abstract: Nonlinear ion-acoustic waves in magnetized plasmas are investigated. In strong magnetic fields they can be described by a Korteweg-de Vries (KdV) type equation. It is shown here that these plane soliton solutions become unstable with respect to bending distortions. Variational principles are derived for the maximum growth rate γ as a function of the transverse wavenumber k of the perturbations. Since the variational principles are formulated in complementary form, the numerical evaluation yields upper and lower bounds for γ. Choosing appropriate test functions and increasing the accuracy of the computations we find very close upper and lower bounds for the γ(k) curve. The results show that the growth rate peaks at a certain value of k and a cut-off kc exists. In the region where the γ(k) curve was not predicted numerically with high accuracy, i.e. near the cut-off, we find very precise analytical estimates. These findings are compared with previous results. For k≥kc, stability with respect to transverse perturbations is proved.

Eigenvalues of the Laplacian on forms

Abstract: Some bounds for eigenvalues of the Laplace operator acting on forms on a compact Riemannian manifold are derived. In case of manifolds without boundary we give upper bounds in terms of the curvature, its covariant derivative and the injectivity radius. For a small geodesic ball upper and lower bounds of eigenvalues in terms of bounds of sectional curvature are given. In (2) Cheng proves a beautiful comparison theorem for the first eigenvalue of the Laplacian Ao on functions for a geodesic ball in a Riemannian manifold, and derives as a consequence, upper bounds for higher eigenvalues of the Laplacian on functions for a compact manifold. These upper bounds are derived by taking the first eigenfunctions with Dirichlet boundary conditions for small balls, extending them by zero to the whole manifold, and estimating the Rayleigh-Ritz quotient of an appropriate linear combination. The same procedure cannot be applied to forms of positive degree since an eigenfunction for a ball satisfying either absolute or relative boundary conditions will not be in the Sobolev space 771 when extended by zero to the whole manifold. In this note a modification of the method of Eichhorn (4) is used to prove that, for certain forms on a ball which vanish on the boundary, it is possible to estimate the Rayleigh-Ritz quotient in terms of geometric quantities. These forms are in H1 when extended by zero and Cheng's argument applied to them gives upper bounds for eigenvalues on a closed manifold. Our result is much less elegant than Cheng's theorem. In the first place we have to require that the geodesic ball is contained within the cut locus. Hence, all estimates of higher eigenvalues depend on the injectivity radius. Secondly, Cheng obtained explicit estimates of eigenvalues of the Laplacian Ao in terms of geometric quantities (cf. (2, Corollaries 2.2, 2.3, Theorem 2.4)), whereas we only say which geometric quantities determine the bounds of eigenvalues but give no estimates of the actual bounds. Explicit estimates could be derived from our method, but the constants are so complicated that we were unable to obtain any useful information from them. It is an interesting question, whether all of the geometric quantities which appear in our estimates (sectional curvature, injectivity radius, the bounds for the Kern tensor R^) are really necessary. We do not know the answer for closed manifolds. However, in §3 we show, following a suggestion of J. Cheeger, that for a geodesic ball of radius smaller than the radius of injectivity, eigenvalues of the Laplacian Ap can be estimated from above and below in terms of bounds of sectional curvature. It is a pleasure to thank J. Cheeger for suggestions which led to improvement of this paper.

Bounds on the time for parallel RAM's to compute simple functions

Abstract: We prove that a parallel RAM with no write conflicts allowed requires Ω(log n) steps to compute the Boolean or of n bits stored in the first n global memory cells. We first argue that this result is subtler than it appears, and in fact the “obvious” lower bound of log2n steps can be beaten.

The direct reaction field hamiltonian: Analysis of the dispersion term and application to the water dimer

Abstract: The induction and dispersion terms obtained from quantum-mechanical calculations with a direct reaction field hamiltonian are compared to second order perturbation theory expressions. The dispersion term is shown to give an upper bound which is a generalization of Alexander's upper bound. The model is illustrated by a calculation on the interactions in the water dimer. The long range Coulomb, induction and dispersion interactions are reasonably reproduced.

Any Discrimination Rule Can Have an Arbitrarily Bad Probability of Error for Finite Sample Size

Abstract: Consider the basic discrimination problem based on a sample of size n drawn from the distribution of (X, Y) on the Borel sets of Rdx {0, 1}. If 0 ?. Thus, any attempt to find a nontrivial distribution-free upper bound for Rn will fail, and any results on the rate of convergence of Rn to R* must use assumptions about the distribution of (X, Y).

Information rates and power spectra of digital codes

Abstract: The encoding of independent data symbols as a sequence of discrete amplitude, real variables with given power spectrum is considered. The maximum rate of such an encoding is determined by the achievable entropy of the discrete sequence with the given constraints. An upper bound to this entropy is expressed in terms of the rate distortion function for a memoryless finite alphabet source and mean-square error distortion measure. A class of simple dc-free power spectra is considered in detail, and a method for constructing Markov sources with such spectra is derived. It is found that these sequences have greater entropies than most codes with similar spectra that have been suggested earlier, and that they often come close to the upper bound. When the constraint on the power spectrum is replaced by a constraint On the variance of the sum of the encoded symbols, a stronger upper bound to the rate of dc-free codes is obtained. Finally, the optimality of the binary biphase code and of the ternary bipolar code is decided.

A general Chebyshev complex function approximation procedure and an application to beamforming

Abstract: A new computational technique is described for the Chebyshev, or minimax, approximation of a given complex valued function by means of linear combinations of given complex valued basis functions. The domain of definition of all functions can be any finite set whatever. Neither the basis functions nor the function approximated need satisfy any special hypotheses beyond the requirement that they be defined on a common domain. Theoretical upper and lower bounds on the accuracy of the computed Chebyshev error are derived. These bounds permit both a priori and a posteriori error assessments. Efforts to extend the method to functions whose domain of definition is a continuum are discussed. An application is presented involving ’’re‐shading’’ a 50‐ element antenna array to minimize the effects of a 10% element failure rate, while maintaining full steering capability and mainlobe beamwidth.

Probability distributions for the strength of fibrous materials under local load sharing i: two-level failure and edge effects

Abstract: The focus of this paper is on obtaining a conservative but tight bound on the probability distribution for the strength of a fibrous material. The model is the chain-of-bundles probability model, and local load sharing is assumed for the fiber elements in each bundle. The bound is based upon the occurrence of two or more adjacent broken fiber elements in a bundle. This event is necessary but not sufficient for failure of the material. The bound is far superior to a simple weakest link bound based upon the failure of the weakest fiber element. For large materials, the upper bound is a Weibull distribution, which is consistent with experimental observations. The upper bound is always conservative, but its tightness depends upon the variability in fiber element strength and the volume of the material. In cases where the volume of material and the variability in fiber strength are both small, the bound is believed to be virtually the same as the true distribution function for material strength. Regarding edge effects on composite strength, only when the number of fibers is very small is a correction necessary to reflect the load-sharing irregularities at the edges of the bundle.

Lower bounds for increasing complexity of derivations after cut elimination

Abstract: We denote by C k * the formula . In this paper for all k there is constructed a derivation of C k * with cut, the number of sequents in which depends linearly on k. On the other hand, it is impossible to give an upper bound which is a Kalmar elementary function of k for the number of sequents in any derivation of the formula C k * without cuts, or for the number of disjunctions in a refutation by the method of resolutions of systems of disjunctions corresponding to the negation of the formula C k * . In particular, one can construct a derivation with cut of the formula C 6 * , in which there is contained no more than 253 sequents, but in seeking a derivation of C 6 * by the method of resolutions it is necessary to construct more than 1019200 disjunctions. With the help of Skolemization and taking out of quantifiers with respect to the formula C k * there is constructed a formula ∃v0B k + (v0), which satisfies the following conditions: 1) one can construct a derivation with cuts of the formula ∃v0B k + (v0) in the constructive predicate calculus, the number of sequents in which depends linearly on k; 2) it is impossible to give an upper bound which is a Kalmar elementary function of k of the length of a term t such that the formula B k + (t) is derivable.

The equation of state of Mg0.6Fe0.4O to 200 GPa

Abstract: New Hugoniot data on polycrystalline (avg.porosity 6.9%) samples of the magnesiowustite Mg_(0.6)Fe_(0.4)O are presented, covering the pressure range up to 200 GPa. When our data are fit by a single 3rd order Eulerian Hugoniot with K_0 constrained to its ultrasonic value of 161.5 GPa, the required isentropic pressure derivative K_0′ is 4.37 +/− 0.37. This value is significantly lower than the ultrasonic one of 6.18; existing isothermal compression data, however, are in agreement with our value rather than the ultrasonic one. Our data are adequately explained without phase transitions. There is some marginal evidence for a possible phase transition around 120 GPa. If such a transition indeed occurs it is probably of small volume change compared to the transition observed in FeO; we place an extreme upper bound of 3% on the density change such a transformation could involve and still be consistent with the data. Contrary to earlier hypotheses, we believe that a phase transition in magnesiowustite is not a likely explanation of the seismic effects in the D″ region of the lower mantle. The wustite transition may be a more complex phenomenon than initially supposed — perhaps an effect of nonstoichiometry localized to the iron-rich end of the solid solution series.