Vague numbers

If there are vague numbers, it would be easier to use numbers as semantic values in a treatment of vagueness while avoiding precise cut-off points. When we assign a particular statement a range of values (less than 1 and greater than 0) there is no precise sharp cut-off point that locates the greatest lower bound or the least upper bound of the interval, I should like to say. Is this possible? “Vague Numbers” stands for awareness of the problem. I do not present a serious theory of vague numbers. I sketch some reasons for using a many-value semantics. These reasons refer to my earlier treatments of determinacy and definitions of higher-order borderline cases. I also sketch how definitions of independence use the determinacy operator. The distinction between actually assigned values and values whose assignments are acceptable helps avoid unwanted precise cut-off points.     

Qubit complexity of continuous problems

The number of qubits used by a quantum algorithm will be a crucial computational resource for the foreseeable future. We show how to obtain the classical query complexity for continuous problems. We then establish a simple formula for a lower bound on the qubit complexity in terms of the classical query complexity.     

Quantum Transport in Degenerate Systems

Transport in nonequilibrium degenerate quantum systems is investigated. The transfer rate depends on the parameters of the system. In this paper we investigate the dependence of the flow (transfer rate) on the angle between “bright” vectors (which define the interaction of the system with the environment). We show that in some approximation for the system under investigation the flow is proportional to the cosine squared of the angle between the “bright” vectors. Earlier the author has shown that in this degenerate quantum system excitation of nondecaying quantum “dark” states is possible; moreover, the effectiveness of this process is proportional to the sine squared of the angle between the “bright” vectors (this phenomenon was discussed as a possible model of excitation of quantum coherence in quantum photosynthesis). Thus quantum transport and excitation of dark states are competing processes; “dark” states can be considered as a result of leakage of quantum states in a quantum thermodynamic machine which performs the quantum transport.     

Adaptive energy efficient sensor scheduling for wireless sensor networks

In this paper, we consider an adaptive energy efficient sensor scheduling mechanism. We consider a wireless sensor network where the sink sends queries form time to time, and sensors are equipped with one or more sensing components. Our goal is to design an adaptive sensor scheduling mechanism to choose sets of active sensors to work alternatively such that different types of queries are served, the global connectivity requirements can be met, and network lifetime is maximized. A connected dominating set (CDS) based localized mechanism is proposed. Initially, a basic backbone is constructed, then when a query is issued, new sensors are activated locally such that to meet the requirements of the query and global connectivity. When a query expires, some sensors return to sleep and the CDS is restored. Our simulation results show that the solution is effective and it improved network lifetime.     

Preprocessing chains for fast dihedral rotations is hard or even impossible

We examine a computational geometric problem concerning the structure of polymers. We model a polymer as a polygonal chain in three dimensions. Each edge splits the polymer into two subchains, and a dihedral rotation rotates one of these subchains rigidly about the edge. The problem is to determine, given a chain, an edge, and an angle of rotation, if the motion can be performed without causing the chain to self-intersect. An Ω(nlogn) lower bound on the time complexity of this problem is known.We prove that preprocessing a chain of n edges and answering n dihedral rotation queries is 3 -hard, giving strong evidence that Ω(n²) preprocessing is required to achieve sublinear query time in the worst case. For dynamic queries, which also modify the chain if the requested dihedral rotation is feasible, we show that answering n queries is by itself 3 -hard, suggesting that sublinear query time is impossible after any amount of preprocessing.     

Binary Representation of Coordinate and Momentum in Quantum Mechanics

To simulate a quantum system with continuous degrees of freedom on a quantum computer based on qubits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to binary observables. We consider this problem based on expanding quantum observables in series in powers of two, analogous to the binary representation of real numbers. The coefficients of the series (“digits”) are therefore orthogonal projectors. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary expansion of quantum observables automatically leads to renormalization of some divergent integrals and series (giving them finite values).     

Blister Patterns and Energy Minimization in Compressed Thin Films on Compliant Substrates

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit that compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is therefore natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the elastic deformation of the film and substrate and the geometry of the blistered region. It involves three small parameters: the nondimensionalized thickness of the film, the compliance ratio of the film/substrate pair, and the mismatch strain. In formulating the problem, we use a small‐slope (Föppl–von Kármán) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a one‐dimensional version of the problem, we obtain “matching” upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. The upper bound is straightforward and familiar: it is achieved by periodic blistering on a specific length scale. The lower bound is more subtle, since it must be proved without any assumption on the geometry of the blistered region. For a two‐dimensional version of the problem, our results are less complete. Our upper and lower bounds only “match” in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The lower bound is an easy consequence of our one‐dimensional analysis. The upper bound considers a two‐dimensional lattice of blisters and uses ideas from the literature on the folding or “crumpling” of a confined elastic sheet. Our main two‐dimensional result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters. © 2015 Wiley Periodicals, Inc.     

Quantum lower bounds by entropy numbers

We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the nth minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional L_p spaces and of Sobolev embeddings.     

Measurable versions of the LS category on laminations

We give two new versions of the LS category for the set-up of measurable laminations defined by Bermúdez. Both of these versions must be considered as “tangential categories”. The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Macías Virgós. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the Λ-category, depends on the choice of a transverse invariant measure Λ. We show that both of these “tangential categories” satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, existence of a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function. Also, we show possible applications of these invariants to variational problems.     

Effective Search Problems

The task of computing a function F with the help of an oracle X can be viewed as a search problem where the cost measure is the number of queries to X. We ask for the minimal number that can be achieved by a suitable choice of X and call this quantity the query complexity of F. This concept is suggested by earlier work of Beigel, Gasarch, Gill, and Owings on “Bounded query classes”. We introduce a fault tolerant version and relate it with Ulam's game. For many natural classes of functions F we obtain tight upper and lower bounds on the query complexity of F. Previous results like the Nonspeedup Theorem and the Cardinality Theorem appear in a wider perspective. Mathematics Subject Classification: 03D20, 68Q15, 68R05.     

Convergence of finite element approximations of large eddy motion

Fluid motion in many applications occurs at higher Reynolds numbers. In these applications dealing with turbulent flow is thus inescapable. One promising approach to the simulation of the motion of the large structures in turbulent flow is large eddy simulation in which equations describing the motion of local spatial averages of the fluid velocity are solved numerically. This report considers “numerical errors” in LES. Specifically, for one family of space filtered flow models, we show convergence of the finite element approximation of the model and give an estimate of the error. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 689–710, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10027.     

Heuristics for selecting robust database structures with dynamic query patterns

The success of a company increasingly depends on timely information (internal or external) being available to the right person at the right time for crucial managerial decision-making. Achieving such a “right time/right place” duet depends directly on database performance. A database system has been a core component that supports modern business system such as enterprise resource planning (ERP) system that integrates and supports all enterprise processes including product designing and engineering, manufacturing, and other business functions to achieve highest efficiency and effectiveness of operations. We develop and demonstrate through a proof-of-concept case study, a new “query-driven” heuristics for database design that seeks to identify database structures that perform robustly in dynamic settings with dynamic queries. Our focus is the design of efficient structures to process read-only queries in complex environments. Our heuristics begins with detailed analysis of relationships between diverse queries and the performance of different database structures. These relationships are then used in a series of steps that identify “robust” database structures that maintain high performance levels for a wide range of query patterns. We conjecture that our heuristics can facilitate efficient operations and effective decision-making of companies in today’s dynamic environment.     

On simulating the quantum and classical branching programs

The complexity classes defined on the basis of branching programs are considered. Some basic relations are established between the complexity classes defined by the probabilistic and quantum branching programs (measure-once, as well as measure-many), computing with bounded or unbounded error. To prove these relations, we developed a method of “linear simulation” of a quantum branching program and a method of “quantum simulation” of a probabilistic branching program.     

On the Discrete Spectrum of Generalized Quantum Tubes

The spectrum (of the Dirichlet Laplacian) of non-compact, non-complete Riemannian manifolds is much less understood than their compact counterparts. In particular it is often not even known whether such a manifold has any discrete spectra. In this article, we will prove that a certain type of non-compact, non-complete manifold called the quantum tube has non-empty discrete spectrum. The quantum tube is a tubular neighborhood built about an immersed complete manifold in Euclidean space. The terminology of “quantum” implies that the geometry of the underlying complete manifold can induce discrete spectra – hence quantization. We will show how the Weyl tube invariants appear in determining the existence of discrete spectra. This is an extension and generalization, on the geometric side, of the previous work of the author on the “quantum layer.”     

Box-Trees and R-Trees with Near-Optimal Query Time

Abstract. A box-tree is a bounding-volume hierarchy that uses axis-aligned boxes as bounding volumes. The query complexity of a box-tree with respect to a given type of query is the maximum number of nodes visited when answering such a query. We describe several new algorithms for constructing box-trees with small worst-case query complexity with respect to queries with axis-parallel boxes and with points. We also prove lower bounds on the worst-case query complexity for box-trees, which show that our results are optimal or close to optimal. Finally, we present algorithms to convert box-trees to R-trees, resulting in R-trees with (almost) optimal query complexity.     

Chain rule for conic derivatives

For all “nice” definitions of differentiability, the Chain Rule should be valid. We show that the Chain Rule remains true for some wide class of definitions of differentiability if one considers as approximative mappings (derivatives) not just continuous linear, but positively homogeneous mappings satisfying certain topological conditions (which are fulfilled for continuous linear mappings). For brevity, we call such derivatives conic. We will give corollaries for the case of conic differentiation of mappings between normed spaces, especially for the case of Fréchet conic differentiation and compact conic differentiation.     

Global-in-time existence of solutions to the multiconfiguration time-dependent Hartree–Fock equations: A sufficient condition

The multiconfiguration time-dependent Hartree–Fock (MCTDHF for short) system is an approximation of the linear many-particle Schrödinger equation with a binary interaction potential by nonlinear “one-particle” equations. MCTDHF methods are widely used for numerical calculations of the dynamics of few-electron systems in quantum physics and quantum chemistry, but the time-dependent case still poses serious open problems for the analysis, e.g. in the sense that global-in-time existence of solutions is not proved yet. In this letter we present the first result ever where global existence is proved under a condition on the initial datum that it has to be somewhat close to the “ground state”.     

A notion of polyconvex function on a surface suggested by nonlinear shell theory

Combining the definitions set forth by J. Ball in 1977 and by J. Ball, J.C. Currie, and P.J. Olver in 1981, we propose in this Note a definition of a “polyconvex function on a surface”. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, which eventually allows to establish the existence of minimizers.     

Nesterenko’s criterion when the small linear forms oscillate

In this paper we generalize Nesterenko’s criterion to the case where the small linear forms have an oscillating behaviour (for instance given by the saddle point method). This criterion provides both a lower bound for the dimension of the vector space spanned over the rationals by a family of real numbers and a measure of simultaneous approximation to these numbers (namely, an upper bound for the irrationality exponent if 1 and only one other number are involved). As an application, we prove an explicit measure of simultaneous approximation to ζ(5), ζ(7), ζ(9), and ζ(11), using Zudilin’s proof that at least one of these numbers is irrational.