Theoretic Quantities Research Articles

Large-Nestimates of universal amplitudes of theCPN−1theory and comparison with aS=12square-lattice model with competing four-spin interactions

We present computations of certain finite-size scaling functions and universal amplitude ratios in the large-$N$ limit of the $\mathbb{C}{\mathbb{P}}^{N\ensuremath{-}1}$ field theory. We pay particular attention to the uniform susceptibility, the spin stiffness, and the specific heat. Field theoretic arguments have shown that the long-wavelength description of the phase transition between the N\'eel and valence-bond solid states in square lattice $S=1/2$ antiferromagnets is expected to be the noncompact $\mathbb{C}{\mathbb{P}}^{1}$ field theory. We provide a detailed comparison between our field theoretic calculations and quantum Monte Carlo data close to the N\'eel-VBS transition on a $S=1/2$ square-lattice model with competing four-spin interactions (the JQ model).

Simplifying additivity problems using direct sum constructions

We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy, and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds if and only if it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This provides a simple and general tool for lifting additivity results.

Input Selection for Modeling and Diagnostics With Application to Diesel Engines

A theoretical and experimental approach to the use of information theory in input space selection for modeling and diagnostic applications is examined. The assumptions and test cases used throughout the paper are specifically tailored to diesel engine diagnostic and modeling applications. This work seeks to quantify the amount of information about an output contained within an input space. The information theoretic quantity, conditional entropy, is shown to be an accurate predictor of model and diagnostic algorithm performance and therefore is a good choice for an input vector selection metric. Methods of estimating conditional entropy from collected data, including the amount of needed data, are also discussed.

The statistics of supersymmetric D-brane models

We investigate the statistics of the phenomenologically important D-brane sector of string compactifications. In particular for the class of intersecting D-brane models, we generalise methods known from number theory to determine the asymptotic statistical distribution of solutions to the tadpole cancellation conditions. Our approach allows us to compute the statistical distribution of gauge theoretic observables like the rank of the gauge group, the number of chiral generations or the probability of an SU ( N ) gauge factor. Concretely, we study the statistics of intersecting branes on T 2 and T 4 / Z 2 and T 6 / Z 2 × Z 2 orientifolds. Intriguingly, we find a statistical correlation between the rank of the gauge group and the number of chiral generations. Finally, we combine the statistics of the gauge theory sector with the statistics of the flux sector and study how distributions of gauge theoretic quantities are affected.

Local correlation measures in atomic systems

The phenomenon of electron correlation in atomic systems is examined and compared from the statistical, information theoretic, and energetic perspectives. Local correlation measures, based on the correlation coefficient, information entropies, and idempotency measure, are compared to the correlation energy density. Analysis of these local measures reveals that the chemically significant valence region is responsible for the behavior of their respective global measures in contrast to the correlation energy density which has large contributions to the correlation energy from both the core and valence regions. These results emphasize the difference in the mechanisms inherent in the different perspectives, the similarity between the statistical, information entropic, and idempotency views, and provides further evidence for the use of information theoretic based quantities in studies of electron correlation.

Fisher information, Wehrl entropy, and Landau diamagnetism

Using information theoretic quantities like the Wehrl entropy and Fisher's information measure we study the thermodynamics of the problem leading to Landau's diamagnetism, namely, a free spinless electron in a uniform magnetic field. We reveal a new Fisher-uncertainty relation, derived via the Cramer-Rao inequality, that involves phase space localization and energy fluctuations.

Wootters? distance revisited: a new distinguishability criterium

The notion of distinguishability between quantum states has shown to be fundamental in the frame of quantum information theory. In this paper we present a new distinguishability criterium by using a information theoretic quantity: the Jensen-Shannon divergence (JSD). This quantity has several interesting properties, both from a conceptual and a formal point of view. Previous to define this distinguishability criterium, we review some of the most frequently used distances defined over quantum mechanics' Hilbert space. In this point our main claim is that the JSD can be taken as a unifying distance between quantum states.

Complex multiplication of exactly solvable Calabi–Yau varieties

We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi–Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi–Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.

Mathematical properties of the jpeg2000 wavelet filters

The LeGall 5/3 and Daubechies 9/7 filters have risen to special prominence because they were selected for inclusion in the JPEG2000 standard. We determine their key mathematical features: Riesz bounds, order of approximation, and regularity (Hölder and Sobolev). We give approximation theoretic quantities such as the asymptotic constant for the L2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection. We also derive new asymptotic error formulae that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet. The Daubechies 9/7 stands out because it is very close to orthonormal, but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments.

Open Wilson lines and group theory of noncommutative Yang-Mills theory in two dimensions

The correlation functions of open Wilson line operators in two-dimensional Yang-Mills theory on the noncommutative torus are computed exactly. The correlators are expressed in two equivalent forms. An instanton expansion involves only topological numbers of Heisenberg modules and enables extraction of the weak-coupling limit of the gauge theory. A dual algebraic expansion involves only group theoretic quantities, winding numbers and translational zero modes, and enables analysis of the strong-coupling limit of the gauge theory and the high-momentum behaviour of open Wilson lines. The dual expressions can be interpreted physically as exact sums over contributions from virtual electric dipole quanta.

Information content for quantum states

A method for representing probabilistic aspects of quantum systems by means of a density function on the space of pure quantum states is introduced. In particular, a maximum entropy argument allows us to obtain a natural density function that only reflects the information provided by the density matrix. This result is applied to derive the Shannon entropy of a quantum state. The information theoretic quantum entropy thereby obtained is shown to have the desired concavity property, and to differ from the conventional von Neumann entropy. This is illustrated explicitly for a two-state system.

Refined algebraic quantization in the oscillator representation of SL(2, ℝ)

We investigate refined algebraic quantization (RAQ) with group averaging in a constrained Hamiltonian system with unreduced phase space T*R4 and gauge group SL(2, ℝ). The reduced phase space ℳ is connected and contains four mutually disconnected “regular” sectors with topology R×S1, but these sectors are connected to each other through an exceptional set, where ℳ is not a manifold and where ℳ has non-Hausdorff topology. The RAQ physical Hilbert space Hphys decomposes as Hphys≃⊕Hi, where the four subspaces Hi naturally correspond to the four regular sectors of ℳ. The RAQ observable algebra Aobs, represented on Hphys, contains natural subalgebras represented on each Hi. The group averaging takes place in the oscillator representation of SL(2, ℝ) on L2(R2,2), and ensuring convergence requires a subtle choice for the test state space: the classical analog of this choice is to excise from ℳ the exceptional set while nevertheless retaining information about the connections between the regular sectors. A quantum theory with the Hilbert space Hphys and a finitely generated observable subalgebra of Aobs is recovered through both Ashtekar’s algebraic quantization and Isham’s group theoretic quantization.

A tight upper bound on the gain of linear and nonlinear predictors for stationary stochastic processes

One of the striking questions in prediction theory is this: is there a chance to predict future values of a given signal? Usually, we design a predictor for a special signal or problem and then measure the resulting prediction quality. If there is no a priori knowledge on the optimal predictor, the achieved prediction gain will depend strongly of the prediction model used. To cope with this lack of knowledge, a theorem on the maximum achievable prediction gain of stationary signals is presented. This theorem provides the foundation for estimating a quality goal for the predictor design, independent of a special predictor implementation (linear or nonlinear). As usual, the prediction gain is based on the mean square error (MSE) of the predicted signal. The achievable maximum of the prediction gain is calculated using an information theoretic quantity known as the mutual information. In order to obtain the gain, we use a nonparametric approach to estimate the maximum prediction gain based on the observation of one specific signal. We illustrate this by means of well-known example signals and show an application to load forecasting. An estimation algorithm for the prediction gain has been implemented and used in the experimental part of the paper.

Generalized redundancies for time series analysis

Extensions to various information theoretic quantities used for nonlinear time series analysis are discussed, as well as their relationship to the generalized correlation integral. It is shown that calculating redundancies from the correlation integral can be more accurate and more efficient than direct box counting methods. It is also demonstrated that many commonly used nonlinear statistics have information theory based analogues. Furthermore, the relationship between the correlation integral and information theoretic statistics allows us to define ``local'' versions of many information theory based statistics; including a local version of the Kolmogorov-Sinai entropy, which gives an estimate of the local predictability.

On the stochastic complexity of learning realizable and unrealizable rules

The problem of learning from examples in an average case setting is considered. Focusing on the stochastic complexity, an information theoretic quantity measuring the minimal description length of the data given a class of models, we find rigorous upper and lower bounds for this quantity under various conditions. For realizable problems, where the model class used is sufficiently rich to represent the function giving rise to the examples, we find tight upper and lower bounds for the stochastic complexity. In this case, bounds on the prediction error follow immediately using the methods of Haussler et al. (1994a). For unrealizable learning we find a tight upper bound only in the case of learning within a space of finite VC dimension. Moreover, we show in the latter case that the optimal method for prediction maynot be the same as that for data compression, even in the limit of an infinite amount of training data, although the two problems (i.e. prediction and compression) are asymptotically equivalent in the realizable case. This result may bear consequences for many of the widely used model selection methods.

On the Stochastic Complexity of Learning Realizable and Unrealizable Rules

The problem of learning from examples in an average case setting is considered. Focusing on the stochastic complexity, an information theoretic quantity measuring the minimal description length of the data given a class of models, we find rigorous upper and lower bounds for this quantity under various conditions. For realizable problems, where the model class used is sufficiently rich to represent the function giving rise to the examples, we find tight upper and lower bounds for the stochastic complexity. In this case, bounds on the prediction error follow immediately using the methods of Haussler et al. (1994a). For unrealizable learning we find a tight upper bound only in the case of learning within a space of finite VC dimension. Moreover, we show in the latter case that the optimal method for prediction may not be the same as that for data compression, even in the limit of an infinite amount of training data, although the two problems (i.e. prediction and compression) are asymptotically equivalent in the realizable case. This result may bear consequences for many of the widely used model selection methods.

Rate-distortion function when side-information may be present at the decoder

A discrete memoryless source {X/sub k/} is to be coded into a binary stream of rate R bits/symbol such that {X/sub k/} can be recovered with minimum possible distortion. The system is to be optimized for best performance with two decoders, one of which has access to side-information about the source. For given levels of average distortion for these two decoders, the minimum achievable rate R (in the usual Shannon theory sense) is given as a per-letter minimization of information theoretic quantities. The problem is solved for two cases: where the encoder is and is not informed of the side-information. >

Covariation of mutations in the V3 loop of human immunodeficiency virus type 1 envelope protein: an information theoretic analysis.

The V3 loop of the human immunodeficiency virus type 1 (HIV-1) envelope protein is a highly variable region that is both functionally and immunologically important. Using available amino acid sequences from the V3 region, we have used an information theoretic quantity called mutual information, a measure of covariation, to quantify dependence between mutations in the loop. Certain pairs of sites, including non-contiguous sites along the sequence, do not have independent mutations but display considerable, statistically significant, covarying mutations as measured by mutual information. For the pairs of sites with the highest mutual information, specific amino acids were identified that were highly predictive of amino acids in the linked site. The observed interdependence between variable sites may have implications for structural or functional relationships; separate experimental evidence indicates functional linkage between some of the pairs of sites with high mutual information. Further specific mutational studies of the V3 loop's role in determining viral phenotype are suggested by our analyses. Also, the implications of our results may be important to consider for V3 peptide vaccine design. The methods used here are generally applicable to the study of variable proteins.

Information theoretic analysis of action potential trains. I. Analysis of correlation between two neurons.

A crosscorrelational method of action potential trains has been proposed, based on information theory. Two information theoretic quantities, mutual information and channel capacity, were calculated from a pair of action potential trains for detecting a crosscorrelation and estimating synaptic connectivity. The method was compared with conventional ones, using action potential trains obtained by the simulation of a neuron model. This method was shown to have the advantages to more easily find a weak but significant crosscorrelation and to give better estimation of synaptic connectivity independent of the firing probability of a presynaptic neuron.

INDUCTIVE ALGORITHMS ON FINITE TREES

ABSTRACT Many algorithms for calculating graph theoretic quantities on trees rely implicitly on an inductive procedure for calculating a certain function defined on the vertices of the tree, after which the function is used to determine the quantity. Examples of such quantities are the diameter, 1-centre, core and cutting centre. Each of the latter can be calculated in linear time, but in the last three cases this is far from obvious, and they have been the subjects of research papers in recent years. We present a unified treatment of a class of algorithms, containing amongst them algorithms for the above problems, which proceed in a sense by a method analagous to that of difference equations on an n-dimensional integer lattice. An analysis of the time complexity is given, and applications are made to the calculations of the above quantities, as well as to the calculation of the vertex v such that if the tree is rooted at v, then the number of maximal antichains is maximal (or minimal).