Entropy Space Research Articles

Conical singularity and quantum corrections to the entropy of a black hole

For a general finite temperature different from the Hawking one there appears a well known conical singularity in the Euclidean classical solution of gravitational equations. The method of regularizing the cone by a regular surface is used to determine the curvature tensors for such a metric. This allows one to calculate the one-loop matter effective action and the corresponding one-loop quantum corrections to the entropy in the framework of the path integral approach of Gibbons and Hawking. The two-dimensional (2D) and four-dimensional cases are considered. The entropy of Rindler space is shown to be divergent logarithmically in two dimensions and quadratically in four dimensions that coincides with results obtained earlier. For the eternal 2D black hole we observe a finite, dependent on the mass, correction to the entropy. The entropy of the 4D Schwarzschild black hole is shown to possess an additional (in comparison with the 4D Rindler space) logarithmically divergent correction which does not vanish in the limit of infinite mass of the black hole. We argue that infinities of the entropy in four dimensions are renormalized by the renormalization of the gravitational coupling.

Finite moment problems: geometry of the data space and conditioned maximum entropy solutions

Article Finite moment problems: geometry of the data space and conditioned maximum entropy solutions was published on January 1, 1995 in the journal Journal of Inverse and Ill-posed Problems (volume 3, issue 3).

Understanding fully developed turbulence: Some new ideas

The present paper is a contribution within the E(∞) Cantorian space theory. After a brief review of the historical development of length and time measurements, we consider conventional definitions and methods of calculation of vector and fractal dimensions. A definition of emptiness as zero entropy is discussed.Treating the `fabric' of space-time as non-permanent, we consider other possible ways of filling emptiness. This leads to a definition of range dimension (p), the properties of which are then discussed. p=1/2 and p=1/4 range dimensions are linked to p=1 (conventional) range dimensions in fluid environments using differential velocities and accelerations. Applications of p=2 range dimensions to quantum events are also considered, and eclectic evidence for the existence of range dimensions is presented.The bijection relationship, defining space-filling fractal dimension (dc,n) is extended to cover the range 1<n<4 vector dimensions, and a physical interpretation in terms of a characteristic length ratio (ln/l0) is developed. The n-dimensional term (ln/l0)n is then closely correlated to the ratio (l1/l0)dc,n. The bijection relationship is extended to other range dimensions.An approximate empirical relationship is now presented, which provides a basis for defining conventional (p=1) range dimensional space, by introducing Nn, the number of elements needed to set up 1–4 vector dimensions. Two other ways of obtaining approximately the same number of elements are also given, one of which leads to the concept of a p=2 range dimensional substrate for conventional (p=1) range dimensions. It is found that a single track in p=1 range dimensions needs nearly p=2 range dimensions to support it. The required value then falls as n increases, equaling approximately unity for n=3 vector dimensions. A second empirical equation defining the same value of Nn for 3<n<4 vector dimensions, based on space-filling fractal dimensions for p=2 range dimensions is also presented.A fractal form for conventional space is then defined, characterised by tiny elements of negative entropy (in log space). This definition is then readily amalgamated with dynamically generated space, and the latter is shown to correlate with the classical gas equation and Hooke's law for solids. Based on the new definition of space, a proof of both the well-known inverse square law and Hubble's astronomical law is given.A relationship between the number of events in p=1andp=2 range dimensions is next established, and the effect of `information loss' corresponding to the reduction in the required value of range dimension as vector dimension (n) increases is tentatively linked to wave functions, for n=4 vector dimensions. The corresponding `useful' information is then shown to be contained in the exponent field, which is of the form {2f(e−1)pdc,4}, where f≈1,p=2 range dimensions, and e (within the exponent!) equals the numerical value of the exponent. This field is then sub-divided to various extents, and the corresponding entropies are used to determine sets of distinguishable arrangements N. These sets are successfully compared with `magic' numbers in atomic and nuclear physics, and with preferred sodium atom groupings. A prediction of the number of occupied energy levels in atomic electron shell theory is successfully carried out.Finally, combining `useful' and `hidden' information the p=2 range dimension substrate is re-created by allowing matter to collapse and `drain away' from its p=2 range dimensional space base. Light propagation through space having p=2 range dimensions is then discussed.The paper concludes with a discussion of various ideas implicit in the text, though not necessarily directly obvious. The interdependence of fractal, range and vector dimensions is first considered, followed by a review of the universal application of range dimensions. The paper ends with a study of the fractal nature of space and its justification.

Local moduli of continuity for some classes of gaussian processes

We study local behavior of sample paths of Gaussian processes. The purpose is to find a connection between local sample moduli of a Gaussian process at a certain point of the parameter space and metric entropy of the set generated by the process in the corresponding Hilbert space. For certain classes of Gaussian processes, our results provide a unifying approach to finding local moduli of continuity

Global complexity and essential simplicity: A conjectural picture of the boundary of chaos for smooth endomorphisms of the interval

Mixing a symbolic approach to the dynamics of the period-doubling operators, recent results by Sullivan on the renormalization for real analytic maps, and some confidence, a global picture emerges for the structure of the boundary of positive topological entropy in spaces of smooth endomorphisms of the interval.

Some notes on entropy measures. II. The relative entropy matrix and the positivity constraint

Based on the maximum entropy concept with the number of “bins” (class intervals), N, maximized in terms of residual errors in sampling of the system, the residual entropy matrix (REM), which is intrinsically positive definite, is sampled uniformly at N equal intervals in entropy space. The mandatory constraint of positivity shows that the finitely sampled REM then provides estimates on the magnitudes of “missing” components of the autocorrelation sequence describing REM. If the mandatory constraint is relaxed to its limit, the resulting prediction for REM is identical to Burg's (1967) maximum entropy algorithm. The advantages are that assessments can be made: (i) whether the “missing” components are known accurately enough that no further measurements are needed; (ii) the N equal intervals of the entropy measure are sufficient to provide resolution on the REM for each and every sample; or (iii) whether finer interval resolution is needed to extract information for a particular sample.

On a natural connection between the entropy spaces and Hardy space 𝑅𝑒𝐻¹

In 1983 B. Korenblum [7, 8] introduced a class of Banach function spaces associated with the notion of entropy (we will call these spaces and their norms entropy spaces and entropy norms, respectively). Entropy spaces were used in [8] as a tool for proving a new convergence test for Fourier series which includes classical tests of Dirichlet-Jordan and Dini-Lipschitz. In this paper we construct natural linear operators from the entropy spaces to Hardy space Re H 1 {\text {Re}}{H^1} [5, 6]. In fact, these operators define multiplier type bounded embeddings of entropy spaces to Re H 1 {\text {Re}}{H^1} . As a corollary we obtain a growth condition for Fourier coefficients of a continuous periodic function of bounded entropy norm (as announced in [4]). In particular, we show that if f f is a continuous periodic function of bounded Shannon entropy norm (resp. of bounded Lipschitz entropy norm with exponent q , 0 &gt; q &gt; 1 q,0 &gt; q &gt; 1 ), and { a n } n ∈ Z {\left \{ {{a_n}} \right \}_{n \in {\mathbf {Z}}}} are the Fourier coefficients of f f , then ∑ n ≠ 0 | a n ( log(|n|)/n ) | &gt; ∞ \sum \nolimits _{n \ne 0} {|{a_n}({\text {log(|n|)/n}})| &gt; \infty } (resp. ∑ n ≠ 0 | a n | / |n| ) q &gt; ∞ \sum \nolimits _{n \ne 0} {|{a_n}|/{\text {|n|}}{)^q} &gt; \infty } ). In §4 we study the relationship between the dual spaces of entropy spaces and space B.M.O. of functions of bounded mean oscillation. In §5 we conjecture that Re ⁡ H 1 \operatorname {Re} {H^1} is a direct limit of the entropy spaces in an appropriate sense.

Rigorous relationships among quantum-mechanical kinetic energy and atomic information entropies: Upper and lower bounds.

An uncertainty-type lower bound [I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975)] to the information-entropy sum in complementary spaces has recently been reformulated by Gadre et al. [Phys. Rev. A 32, 2602 (1985)] in terms of the respective one-particle probability densities. This bound has been exploited to derive rigorous upper as well as lower bounds to the information entropies and their sum in terms of the corresponding second moments of their distributions. Thus the present work establishes a direct connection, as suggested by Sears, Parr, and Dinur [Israel J. Chem. 19, 165 (1980)], between the quantum-mechanical kinetic energy and information entropy in position space. It has also been demonstrated that given at least one arbitrary moment-type constraint in each space, it is possible to derive an upper bound to the information entropy sum in complementary spaces.

An Entropy-Minimum Aspect of the Equivalent Linearization of a Nonlinear Stochastic Differential Equation

For a diffusion process determined by a stochastic differential equation with a constant diffusion, we introduce a relative (Kullback) entropy in a function space. We show minimization of this entropy yields the so-called Equivalent (Statistical) Linearization method which gives the best Gaussian solution.

Nonequilibrium entropy, Lyapounov variables, and ergodic properties of classical systems

We discuss the problem of defining (nonequilibrium) entropy in terms of the concepts of mechanics and of reconciling its monotonic increase with the Hamiltonian evolution of the dynamical system. This leads to investigating necessary and sufficient conditions for the existence of monotonically increasing quantities or the so-called Lyapounov variables of classical systems. It is found that the condition of "mixing" is necessary and the property of being K-flow is sufficient for the existence of a Lyapounov variable. The significance of the study of Lyapounov variables for the elucidation of the fundamental questions of statistical mechanics is briefly discussed. It is seen that every Lyapounov variable must fail to commute with at least some of the operators of multiplication by phase space functions. The uncertainty relations implied by this necessary noncommutativity would then set a limit on the simultaneous determination of entropy and trajectories in phase space. These considerations thus support and sharpen the view that the thermodynamical and the (microscopic) dynamical descriptions of classical systems could be consistently reconciled as being complementary descriptions analogous to the complementary descriptions encountered in quantum mechanics.

Nonnegative entropy measures of multivariate symmetric correlations

A study of nonnegativity “in general” in the symmetric (correlative) entropy space as well as discussions of some related problems is presented. The main result is summarized as Theorems 4.1 and 5.3, which give the necessary and sufficient condition for an element of the symmetric (correlative) entropy space to be nonnegative. In particular, Theorem 4.1 may be regarded as establishing a mathematical foundation for information-theoretic analysis of multivariate symmetric correlation. On the basis of these results, we propose a “hierarchical structure” of probabilistic dependence relations where it is shown that any symmetric correlation associated with a nonnegative entropy is decomposed into pairwise conditional and/or nonconditional correlations. A systematic duality existing in the set of nonnegative entropies is also considerably clarified.

Linear dependence structure of the entropy space

A study of linear dependence relations among Shannon's and McGill's (multiple) mutual informations as well as the lattice—theoretic description of them is presented. The concept of entropy vector is defined as a functional on the set of admissible probability distributions by which the problem can be transformed to that of investigating the algebraic structure of the (correlative) entropy space. First, we give the bases of this space as well as several basis transformations and thus determine the dimension. Next, a set of admissible values which McGill's informations can simultaneously take is obtained by establishing the elementary distributions, where, as a by-product, certain nonlinear dependence property is clarified. Finally, considerations on the duality of the lattice of random variables lead to the concept of dual (multiple) mutual informations, from which the duals of several theorems are derived.

A contribution to the problem of feature selection with similarity functionals in pattern recognition

In dealing with the problem of selecting characteristic features for pattern recognition, this paper discusses the evaluation of feature quality or “goodness” by means of a similarity functional. The suggestion by P. M. Lewis 1 to use the trans-information functional is considered more closely, and an experimental method for estimating feature goodness is specified. Characterization of a pattern set by features is interpreted as a transmission system, and the requirement of high recognition reliability is shown to be incompatible with the requirement of low feature space entropy. Applicability of this method is outlined briefly at the end.

Forensic Entropy Analysis of Microsoft Windows Storage Volumes

The use of file or volume encryption as a counter-forensic technique depends on the ability to plausibly deny the presence of such encrypted data. Establishing the likely presence of encrypted data is highly desirable for forensic investigations. We claim that the current or previous existence of encrypted volumes can be derived from studying file and volume entropy characteristics using knowledge of the development of volume entropy over time. To validate our hypothesis, we have examined several versions of the Microsoft Windows operating system platform over a simulated installation life-cycle and established file and volume entropy metrics. Similarly we verified the hypothesis that the ageing through regular use of an installation is identifiable through entropy fingerprint analysis. The results obtained and tests devised allow the rapid identification of several volume-level operations and also detect anomalous slack space entropy indicative of the use of encryption techniques.

Epsilon Entropy of Stochastic Processes

This paper introduces the concept of epsilon-delta entropy for "probabilistic metric spaces." The concept arises in the study of efficient data transmission, in other words, in "Data Compression." In a case of particular interest, the space is the space of paths of a stochastic process, for example $L_2\lbrack 0, 1\rbrack$ under the probability distribution induced by a mean-continuous process on the unit interval. For any epsilon and delta both greater than zero, the epsilon-delta entropy of any probabilistic metric space is finite. However, when delta is zero, the resulting entropy, called simply the epsilon entropy of the space, can be infinite. We give a simple condition on the eigenvalues of a process on $L_2\lbrack 0, 1\rbrack$ such that any process satisfying that condition has finite epsilon entropy for any epsilon greater than zero. And, for any set of eigenvalues not satisfying the given condition, we produce a mean-continuous process on the unit interval having infinite epsilon entropy for every epsilon greater than zero. The condition is merely that $\sum n\sigma_n^2$ be finite, where $\sigma_1^2 \geqq \sigma_2^2 \geqq \cdots$ are the eigenvalues of the process.

Electrical pulse communication systems. Part 2: Message encoding and signal formation in pulse systems

Five specifications should be known of an information source: dimensionality; maximum information content and peak rate of information production; average rate of information production auto-correlation function; and statistical fine structure of the source. Various sources are discussed with reference to the specifications. Message encoding is split into two distinct operations : space matching and entropy matching. The former process has to reduce the dimensionality of the message space to match the three-dimensional signal space, the latter has to remove redundancy. Signal formation involves sampling, quantization and pulse modulation. The typical pulse waveforms are compared and their frequency spectra are listed. Signal encoding is considered as a linear transformation of the signal space into the channel space for the sole purpose of matching it to the requirements of a noisy channel of limited bandwidth.