Conditional Entropy of Partitions on Quantum Logic

A construction of conditional entropy of partitions on quantum logic is given,and the properties ofconditional entropy are investigated.

Attribute Reduction of Hybrid Decision Information Systems Based on Fuzzy Conditional Information Entropy

The presence of numerous uncertainties in hybrid decision information systems(HDISs)renders attribute reduction a formidable task.Currently available attribute reduction algorithms,including those based on Pawlak attribute importance,Skowron discernibility matrix,and information entropy,struggle to effectively manages multiple uncertainties simultaneously in HDISs like the precise measurement of disparities between nominal attribute values,and attributes with fuzzy boundaries and abnormal values.In order to address the aforementioned issues,this paper delves into the study of attribute reduction withinHDISs.First of all,a novel metric based on the decision attribute is introduced to solve the problem of accurately measuring the differences between nominal attribute values.The newly introduced distance metric has been christened the supervised distance that can effectively quantify the differences between the nominal attribute values.Then,based on the newly developed metric,a novel fuzzy relationship is defined from the perspective of“feedback on parity of attribute values to attribute sets”.This new fuzzy relationship serves as a valuable tool in addressing the challenges posed by abnormal attribute values.Furthermore,leveraging the newly introduced fuzzy relationship,the fuzzy conditional information entropy is defined as a solution to the challenges posed by fuzzy attributes.It effectively quantifies the uncertainty associated with fuzzy attribute values,thereby providing a robust framework for handling fuzzy information in hybrid information systems.Finally,an algorithm for attribute reduction utilizing the fuzzy conditional information entropy is presented.The experimental results on 12 datasets show that the average reduction rate of our algorithm reaches 84.04%,and the classification accuracy is improved by 3.91%compared to the original dataset,and by an average of 11.25%compared to the other 9 state-of-the-art reduction algorithms.The comprehensive analysis of these research results clearly indicates that our algorithm is highly effective in managing the intricate uncertainties inherent in hybrid data.

A Knowledge Reduction Algorithm Based on Conditional Entropy

Rough Set is a valid mathematical theory developed in recent years, which has been applied successfully in such fields as machine learning, data mining, intelligent data analyzing and control algorithm acquiring. In this paper, the authors discuss the reduction of knowledge using conditional entropy in rough set theory. First, the changing tendency of the conditional entropy of decision attributes giving condition attributes is studied from the viewpoint of information. Next, a new reduction algorithm based on conditional entropy is developed. Furthermore, our simulation results show that the algorithm can find the minimal reduction in most cases.

Information Entropy Measures for Stand Structural Diversity:Joint Entropy

Structural diversity is the key attribute of a stand. A set of biodiversity measures in ecology was introduced in forest management for describing stand structure, of which Shannon information entropy (Shannon index) has been the most widely used measure of species diversity. It is generally thought that tree size diversity could serve as a good proxy for height diversity. However, tree size diversity and height diversity for stand structure is not completely consistent. Stand diameter cannot reflect height information completely. Either tree size diversity or height diversity is one-dimensional information entropy measure. This paper discussed the method of multiple-dimensional information entropy measure with the concept of joint entropy. It is suggested that joint entropy is a good measure for describing overall stand structural diversity.

Upper Estimates on the Higher-dimensional Multifractal Spectrum of Local Entropy

We discuss the problem of higher-dimensional multifractal spectrum of local entropy for arbitrary invariant measures. By utilizing characteristics of a dynamical system, namely, higher-dimensional entropy capacities and higher-dimensional correlation entropies, we obtain three upper estimates on the hlgher-dimensional multifractal spectrum of local entropies. We also study the domain of higher-dimensional multifractai spetrum of entropies.

DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL

This paper deals with the existence and the asymptotic behavior of discontinu- ous traveling wave entropy solutions for a modified Allen-Cahn model, in which, the usual Ficken-based model for phase transition is replaced with a more physical model with nonSn- ear diffusive flux. The discontinuous traveling waves correspond to the discontinuous phase transition phenomena.

Fusing Supervised and Unsupervised Measures for Attribute Reduction

It is well-known that attribute reduction is a crucial action of rough set.The significant characteristic of attribute reduction is that it can reduce the dimensions of data with clear semantic explanations.Normally,the learning performance of attributes in derived reduct is much more crucial.Since related measures of rough set dominate the whole process of identifying qualified attributes and deriving reduct,those measures may have a direct impact on the performance of selected attributes in reduct.However,most previous researches about attribute reduction take measures related to either supervised perspective or unsupervised perspective,which are insufficient to identify attributes with superior learning performance,such as stability and accuracy.In order to improve the classification stability and classification accuracy of reduct,in this paper,a novel measure is proposed based on the fusion of supervised and unsupervised perspectives:(1)in terms of supervised perspective,approximation quality is helpful in quantitatively characterizing the relationship between attributes and labels;(2)in terms of unsupervised perspective,conditional entropy is helpful in quantitatively describing the internal structure of data itself.In order to prove the effectiveness of the proposed measure,18 University of CaliforniaIrvine(UCI)datasets and 2 Yale face datasets have been employed in the comparative experiments.Finally,the experimental results show that the proposed measure does well in selecting attributes which can provide distinguished classification stabilities and classification accuracies.

THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS

In this paper,we study the shock waves for a mixed-type system from chemotaxis.We are concerned with the jump conditions for the left state which is located in the elliptical region and the right state in the hyperbolic region.Under the generalized entropy conditions,we find that there are different shock wave structures for different parameters.To guarantee the uniqueness of the solutions,we obtain the admissible shock waves which satisfy the generalized entropy condition in both parameters.Finally,we construct the Riemann solutions in some solvable regions.

Statistical segmentation model on lattices

To reduce the difficulty of implementation and shorten the runtime of the traditional Kim-Fisher model, an entirely discrete Kim-Fisher-like model on lattices is proposed. The discrete model is directly built on the lattices, and the greedy algorithm is used in the implementation to continually decrease the energy function. First, regarding the gray values in images as discrete-valued random variables makes it possible to make a much simpler estimation of conditional entropy. Secondly, a uniform method within the level set framework for two-phase and multiphase segmentations without extension is presented. Finally, a more accurate approximation to the curve length on lattices with multi-labels is proposed. The experimental results show that, compared with the continuous Kim-Fisher model, the proposed model can obtain comparative results, while the implementation is much simpler and the runtime is dramatically reduced.

Comprehensive security risk factor identification for small reservoirs with heterogeneous data based on grey relational analysis model

Identification of security risk factors for small reservoirs is the basis for implementation of early warning systems.The manner of identification of the factors for small reservoirs is of practical significance when data are incomplete.The existing grey relational models have some disadvantages in measuring the correlation between categorical data sequences.To this end,this paper introduces a new grey relational model to analyze heterogeneous data.In this study,a set of security risk factors for small reservoirs was first constructed based on theoretical analysis,and heterogeneous data of these factors were recorded as sequences.The sequences were regarded as random variables,and the information entropy and conditional entropy between sequences were measured to analyze the relational degree between risk factors.Then,a new grey relational analysis model for heterogeneous data was constructed,and a comprehensive security risk factor identification method was developed.A case study of small reservoirs in Guangxi Zhuang Autonomous Region in China shows that the model constructed in this study is applicable to security risk factor identification for small reservoirs with heterogeneous and sparse data.

SUB-ADDITIVE PRESSURE ON A BOREL SET

The goal of this paper is to investigate topological conditional pressure of a continuous transformation as defined for sub-additive potentials. This study presents a vari- ational inequality for sub-additive topological conditional pressure on a closed subset, which is the other form of the variational principle for the sub-additive topological pressure pre- sented by Cao, Feng, and Huang in [9]. Moreover, under some additional assumptions, this result can be generalized to the non-compact case.

Information Theory Model for Radiation

Information based models for radiation emitted by a Black Body which passes through a scattering medium are analyzed. In the limit, when there is no scattering this model reverts to the Black Body Radiation Law. The advantage of this mathematical model is that it includes the effect of the scattering of the radiation between source and detector. In the case when the exact form of the scattering mechanism is not known a model using a single scattering parameter is derived. A simple version of this model is derived which is useful for analyzing large data.

Generalized Riemann problem for gas dynamic combustion

The generalized Riemann problem for gas dynamic combustion in a neighborhood of the origin t 0 in the （x, t） plane is considered. Under the modified entropy conditions, the unique solutions are constructed, which are the limits of the selfsimilar Zeldovich-von Neumann-Dring （ZND） combustion model. The results show that, for some cases, there are intrinsical differences between the structures of the perturbed Riemann solutions and the corresponding Riemann solutions. Especially, a strong detonation in the corresponding Riemann solution may be transformed into a weak deflagration coalescing with the pre-compression shock wave after perturbation. Moreover, in some cases, although no combustion wave exists in the corresponding Riemann solution, the combustion wave may occur after perturbation, which shows the instability of the unburnt gases.

Nonlinear Analysis of Physiological Time Series

The heart rate variability could be explained by a low-dimensional governing mechanism. There has been increasing interest in verifying and understanding the coupling between the respiration and the heart rate. In this paper we use the nonlinear detection method to detect the nonlinear deterministic component in the physiological time series by a single variable series and two variables series respectively, and use the conditional information entropy to analyze the correlation between the heart rate, the respiration and the blood oxygen concentration. The conclusions are that there is the nonlinear deterministic component in the heart rate data and respiration data, and the heart rate and the respiration are two variables originating from the same underlying dynamics.

Solutions to a hyperbolic system of conservation laws on two boundaries

This paper studies the interaction of elementary waves including delta-shock waves on two boundaries for a hyperbolic system of conservation laws. The solutions of the initialboundary value problem for the system are constructively obtained. In the problem the initialboundary data are in piecewise constant states.

A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition

The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entropy condition.Until 2020,for one-dimensional gas dynamics equations,the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate(AA)flux and the entropy conservative(EC)flux developed by Maire et al.was used.This hybridized CL scheme satisfies the entropy condition;however,it is under-entropic in the part zones of rarefaction waves.Moreover,the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves.Another disadvantage of this scheme is that it is of only first-order accuracy.In this paper,we firstly construct a modified entropy conservative(MEC)flux which can damp effectively numerical oscillations in simulating strong rarefaction waves.Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows.This new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic,we propose using the third-order TVD-type Runge-Kutta time discretization method.Based on the new hybridized flux,we develop the second-order CL scheme that satisfies the entropy condition.Finally,the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.

ON ENTROPY CONDITIONS OF HIGH RESOLUTION SCHEMES FOR SCALAR CONSERVATION LAWS

Presents a method of proof which improves the estimates of entropy production for general total variation diminishing (TVD) schemes. Elements of the general theory of TVD schemes; Basis for obtaining the entropy inequality of a class of second order resolution-TVD schemes for strict convex conservation laws; Definition of discrete entropy inequality.

Sharp heat kernel bounds and entropy in metric measure spaces

We establish the sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying the RCD(0, N)(equivalently, RCD~*(0, N), condition with N∈N\ {1} and having the maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the(minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in the Riemannian manifolds, and some of them appear more explicit and sharper than the ones in metric measure spaces obtained recently by Jiang et al.(2016).

Tree-Indexed Markov Chains in Random Environment and Some of Their Strong Limit Properties

In this paper,the authors first introduce the tree-indexed Markov chains in random environment,which takes values on a general state space.Then,they prove the existence of this stochastic process,and develop a class of its equivalent forms.Based on this property,some strong limit theorems including conditional entropy density are studied for the tree-indexed Markov chains in random environment.

CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX

The authors give the first convergence proof for the Lax-Friedrichs finite differencescheme for non-convex genuinely nonlinear scalar conservation laws of the formu_t + f(k(x, t), u)_x = 0,where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t)plane. In contrast to most of the existing literature on problems with discontinuouscoefficients, here the convergence proof is not based on the singular mapping approach,but rather on the div-curl lemma (but not the Young measure) and a Lax type en-tropy estimate that is robust with respect to the regularity of k(x, t). Following [14],the authors propose a definition of entropy solution that extends the classical Kruzkovdefinition to the situation where k(x, t) is piecewise Lipschitz continuous in the (x, t)plane, and prove the stability (uniqueness) of such entropy solutions, provided that theflux function satisfies a so-called crossng condition, and that strong traces of the solu-tion exist along the curves where k(x, t) is discontinuous. It is shown that a convergentsubsequence of approximations produced by the Lax-Friedrichs scheme converges tosuch an entropy solution, implying that the entire computed sequence converges.