THE HAUSDORFF CENTRED MEASURE OF THE SYMMETRY CANTOR SETS

Let 0<A≤1/3 ,K(λ) be the attractor of an iterated function system {ψ1,ψ2} on the line, where 1(x)= AT, ψ1(x) = 1-λ+λx, x∈[0,1]. We call K(λ) the symmetry Cantor sets. In this paper, we obtained the exact Hausdorff Centred measure of K(λ).

THE INTEGRAL FORMULA FOR CALCULATINGTHE HAUSDORFF MEASURE OF SOME FRACTAL SETS

It is important to calculate the Hausdorff dimension and the Hausdorff mesure respect to this dimension for some fractal sets. By using the usual method of “Mass Distribution”, we can only calculate the Hausdorff dimension. In this paper, we will construct an integral formula by using lower inverse s-density and then use it to calculate the Hausdorff measures for some fractional dimensional sets.

Einstein-Rosen Bridge (ER), Einstein-Podolsky-Rosen Experiment (EPR) and Zero Measure Rindler-KAM Cantorian Spacetime Geometry (ZMG) Are Conceptually Equivalent

By viewing spacetime as a transfinite Turing computer, the present work is aimed at a generalization and geometrical-topological reinterpretation of a relatively old conjecture that the wormholes of general relativity are behind the physics and mathematics of quantum entanglement theory. To do this we base ourselves on the comprehensive set theoretical and topological machinery of the Cantorian-fractal E-infinity spacetime theory. Going all the way in this direction we even go beyond a quantum gravity theory to a precise set theoretical understanding of what a quantum particle, a quantum wave and quantum spacetime are. As a consequence of all these results and insights we can reason that the local Casimir pressure is the difference between the zero set quantum particle topological pressure and the empty set quantum wave topological pressure which acts as a wormhole “connecting” two different quantum particles with varying degrees of entanglement corresponding to varying degrees of emptiness of the empty set (wormhole). Our final result generalizes the recent conceptual equation of Susskind and Maldacena ER = EPR to become ZMG = ER = EPR where ZMG stands for zero measure Rindler-KAM geometry (of spacetime). These results were only possible because of the ultimate simplicity of our exact model based on Mauldin-Williams random Cantor sets and the corresponding exact Hardy’s quantum entanglement probability P(H) = where is the Hausdorff dimension of the topologically zero dimensional random Cantor thin set, i.e. a zero measure set and . On the other hand the positive measure spatial separation between the zero sets is a fat Cantor empty set possessing a Hausdorff dimension equal while its Menger-Urysohn topological dimension is a negative value equal minus one. This is the mathematical quintessence of a wormhole paralleling multiple connectivity in classical topology. It is both physically there because of the positive measure and not there because of the negative topological dimension.

Set Valued Vitali-Hahn-Saks-Nikodym Theorem

In this paper, we establish the set valued Vitali-Hahn-Saks-Nikodym Theorem.

APPROXIMATION OF SPHERICAL MEANS OF MULTIPLE FOURIER INTEGRALS AND FOURIER SERIES ON SET OF FULL MEASURE

In this paper we consider the approximation for functions in some subspaces of L^2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.

New Solution and Application of the Theory of "Acting According to Time"

From the perspective of theoretical analysis,based on the annotation of ancient doctors on the theory of"acting according to time,"this article puts forward the view of "time";that is,the impact of the social environment on the occurrence and development of diseases.In addition,it points out that the characteristics of modem diseases are of"multiple heat syndromes and multiple repletion pattern."Based on this characteristic,the use of heat-clearing drugs in the clinical stage would have a good eflect.It has been suggested that contemporary doctors should not only inherit the theories from predecessors,but also think diligently and innovate bravely along with the current environment in order to ensure that the theories ofTCM are constantly full of vitality to better serve the clinical aspect.

Controllability of the Heat Equation with a Control Acting on a Measurable Set

The paper deals with the controllability of a heat equation. It is well-known that the heat equation yt - △y = uxE in （0, T） × Ω with homogeneous Dirichlet boundary conditions is null controllable for any T 〉 0 and any open nonempty subset E of Ω. In this note, the author studies the case that E is an arbitrary measurable set with positive measure.

REMARKS OF DIFFERENTIAL EQUATIONS ON CLOSED SUBSETS IN A BANACH SPACE

We give some extensions of Monch-Harton inequalities with respect to measures of noncompactness. As an example of the application, we obtain two existencetheorems of solutions for Cauchy problems of differential equations on closed setsunder weaker compactness conditions.

Multi-criteria decision making method based on improved cosine similarity measure with interval neutrosophic sets

Purpose–The fuzziness and complexity of evaluation information are common phenomenon in practical decision-making problem,interval neutrosophic sets(INSs)is a power tool to deal with ambiguous information.Similarity measure plays an important role in judging the degree between ideal and each alternative in decision-making process,the purpose of this paper is to establish a multi-criteria decision-making method based on similarity measure under INSs.Design/methodology/approach–Based on an extension of existing cosine similarity,this paper first introduces an improved cosine similarity measure between interval neutosophic numbers,which considers the degrees of the truth membership,the indeterminacy membership and the falsity membership of the evaluation values.And then a multi-criteria decision-making method is established based on the improved cosine similarity measure,in which the ordered weighted averaging(OWA)is adopted to aggregate the neutrosophic information related to each alternative.Finally,an example on supplier selection is given to illustrate the feasibility and practicality of the presented decision-making method.Findings–In the whole process of research and practice,it was realized that the application field of the proposed similarity measure theory still should be expanded,and the development of interval number theory is one of further research direction.Originality/value–The main contributions of this paper are as follows:this study presents an improved cosine similarity measure under INSs,in which the weights of the three independent components of an interval number are taken into account;OWA are adopted to aggregate the neutrosophic information related to each alternative;and a multi-criteria decision-making method using the proposed similarity is developed under INSs.

Hausdorff measure of sets of finite type of one-sided symbolic space

Some estimation formulae and a computation formula for the hausdorff meausre fo the sets of finite type of the one-sided symbolic space are given

Loopy belief propagation based data association for extended target tracking

The data association problem of multiple extended target tracking is very challenging because each target may generate multiple measurements.Recently,the belief propagation based multiple target tracking algorithms with high efficiency have been a research focus.Different from the belief propagation based Extended Target tracking based on Belief Propagation(ET-BP)algorithm proposed in our previous work,a new graphical model formulation of data association for multiple extended target tracking is proposed in this paper.The proposed formulation can be solved by the Loopy Belief Propagation(LBP)algorithm.Furthermore,the simplified measurement set in the ET-BP algorithm is modified to improve tracking accuracy.Finally,experiment results show that the proposed algorithm has better performance than the ET-BP and joint probabilistic data association based on the simplified measurement set algorithms in terms of accuracy and efficiency.Additionally,the convergence of the proposed algorithm is verified in the simulations.

Approximating the Spectral Abscissa for Switched Linear Systems via Coordinate Transformations

In this paper,an approach of square coordinate transformation is proposed to approximate the spectral abscissa for continuous-time switched linear systems.By applying elementary transformations iteratively,a series of minimums of least μ1 matrix set measures are obtained,which are utilized to approximate the spectral abscissa of the switched system.The approach is developed into tractable numerical algorithms that provide upper bound estimates of the spectral abscissa.Numerical simulations show the effectiveness of the proposed method.

GENERALIZED INTUITIONISTIC FUZZY HYBRID CHOQUET AVERAGING OPERATORS

In this study a new hybrid aggregation operator named as the generalized intuitionistic fuzzy hybrid Choquet averaging（GIFHCA） operator is defined.Meantime,some desirable properties are studied, and several important cases are examined.Furthermore,we define the generalized Shapley GIFHCA （GS-GIFHCA） operator,which does not only overall consider the importance of elements and their ordered positions,but also globally reflect the correlations among them and their ordered positions.In order to simplify the complexity of solving a fuzzy measure,we further define the generalizedλ-Shapley GIFHCA（GλS-GIFHCA） operator.

Can Society's Happiness Be Measured by a Set of Figures?

Knowing if you're happy could soon be easier than you think in China. The government department respon-sible for statistics has planned to issue an official "happiness index," which reportedly will measure the level of people's well being by a set of figures. The National Bureau of Statistics announced in September that in the

Bernoulli particle flter with observer altitude for maritime radiation source tracking in the presence of measurement uncertainty

For maritime radiation source target tracking in particular electronic counter measures（ECM）environment,there exists two main problems which can deteriorate the tracking performance of traditional approaches.The frst problem is the poor observability of the radiation source.The second one is the measurement uncertainty which includes the uncertainty of the target appearing/disappearing and the detection uncertainty（false and missed detections）.A novel approach is proposed in this paper for tracking maritime radiation source in the presence of measurement uncertainty.To solve the poor observability of maritime radiation source target,using the radiation source motion restriction,the observer altitude information is incorporated into the bearings-only tracking（BOT）method to obtain the unique target localization.Then the two uncertainties in the ECM environment are modeled by the random fnite set（RFS）theory and the Bernoulli fltering method with the observer altitude is adopted to solve the tracking problem of maritime radiation source in such context.Simulation experiments verify the validity of the proposed approach for tracking maritime radiation source,and also demonstrate the superiority of the method compared with the traditional integrated probabilistic data association（IPDA）method.The tracking performance under different conditions,particularly those involving different duration of radiation source opening and switching-off,indicates that the method to solve our problem is robust and effective.