ON UNIQUENESS,EXISTENCE AND OBJECTIVITY OF S-R DECOMPOSITION THEOREM

For a physically possible deformation field of a continuum, the deformation gradient function F can be decomposed into direct sum of a symmetric tensor S and on orthogonal tensor R, which is called S-R decomposition theorem. In this paper, the S-R decomposition unique existence theorem is proved, by employing matrix and tensor method. Also, a brief proof of its objectivity is given.

A Note on the Filtered Decomposition Theorem

We generalize the logarithmic decomposition theorem of Deligne–Illusie to a filtered version.There are two applications.The easier one provides a mod p proof for a vanishing theorem in characteristic zero.The deeper one gives rise to a positive characteristic analogue of a theorem of Deligne on the mixed Hodge structure attached to complex algebraic varieties.

Almansi-Type Decomposition Theorem for Bi-k-regular Functions in the Clifford Algebra Cl_(2n+2,0)(R)

Almansi-type decomposition theorem for bi-k-regular functions defined in a star-like domainΩ⊆R^(n+1)×R^(n+1)centered at the origin with values in the Clifford algebra Cl_(2n+2,0)(R)is proved.As a corollary,Almansi-type decomposition theorem for biharmonic functions of degree k is given.

On the Decomposition Theorems for C^(*)-Algebras

Elliott dimension drop interval algebra is an important class among all C^*-algebras in the classification theory.Especially,they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth.In this paper,the authors will show two decomposition theorems related to the Elliott dimension drop interval algebra.Their results are key steps in classifying all AH algebras with the ideal property of no dimension growth.

Decomposition Theorems and Conjugate Pair in D_K Spaces

Let K be a right-continuous and nondecreasing function. A function f analytic in the unit disk D belongs to the space OK if ∫D｜f＇（z）｜2K（1 - ｜z｜2）dA（z） 〈 ∞. Decomposition theorems for OK spaces are established in this paper. As an application, we obtain a characterization of interpolation by functions in DK spaces. Furthermore, we characterize functions in DK spaces by conjugate pairs.

THEOREMS OF INTERVAL FUZZY SET AND ITS OPERATION RULES

Although the concept of interval fuzzy set and its properties have been defined, its three theorems and their effectiveness are not proved. Therefore, the knowledge presentation and its operation rules of interval fuzzy set are studied firstly, and then the cut set of interval fuzzy set is proposed. Moreover, the decomposition theo- rem, the representation theorem and the extension theorem of interval fuzzy set are presented. Finally, examples are given to demonstrate that the classical fuzzy set is a special case of interval fuzzy set and interval fuzzy set is an effective expansion of the classical fuzzy set.

Radon-Nikodym Theorem of Signed Additive Fuzzy Measure

This is subsequent of , by using the theory of additive fuzzy measure and signed additive fuzzy measure , we prove the Radon_Nikodym Theorem and Lebesgue decomposition Theorem of signed additive fuzzy measure.

General Operational Protocol for Coherence. Central Limit Theorem as Approximation

A general operational protocol which provides permanent macroscopic coherence of the response of any stable complex system put in an ever-changing environment is proposed. It turns out that the coherent response consists of two parts: 1) a specific discrete pattern, called by the author homeostatic one, whose characteristics are robust to the statistics of the environment;2) the rest part of the response forms a stationary homogeneous process whose coarse-grained structure obeys universal distribution which turns out to be scale-invariant. It is demonstrated that, for relatively short time series, a measurement, viewed as a solitary operation of coarse-graining, superimposed on the universal distribution results in a rich variety of behaviors ranging from periodic-like to stochastic-like, to a sequences of irregular fractal-like objects and sequences of random-like events. The relevance of the Central Limit theorem applies to the latter case. Yet, its application is still an approximation which holds for relatively short time series and for specific low resolution of the measurement equipment. It is proven that the asymptotic behavior in each and every of the above cases is provided by the recently proven decomposition theorem.

Could Long-Term Stability Last Forever?

The subject of the present paper is to prove that the recently introduced conjecture of boundedness puts a ban over the view of stability as asymptotic property. This result comes in sharp contrast with the prescription of the traditional thermodynamics and statistical physics which consider the existence of equilibrium as asymptotic property of all systems. The difference commences from the use of infinitesimal calculus as the basic implement for modelling by the latter while the primary premise of the conjecture of boundedness is sustaining the energy/matter/information permanently bounded and finite. The latter property overrules the infinitesimal calculus as the major implement of modelling because, among all, it is proven that the traditional one suffers unsoluble difficulties.

Signed Additive Fuzzy Measure

In this paper, we introduce the concept of signed additive fuzzy measure on a class of fuzzy sets, then, on certain condition, a series of decomposition theorems of signed additive fuzzy measure are proved.

RITT-WU'S CHARACTERISTIC SET METHOD FOR ORDINARY DIFFERENCE POLYNOMIAL SYSTEMS WITH ARBITRARY ORDERING

In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.

ON ALTERNATIVE OPTIMAL SOLUTIONS TO QUASIMONOTONIC PROGRAMMING WITH LINEAR CONSTRAINTS

In this paper, the nonlinear programming problem with quasimonotonic （ both quasiconvex and quasiconcave ）objective function and linear constraints is considered. With the decomposition theorem of polyhedral sets, the structure of optimal solution set for the programming problem is depicted. Based on a simplified version of the convex simplex method, the uniqueness condition of optimal solution and the computational procedures to determine all optimal solutions are given, if the uniqueness condition is not satisfied. An illustrative example is also presented.

Some Improvement on Hilbert's Inequality and Its Application

A new improvement of Hilbert's inequality for double series can be establishedby means of a strengthened Cauchy's inequality. As application, a quite sharp result onFejer-Riesz's inequality is obtained.

Two Generating Mappings ω_L and l_L in Complete Lattices TVS and LFTVS

In this paper,we give a new decomposition theorem of L-fuzzy sets. Then we use this theorem to discuss the relations between the crisp topological vector spaces and the L-fuzzy topological vector spaces[1],and we obtain some interesting properties.

New Approach to Deep Miniaturization: A Way to Overcoming Moore’s Law

The matter about some far-going consequences commencing from the replacement of one of the basic principles of the traditional physics that is the principle of locality, with the recently put forward principle of boundedness is considered. It is proven that the thermodynamic theory which is explicitly grounded on the principle of locality, suffers inherent contradiction which roots lay down to the principle of locality. The way to overcome it goes via the replacement of the principle of locality with the recently put forward by the author principle of boundedness. In turn, the latter gives rise not only to a fundamentally novel notion for a number of ideas but also it yields replacement of the proportionality between the software and hardware components with a new non-extensive approach to the matter. It is proven that the famous Moore’s law stands in new reading both in its support and the way to overcome its limitations. Apart from its role for the technical applications, the present considerations stand also as a methodological example how the role of the basics of any theory affects practical rules such as the Moore’s law.

Use and Abuse of General Criterion for Discrimination of Causal Correlations from Individual Peculiarities and Provisional Correlations

A fundamentally novel approach to the issue about existence of a general criterion for autonomous discrimination of causal correlations from individual peculiarities and provisional correlations at stable complex systems is put forward. It is grounded on a recently proven by the author decomposition theorem whose subject has no cross section with the subject of the Central Limit Theorem. The fundamental advantage of that criterion lies in insensitivity to the details of the underlying dynamics and to the details of the hierarchical structure regardless of the nature of the corresponding system. It holds in an unspecified ever-changing environment. It also holds when information is incomplete and/or uncertain. Another advantage of the criterion is the ability to forecast a change in a system. The limitation of the criterion is substantiated as a ban over predictability whether that change would develop in an adaptation or in destruction. It is worth noting that while the criterion itself holds in the frame of the recently proposed theory of boundedness, the ban over prediction of the nature of a change is model-free.

Is Evolution a Causal, Yet Not-Predetermined Process?

It is demonstrated that “survival of the fittest” approach suffers fundamental flaw planted in its very goal: reaching a uniform state starting from a minor random event. Simple considerations prove that a generic property of any such state is its global instability. That is why a new approach to the evolution is put forward. It conjectures equilibrium for systems put in an ever-changing environment. The importance of this issue lies in the view that an ever-changing environment is much closer to the natural environment where the biological species live in. The major goal of the present paper is to demonstrate that a specific form of dynamical equilibrium among certain mutations is established in each and every stable in a long-run system. Major result of our considerations is that neither mutation nor either kind dominates forever because a temporary dynamical equilibrium is replaced with another one in the time course. It will be demonstrated that the evolution of those pieces of equilibrium is causal, yet not predetermined process.

DECOMPOSITION OF A CLASS OF FUNCTIONALS AND THE PREDICTABLE REPRESENTATION THEOREM ON BANACH SPACES

Let B be a separable real Banach space and X(t) be a symmetric conservative diffusionprocess taking values in B. In this paper, we decompose the functional u(X(t),t) into a sumof a square integrable martingale and a regular 0-quadratic variation process. On this basis, weestablish the predictable representation theorem of X(t).

Some properties on G-evaluation and its applications to G-martingale decomposition

In this article, a sublinear expectation induced by G-expectation is introduced, which is called G- evaluation for convenience. As an application, we prove that for any ξ∈ L β G (Ω T ) with some β > 1 the martingale decomposition theorem under G-expectaion holds, and that any β > 1 integrable symmetric G-martingale can be represented as an Ito integral w.r.t. G-Brownian motion. As a byproduct, we prove a regularity property for G-martingales: Any G-martingale {M t } has a quasi-continuous version.

A direct product decomposition of QMV algebras

We study the direct product decomposition of quantum many-valued algebras (QMV algebras) which generalizes the decomposition theorem of ortholattices (orthomodular lattices).In detail,for an idempo- tent element of a given QMV algebra,if it commutes with every element of the QMV algebra,it can induce a direct product decomposition of the QMV algebra.At the same time,we introduce the commutant C(S) of a set S in a QMV algebra,and prove that when S consists of idempotent elements,C(S) is a subalgebra of the QMV algebra.This also generalizes the cases of orthomodular lattices.