Representation Theorem of Fuzzy Rough Set

This paper gives the definition of λ-cut sets and studies the structure of fuzzy rough sets. Based on the concept of rough sets, this paper proposes the representation theorem of fuzzy rough sets.

An elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor

Based on the Cayley-Hamilton theorem and fixed-point method,we provide an elementary proof for the representation theorem of analytic isotropic tensor functions of a second-order tensor in a three-dimensional(3D)inner-product space,which avoids introducing the generating function and Taylor series expansion.The proof is also extended to any finite-dimensional inner-product space.

Representation theorem and viability property for multidimensional BSDEs and their applications

The representation theorem and the viability property for backward stochastic differential equations(BSDEs)require further exploration,given their widespread use in both theory and practical applications.In this study,we present a positive answer to the long-standing open question of whether the representation theorem still holds in the L^(2)-sense under the standard assumptions of square integrability and Lipschitzian continuity on the generators of BSDEs.In the process,the multidimensional case is considered.Subsequently,based on the representation theorem,we obtain a necessary and sufficient condition for the viability property of the BSDEs under standard conditions on the generators.This removes the requirement for the generator to possess the properties of stronger integrability and continuity with respect to time variables.As an application of these results,we conduct various types of comparisons and converse comparisons for the solutions of multidimensional BSDEs,and several properties of the multidimensional g-expectation are obtained.

Representation Theorem for Generators of Quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation（BSDE）, whose generator has a quadratic growth in z. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic g-expectation.

Representation Theorems for Generators of BSDEs in L^p Spaces

In this paper, we prove that the generator g of a class of backward stochastic differential equations （BSDEs） can be represented by the solutions of the corresponding BSDEs at point （t, y, z）, when the terminal data is in Lp spaces, for 1 〈 p ≤ 2.

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).

FREUD-BUTZER-HAHN TYPE QUANTITATIVE THEOREMS FOR PROBABILISTIC REPRESENTATIONS OF(C_0)OPERATOR SEMIGROUPS

In this paper,we shall exploit the Freud method in the Classical operator approximation theory to im- prove known quantitative estimalions with an emphasis on Calculah' on the generic constants.

MARTINGALE REPRESENTATION AND LOGARITHMIC-SOBOLEV INEQUALITY FOR THE FRACTIONAL ORNSTEIN-UHLENBECK MEASURE

In this paper,we consider the measure determined by a fractional OrnsteinUhlenbeck process.For such a measure,we establish an explicit form of the martingale representation theorem and consequently obtain an explicit form of the Logarithmic-Sobolev inequality.To this end,we also present the integration by parts formula for such a measure,which is obtained via its pull back formula and the Bismut method.

INVARIANT REPRESENTATION FOR STOCHASTIC DIFFERENTIAL OPERATOR BY BSDES WITH UNIFORMLY CONTINUOUS COEFFICIENTS AND ITS APPLICATIONS

In this paper, we prove that a kind of second order stochastic differential op- erator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of the representation for the uniformly continuous generator. With the help of this representation, we obtain the corresponding converse comparison theorem for the BSDEs with uniformly continuous coefficients, and get some equivalent relationships between the properties of the generator g and the associated solutions of BSDEs. Moreover, we give a new proof about g-convexity.

The Representation Problems of Conjugate Spaces of l0（{Xi}） Type F-normed Spaces

Extending the results of an article published in(Acta Mathematica Sinica(2016,59(4)) by the author, for a sequence of normed spaces {Xi}, the representation problem of conjugate spaces of some l^0({X_i}) type F-normed spaces are studied in this paper. The algebraic representation continued equalities l^0({X_i}) * A=c_(00)~0({X_i}) * A= c_(00)({X_i~*}),(l^0(X))~* A=(c^0(X) )~* A=(c_0~0(X))~* A=(c_(00)~0(X))~* A= c_(00)(X~*)are obtained in the first part. Under weak-star topology, the topological representation c_(00)~0({X_i}) ~*, w~* = c_(00)~0({X_i~*}) is obtained in the second part. For the sequence of inner product spaces and number fields with the usual topology, the concrete forms of the basic representation theorems are obtained at last.

Multiple optimal solutions to a sort of nonlinear optimization problem

The optimization problem is considered in which the objective function is pseudolinear（both pseudoconvex and pseudoconcave） and the constraints are linear. The general expression for the optimal solutions to the problem is derived with the representation theorem of polyhedral sets, and the uniqueness condition of the optimal solution and the computational procedures to determine all optimal solutions （if the uniqueness condition is not satisfied ） are provided. Finally, an illustrative example is also given.

FIfth-ORDER ISOTROPIC DESCARTES TENSOR

Fifth-order isotropic descartes tensor and its existence theorem and representation problems are researched, then a general representation formula of fifth-order isotropic descartes tensor is got.

Enhanced Constitutive Theories for Classical Thermoviscoelastic Polymeric Fluids

This paper presents ordered rate nonlinear constitutive theories for thermoviscoelastic fluids based on Classical Continuum Mechanics (CCM). We refer to these fluids as classical thermoviscoelastic polymeric fluids. The conservation and balance laws of CCM constitute the core of the mathematical model. Constitutive theories for the Cauchy stress tensor are derived using the conjugate pair in the entropy inequality, additional desired physics, and the representation theorem. The constitutive theories for the Cauchy stress tensor consider convected time derivatives of Green’s strain tensor or the Almansi strain tensor up to order n and the convected time derivatives of the Cauchy stress tensor up to order m. The resulting constitutive theories of order (m, n) are based on integrity and are valid for dilute as well as dense polymeric, compressible, and incompressible fluids with variable material coefficients. It is shown that Maxwell, Oldroyd-B, and Giesekus constitutive models can be described by a single constitutive theory. It is well established that the currently used Maxwell and Oldroyd-B models predict zero normal stress perpendicular to the flow direction. It is shown that this deficiency is a consequence of not retaining certain generators and invariants from the integrity (complete basis) in the constitutive theory and can be corrected by including additional generators and invariants in the constitutive theory. Similar improvements are also suggested for the Giesekus constitutive model. Model problem studies are presented for BVPs consisting of fully developed flow between parallel plates and lid-driven cavities utilizing the new constitutive theories for Maxwell, Oldroyd-B, and Giesekus fluids. Results are compared with those obtained from using currently used constitutive theories for the three polymeric fluids.

On existence and uniqueness of solutions to uncertain backward stochastic differential equations

This paper is concerned with a class of uncertain backward stochastic differential equations （UBSDEs） driven by both an m-dimensional Brownian motion and a d-dimensional canonical process with uniform Lipschitzian coefficients. Such equations can be useful in mod- elling hybrid systems, where the phenomena are simultaneously subjected to two kinds of un- certainties： randomness and uncertainty. The solutions of UBSDEs are the uncertain stochastic processes. Thus, the existence and uniqueness of solutions to UBSDEs with Lipschitzian coeffi- cients are proved.

Ito Formula for Integral Processes Related to Space-Time Levy Noise

In this article, we give a new proof of the Itôformula for some integral processes related to the space-time Lévy noise introduced in [1] [2] as an alternative for the Gaussian white noise perturbing an SPDE. We discuss two applications of this result, which are useful in the study of SPDEs driven by a space-time Lévy noise with finite variance: a maximal inequality for the p-th moment of the stochastic integral, and the Itôrepresentation theorem leading to a chaos expansion similar to the Gaussian case.

A New Representation for Second Order Stochastic Integral-differential Operators and Its Applications

In this paper, we＇ll prove new representation theorems for a kind of second order stochastic integral- differential operator by stochastic differential equations （SDEs） and backward stochastic differential equations （BSDEs） with jumps, and give some applications.

Generalized Cointegrated Time Series: Representation and Estimation

Cointegration analysis is often of special interest in the analysis of nonstationary vector valued time series, especially for the vector with higher order integrated components. In this paper we present the representation theorem for a generalized cointegrated system, and formulate the generalized error correction model(GECM). We also provide the modified EG two step procedure to estimate the parameters in a GECM. The paper extends some important results of cointegration into the analysis of generalized cointegration, and is of great importance in the analysis of nonstationary time series.

Some New Results on Transition Probability

In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space （E, δ）. Our main tools such as Egoroff＇s Theorem, Vitali-Hahn-Saks＇s Theorem and the theory of atomic set and well- posedness of measure are also very interesting and fashionable.

Agent's Optimal Compensation Under Inflation Risk by Using Dynamic Contract Model

This paper studies the problem of principal-agent with moral hazard in continuous time.The firm’s cash flow is described by geometric Brownian motion(hereafter GBM).The agent affects the drift of the firm’s cash flow by her hidden effort.Meanwhile,the firm rewards the agent with corresponding compensation and equity which depend on the output.The model extends dynamic optimal contract theory to an inflation environment.Firstly,the authors obtain the dynamic equation of the firm’s real cash flow under inflation by using the It?formula.Then,the authors use the martingale representation theorem to obtain agent’s continuation value process.Moreover,the authors derive the Hamilton-Jacobi-Bellman(HJB)equation of investor’s value process,from which the authors derive the investors’scaled value function by solving the second-order ordinary differential equation.Comparing with He;,the authors find that inflation risk affects the agent’s optimal compensation depending on the firm’s position in the market.

On the Seneta Sequences

In this paper, we prove some properties of the Seneta sequences and functions, and in particular we prove a representation theorem in the Karamata sense for the sequences from the Seneta class SOc.