scholarly journals A New Exceptional Family of Elements and Solvability of General Order Complementarity Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Na Huang ◽  
Changfeng Ma
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Complementarity Problems ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Exceptional Family Of Elements ◽  
Exceptional Family ◽  
General Order ◽  
Sufficient And Necessary Condition

By using the concept of exceptional family, we propose a sufficient condition of a solution to general order complementarity problems (denoted by GOCP) in Banach space, which is weaker than that in Németh, 2010 (Theorem 3.1). Then we study some sufficient conditions for the nonexistence of exceptional family for GOCP in Hilbert space. Moreover, we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone general order complementarity problems.

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

The necessary and sufficient condition for clustering of nodes based on the signs of connections in generalized signed networks

2019 ◽  
Vol 33 (10) ◽  
pp. 1950086
Author(s):  
Qi Wang ◽  
Yinhe Wang ◽  
Zilin Gao ◽  
Lili Zhang ◽  
Wenli Wang
Keyword(s):  
Clustering Algorithms ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Signed Networks ◽  
Clustering Problem ◽  
Structural Hole ◽  
Signed Network ◽  
Necessary And Sufficient ◽  
Sufficient And Necessary Condition

This paper investigates the clustering problem for the generalized signed networks. By rigorous derivations, a sufficient and necessary condition for clustering of the nodes in generalized signed networks is proposed in this paper. In order to obtain this condition, the concept of friends group is first introduced for the nodes based on their links’ sign. Then, the unprivileged network is also defined in this paper by employing the concepts of structural hole and broker. Compared with the existing clustering algorithms, the outstanding advantage in this paper is that only the positive or negative (especially, or zero) sign of the links is required regardless of their density or sparsity. We have proved mathematically that a generalized signed network is classifiable if and only if it is an unprivileged network. Finally, two examples associated with numerical simulations are proposed to generate the unprivileged networks.

A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

2008 ◽  
Vol 24 (3) ◽  
pp. 823-828 ◽  
Author(s):  
Henghsiu Tsai ◽  
Kung-Sik Chan
Keyword(s):  
Generating Function ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Conditional Variance ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Economic Statistics ◽  
Absolute Monotonicity ◽  
Necessary And Sufficient ◽  
The Garch Model

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH(p,q) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH(p,q) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH(p,q) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

scholarly journals Hyponormality on general Bergman spaces

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5737-5741 ◽  
Author(s):  
Houcine Sadraoui
Keyword(s):  
Hilbert Space ◽  
Unit Disk ◽  
Toeplitz Operators ◽  
Bounded Operator ◽  
Bergman Spaces ◽  
Weighted Bergman Spaces ◽  
Necessary Condition ◽  
Sufficient Condition

A bounded operator T on a Hilbert space is hyponormal if T*T-TT* is positive. We give a necessary condition for the hyponormality of Toeplitz operators on weighted Bergman spaces, for a certain class of radial weights, when the symbol is of the form f+g?, where both functions are analytic and bounded on the unit disk. We give a sufficient condition when f is a monomial.

scholarly journals Characterizations of Double Commutant Property on B H

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Chaoqun Chen ◽  
Fangyan Lu ◽  
Cuimei Cui ◽  
Ling Wang
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Necessary Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Double Commutant ◽  
Sufficient And Necessary Condition

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

On the Diffeomorphisms Between Banach and Hilbert Spaces

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Dariusz Idczak ◽  
Andrzej Skowron ◽  
Stanislaw Walczak
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Differential Equations ◽  
Existence Theorem ◽  
Hilbert Spaces ◽  
Mountain Pass Theorem ◽  
Sufficient Conditions ◽  
Mountain Pass ◽  
Final Part

AbstractIn this paper, we give some sufficient conditions for f : X → H to be a diffeomorphism, where X is a Banach space and H is a Hilbert space. The proof of the result is based on the mountain pass theorem. Using this result, in the final part of the paper, we prove an existence theorem for some class of integro-differential equations.

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