scholarly journals Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

2007 ◽  
Vol 83 (3) ◽  
pp. 297-326 ◽  
Author(s):  
Sergey S. Ajiev
Keyword(s):  
Sufficient Conditions ◽  
Dirac Operators ◽  
Coercivity Conditions ◽  
Embedding Theorems ◽  
Lebesgue Spaces ◽  
Root Model ◽  
Quadratic Estimates ◽  
Resolvent Approach ◽  
Stability Results ◽  
Square Root Model

AbstractSeveral rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L2 to Lp are established. As consequences, we obtain some embedding theorems, quadratic estimates and Littlewood–Paley theorems in terms of this calculus in Lebesgue spaces. Some further generalizations, utilised in Part II devoted to applications, which include the Kato square root model, are discussed. We use resolvent approach and show the irrelevance of the semigroup one. Auxiliary results include a high order counterpart of the Hilbert identity, the derivation of new forms of ‘off-diagonal’ estimates, and the study of the structure of the model in Lebesgue spaces and its interpolation properties. In particular, some coercivity conditions for forms in Banach spaces are used as a substitution of the ellipticity ones. Attention is devoted to the relations between the properties of perturbed and unperturbed generalized Dirac operators. We do not use any stability results.

The M/M/1 queue with mass exodus and mass arrivals when empty

1997 ◽  
Vol 34 (01) ◽  
pp. 192-207 ◽  
Author(s):  
Anyue Chen ◽  
Eric Renshaw
Keyword(s):  
Sufficient Conditions ◽  
High Order ◽  
Necessary And Sufficient Conditions ◽  
Positive Recurrence ◽  
Powerful Technique ◽  
Resolvent Approach ◽  
Necessary And Sufficient ◽  
Direct Attack

An M/M/1 queue is subject to mass exodus at rate β and mass immigration at rate when idle. A general resolvent approach is used to derive occupation probabilities and high-order moments. This powerful technique is not only considerably easier to apply than a standard direct attack on the forward p.g.f. equation, but it also implicitly yields necessary and sufficient conditions for recurrence, positive recurrence and transience.

scholarly journals Potential Operators on Cones of Nonincreasing Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Alexander Meskhi ◽  
Ghulam Murtaza
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Product Type ◽  
Necessary And Sufficient Conditions ◽  
Lebesgue Spaces ◽  
Potential Operators ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators(Iαf)(x)=∫0∞(f(t)/|x−t|1−α)dtand(ℐα1,α2f)(x,y)=∫0∞∫0∞(f(t,τ)/|x−t|1−α1|y−τ|1−α2)dtdτon the cone of nonincreasing functions are derived. In the case ofℐα1,α2, we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.

scholarly journals Uniqueness and Stability Results on Non-local Stochastic Random Impulsive Integro-Differential Equations

2021 ◽  
Vol 2 (3) ◽  
pp. 9-20
Author(s):  
VARSHINI S ◽  
BANUPRIYA K ◽  
RAMKUMAR K ◽  
RAVIKUMAR K
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Local Conditions ◽  
Non Local ◽  
Inequality Techniques ◽  
Stability Results

The paper is concerned with stochastic random impulsive integro-differential equations with non-local conditions. The sufficient conditions guarantees uniqueness of mild solution derived using Banach fixed point theorem. Stability of the solution is derived by incorporating Banach fixed point theorem with certain inequality techniques.

ASYMPTOTIC HYPERSTABILITY OF A CLASS OF NEURAL NETWORKS

1999 ◽  
Vol 09 (02) ◽  
pp. 95-98 ◽  
Author(s):  
ANKE MEYER-BÄSE
Keyword(s):  
Neural Networks ◽  
Recurrent Neural Networks ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lyapunov Methods ◽  
Network Parameters ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Stability Results

This paper is concerned with the asymptotic hyperstability of recurrent neural networks. We derive based on the stability results necessary and sufficient conditions for the network parameters. The results we achieve are more general than those based on Lyapunov methods, since they provide milder constraints on the connection weights than the conventional results and do not suppose symmetry of the weights.

Embedding theorems and the spectra of certain differential operators

1991 ◽  
Vol 434 (1892) ◽  
pp. 643-656 ◽  
Keyword(s):  
Differential Operators ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Independent Interest ◽  
Embedding Theorems ◽  
Necessary And Sufficient

This paper is concerned with problems of the form n Ʃ k =0 (─1) k ( ρ 2 k y ( k ))( k ) = λ r 2 y on R , ry ∈ L 2 ( R ) and gives conditions on the coefficients sufficient to ensure that the spectrum is discrete; necessary conditions are also given. In certain circumstances, necessary and sufficient conditions for discreteness are given, thus extending the celebrated Molcanov criterion. These results follow from embedding theorems which have independent interest.

Optimal temperature input design for estimation of the Square Root model parameters: parameter accuracy and model validity restrictions

2002 ◽  
Vol 73 (2-3) ◽  
pp. 145-157 ◽  
Author(s):  
Kristel Bernaerts ◽  
Roos D. Servaes ◽  
Steven Kooyman ◽  
Karina J. Versyck ◽  
Jan F. Van Impe
Keyword(s):  
Model Parameters ◽  
Square Root ◽  
Optimal Temperature ◽  
Model Validity ◽  
Input Design ◽  
Root Model ◽  
Square Root Model
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