Some notes on weak subdifferential

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3407-3420 ◽  
Author(s):  
P. Cheraghi ◽  
Ali Farajzadeh ◽  
Gradimir Milovanovic
Keyword(s):  
Global Minimum ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sum Rule ◽  
Subdifferential Operator ◽  
Weak Subdifferential ◽  
Necessary And Sufficient

Some necessary conditions for having nonempty weak subdifferential of a function are presented and the positively homogeneous of the weak subdifferential operator is proved. Necessary and sufficient conditions for achieving a global minimum of a weak subdifferentiable function is stated, as well as a link between subdifferential and the Fr?chet differential with a weak subdifferential. A result about the equality of the fuzzy sum rule inclusion is also investigated. Finally, some examples are included.

Normal and osculating maps for submanifolds of RN

1990 ◽  
Vol 114 (1-2) ◽  
pp. 39-55 ◽  
Author(s):  
I. Cattaneo Gasparini ◽  
G. Romani
Keyword(s):  
Isometric Immersion ◽  
Normal Bundle ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Gauss Map ◽  
Necessary And Sufficient Conditions ◽  
Totally Geodesic ◽  
Precise Formulation ◽  
Parallel Case ◽  
Necessary And Sufficient

SynopsisLet Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .

AN W ALGEBRAS AND CLASSIFICATION

1992 ◽  
Vol 07 (36) ◽  
pp. 3419-3423
Author(s):  
LIU CHAO ◽  
BOYU HOU
Keyword(s):  
Lie Algebra ◽  
Regular Element ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arbitrary Degree ◽  
Wznw Model ◽  
Necessary And Sufficient

The necessary and sufficient conditions for the existence of a regular element of arbitrary degree under arbitrary integral gradation of the Lie algebra g is presented. Such elements, while chosen as constraints in WZNW model, give rise to a W-algebra. It is then found that there might be some isomorphic relations between different W-algebras. The necessary conditions for such isomorphisms to appear are also given. Up to the A4 cases these conditions are checked to be sufficient.

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.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

scholarly journals Mixed-Norm Amalgam Spaces and Their Predual

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 74
Author(s):  
Houkun Zhang ◽  
Jiang Zhou
Keyword(s):  
Duality Theory ◽  
Fractional Integral ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Mixed Norm ◽  
Fractional Integral Operators ◽  
Primal Space ◽  
Necessary And Sufficient ◽  
Amalgam Spaces

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

Center Problem for Quasi-Homogeneous Polynomial Systems with a Given Weight Degree

2018 ◽  
Vol 28 (14) ◽  
pp. 1850174 ◽  
Author(s):  
Yanqin Xiong ◽  
Jianqiang Hu ◽  
Shimin Li ◽  
Jingzheng Li
Keyword(s):  
Homogeneous Polynomial ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Polynomial Systems ◽  
Necessary And Sufficient Conditions ◽  
Center Problem ◽  
Necessary And Sufficient

This paper considers the center problem for quasi-homogeneous polynomial systems with a given weight degree. We provide the necessary conditions such that these systems have a center at the origin. Especially, we present the necessary and sufficient conditions on the existence of a center for some class of such systems.

scholarly journals Maximal topologies

1973 ◽  
Vol 15 (3) ◽  
pp. 279-290 ◽  
Author(s):  
Asit Baran Raha
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Compact Spaces ◽  
Other Hand ◽  
Countably Compact ◽  
Countably Compact Spaces ◽  
Necessary And Sufficient

This article is devoted to studying maximal π spaces where π = Lindelöf, countably compact, connected, lightly compact or pseudocompact. Necessary and sufficient conditions for Lindelöf or countably compact spaces to be maximal Lindelöf or maximal countably compact have been obtained. On the other hand only necessary conditions for maximal π spaces have been deduced where π = connected, lightly compact or pseudocompact.

INCREASING SPOT RATES OF INTEREST: STRUCTURE OF THE PRICE OF A DEFAULT FREE DISCOUNT BOND

2002 ◽  
Vol 05 (03) ◽  
pp. 321-332 ◽  
Author(s):  
SALVADOR CRUZ RAMBAUD ◽  
MARÍA DEL CARMEN VALLS MARTÍNEZ
Keyword(s):  
Interest Rate ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Auxiliary Function ◽  
Necessary And Sufficient Conditions ◽  
Bond Price ◽  
Logarithmic Density ◽  
Generalized Derivative ◽  
Necessary And Sufficient ◽  
Final Amount

This paper aims to establish a set of necessary and sufficient conditions that the expression of a default free bond must verify in a situation in which spot rates of interest increase. In this case, if the agent divides the period of his investment, that is, if he disinvests and, immediately, re-invests his capital in different issues of bonds, can obtain an advantage in the final amount. In order to do this, necessary conditions include an approximation of the forward interest rate, that is to say the logarithmic density, its limits to +∞ and -∞ and the second generalized derivative of the bond price. Sufficient conditions involve an auxiliary function using these limits and, finally, necessary and sufficient conditions are related with the logarithm of the bond price.

scholarly journals Necessary and sufficient conditions on weighted multilinear fractional integral inequality

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zhou Yongliang ◽  
Deng Yangkendi ◽  
Wu Di ◽  
Yan Dunyan
Keyword(s):  
Integral Inequality ◽  
Sobolev Inequality ◽  
Fractional Integral ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Integral Inequalities ◽  
Multilinear Fractional Integral ◽  
Sufficient And Necessary Conditions ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>We consider certain kinds of weighted multi-linear fractional integral inequalities which can be regarded as extensions of the Hardy-Littlewood-Sobolev inequality. For a particular case, we characterize the sufficient and necessary conditions which ensure that the corresponding inequality holds. For the general case, we give some sufficient conditions and necessary conditions, respectively.</p>

scholarly journals Necessary and Sufficient Conditions for Set-Valued Maps with Set Optimization

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Abdessamad Oussarhan ◽  
Ikram Daidai
Keyword(s):  
Optimality Conditions ◽  
Existence Of Solutions ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Set Optimization ◽  
Contingent Derivative ◽  
Necessary And Sufficient

Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of S-derivative and contingent derivative. Sufficient conditions for the existence of solutions are shown for set-valued maps under generalized quasiconvexity assumptions.

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