scholarly journals THE MAIN HAMILTON FUNCTION AND THE NECESSARY CONDITIONS FOR THE EXISTENCE OF A SITUATION IN THE FORM OF THE HAMILTON-JACOBI EQUATIONS

2021 ◽  
Vol 73 (1) ◽  
pp. 32-41
Author(s):  
Sh.D. Makhmudova ◽  
◽  
A.N. Urazgalieva ◽  
Keyword(s):  
Learning Outcomes ◽  
Differential Games ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Analytical Mechanics ◽  
Equilibrium Conditions ◽  
Holistic Understanding ◽  
Theory Of Optimal Control ◽  
Necessary And Sufficient ◽  
Jacobi Equations

Interdisciplinary application of learning outcomes is essential to the field of mathematics, attainable at intersection of math with other subjects – which includes applied tasks When studying a number of technical disciplines, as well as solving applied problems, it is possible to use certain aspects of the theory of optimal control - which is an example of interdisciplinary link. Analytical mechanics, among other disciplines, enables leverage of certain aspects of the theory of differential games, namely, equilibrium conditions in non-coalitional differential games of several players. This article provides studies of the necessary and sufficient conditions for the existence of equilibrium situations, using some concepts of analytical mechanics. In line with Hamilton’s definition, necessary conditions were obtained in the form of Hamilton-Jacobi equations. This form of necessary conditions in differential games of N persons is of interest to students of natural and technical fields. The main goal of the article is to demonstrate interdisciplinary link, an important component of the process of training future engineers for various sectors of the economy. It is necessary for the holistic understanding of the material, so that students of technical specialties of various fields can use it. Proposed work can aid in study of this area of analytical mechanics by university students and young scientists alike.

scholarly journals Noetherian properties in composite generalized power series rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1540-1551
Author(s):  
Jung Wook Lim ◽  
Dong Yeol Oh
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Power Series ◽  
Ordered Monoid ◽  
Power Series Rings ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

Abstract Let ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D. Set \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

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.

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.

Simplicity of iterated Ore extensions

2021 ◽  
pp. 2250052
Author(s):  
Mamta Balodi ◽  
Sumit Kumar Upadhyay
Keyword(s):  
Polynomial Ring ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Commutative Domain ◽  
Necessary And Sufficient ◽  
Commutative Subring

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

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