Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2

1995 ◽  
Vol 39 (1) ◽  
pp. 31-46 ◽  
Author(s):  
Gilles Aubert
Keyword(s):  
Convex Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rank One ◽  
Dimension 2 ◽  
Necessary And Sufficient

scholarly journals Convex functions and the rolling circle criterion

1995 ◽  
Vol 18 (4) ◽  
pp. 799-811
Author(s):  
V. Srinivas ◽  
O. P. Juneja ◽  
G. P. Kapoor
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rolling Circle ◽  
Circle Criterion ◽  
Rotation Theorem ◽  
Distortion Theorems ◽  
Necessary And Sufficient ◽  
Growth And Distortion Theorems

Given0≤R1≤R2≤∞,CVG(R1,R2)denotes the class of normalized convex functionsfin the unit discU, for which∂f(U)satisfies a Blaschke Rolling Circles Criterion with radiiR1andR2. Necessary and sufficient conditions forR1=R2, growth and distortion theorems forCVG(R1,R2)and rotation theorem for the class of convex functions of bounded type, are found.

Quantum Kostka and the rank one problem for 𝔰𝔩2m

2019 ◽  
Vol 19 (1) ◽  
pp. 71-88
Author(s):  
Natalie L. F. Hobson
Keyword(s):  
Vector Bundles ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Chern Classes ◽  
Finitely Generated ◽  
Conformal Blocks ◽  
Rank One ◽  
Necessary And Sufficient ◽  
Nef Cone

Abstract We give necessary and sufficient conditions to specify vector bundles of conformal blocks for 𝔰𝔩2m with rectangular weights which have ranks 0, 1, and larger than 1. We show that the first Chern classes of such rank one bundles determine a finitely generated subcone of the nef cone.

scholarly journals On separation by h-convex functions

2015 ◽  
Vol 62 (1) ◽  
pp. 105-111 ◽  
Author(s):  
Andrzej Olbryś
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Abstract In the present paper, we establish the necessary and sufficient conditions under which two functions can be separated by h-convex function, in the case, when the function h is multiplicative. This result is related to the theorem on separation by convex functions presented in Baron, K. Matkowski, J. Nikodem, K. [A sandwich with convexity, Math. Pannon. 5 (1994), 139 144].

scholarly journals On a Separation Theorem for Delta-Convex Functions

2020 ◽  
Vol 34 (1) ◽  
pp. 133-141
Author(s):  
Andrzej Olbryś
Keyword(s):  
Convex Function ◽  
Partial Order ◽  
Convex Functions ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Separation Theorem ◽  
Necessary And Sufficient Conditions ◽  
Lorentz Cone ◽  
Necessary And Sufficient

AbstractIn the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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