scholarly journals A CHARACTERIZATION OF SELF-ADJOINT OPERATORS DETERMINED BY THE WEAK FORMULATION OF SECOND-ORDER SINGULAR DIFFERENTIAL EXPRESSIONS

2009 ◽  
Vol 51 (2) ◽  
pp. 385-404 ◽  
Author(s):  
MOHAMED EL-GEBEILY ◽  
DONAL O'REGAN
Keyword(s):  
Complete Characterization ◽  
Second Order ◽  
Inner Product ◽  
Type I ◽  
Weak Formulation ◽  
Friedrichs Extension ◽  
Sesquilinear Form ◽  
Adjoint Operators ◽  
Special Case

AbstractIn this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

scholarly journals THE BOUNDARY CONDITIONS DESCRIPTION OF TYPE I DOMAINS

2010 ◽  
Vol 52 (3) ◽  
pp. 619-633 ◽  
Author(s):  
MOHAMED EL-GEBEILY ◽  
DONAL O'REGAN
Keyword(s):  
Boundary Conditions ◽  
The Self ◽  
Inner Product ◽  
Type I ◽  
Singular Case ◽  
Regular Case ◽  
Weak Formulation ◽  
Sesquilinear Form ◽  
Classical Function ◽  
Adjoint Operators

AbstractType I domains are the domains of the self-adjoint operators determined by the weak formulation of formally self-adjoint differential expressions ℓ. This class of operators is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts agrees with the inner product 〈ℓu, v〉. A complete characterisation of the boundary conditions assumed by functions in these domains for second-order differential expressions is given in this paper. In the singular case, the boundary conditions are stated in terms of certain ‘boundary condition’ functions and in the regular case they are given in terms of classical function values.

Quadratic-monomial generated domains from mixed signed, directed graphs

2019 ◽  
Vol 29 (02) ◽  
pp. 279-308
Author(s):  
Michael A. Burr ◽  
Drew J. Lipman
Keyword(s):  
Directed Graphs ◽  
Complete Characterization ◽  
Combinatorial Structure ◽  
Signed Directed Graph ◽  
Odd Cycle ◽  
Combinatorial Classification ◽  
Special Case ◽  
Text Condition

Determining whether an arbitrary subring [Formula: see text] of [Formula: see text] is a normal or Cohen-Macaulay domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. We provide a complete characterization of the normality, normalizations, and Serre’s [Formula: see text] condition for quadratic-monomial generated domains. For a quadratic-monomial generated domain [Formula: see text], we develop a combinatorial structure that assigns, to each quadratic monomial of the ring, an edge in a mixed signed, directed graph [Formula: see text], i.e. a graph with signed edges and directed edges. We classify the normality and the normalizations of such rings in terms of a generalization of the combinatorial odd cycle condition on [Formula: see text]. We also generalize and simplify a combinatorial classification of Serre’s [Formula: see text] condition for such rings and construct non-Cohen–Macaulay rings.

scholarly journals Antinormal composition operators on $ \mbf{\ell^2}$

2008 ◽  
Vol 39 (4) ◽  
pp. 347-352 ◽  
Author(s):  
Gyan Prakash Tripathi ◽  
Nand Lal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Composition Operators ◽  
Complete Characterization ◽  
Inner Product ◽  
Complex Numbers ◽  
Bounded Linear ◽  
Normal Operators

A bounded linear operator $ T $ on a Hilbert space $ H $ is called antinormal if the distance of $ T $ from the set of all normal operators is equal to norm of $ T $. In this paper, we give a complete characterization of antinormal composition operators on $ \ell^2 $, where $ \ell^2 $ is the Hilbert space of all square summable sequences of complex numbers under standard inner product on it.

The complexity of resolution refinements

2007 ◽  
Vol 72 (4) ◽  
pp. 1336-1352 ◽  
Author(s):  
Joshua Buresh-Oppenheim ◽  
Toniann Pitassi
Keyword(s):  
Theorem Proving ◽  
Complete Characterization ◽  
Exponential Separation ◽  
Semantic Resolution ◽  
Special Case

AbstractResolution is the most widely studied approach to propositional theorem proving. In developing efficient resolution-based algorithms, dozens of variants and refinements of resolution have been studied from both the empirical and analytic sides. The most prominent of these refinements are: DP (ordered), DLL (tree), semantic, negative, linear and regular resolution. In this paper, we characterize and study these six refinements of resolution. We give a nearly complete characterization of the relative complexities of all six refinements. While many of the important separations and simulations were already known, many new ones are presented in this paper; in particular, we give the first separation of semantic resolution from general resolution. As a special case, we obtain the first exponential separation of negative resolution from general resolution. We also attempt to present a unifying framework for studying all of these refinements.

Colour perception may optimize biologically relevant surface discriminations – rather than type-I constancy

2001 ◽  
Vol 24 (4) ◽  
pp. 658-659
Author(s):  
Nicola Bruno ◽  
Stephen Westland
Keyword(s):  
Complete Characterization ◽  
Type I ◽  
Colour Perception ◽  
Biologically Relevant

Trichromacy may result from an adaptation to the regularities in terrestrial illumination. However, we suggest that a complete characterization of the challenges faced by colour perception must include changes in surface surround and illuminant changes due to inter-reflections between surfaces in cluttered scenes. Furthermore, our trichromatic system may have evolved to allow the detection of brownish-reddish edibles against greenish backgrounds. [Shepard]

scholarly journals On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials

2006 ◽  
Vol 58 (4) ◽  
pp. 726-767 ◽  
Author(s):  
Yik-Man Chiang ◽  
Mourad E. H. Ismail
Keyword(s):  
Differential Equations ◽  
Special Functions ◽  
Complete Characterization ◽  
Second Order ◽  
Value Distribution ◽  
Value Distribution Theory ◽  
Confluent Hypergeometric Functions ◽  
Number Of Zeros ◽  
Oscillation Of Solutions

AbstractWe show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above “special function approach” can be described by a classical Heine problem for differential equations that admit polynomial solutions.

scholarly journals A complete characterization of disjunctive conic cuts for mixed integer second order cone optimization

2017 ◽  
Vol 24 ◽  
pp. 3-31 ◽  
Author(s):  
Pietro Belotti ◽  
Julio C. Góez ◽  
Imre Pólik ◽  
Ted K. Ralphs ◽  
Tamás Terlaky
Keyword(s):  
Complete Characterization ◽  
Second Order ◽  
Mixed Integer ◽  
Second Order Cone

scholarly journals On the boundary conditions for products of Sturm-Liouville differential operators

2001 ◽  
Vol 32 (3) ◽  
pp. 187-199
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Sesquilinear Form ◽  
Sturm Liouville

In this paper, the second-order symmetric Sturm-Liouville differential expressions $ \tau_1, \tau_2, \ldots, \tau_n $ with real coefficients are considered on the interval $ I = (a,b) $, $ - \infty \le a < b \le \infty $. It is shown that the characterization of singular self-adjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximan domain of the product operators, and is an exact parallel of the regular case. This characterization is an extension of those obtained in [6], [8], [11-12], [14] and [15].

scholarly journals On the domain of selfadjoint extension of the product of Sturm-Liouville differential operators

2003 ◽  
Vol 2003 (11) ◽  
pp. 695-709
Author(s):  
Sobhy El-Sayed Ibrahim
Keyword(s):  
Boundary Conditions ◽  
Differential Operators ◽  
Second Order ◽  
Regular Case ◽  
Maximal Domain ◽  
Selfadjoint Extension ◽  
Sesquilinear Form ◽  
Sturm Liouville

The second-order symmetric Sturm-Liouville differential expressionsτ1,τ2,…,τnwith real coefficients are considered on the intervalI=(a,b),−∞≤a<b≤∞. It is shown that the characterization of singular selfadjoint boundary conditions involves the sesquilinear form associated with the product of Sturm-Liouville differential expressions and elements of the maximal domain of the product operators, and it is an exact parallel of the regular case. This characterization is an extension of those obtained by Everitt and Zettl (1977), Hinton, Krall, and Shaw (1987), Ibrahim (1999), Krall and Zettl (1988), Lee (1975/1976), and Naimark (1968).

scholarly journals A complete folk theorem for finitely repeated games

2020 ◽  
Vol 49 (4) ◽  
pp. 1129-1142
Author(s):  
Ghislain-Herman Demeze-Jouatsa
Keyword(s):  
Repeated Games ◽  
Pure Strategy ◽  
Complete Information ◽  
Repeated Game ◽  
Complete Characterization ◽  
Folk Theorem ◽  
Equilibrium Payoff ◽  
Perfect Monitoring ◽  
Special Case

AbstractThis paper analyzes the set of pure strategy subgame perfect Nash equilibria of any finitely repeated game with complete information and perfect monitoring. The main result is a complete characterization of the limit set, as the time horizon increases, of the set of pure strategy subgame perfect Nash equilibrium payoff vectors of the finitely repeated game. This model includes the special case of observable mixed strategies.

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