Necessary and Sufficient Condition for Uniqueness of the Characteristic Cauchy Problem for Differential Operators with Constant Coefficients

1986 ◽  
Vol 127 (1) ◽  
pp. 309-315 ◽  
Author(s):  
Todor V. Gramchev
Keyword(s):  
Cauchy Problem ◽  
Differential Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Constant Coefficients ◽  
Characteristic Cauchy Problem ◽  
Necessary And Sufficient

NOETHER IDENTITIES OF A DIFFERENTIAL OPERATOR: THE KOSZUL–TATE COMPLEX

2005 ◽  
Vol 02 (05) ◽  
pp. 873-886 ◽  
Author(s):  
G. SARDANASHVILY
Keyword(s):  
Differential Operator ◽  
Fiber Bundle ◽  
Differential Operators ◽  
Chain Complex ◽  
Boundary Operator ◽  
Lagrangian System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Given a generic Lagrangian system, its Euler–Lagrange operator obeys Noether identities which need not be independent, but satisfy first-stage Noether identities, and so on. This construction is generalized to arbitrary differential operators on a smooth fiber bundle. Namely, if a certain necessary and sufficient condition holds, one can associate to a differential operator the exact chain complex with the boundary operator whose nilpotency restarts all the Noether identities characterizing the degeneracy of an original differential operator.

15.—The Measure of Non-compactness of Some Linear Integral Operators

1973 ◽  
Vol 71 (2) ◽  
pp. 167-179
Author(s):  
C. A. Stuart
Keyword(s):  
Compact Operator ◽  
Essential Spectrum ◽  
Differential Operators ◽  
Integral Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Set ◽  
Linear Integral ◽  
Necessary And Sufficient ◽  
Linear Integral Operators

SynopsisThe measure of non-compactness of linear integral operators on the half-line [0, ∞) of a special type is studied. In particular, a necessary and sufficient condition is established for an operator of this type to define a compact operator from L2(0, ∞) into itself. These results are then used to discuss the spectrum of second-order differential operators. A necessary and sufficient condition for the spectrum to be discrete is established together with estimates for the distance of a point in the resolvent set from the essential spectrum.

scholarly journals Microlocal regularity on step two nilpotent Lie groups

1988 ◽  
Vol 31 (1) ◽  
pp. 25-39
Author(s):  
Kenneth G. Miller
Keyword(s):  
Differential Operators ◽  
Pseudodifferential Operators ◽  
Partial Differential Operator ◽  
Irreducible Unitary Representation ◽  
Nilpotent Lie Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Necessary And Sufficient

A necessary and sufficient condition for a homogeneous left invariant partial differential operator P on a nilpotent Lie group G to be hypoelliptic is that π(P) be injective in π for every nontrivial irreducible unitary representation π of G. This was conjectured by Rockland in [18], where it was also proved in the case of the Heisenberg group. The necessity of the condition in the general case was proved by Beals [2] and the sufficiency by Helffer and Nourrigat [4]. In this paper we present a microlocal version of this theorem when G is step two nilpotent. The operator may be homogeneous with respect to any family of dilations on G, not just the natural dilations. We may also consider pseudodifferential operators as well as partial differential operators.

scholarly journals A necessary and sufficient condition for the existence of a unique solution of a discrete boundary value problem

2003 ◽  
Vol 2003 (39) ◽  
pp. 2455-2463 ◽  
Author(s):  
Raghib Abu-Saris ◽  
Wajdi Ahmad
Keyword(s):  
Boundary Value Problem ◽  
Unique Solution ◽  
Boundary Value ◽  
Linear Difference Equation ◽  
Fundamental Question ◽  
Well Posedness ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Constant Coefficients ◽  
Necessary And Sufficient

Akth-order linear difference equation with constant coefficients subject to boundary conditions is considered. A necessary and sufficient condition for the existence of a unique solution for such a boundary value problem is established. The condition established answers a fundamental question for well-posedness and can be easily applied using a simple and computationally tractable algorithm that does not require finding the roots of the associated characteristic equation.

scholarly journals Mannheim Curves in Nonflat 3-Dimensional Space Forms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Wenjing Zhao ◽  
Donghe Pei ◽  
Xinyu Cao
Keyword(s):  
Linear Relationship ◽  
Dimensional Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
De Sitter ◽  
Space Forms ◽  
3 Dimensional ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
The Given

We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian) and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space.

Characterizations of nuclear pseudo-differential operators on ℤ with some applications

2018 ◽  
Vol 13 (4) ◽  
pp. 33
Author(s):  
Majid JamalpourBirgani
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bounded Operators ◽  
Nuclear Operators ◽  
Pseudo Differential Operators ◽  
Necessary And Sufficient ◽  
Adjoint Operators

In this paper, we give necessary and sufficient conditions on the symbols σ, such that the corresponding pseudo-differential operators Tσ from Lp1(ℤ) into Lp2(ℤ), 1 ≤ p1,p2 < ∞, be nuclear. We show that the adjoint operators of the nuclear pseudo-differential operators from Lp′2(ℤ) into Lp′1(ℤ) are nuclear and present a necessary and sufficient condition on the symbols of the nuclear pseudo-differential operators from L2(ℤ) into L2(ℤ) to be self-adjoint. As applications, We get the symbol of the product of the nuclear operators with the bounded operators, and a necessary and sufficient condition on the symbols of nuclear operators is given to be normal.

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