Functions of polynomial growth and domains of holomorphy

1978 ◽  
pp. 224-232
Author(s):  
Peter Pflug
Keyword(s):  
Polynomial Growth ◽  
Domains Of Holomorphy

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
Author(s):  
Philippe Briand ◽  
René Carmona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Polynomial Growth ◽  
Probabilistic Representation ◽  
Existence And Uniqueness Results ◽  
State Variable ◽  
Cahn Equation ◽  
Uniqueness Results ◽  
Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

Heat equation with singular potential and singular data

2005 ◽  
Vol 135 (4) ◽  
pp. 863-886 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović ◽  
D. Rajter-Ćirić
Keyword(s):  
Polynomial Growth ◽  
Singular Potential ◽  
Function Algebra ◽  
Energy Estimates ◽  
Generalized Function ◽  
Type Condition ◽  
Classical Energy ◽  
The Cauchy Problem ◽  
White Noise Calculus ◽  
Singular Data

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

scholarly journals On stable solutions of the biharmonic problem with polynomial growth

2014 ◽  
Vol 270 (1) ◽  
pp. 79-93 ◽  
Author(s):  
Hatem Hajlaoui ◽  
Abdellaziz Harrabi ◽  
Dong Ye
Keyword(s):  
Polynomial Growth ◽  
Biharmonic Problem ◽  
Stable Solutions

On the spectrum and the distribution of singular values of Schrödinger operators with a complex potential

1983 ◽  
Vol 388 (1794) ◽  
pp. 195-218 ◽  
Keyword(s):  
Complex Potential ◽  
Spectral Properties ◽  
Polynomial Growth ◽  
Schrödinger Operators ◽  
Singular Values ◽  
Negative Real Part ◽  
Adjoint Problem ◽  
Schrodinger Operators ◽  
Asymptotic Estimates ◽  
Integrability Conditions

This paper is concerned with spectral properties of the Schrödinger operator ─ ∆+ q with a complex potential q which has non-negative real part and satisfies weak integrability conditions. The problem is dealt with as a genuine non-self-adjoint problem, not as a perturbation of a self-adjoint one, and global and asymptotic estimates are obtained for the corresponding singular values. From these estimates information is obtained about the eigenvalues of the problem. By way of illustration, detailed calculations are given for an example in which the potential has at most polynomial growth.

Domains of Holomorphy

1976 ◽  
pp. 29-67
Author(s):  
H. Grauert ◽  
K. Fritzsche
Keyword(s):  
Domains Of Holomorphy

Semigroups of Polynomial Growth

2020 ◽  
pp. 95-106
Author(s):  
Jan Okniński
Keyword(s):  
Polynomial Growth

scholarly journals Nilpotent Group C*-algebras as Compact Quantum Metric Spaces

2017 ◽  
Vol 60 (1) ◽  
pp. 77-94 ◽  
Author(s):  
Michael Christ ◽  
Marc A. Rieòel
Keyword(s):  
Finite Groups ◽  
Word Length ◽  
Polynomial Growth ◽  
Metric Spaces ◽  
Length Function ◽  
C Algebra ◽  
Pointwise Multiplication ◽  
C Algebras ◽  
Definition Of ◽  
Compact Quantum Metric Space

AbstractLet be a length function on a group G, and let M denote the operator of pointwise multiplication by on l2(G). Following Connes, M𝕃 can be used as a “Dirac” operator for the reduced group C*-algebra (G). It deûnes a Lipschitz seminorm on (G), which defines a metric on the state space of (G). We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-* topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on ûnitely generated nilpotent-by-finite groups.

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