scholarly journals Pointwise Remez inequality

2021 ◽  
Author(s):  
B. Eichinger ◽  
P. Yuditskii
Keyword(s):  
Fixed Point ◽  
Asymptotic Solution ◽  
Lebesgue Measure ◽  
Chebyshev Polynomials ◽  
Extremal Solution ◽  
Fixed Size ◽  
Extremal Polynomials ◽  
Remez Inequality ◽  
Upper Estimate ◽  
Extremal Value

AbstractThe standard well-known Remez inequality gives an upper estimate of the values of polynomials on $$[-1,1]$$ [ - 1 , 1 ] if they are bounded by 1 on a subset of $$[-1,1]$$ [ - 1 , 1 ] of fixed Lebesgue measure. The extremal solution is given by the rescaled Chebyshev polynomials for one interval. Andrievskii asked about the maximal value of polynomials at a fixed point, if they are again bounded by 1 on a set of fixed size. We show that the extremal polynomials are either Chebyshev (one interval) or Akhiezer polynomials (two intervals) and prove Totik–Widom bounds for the extremal value, thereby providing a complete asymptotic solution to the Andrievskii problem.

scholarly journals Hybrid Fixed-Point Fixed-Stress Splitting Method for Linear Poroelasticity

Geosciences ◽  
2019 ◽  
Vol 9 (1) ◽  
pp. 29 ◽  
Author(s):  
Paul M. Delgado ◽  
V. M. Krushnarao Kotteda ◽  
Vinod Kumar
Keyword(s):  
Fixed Point ◽  
Asymptotic Solution ◽  
Relative Error ◽  
Operator Splitting ◽  
Splitting Method ◽  
Approximate Solutions ◽  
Time Step ◽  
Operator Splitting Method ◽  
The Stability ◽  
Stability And Consistency

Efficient and accurate poroelasticity models are critical in modeling geophysical problems such as oil exploration, gas-hydrate detection, and hydrogeology. We propose an efficient operator splitting method for Biot’s model of linear poroelasticity based on fixed-point iteration and constrained stress. In this method, we eliminate the constraint on time step via combining our fixed-point approach with a physics-based restraint between iterations. Three different cases are considered to demonstrate the stability and consistency of the method for constant and variable parameters. The results are validated against the results from the fully coupled approach. In case I, a single iteration is used for continuous coefficients. The relative error decreases with an increase in time. In case II, material coefficients are assumed to be linear. In the single iteration approach, the relative error grows significantly to 40% before rapidly decaying to zero. This is an artifact of the approximate solutions approaching the asymptotic solution. The error in the multiple iterations oscillates within 10 − 6 before decaying to the asymptotic solution. Nine iterations per time step are enough to achieve the relative error close to 10 − 7 . In the last case, the hybrid method with multiple iterations requires approximately 16 iterations to make the relative error 5 × 10 − 6 .

ON FUNCTIONS ATTRACTING POSITIVE ENTROPY

2017 ◽  
Vol 97 (1) ◽  
pp. 69-79 ◽  
Author(s):  
ANNA LORANTY ◽  
RYSZARD J. PAWLAK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Positive Entropy ◽  
Density Point

We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$.

scholarly journals F-evolution algebra

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2637-2652 ◽  
Author(s):  
Uygun Jamilov ◽  
Manuel Ladra
Keyword(s):  
Fixed Point ◽  
Banach Algebra ◽  
Necessary Conditions ◽  
Zero Point ◽  
Limit Set ◽  
Volterra Type ◽  
Quadratic Stochastic Operator ◽  
Stochastic Operator ◽  
Evolution Algebra ◽  
Upper Estimate

We consider the evolution algebra of a free population generated by an F-quadratic stochastic operator. We prove that this algebra is commutative, not associative and necessarily power-associative. We show that this algebra is not conservative, not stationary, not genetic and not train algebra, but it is a Banach algebra. The set of all derivations of the F-evolution algebra is described. We give necessary conditions for a state of the population to be a fixed point or a zero point of the F-quadratic stochastic operator which corresponds to the F-evolution algebra. We also establish upper estimate of the ?-limit set of the trajectory of the operator. For an F-evolution algebra of Volterra type we describe the full set of idempotent elements and the full set of absolute nilpotent elements.

scholarly journals Extension of holomorphic L2-functions with weighted growth conditions

1992 ◽  
Vol 126 ◽  
pp. 141-157 ◽  
Author(s):  
Klas Diederich ◽  
Gregor Herbort
Keyword(s):  
Fixed Point ◽  
Holomorphic Function ◽  
Lebesgue Measure ◽  
Complex Plane ◽  
Pseudoconvex Domain ◽  
Smooth Boundary ◽  
Small Neighborhood ◽  
Growth Conditions

In this article a new contribution to the following question is given: Let Ω ⊂ ⊂ Cn be a bounded pseudoconvex domain with C∞-smooth boundary, q ∈ ∂Ω a fixed point and H a k-dimensional affine complex plane such that q ∈ H and H intersects ∂Ω at q transversally. Let U be a suitably small neighborhood of q, and denote by r a C∞-defining function of Ω on U. Under which conditions on ∂Ω near q is it possible to find an exponent η>0 > 0 such that every holomorphic function f on Ω′ = H ∩Ω∩ U withwhere dλ′ denotes the Lebesgue-measure on H, can be extended to a holomorphic function ^f on Ω ∩ U such that even

scholarly journals Recurrence and fixed points of surface homeomorphisms

1988 ◽  
Vol 8 (8) ◽  
pp. 99-107 ◽  
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lebesgue Measure ◽  
Invariant Chain ◽  
Transitive Set ◽  
Surface Homeomorphisms ◽  
A Chain ◽  
Area Preserving ◽  
Twist Condition

AbstractWe prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L, then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.

scholarly journals Existence and Extremal Solution of Boundary Value Problem for Nonlinear Hybrid Fractional Differential Equation in Banach Algebras

2020 ◽  
Vol 1 ◽  
pp. 23-32
Author(s):  
B.D. Karande ◽  
Pravin M. More
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Boundary Value Problem ◽  
Fractional Differential Equation ◽  
Boundary Value ◽  
Banach Algebras ◽  
Extremal Solution ◽  
Hybrid Fixed Point Theorem ◽  
Fractional Differential ◽  
Nonlinear Hybrid

In this work we study the existence and extremal solution for the boundary value problem of the nonlinear hybrid fractional differential equation by using hybrid fixed point theorem in Banach Algebra due to Dhage’s theorem.

Fibonacci fixed point of renormalization

2000 ◽  
Vol 20 (5) ◽  
pp. 1287-1317 ◽  
Author(s):  
XAVIER BUFF
Keyword(s):  
Fixed Point ◽  
Lebesgue Measure ◽  
Julia Set ◽  
Positive Lebesgue Measure ◽  
Quadratic Polynomials ◽  
Polynomial Maps ◽  
Mathematical Sciences

To study the geometry of a Fibonacci map $f$ of even degree $\ell\geq 4$, Lyubich (Dynamics of quadratic polynomials, I–II. Acta Mathematica178 (1997), 185–297) defined a notion of generalized renormalization, so that $f$ is renormalizable infinitely many times. van Strien and Nowicki (Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps. Preprint, Institute for Mathematical Sciences, SUNY at Stony Brook, 1994) proved that the generalized renormalizations ${\cal R}^{\circ n}(f)$ converge to a cycle $\{f_1,f_2\}$ of order two depending only on $\ell$. We will explicitly relate $f_1$ and $f_2$ and show the convergence in shape of Fibonacci puzzle pieces to the Julia set of an appropriate polynomial-like map.

scholarly journals A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds

1987 ◽  
Vol 7 (3) ◽  
pp. 463-479 ◽  
Author(s):  
Stephan Pelikan ◽  
Edward E. Slaminka
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Fixed Point Index ◽  
Compact Sets ◽  
Point Index ◽  
Area Preserving Maps ◽  
Area Preserving

AbstractThe study of area preserving maps of manifolds has an extensive history in the theory of dynamical systems. One interest has been in the behaviour of such maps near an isolated fixed point. In 1974 Carl Simon proved the existence of an upper bound for the index of an isolated fixed point for Ck area preserving diffeomorphisms of a surface. We extend his result to homeomorphisms of an orientable two manifold. The proof utilizes the notion of free modification, developed by Morton Brown, and enlarges the scope of the problem to the consideration of ‘nice’ measures, i.e. uniformly equivalent to Lebesgue measure on compact sets. By suitably modifying the homeomorphism and the measure, we obtain the following theorem.

scholarly journals Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour

1982 ◽  
Vol 2 (3-4) ◽  
pp. 439-463 ◽  
Author(s):  
Feliks Przytycki
Keyword(s):  
Fixed Point ◽  
Lebesgue Measure ◽  
Bifurcation Parameter ◽  
Two Dimensional ◽  
Open Area ◽  
Dimensional Torus ◽  
Characteristic Exponents ◽  
Almost Everywhere ◽  
Lyapunov Characteristic Exponents ◽  
Real Analytic

AbstractWe find very simple examples of C∞-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.

Solar Internal Rotation and Dynamo Waves: A Two-Dimensional Asymptotic Solution in the Convection Zone

2000 ◽  
Vol 179 ◽  
pp. 379-380
Author(s):  
Gaetano Belvedere ◽  
Kirill Kuzanyan ◽  
Dmitry Sokoloff
Keyword(s):  
Magnetic Field ◽  
Asymptotic Solution ◽  
Internal Rotation ◽  
Convection Zone ◽  
General Idea ◽  
Dynamo Model ◽  
Axially Symmetric ◽  
Dynamo Action ◽  
Regeneration Rate ◽  
Dynamo Waves

Extended abstractHere we outline how asymptotic models may contribute to the investigation of mean field dynamos applied to the solar convective zone. We calculate here a spatial 2-D structure of the mean magnetic field, adopting real profiles of the solar internal rotation (the Ω-effect) and an extended prescription of the turbulent α-effect. In our model assumptions we do not prescribe any meridional flow that might seriously affect the resulting generated magnetic fields. We do not assume apriori any region or layer as a preferred site for the dynamo action (such as the overshoot zone), but the location of the α- and Ω-effects results in the propagation of dynamo waves deep in the convection zone. We consider an axially symmetric magnetic field dynamo model in a differentially rotating spherical shell. The main assumption, when using asymptotic WKB methods, is that the absolute value of the dynamo number (regeneration rate) |D| is large, i.e., the spatial scale of the solution is small. Following the general idea of an asymptotic solution for dynamo waves (e.g., Kuzanyan & Sokoloff 1995), we search for a solution in the form of a power series with respect to the small parameter |D|–1/3(short wavelength scale). This solution is of the order of magnitude of exp(i|D|1/3S), where S is a scalar function of position.

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