scholarly journals Hausdorff dimension of the nondifferentiability set of a convex function

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5827-5831
Author(s):  
Reza Mirzaie
Keyword(s):  
Riemannian Manifold ◽  
Hausdorff Dimension ◽  
Convex Function ◽  
Open Subset ◽  
Upper Bound ◽  
Continuous Convex Function

We find an upper bound for the Hausdorff dimension of the nondifferentiability set of a continuous convex function defined on a Riemannian manifold. As an application, we show that the boundary of a convex open subset of Rn, n ? 2, has Hausdorff dimension at most n - 2.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

A MEAN–VARIANCE BOUND FOR A THREE-PIECE LINEAR FUNCTION

2007 ◽  
Vol 21 (4) ◽  
pp. 611-621 ◽  
Author(s):  
Karthik Natarajan ◽  
Zhou Linyi
Keyword(s):  
Convex Function ◽  
Closed Form ◽  
Linear Function ◽  
Upper Bound ◽  
Random Variable ◽  
Expected Value ◽  
Variance Bound ◽  
The Mean ◽  
Mean Variance

In this article, we derive a tight closed-form upper bound on the expected value of a three-piece linear convex function E[max(0, X, mX − z)] given the mean μ and the variance σ2 of the random variable X. The bound is an extension of the well-known mean–variance bound for E[max(0, X)]. An application of the bound to price the strangle option in finance is provided.

Some results on harmonic and bi-harmonic maps

2017 ◽  
Vol 14 (07) ◽  
pp. 1750098 ◽  
Author(s):  
Ahmed Mohammed Cherif
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Convex Function ◽  
Open Problem ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Complete Riemannian Manifold ◽  
Strongly Convex Function ◽  
Strongly Convex ◽  
Homothetic Vector

In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary [Formula: see text] to Riemannian manifold [Formula: see text] is necessarily constant with [Formula: see text] admitting a strongly convex function [Formula: see text] such that [Formula: see text] is a Jacobi-type vector field (or [Formula: see text] admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.

MIXED MULTIFRACTAL ANALYSIS FOR FUNCTIONS: GENERAL UPPER BOUND AND OPTIMAL RESULTS FOR VECTORS OF SELF-SIMILAR OR QUASI-SELF-SIMILAR OF FUNCTIONS AND THEIR SUPERPOSITIONS

Fractals ◽  
2016 ◽  
Vol 24 (04) ◽  
pp. 1650039 ◽  
Author(s):  
MOURAD BEN SLIMANE ◽  
ANOUAR BEN MABROUK ◽  
JAMIL AOUIDI
Keyword(s):  
Hausdorff Dimension ◽  
Upper Bound ◽  
Multifractal Analysis ◽  
Hölder Functions ◽  
Single Function ◽  
Cascade Models ◽  
Self Similar ◽  
Wavelet Decompositions ◽  
Vector Formula ◽  
Set Of Points

Mixed multifractal analysis for functions studies the Hölder pointwise behavior of more than one single function. For a vector [Formula: see text] of [Formula: see text] functions, with [Formula: see text], we are interested in the mixed Hölder spectrum, which is the Hausdorff dimension of the set of points for which each function [Formula: see text] has exactly a given value [Formula: see text] of pointwise Hölder regularity. We will conjecture a formula which relates the mixed Hölder spectrum to some mixed averaged wavelet quantities of [Formula: see text]. We will prove an upper bound valid for any vector of uniform Hölder functions. Then we will prove the validity of the conjecture for self-similar vectors of functions, quasi-self-similar vectors and their superpositions. These functions are written as the superposition of similar structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions.

scholarly journals Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

2020 ◽  
Vol 18 (1) ◽  
pp. 1-9
Author(s):  
Carlo Mariconda ◽  
Giulia Treu
Keyword(s):  
Convex Function ◽  
Calculus Of Variations ◽  
Open Subset ◽  
Lipschitz Function ◽  
Lavrentiev Phenomenon ◽  
Smooth Functions ◽  
Bounded Open Subset ◽  
Admissible Functions

Abstract We consider the classical functional of the Calculus of Variations of the form $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

More bounds on the expectation of a convex function of a random variable

1972 ◽  
Vol 9 (4) ◽  
pp. 803-812 ◽  
Author(s):  
Ben-Tal A. ◽  
E. Hochman
Keyword(s):  
Convex Function ◽  
Upper Bound ◽  
Random Variable ◽  
Independent Random Variables ◽  
Multivariate Distributions ◽  
Absolute Deviation ◽  
Additional Information ◽  
Wait And See ◽  
The Mean ◽  
Moment Spaces

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T).The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.

scholarly journals Another Aspect of Triangle Inequality

2011 ◽  
Vol 2011 ◽  
pp. 1-5
Author(s):  
Kichi-Suke Saito ◽  
Runling An ◽  
Hiroyasu Mizuguchi ◽  
Ken-Ichi Mitani
Keyword(s):  
Convex Function ◽  
Triangle Inequality ◽  
Unit Interval ◽  
Usual Sense ◽  
Continuous Convex Function

We introduce the notion of ψ-norm by considering the fact that an absolute normalized norm on C2 corresponds to a continuous convex function ψ on the unit interval [0,1] with some conditions. This is a generalization of the notion of q-norm introduced by Belbachir et al. (2006). Then we show that a ψ-norm is a norm in the usual sense.

scholarly journals Extension of Positive Currents with Special Properties of Monge-Ampère Operators

2013 ◽  
Vol 113 (1) ◽  
pp. 108 ◽  
Author(s):  
Ahmad L. al Abdulaali
Keyword(s):  
Convex Function ◽  
Open Subset ◽  
Hausdorff Measure ◽  
Negative Current ◽  
Positive Currents ◽  
Monge Ampere

In this paper we study the extension of currents across small obstacles. Our main results are: 1) Let $A$ be a closed complete pluripolar subset of an open subset $\Omega$ of $\mathsf{C}^n$ and $T$ be a negative current of bidimension $(p,p)$ on $\Omega\setminus A$ such that $dd^{c}T\geq-S$ on $\Omega\setminus A$ for some positive plurisubharmonic current $S$ on $\Omega$. Assume that the Hausdorff measure $\mathscr{H}_{2p}(A\cap \overline{\operatorname{Supp} T})=0$. Then $\widetilde{T}$ exists. Furthermore, the current $R= \widetilde{dd^{c}T}-{dd}^{c} \widetilde{T}$ is negative supported in $A$. 2) Let $u$ be a positive strictly $k$-convex function on an open subset $\Omega$ of $\mathsf{C}^n$ and set $A=\{z\in\Omega:u(z)=0\}$. Let $T$ be a negative current of bidimension $(p,p)$ on $\Omega\setminus A$ such that $dd^{c}T\geq -S$ on $\Omega\setminus A$ for some positive plurisubharmonic (or $dd^{c}$-negative) current $S$ on $\Omega$. If $p\geq k+1$, then $\widetilde{T}$ exists. If $p\geq k+2$, $dd^{c}S\leq 0$ and $u$ of class $\mathscr{C}^{2}$, then $\widetilde{dd^{c}T}$ exists and $\widetilde{dd^{c}T}= dd^{c}\widetilde{T}$.

More bounds on the expectation of a convex function of a random variable

1972 ◽  
Vol 9 (04) ◽  
pp. 803-812 ◽  
Author(s):  
Ben-Tal A. ◽  
E. Hochman
Keyword(s):  
Convex Function ◽  
Upper Bound ◽  
Random Variable ◽  
Independent Random Variables ◽  
Multivariate Distributions ◽  
Absolute Deviation ◽  
Additional Information ◽  
Wait And See ◽  
The Mean ◽  
Moment Spaces

Jensen gave a lower bound to Eρ(T), where ρ is a convex function of the random vector T. Madansky has obtained an upper bound via the theory of moment spaces of multivariate distributions. In particular, Madansky's upper bound is given explicitly when the components of T are independent random variables. For this case, lower and upper bounds are obtained in the paper, which uses additional information on T rather than its mean (mainly its expected absolute deviation about the mean) and hence gets closer to Eρ(T). The importance of having improved bounds is illustrated through a nonlinear programming problem with stochastic objective function, known as the “wait and see” problem.

ESTIMATE OF THE HAUSDORFF MEASURE OF THE SIERPINSKI TRIANGLE

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 137-148
Author(s):  
PÉTER MÓRA
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Upper Bound ◽  
Open Problem ◽  
Dimensional Hausdorff Measure ◽  
Sierpinski Triangle ◽  
The One

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

Sign in / Sign up

Export Citation Format

Share Document