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Instability of the eikonal equation and shape from shading

Ian Barnes, Kewei Zhang (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In the shape from shading problem of computer vision one attempts to recover the three-dimensional shape of an object or landscape from the shading on a single image. Under the assumptions that the surface is dusty, distant, and illuminated only from above, the problem reduces to that of solving the eikonal equation |Du|=f on a domain in ℝ 2 . Despite various existence and uniqueness theorems for smooth solutions, we show that this problem is unstable, which is catastrophic for general numerical algorithms. ...

Integrable functions for the Bernoulli measures of rank 1

Hamadoun Maïga (2010)

Annales mathématiques Blaise Pascal

In this paper, following the p -adic integration theory worked out by A. F. Monna and T. A. Springer [4, 5] and generalized by A. C. M. van Rooij and W. H. Schikhof [6, 7] for the spaces which are not σ -compacts, we study the class of integrable p -adic functions with respect to Bernoulli measures of rank 1 . Among these measures, we characterize those which are invertible and we give their inverse in the form of series.

Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type

Silvia I. Hartzstein, Beatriz E. Viviani (2002)

Commentationes Mathematicae Universitatis Carolinae

In the setting of spaces of homogeneous-type, we define the Integral, I φ , and Derivative, D φ , operators of order φ , where φ is a function of positive lower type and upper type less than 1 , and show that I φ and D φ are bounded from Lipschitz spaces Λ ξ to Λ ξ φ and Λ ξ / φ respectively, with suitable restrictions on the quasi-increasing function ξ in each case. We also prove that I φ and D φ are bounded from the generalized Besov B ˙ p ψ , q , with 1 ≤ p , q < ∞ , and Triebel-Lizorkin spaces F ˙ p ψ , q , with 1 < p , q < ∞ , of order ψ to those of order φ ψ and ψ / φ respectively,...

Integral inequalities involving generalized Erdélyi-Kober fractional integral operators

Dumitru Baleanu, Sunil Dutt Purohit, Jyotindra C. Prajapati (2016)

Open Mathematics

Using the generalized Erdélyi-Kober fractional integrals, an attempt is made to establish certain new fractional integral inequalities, related to the weighted version of the Chebyshev functional. The results given earlier by Purohit and Raina (2013) and Dahmani et al. (2011) are special cases of results obtained in present paper.

Integral of Complex-Valued Measurable Function

Keiko Narita, Noboru Endou, Yasunari Shidama (2008)

Formalized Mathematics

In this article, we formalized the notion of the integral of a complex-valued function considered as a sum of its real and imaginary parts. Then we defined the measurability and integrability in this context, and proved the linearity and several other basic properties of complex-valued measurable functions. The set of properties showed in this paper is based on [15], where the case of real-valued measurable functions is considered.MML identifier: MESFUN6C, version: 7.9.01 4.101.1015

Integral of Real-Valued Measurable Function 1

Yasunari Shidama, Noboru Endou (2006)

Formalized Mathematics

Based on [16], authors formalized the integral of an extended real valued measurable function in [12] before. However, the integral argued in [12] cannot be applied to real-valued functions unconditionally. Therefore, in this article we have formalized the integral of a real-value function.

Integral operators and weighted amalgams

C. Carton-Lebrun, H. Heinig, S. Hofmann (1994)

Studia Mathematica

For large classes of indices, we characterize the weights u, v for which the Hardy operator is bounded from ℓ q ̅ ( L v p ̅ ) into ℓ q ( L u p ) . For more general operators of Hardy type, norm inequalities are proved which extend to weighted amalgams known estimates in weighted L p -spaces. Amalgams of the form ℓ q ( L w p ) , 1 < p,q < ∞ , q ≠ p, w ∈ A p , are also considered and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator and local maximal operator in these spaces are obtained.

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