Displaying similar documents to “A discrepancy principle for Tikhonov regularization with approximately specified data”

On some properties of three-dimensional minimal sets in 4

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in 4 around a 𝕐 -point and the existence of a point of particular type of a Mumford-Shah minimal set in 4 , which is very close to a 𝕋 . This will give a local description of minimal sets of dimension 3 in 4 around a singular point and a property of Mumford-Shah minimal sets in 4 .

Zero-set property of o-minimal indefinitely Peano differentiable functions

Andreas Fischer (2008)

Annales Polonici Mathematici

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Given an o-minimal expansion ℳ of a real closed field R which is not polynomially bounded. Let denote the definable indefinitely Peano differentiable functions. If we further assume that ℳ admits cell decomposition, each definable closed subset A of Rⁿ is the zero-set of a function f:Rⁿ → R. This implies approximation of definable continuous functions and gluing of functions defined on closed definable sets.

Exceptional sets in Waring's problem: two squares and s biquadrates

Lilu Zhao (2014)

Acta Arithmetica

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Let R s ( n ) denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When s = 3 or 4, it is established that the anticipated asymptotic formula for R s ( n ) holds for all n X with at most O ( X ( 9 - 2 s ) / 8 + ε ) exceptions.

Renormings of c 0 and the minimal displacement problem

Łukasz Piasecki (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this paper is to show that for every Banach space ( X , · ) containing asymptotically isometric copy of the space c 0 there is a bounded, closed and convex set C X with the Chebyshev radius r ( C ) = 1 such that for every k 1 there exists a k -contractive mapping T : C C with x - T x > 1 1 / k for any x C .

Linear maps preserving A -unitary operators

Abdellatif Chahbi, Samir Kabbaj, Ahmed Charifi (2016)

Mathematica Bohemica

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Let be a complex Hilbert space, A a positive operator with closed range in ( ) and A ( ) the sub-algebra of ( ) of all A -self-adjoint operators. Assume φ : A ( ) onto itself is a linear continuous map. This paper shows that if φ preserves A -unitary operators such that φ ( I ) = P then ψ defined by ψ ( T ) = P φ ( P T ) is a homomorphism or an anti-homomorphism and ψ ( T ) = ψ ( T ) for all T A ( ) , where P = A + A and A + is the Moore-Penrose inverse of A . A similar result is also true if φ preserves A -quasi-unitary operators in both directions such that there...

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

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Let L ( H ) denote the algebra of operators on a complex infinite dimensional Hilbert space H . For A , B L ( H ) , the generalized derivation δ A , B and the elementary operator Δ A , B are defined by δ A , B ( X ) = A X - X B and Δ A , B ( X ) = A X B - X for all X L ( H ) . In this paper, we exhibit pairs ( A , B ) of operators such that the range-kernel orthogonality of δ A , B holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of Δ A , B with respect to the wider class of unitarily invariant...

Best constants for the isoperimetric inequality in quantitative form

Marco Cicalese, Gian Paolo Leonardi (2013)

Journal of the European Mathematical Society

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We prove some results in the context of isoperimetric inequalities with quantitative terms. In the 2 -dimensional case, our main contribution is a method for determining the optimal coefficients c 1 , ... , c m in the inequality δ P ( E ) k = 1 m c k α ( E ) k + o ( α ( E ) m ) , valid for each Borel set E with positive and finite area, with δ P ( E ) and α ( E ) being, respectively, the 𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡 and the 𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦 of E . In n dimensions, besides proving existence and regularity properties of minimizers for a wide class of 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠 including the lower semicontinuous extension of δ P ( E ) α ( E ) 2 , we...

H calculus and dilatations

Andreas M. Fröhlich, Lutz Weis (2006)

Bulletin de la Société Mathématique de France

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We characterise the boundedness of the H calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if - A generates a bounded analytic C 0 semigroup ( T t ) on a UMD space, then the H calculus of A is bounded if and only if ( T t ) has a dilation to a bounded group on L 2 ( [ 0 , 1 ] , X ) . This generalises a Hilbert space result of C.LeMerdy. If X is an L p space we can choose another L p space in place of L 2 ( [ 0 , 1 ] , X ) .

Product property for capacities in N

Mirosław Baran, Leokadia Bialas-Ciez (2012)

Annales Polonici Mathematici

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The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the L-capacity, has the following product property: C ν ( E × E ) = m i n ( C ν ( E ) , C ν ( E ) ) , where E j and ν j are respectively a compact set and a norm in N j (j = 1,2), and ν is a norm in N + N , ν = ν₁⊕ₚ ν₂ with some 1 ≤ p ≤ ∞. For a convex subset E of N , denote by C(E) the standard L-capacity and by ω E the minimal width of E, that is, the minimal Euclidean distance between two supporting hyperplanes...

Positivity and anti-maximum principles for elliptic operators with mixed boundary conditions

Catherine Bandle, Joachim von Below, Wolfgang Reichel (2008)

Journal of the European Mathematical Society

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We consider linear elliptic equations - Δ u + q ( x ) u = λ u + f in bounded Lipschitz domains D N with mixed boundary conditions u / n = σ ( x ) λ u + g on D . The main feature of this boundary value problem is the appearance of λ both in the equation and in the boundary condition. In general we make no assumption on the sign of the coefficient σ ( x ) . We study positivity principles and anti-maximum principles. One of our main results states that if σ is somewhere negative, q 0 and D q ( x ) d x > 0 then there exist two eigenvalues λ - 1 , λ 1 such the positivity...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set Ω in n (or in a Riemannian manifold) and a partition of Ω by k open sets D j , we consider the quantity 𝚖𝚊𝚡 j λ ( D j ) where λ ( D j ) is the ground state energy of the Dirichlet realization of the Laplacian in D j . If we denote by k ( Ω ) the infimum over all the k -partitions of 𝚖𝚊𝚡 j λ ( D j ) , a minimal k -partition is then a partition which realizes the infimum. When k = 2 , we find the two nodal domains of a second eigenfunction, but the analysis of higher k ’s is non trivial and quite interesting. In this...

Moser's Inequality for a class of integral operators

Finbarr Holland, David Walsh (1995)

Studia Mathematica

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Let 1 < p < ∞, q = p/(p-1) and for f L p ( 0 , ) define F ( x ) = ( 1 / x ) ʃ 0 x f ( t ) d t , x > 0. Moser’s Inequality states that there is a constant C p such that s u p a 1 s u p f B p ʃ 0 e x p [ a x q | F ( x ) | q - x ] d x = C p where B p is the unit ball of L p . Moreover, the value a = 1 is sharp. We observe that F = K 1 f where the integral operator K 1 has a simple kernel K. We consider the question of for what kernels K(t,x), 0 ≤ t, x < ∞, this result can be extended, and proceed to discuss this when K is non-negative and homogeneous of degree -1. A sufficient condition on K is found for...

A localization property for B p q s and F p q s spaces

Hans Triebel (1994)

Studia Mathematica

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Let f j = k a k f ( 2 j + 1 x - 2 k ) , where the sum is taken over the lattice of all points k in n having integer-valued components, j∈ℕ and a k . Let A p q s be either B p q s or F p q s (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on n . The aim of the paper is to clarify under what conditions f j | A p q s is equivalent to 2 j ( s - n / p ) ( k | a k | p ) 1 / p f | A p q s .

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

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Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

On the real X -ranks of points of n ( ) with respect to a real variety X n

Edoardo Ballico (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let  X n be an integral and non-degenerate m -dimensional variety defined over . For any P n ( ) the real X -rank r X , ( P ) is the minimal cardinality of S X ( ) such that P S . Here we extend to the real case an upper bound for the X -rank due to Landsberg and Teitler.

Computing the greatest 𝐗 -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector x is said to be an eigenvector of a square max-min matrix A if A x = x . An eigenvector x of A is called the greatest 𝐗 -eigenvector of A if x 𝐗 = { x ; x ̲ x x ¯ } and y x for each eigenvector y 𝐗 . A max-min matrix A is called strongly 𝐗 -robust if the orbit x , A x , A 2 x , reaches the greatest 𝐗 -eigenvector with any starting vector of 𝐗 . We suggest an O ( n 3 ) algorithm for computing the greatest 𝐗 -eigenvector of A and study the strong 𝐗 -robustness. The necessary and sufficient conditions for strong 𝐗 -robustness are introduced...