Variation of Calderón–Zygmund operators with matrix weight

2020 ◽  
pp. 2050062
Author(s):  
Xuan Thinh Duong ◽  
Ji Li ◽  
Dongyong Yang
Keyword(s):  
Convex Body ◽  
Modulus Of Continuity ◽  
Weak Type ◽  
Type Estimate ◽  
Variation Formula ◽  
Dini Condition ◽  
Zygmund Operator ◽  
The Matrix ◽  
Matrix Formula

Let [Formula: see text], [Formula: see text] and [Formula: see text] be a matrix [Formula: see text] weight. In this paper, we introduce a version of variation [Formula: see text] for matrix Calderón–Zygmund operators with modulus of continuity satisfying the Dini condition. We then obtain the [Formula: see text]-boundedness of [Formula: see text] with norm [Formula: see text] by first proving a sparse domination of the variation of the scalar Calderón–Zygmund operator, and then providing a convex body sparse domination of the variation of the matrix Calderón–Zygmund operator. The key step here is a weak type estimate of a local grand maximal truncated operator with respect to the scalar Calderón–Zygmund operator.

Weighted inequalities for some integral operators with rough kernels

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
María Riveros ◽  
Marta Urciuolo
Keyword(s):  
Weak Type ◽  
Integral Operators ◽  
Degree Zero ◽  
Weighted Inequalities ◽  
Rough Kernels ◽  
Type Estimate ◽  
Dini Condition ◽  
Weighted Bmo ◽  
Weak Type Estimates ◽  
Invertible Matrices

AbstractIn this paper we study integral operators with kernels $$K(x,y) = k_1 (x - A_1 y) \cdots k_m \left( {x - A_m y} \right),$$ $$k_i \left( x \right) = {{\Omega _i \left( x \right)} \mathord{\left/ {\vphantom {{\Omega _i \left( x \right)} {\left| x \right|}}} \right. \kern-\nulldelimiterspace} {\left| x \right|}}^{{n \mathord{\left/ {\vphantom {n {q_i }}} \right. \kern-\nulldelimiterspace} {q_i }}}$$ where Ωi: ℝn → ℝ are homogeneous functions of degree zero, satisfying a size and a Dini condition, A i are certain invertible matrices, and n/q 1 +…+n/q m = n−α, 0 ≤ α < n. We obtain the appropriate weighted L p-L q estimate, the weighted BMO and weak type estimates for certain weights in A(p, q). We also give a Coifman type estimate for these operators.

Complex Chi-Squares: Matrix Notation and Calculation

1972 ◽  
Vol 30 (3) ◽  
pp. 743-746 ◽  
Author(s):  
Edward F. Gocka
Keyword(s):  
Computational Algorithms ◽  
Behavioral Sciences ◽  
Matrix Notation ◽  
Standard Formula ◽  
The Matrix ◽  
Matrix Formula

A matrix formula available for the calculation of complex chi-squares allows several computational variations, each of which requires fewer steps than the standard formula. However, neither the matrix formula nor the associated computational algorithms have been given adequate exposure in statistical texts for the behavioral sciences. This paper reintroduces the formula, expands the notation, and shows how several computational variations can be derived.

Rank, trace, eigenvalues and norms of a structured matrix

2019 ◽  
Vol 12 (06) ◽  
pp. 2040015
Author(s):  
Ahmet İpek
Keyword(s):  
Real Sequence ◽  
Simple Property ◽  
Structured Matrix ◽  
The Matrix ◽  
Matrix Formula ◽  
Appl Math ◽  
Real Matrices

The paper deals with rank, trace, eigenvalues and norms of the matrix [Formula: see text], where [Formula: see text] are ith components of any real sequence [Formula: see text]. A result in this paper is that the Euclidean and spectral norms of the matrix [Formula: see text] is [Formula: see text]. This is a generalization of the main result by Solak [Appl. Math. Comput. 232 (2014) 919–921], with the proof based on a simple property of norms of real matrices.

PERTURBATIVE SOLUTIONS OF DIFFERENTIAL EQUATIONS IN LIE GROUPS

2005 ◽  
Vol 02 (01) ◽  
pp. 111-125 ◽  
Author(s):  
PAOLO ANIELLO
Keyword(s):  
Differential Equations ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Similarity Transformation ◽  
Perturbative Expansion ◽  
Analytic Perturbation ◽  
The Matrix ◽  
Matrix Formula

We show that, given a matrix Lie group [Formula: see text] and its Lie algebra [Formula: see text], a 1-parameter subgroup of [Formula: see text] whose generator is the sum of an unperturbed matrix Â0 and an analytic perturbation Â♢(λ) can be mapped — under suitable conditions — by a similarity transformation depending analytically on the perturbative parameter λ, onto a 1-parameter subgroup of [Formula: see text] generated by a matrix [Formula: see text] belonging to the centralizer of Â0 in [Formula: see text]. Both the similarity transformation and the matrix [Formula: see text] can be determined perturbatively, hence allowing a very convenient perturbative expansion of the original 1-parameter subgroup.

On the computation of the true‐amplitude weighting functions

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1648-1651 ◽  
Author(s):  
Jianguo Sun ◽  
Dirk Gajewski
Keyword(s):  
Reflection Coefficient ◽  
Weighting Function ◽  
The Other ◽  
Phase Factor ◽  
Weighting Functions ◽  
Kirchhoff Type ◽  
New Development ◽  
The Matrix ◽  
Matrix Formula

True‐amplitude migration is a new development of Kirchhoff‐type migration. By using a proper weighting function, called true‐amplitude weighting function, in the modified diffraction‐stack operator, one can obtain a reconstructed source pulse proportional to the reflection coefficient of the target reflector. According to Schleicher et al. (1993), the true‐amplitude weighting function is complex and consists of two parts. One part is the module of the complex weighting function, given by means of a 2 × 2 ray transformation submatrix called [Formula: see text]. The other is the phase factor of the complex weighting function, determined by the number of ray‐branch caustics. Thus, to obtain the true‐amplitude weighting function, one should compute the matrix [Formula: see text] as well as the number of ray‐branch caustics.

Approximation by max-product sampling Kantorovich operators with generalized kernels

2019 ◽  
pp. 1-26 ◽  
Author(s):  
Lucian Coroianu ◽  
Danilo Costarelli ◽  
Sorin G. Gal ◽  
Gianluca Vinti
Keyword(s):  
Uniform Convergence ◽  
Real Axis ◽  
Modulus Of Continuity ◽  
Higher Dimensions ◽  
Jackson Type ◽  
Convergence Properties ◽  
Type Estimate ◽  
Quantitative Estimates ◽  
Uniform Modulus Of Continuity ◽  
Kantorovich Operators

In a recent paper, for max-product sampling operators based on general kernels with bounded generalized absolute moments, we have obtained several pointwise and uniform convergence properties on bounded intervals or on the whole real axis, including a Jackson-type estimate in terms of the first uniform modulus of continuity. In this paper, first, we prove that for the Kantorovich variants of these max-product sampling operators, under the same assumptions on the kernels, these convergence properties remain valid. Here, we also establish the [Formula: see text] convergence, and quantitative estimates with respect to the [Formula: see text] norm, [Formula: see text]-functionals and [Formula: see text]-modulus of continuity as well. The results are tested on several examples of kernels and possible extensions to higher dimensions are suggested.

scholarly journals Polar degrees and closest points in codimension two

2019 ◽  
Vol 18 (05) ◽  
pp. 1950095
Author(s):  
Martin Helmer ◽  
Bernt Ivar Utstøl Nødland
Keyword(s):  
Toric Variety ◽  
Euclidean Distance ◽  
Integer Matrix ◽  
Codimension Two ◽  
Geometric Methods ◽  
The Matrix ◽  
Matrix Formula ◽  
Combinatorial Methods

Suppose that [Formula: see text] is a toric variety of codimension two defined by an [Formula: see text] integer matrix [Formula: see text], and let [Formula: see text] be a Gale dual of [Formula: see text]. In this paper, we compute the Euclidean distance degree and polar degrees of [Formula: see text] (along with other associated invariants) combinatorially working from the matrix [Formula: see text]. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than the analogous combinatorial methods using the matrix [Formula: see text] in the codimension two case.

On the relation between torsion submodule and Fitting ideals

2021 ◽  
pp. 2250173
Author(s):  
S. Hadjirezaei
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Principal Ideal ◽  
Index Set ◽  
Constructive Description ◽  
Regular Ideal ◽  
Fitting Ideal ◽  
The Matrix ◽  
Matrix Formula ◽  
The Ideal

Let [Formula: see text] be a commutative ring and let [Formula: see text] be a submodule of [Formula: see text] which consists of columns of a matrix [Formula: see text] with [Formula: see text] for all [Formula: see text], [Formula: see text], where [Formula: see text] is an index set. For every [Formula: see text], let I[Formula: see text] be the ideal generated by subdeterminants of size [Formula: see text] of the matrix [Formula: see text]. Let [Formula: see text]. In this paper, we obtain a constructive description of [Formula: see text] and we show that when [Formula: see text] is a local ring, [Formula: see text] is free of rank [Formula: see text] if and only if I[Formula: see text] is a principal regular ideal, for some [Formula: see text]. This improves a lemma of Lipman which asserts that, if [Formula: see text] is the [Formula: see text]th Fitting ideal of [Formula: see text] then [Formula: see text] is a regular principal ideal if and only if [Formula: see text] is finitely generated free and [Formula: see text] is free of rank [Formula: see text]

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