Mixed curvature measures and a translative integral formula

1996 ◽  
Vol 28 (2) ◽  
pp. 341-341 ◽  
Author(s):  
Jan Rataj
Keyword(s):  
Integral Formula ◽  
Convex Bodies ◽  
Curvature Measure ◽  
Positive Reach ◽  
Rectifiable Currents ◽  
Curvature Measures ◽  
Dimension 2 ◽  
Image Position ◽  
Almost All ◽  
Sets Of Positive Reach

Let X, Y be two sets of positive reach in ℝd. The translative integral formula says that, for 0 ≦ k ≦ d − 1 and bounded Borel subsets A, B ε ℝd, where is the curvature measure (of order k) of X and is the mixed curvature measure of the sets X, Y and order r, S [1]. The mixed curvature measures are introduced by means of rectifiable currents, which leads to a relatively simple proof of (1). The proof needs an additional assumption on X, Y assuring that also reach (X ∩ Yz) > 0 for almost all z. This assumption is satisfied automatically for convex bodies, in dimension 2, or for sets with a sufficiently smooth boundary. Using the additivity of mixed curvature measures, (1) can be extended to unions of sets of positive reach.

Mixed curvature measures and a translative integral formula

1996 ◽  
Vol 28 (02) ◽  
pp. 341
Author(s):  
Jan Rataj
Keyword(s):  
Integral Formula ◽  
Convex Bodies ◽  
Curvature Measure ◽  
Positive Reach ◽  
Rectifiable Currents ◽  
Curvature Measures ◽  
Dimension 2 ◽  
Image Position ◽  
Almost All ◽  
Sets Of Positive Reach

Let X, Y be two sets of positive reach in ℝ d . The translative integral formula says that, for 0 ≦ k ≦ d − 1 and bounded Borel subsets A, B ε ℝ d , where is the curvature measure (of order k) of X and is the mixed curvature measure of the sets X, Y and order r, S [1]. The mixed curvature measures are introduced by means of rectifiable currents, which leads to a relatively simple proof of (1). The proof needs an additional assumption on X, Y assuring that also reach (X ∩ Yz ) > 0 for almost all z. This assumption is satisfied automatically for convex bodies, in dimension 2, or for sets with a sufficiently smooth boundary. Using the additivity of mixed curvature measures, (1) can be extended to unions of sets of positive reach.

Estimates of Zeros of a Polynomial

1962 ◽  
Vol 58 (2) ◽  
pp. 229-234 ◽  
Author(s):  
L. Mirsky
Keyword(s):  
Integral Formula ◽  
Transcendental Numbers ◽  
Image Position ◽  
Complex Coefficients

Throughout this note we shall consider a fixed polynomial with complex coefficients and of degree n ≥ 2. Its zeros will be denoted by ξ1, ξ2, …, ξn where the numbering is such that Making use of Jensen's integral formula, Mahler (4) showed that, for l ≥ k < n, A slightly weaker result had been established by Feldman in an earlier publication (2). Mahler's inequality (1) is of importance in the study of transcendental numbers, and our first object is to sharpen his bound by proving the following result.

Analytic functions of finite dimensional linear transformations

1959 ◽  
Vol 55 (1) ◽  
pp. 51-61 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Power Series ◽  
Integral Formula ◽  
Characteristic Root ◽  
Interpolation Formula ◽  
General Function ◽  
Linear Transformations ◽  
Multiple Characteristic ◽  
Characteristic Roots ◽  
Image Position ◽  
Necessary And Sufficient

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.

scholarly journals Subspace concentration of dual curvature measures of symmetric convex bodies

2018 ◽  
Vol 109 (3) ◽  
pp. 411-429 ◽  
Author(s):  
Károly J. Böröczky ◽  
Martin Henk ◽  
Hannes Pollehn
Keyword(s):  
Convex Bodies ◽  
Symmetric Convex ◽  
Curvature Measures

Operateur de Stokes Dans des Espaces de Sobolev a Poids sur des Domaines Anguleux

1982 ◽  
Vol 34 (4) ◽  
pp. 853-882 ◽  
Author(s):  
Monique Dauge
Keyword(s):  
Dimension 2 ◽  
Image Position

Rappelons que l'opérateur de Stokes associe au couple vitesse-pression (, p) le couple force-divergence (, g) par:Nous nous plaçons ici en dimension 2. . Voici l'écriture matricielle de l'opérateur:ce que nous noterons: .Ce système est elliptique au sens de [1] sur un domaine Ω avec les conditions complémentaires: (Dirichlet); c'est donc ce probléme que nous étudions ici.

On a Stein And Weiss Property of the Conjugate Function

1990 ◽  
Vol 42 (1) ◽  
pp. 109-125
Author(s):  
Nakhlé Asmar
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Conjugate Function ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Character Group ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Almost All

(1.1) The conjugate function on locally compact abelian groups. Let G be a locally compact abelian group with character group Ĝ. Let μ denote a Haar measure on G such that μ(G) = 1 if G is compact. (Unless stated otherwise, all the measures referred to below are Haar measures on the underlying groups.) Suppose that Ĝ contains a measurable order P: P + P ⊆P; PU(-P)= Ĝ; and P⋂(—P) =﹛0﹜. For ƒ in ℒ2(G), the conjugate function of f (with respect to the order P) is the function whose Fourier transform satisfies the identity for almost all χ in Ĝ, where sgnP(χ)= 0, 1, or —1, according as χ =0, χ ∈ P\\﹛0﹜, or χ ∈ (—P)\﹛0﹜.

scholarly journals An infinite integral formula

1939 ◽  
Vol 6 (2) ◽  
pp. 75-77
Author(s):  
C. G. Lambe
Keyword(s):  
Integral Formula ◽  
The Other ◽  
Infinite Integral ◽  
Image Position

§ 1. The object of this note is to discuss the formulathe integral being supposed convergent for certain ranges of values of x and z. The contour is such that the poles of Γ(– s)lie to its right and the other poles of the integrand to its left. It will be seen that all the Pincherle-Mellin-Barnes integrals are particular cases of this formula.

A non-Markovian birth process with logarithmic growth

1975 ◽  
Vol 12 (04) ◽  
pp. 673-683
Author(s):  
G. R. Grimmett
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Birth Process ◽  
Image Position ◽  
Almost All ◽  
Increasing Function ◽  
Logarithmic Growth

I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 &lt; λ &lt; 1, is an example of such a process.

Numbers badly approximate by fractions with prime denominator

1995 ◽  
Vol 118 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Continued Fraction ◽  
Arithmetic Progressions ◽  
The Other ◽  
Strong Result ◽  
Infinitely Many Solutions ◽  
Continued Fraction Expansion ◽  
Primes In Arithmetic Progressions ◽  
Prime Variable ◽  
Image Position ◽  
Almost All

We write ‖x‖ to denote the least distance from x to an integer, and write p for a prime variable. Duffin and Schaeffer [l] showed that for almost all real α the inequalityhas infinitely many solutions if and only ifdiverges. Thus f(x) = (x log log (10x))−1 is a suitable choice to obtain infinitely many solutions for almost all α. It has been shown [2] that for all real irrational α there are infinitely many solutions to (1) with f(p) = p−/13. We will show elsewhere that the exponent can be increased to 7/22. A very strong result on primes in arithmetic progressions (far stronger than anything within reach at the present time) would lead to an improvement on this result. On the other hand, it is very easy to find irrational a such that no convergent to its continued fraction expansion has prime denominator (for example (45– √10)/186 does not even have a square-free denominator in its continued fraction expansion, since the denominators are alternately divisible by 4 and 9).

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