Displaying similar documents to “On some identities involving spherical means”

On some identities involving spherical means

Gianfranco Cimmino (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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For every positive definite quadratic form in n variables the reciprocal of the square root of the discriminant is equal to the arithmetic mean of the values assumed by the form on the n - 1 sphere centered at 0 and with radius 1 raised to the ( - n 2 )-th. power. Various consequences are deduced from this, in particular a simplification of some calculations from which one obtains the possibility of solving linear systems using spherical means rather than determinants.

The number of minimum points of a positive quadratic form

G. L. Watson

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CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...

Nodal curves in 3 ( )

Edoardo Ballico, Paolo Oliverio (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Siano d , g , t interi con 0 t g ; se esiste in 3 ( ) una curva connessa, non singolare di grado d e genere g , allora esiste in 3 ( ) una curva irriducibile di grado d , genere aritmetico g e t nodi.

An approximation property of quadratic irrationals

Takao Komatsu (2002)

Bulletin de la Société Mathématique de France

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Let α > 1 be irrational. Several authors studied the numbers m ( α ) = inf { | y | : y Λ m , y 0 } , where m is a positive integer and Λ m denotes the set of all real numbers of the form y = ϵ 0 α n + ϵ 1 α n - 1 + + ϵ n - 1 α + ϵ n with restricted integer coefficients | ϵ i | m . The value of 1 ( α ) was determined for many particular Pisot numbers and m ( α ) for the golden number. In this paper the value of  m ( α ) is determined for irrational numbers  α , satisfying α 2 = a α ± 1 with a positive integer a .

Why Jordan algebras are natural in statistics: quadratic regression implies Wishart distributions

G. Letac, J. Wesołowski (2011)

Bulletin de la Société Mathématique de France

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If the space 𝒬 of quadratic forms in n is splitted in a direct sum 𝒬 1 ... 𝒬 k and if X and Y are independent random variables of n , assume that there exist a real number a such that E ( X | X + Y ) = a ( X + Y ) and real distinct numbers b 1 , . . . , b k such that E ( q ( X ) | X + Y ) = b i q ( X + Y ) for any q in 𝒬 i . We prove that this happens only when k = 2 , when n can be structured in a Euclidean Jordan algebra and when X and Y have Wishart distributions corresponding to this structure.

L p -improving properties of certain singular measures on the Heisenberg group

Pablo Rocha (2022)

Mathematica Bohemica

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Let μ A be the singular measure on the Heisenberg group n supported on the graph of the quadratic function ϕ ( y ) = y t A y , where A is a 2 n × 2 n real symmetric matrix. If det ( 2 A ± J ) 0 , we prove that the operator of convolution by μ A on the right is bounded from L ( 2 n + 2 ) ( 2 n + 1 ) ( n ) to L 2 n + 2 ( n ) . We also study the type set of the measures d ν γ ( y , s ) = η ( y ) | y | - γ d μ A ( y , s ) , for 0 γ < 2 n , where η is a cut-off function around the origin on 2 n . Moreover, for γ = 0 we characterize the type set of ν 0 .

Repdigits in generalized Pell sequences

Jhon J. Bravo, Jose L. Herrera (2020)

Archivum Mathematicum

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For an integer k 2 , let ( n ) n be the k - generalized Pell sequence which starts with 0 , ... , 0 , 1 ( k terms) and each term afterwards is given by the linear recurrence n = 2 n - 1 + n - 2 + + n - k . In this paper, we find all k -generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence ( P n ( 2 ) ) n . ...

A Note on VLO Functions

Francesca Angrisani, Giacomo Ascione (2018)

Rendiconto dell’Accademia delle Scienze Fisiche e Matematiche

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Inspired by a result from Leibov, we find that the supremum defining the B L O norm in [ 0 , 1 ] is actually attained by a specific sub-interval of [ 0 , 1 ] for f V L O ( [ 0 , 1 ] )

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Let G be a group and m be an integer greater than or equal to 2 . G is said to be m -permutable if every product of m elements can be reordered at least in one way. We prove that, if G has a centre of finite index z , then G is ( 1 + [ z / 2 ] ) -permutable. More bounds are given on the least m such that G is m -permutable.

On the unit group of a semisimple group algebra 𝔽 q S L ( 2 , 5 )

Rajendra K. Sharma, Gaurav Mittal (2022)

Mathematica Bohemica

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We give the characterization of the unit group of 𝔽 q S L ( 2 , 5 ) , where 𝔽 q is a finite field with q = p k elements for prime p > 5 , and S L ( 2 , 5 ) denotes the special linear group of 2 × 2 matrices having determinant 1 over the cyclic group 5 .

A remark on complex powers of analytic functions

Giuseppe Zampieri (1985)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Sia K n un compatto, f 0 una funzione analitica all'intorno di K , ed m la massima molteplicità in K degli zeri di f ; si prova che la potenza f λ ( λ , R e λ > 1 m ) è integrabile in K . L'estensione meromorfa dell'applicazione λ f λ da R e λ > 0 a tutto (con valori in 𝒟 ( K ) anziché in L 1 ( K ) ) era già stata provata in [1] e [2].

𝐋 -embedding, Amalgamation and 𝐋 -elementary equivalence

Daniele Mundici (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Ogni logica L genera canonicamente la L -equivalenza L e la L -immersione 𝐿 proprio come la logica del primo ordine genera l’equivalenza elementare e l’immersione elementare . Astraendo da L , è interessante studiare in sè relazioni d’equivalenza e di immersione generali tra strutture. Mostriamo che esiste una corrispondenza biunivoca tra relazioni d’equivalenza con la proprietà di Robinson e relazioni d’immersione con la proprietà di Amalgamazione Forte ( A P + ). Caratterizziamo algebricamente...

A note on singular and degenerate abstract equations

Angelo Favini (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si considera l’equazione astratta B A 1 u + A 0 u = h , dove A i ( i = 0 , 1 ) e B sono convenienti operatori lineari chiusi fra spazi di Banach, A i non è necessariamente invertibile, e A 0 , A 1 non commutano con B . Si studiano esistenza ed unicità delle soluzioni. Si indicano alcune applicazioni a certe equazioni differenziali degeneri o singolari.

On the Configuration Spaces of Grassmannian Manifolds

Sandro Manfredini, Simona Settepanella (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let h i ( k , n ) be the i -th ordered configuration space of all distinct points H 1 , ... , H h in the Grassmannian G r ( k , n ) of k -dimensional subspaces of n , whose sum is a subspace of dimension i . We prove that h i ( k , n ) is (when non empty) a complex submanifold of G r ( k , n ) h of dimension i ( n - i ) + h k ( i - k ) and its fundamental group is trivial if i = m i n ( n , h k ) , h k n and n &gt; 2 and equal to the braid group of the sphere P 1 if n = 2 . Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1 .

Equivalence of measures of smoothness in L p ( S d - 1 ) , 1 < p < ∞

F. Dai, Z. Ditzian, Hongwei Huang (2010)

Studia Mathematica

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Suppose Δ̃ is the Laplace-Beltrami operator on the sphere S d - 1 , Δ ρ k f ( x ) = Δ ρ Δ ρ k - 1 f ( x ) and Δ ρ f ( x ) = f ( ρ x ) - f ( x ) where ρ ∈ SO(d). Then ω m ( f , t ) L p ( S d - 1 ) s u p Δ ρ m f L p ( S d - 1 ) : ρ S O ( d ) , m a x x S d - 1 ρ x · x c o s t and K ̃ ( f , t m ) p i n f f - g L p ( S d - 1 ) + t m ( - Δ ̃ ) m / 2 g L p ( S d - 1 ) : g ( ( - Δ ̃ ) m / 2 ) are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for L p ( S d - 1 ) given in this paper plays a significant role in the proof.

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

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In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Lower bound for class numbers of certain real quadratic fields

Mohit Mishra (2023)

Czechoslovak Mathematical Journal

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Let d be a square-free positive integer and h ( d ) be the class number of the real quadratic field ( d ) . We give an explicit lower bound for h ( n 2 + r ) , where r = 1 , 4 . Ankeny and Chowla proved that if g > 1 is a natural number and d = n 2 g + 1 is a square-free integer, then g h ( d ) whenever n > 4 . Applying our lower bounds, we show that there does not exist any natural number n > 1 such that h ( n 2 g + 1 ) = g . We also obtain a similar result for the family ( n 2 g + 4 ) . As another application, we deduce some criteria for a class group of prime power order to be...

L p , q spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the L p , q spaces................................. 114. Integral representation of bounded linear functionals on L p , q ( B ) ........ 235. Examples in L p , q theory...................................................................................

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 .