Displaying similar documents to “On second order Thom-Boardman singularities”

Thom polynomials and Schur functions: the singularities I I I 2 , 3 ( - )

Özer Öztürk (2010)

Annales Polonici Mathematici


We give a closed formula for the Thom polynomials of the singularities I I I 2 , 3 ( - ) in terms of Schur functions. Our computations combine the characterization of the Thom polynomials via the “method of restriction equations” of Rimányi et al. with the techniques of Schur functions.

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables

Janusz Gwoździewicz, Arkadiusz Płoski (2005)

Colloquium Mathematicae


For every polynomial F in two complex variables we define the Łojasiewicz exponents p , t ( F ) measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents p , t ( F ) in terms of the local invariants of singularities of the pencil of projective curves associated with F.

μ -constant monodromy groups and marked singularities

Claus Hertling (2011)

Annales de l’institut Fourier


μ -constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo ± id . Second, marked singularities are defined and global moduli...

A formula for Jack polynomials of the second order

Francisco J. Caro-Lopera, José A. Díaz-García, Graciela González-Farías (2007)

Applicationes Mathematicae


This work solves the partial differential equation for Jack polynomials C κ α of the second order. When the parameter α of the solution takes the values 1/2, 1 and 2 we get explicit formulas for the quaternionic, complex and real zonal polynomials of the second order, respectively.

Singularities of k-tuples of vector fields

Bronisław Jakubczyk, Feliks Przytycki


CONTENTSIntroduction............................................................................51. The main ideas and results................................................62. H n , k -invariant subsets of n , k .........................223. Reduction to germs of differential 1-forms........................354. The case k ≥ 2n-3. Proof of Theorem A...........................445. The case n = 3, k = 2.......................................................46Appendix. Connections with control theory...........................59List...

Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials

Stanislaw Lewanowicz (2002)

Applicationes Mathematicae


Let P k be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients a k in f = k a k P k . A systematic use of the basic properties (including some nonstandard ones) of the polynomials P k results in obtaining a low order of the recurrence.

Singularities of Maxwell’s system in non-hilbertian Sobolev spaces

Wided Chikouche, Serge Nicaise (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze


We study the regularity of the solution of the regularized electric Maxwell problem in a polygonal domain with data in L p ( Ω ) 2 . Using a duality method, we prove a decomposition of the solution into a regular part in the non-Hilbertian Sobolev space W 2 , p ( Ω ) 2 and an explicit singular one.

Springer fiber components in the two columns case for types A and D are normal

Nicolas Perrin, Evgeny Smirnov (2012)

Bulletin de la Société Mathématique de France


We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N 2 = 0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications


We consider the space of ultradifferentiable functions with compact supports and the space of polynomials on . A description of the space ( ) of polynomial ultradistributions as a locally convex direct sum is given.

Approximation by weighted polynomials in k

Maritza M. Branker (2005)

Annales Polonici Mathematici


We apply pluripotential theory to establish results in k concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections Σ k a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici


Let K be any subset of N . We define a pluricomplex Green’s function V K , θ for θ-incomplete polynomials. We establish properties of V K , θ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of N , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on K s u p p ( d d c V K , θ ) N . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute s u p p ( d d c V K , θ ) N when K is a compact...

The Bourgain algebra of the disk algebra A(𝔻) and the algebra QA

Joseph Cima, Raymond Mortini (1995)

Studia Mathematica


It is shown that the Bourgain algebra A ( ) b of the disk algebra A() with respect to H ( ) is the algebra generated by the Blaschke products having only a finite number of singularities. It is also proved that, with respect to H ( ) , the algebra QA of bounded analytic functions of vanishing mean oscillation is invariant under the Bourgain map as is A ( ) b .

Smooth double subvarieties on singular varieties, III

M. R. Gonzalez-Dorrego (2016)

Banach Center Publications


Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that rC = S ∩ F, r ∈ ℕ, where S and F are two surfaces and all the singularities of F are of the form z ³ = x 3 s - y 3 s , s ∈ ℕ. We prove that C can never pass through such kind of singularities of a surface, unless r = 3a, a ∈ ℕ. We study multiplicity-r structures on varieties r ∈ ℕ. Let Z be a reduced irreducible nonsingular (n-1)-dimensional variety such that rZ = X ∩ F, where X is a normal n-fold, F...

The factorization of f ( x ) x n + g ( x ) with f ( x ) monic and of degree 2 .

Joshua Harrington, Andrew Vincent, Daniel White (2013)

Journal de Théorie des Nombres de Bordeaux


In this paper we investigate the factorization of the polynomials f ( x ) x n + g ( x ) [ x ] in the special case where f ( x ) is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that f ( x ) is monic and linear.

Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica


Let · be the uniform norm in the unit disk. We study the quantities M n ( α ) : = inf ( z P ( z ) + α - α ) where the infimum is taken over all polynomials P of degree n - 1 with P ( z ) = 1 and α > 0 . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that inf α > 0 M n ( α ) = 1 / n . We find the exact values of M n ( α ) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

Deformed Heisenberg algebra with reflection and d -orthogonal polynomials

Fethi Bouzeffour, Hanen Ben Mansour, Ali Zaghouani (2017)

Czechoslovak Mathematical Journal


This paper is devoted to the study of matrix elements of irreducible representations of the enveloping deformed Heisenberg algebra with reflection, motivated by recurrence relations satisfied by hypergeometric functions. It is shown that the matrix elements of a suitable operator given as a product of exponential functions are expressed in terms of d -orthogonal polynomials, which are reduced to the orthogonal Meixner polynomials when d = 1 . The underlying algebraic framework allowed a systematic...

Every compact set in 𝐂 n is a good compact set

Jan Erik Björk (1970)

Annales de l'institut Fourier


Let K be an compact subset of an open set V in C n . We show the existence of an open neighborhood U of K satisfying the following condition : if f is holomorphic in V and if there exists a sequence of polynomials which approximate f uniformly in some open neighborhood U f of K , there exists a sequence of polynomial which approximate f uniformly in U .

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&amp;#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici


Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that | f | C | f | ϱ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than 1 - R ( n , d ) - 1 with R ( n , d ) = d ( 3 d - 3 ) n - 1 .