Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f^′ and k^′ which are homotopic as maps ofM-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X₁∪X₂).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.     

On some conformally invariant fully nonlinear equationsSur certaines équations completement nonlinéaires invariantes par transformation conforme

We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7]. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 305–310.     

S-arrangements avec répétitionsS-arrangements with repetitions

We study natural correspondances between the set of arrangements of subsets with repetitions and families of objects met in various areas of mathematics (walks in lattices, pre-order) and computer science (formal languages, Petri nets, Artificial Intelligence), using the framework of formal languages. For n=2, they are enumerated by Delannoy numbers. To cite this article: S.R. Schwer, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 261–266.     

Periodic distributions and non-Liouville numbers

In an earlier paper (Translation-invariant linear forms and a formula for the Dirac measure, J. Functional Analysis8 (1971), 173–188) the author proved that the ratio $\text{α}\text{β}$ of two real numbers is a non-Liouville number if and only if there exist two Schwartz distributions A and B with compact supports on the real line such that δ′ = A ? τ_αA + B ? τ_βB. The present paper presents (in Section 3) a completely new and more elementary proof of this result (stated fully as Theorems 1 and 2 at the end of Section 1) based on some fundamental properties of the mapping H₀ = ? Δ and H = H₀ + T, where T which are established in Section 2. Further connections with Diophantine approximation (badly approximable numbers and Roth's Theorem) are presented in Section 4 where it is proved that the orders of the distributions A and B are always ?2 (Theorem 3) and almost always =3 (Theorem 4). Section 5 contains some partial results (Theorem 5 and Corollary 2 of Theorem 6) on the analogous question (as yet unsettled) for the Banach space C(T) of all continuous functions on the circle group T, and connects this problem with Sidon sets of integers.     

Generalized <Emphasis Type="BoldItalic">r</Emphasis>-Lah numbers

In this paper, we consider a two-parameter polynomial generalization, denoted by \(\mathcal {G}_{a,b}(n,k;r)\), of the r-Lah numbers which reduces to these recently introduced numbers when a = b = 1. We present several identities for \(\mathcal {G}_{a,b}(n,k;r)\) that generalize earlier identities given for the r-Lah and r-Stirling numbers. We also provide combinatorial proofs of some earlier identities involving the r-Lah numbers by defining appropriate sign-changing involutions. Generalizing these arguments yields orthogonality-type relations that are satisfied by \(\mathcal {G}_{a,b}(n,k;r)\).     

Two, more readily computable, equivariant Nielsen numbers II

In this paper, we present and make computations of two equivariant Nielsen type numbers NG(f(H),k(H)) and NG(f(H),k(H)). The second one is new, while the first one extends and clarifies one given earlier by the author and Jianhan Guo. Both numbers were defined here in terms of Nielsen theory of M-ads introduced in the prequel to this work. The theory of M-ads is also used to give both upper and lower bounds on our numbers, and to make specific computations. Our numbers moreover, fit together in the same way that the two Nielsen type periodic point numbers NPn(f) and NΦn(f) fit together. In particular, we show that NG(f(H),k(H)) is greater than or equal to a sum of numbers of the form NG(f(K),k(K)), and give conditions for equality and Möbius inversion. The periodic point theory results are then seen to follow from what are actually generalizations of them.We work with both fixed point, and coincidence point classes, in the context of a category with essentiality which we introduced in the prequel on M-ads. It is intended that this paper be read in tandem with said prequel.     

Chern numbers for two families of noncommutative Hopf fibrations

We consider noncommutative line bundles associated with the Hopf fibrations of SU_q(2) over all Podle? spheres and with a locally trivial Hopf fibration of S³_pq. These bundles are given as finitely generated projective modules associated via 1-dimensional representations of U(1) with Galois-type extensions encoding the principal fibrations of SU_q(2) and S³_pq. We show that the Chern numbers of these modules coincide with the winding numbers of representations defining them. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the projectors FF and FF

A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF^† and F^†F, determined by a complex square matrix F and its Moore-Penrose inverse F^†. Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benítez and Rako?evi? [J. Benítez, V. Rako?evi?, Matrices A such that AA^† − A^†A are nonsingular, Appl. Math. Comput. 217 (2010) 3493-3503]. Further characteristics of FF^† and F^†F, with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF^† and F^†F, the paper develops a general approach whose applicability extends far beyond the characteristics provided therein.     

Frontière de Martin des marches aléatoires sur certains hypergroupes d-dimensionnelsMartin boundary of random walks on certain d-dimensional hypergroups

We give an integral representation formula for harmonic functions of Markov chains on $\text{N}{}^{\mathrm{d}}$ and $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$ which transition probability is invariant by translations of a hypergroup, product of polynomial hypergroups for $\text{N}{}^{\mathrm{d}}$ and product of Sturm–Liouville hypergroups for $\text{R}{}_{\mathrm{+}}{}^{\mathrm{d}}$. To cite this article: L. Godefroy, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1029–1034.     

Complex Pisot numbers of small modulus

We use an algorithm of Chamfy to determine all complex Pisot numbers of modulus less than 1.17. To cite this article: D. Garth, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Bigraded Betti numbers of certain simple polytopes

The bigraded Betti numbers β ^?i,2j (P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers β ^?i,2j (P) reflect the combinatorial structure of P, as well as the topological structure of the corresponding moment-angle manifold Z _P ; thus, they find numerous applications in combinatorial commutative algebra and toric topology. We calculate certain bigraded Betti numbers of the type β ^?i,2(i+1) for associahedra and apply the calculation of bigraded Betti numbers for truncation polytopes to study the topology of their moment-angle manifolds. Presumably, for these two series of simple polytopes, the numbers β ^?i,2j (P) attain their minimum and maximum values among all simple polytopes P of fixed dimension with a given number of facets.     

General formulas for the smoothed analysis of condition numbers

We provide estimates on the volume of tubular neighborhoods around a subvariety Σ of real projective space, intersected with a disk of radius σ. The bounds are in terms of σ, the dimension of the ambient space, and the degree of equations defining Σ. We use these bounds to obtain smoothed analysis estimates for some conic condition numbers. To cite this article: P. Bürgisser et al., C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Pinceau intégral enlacéLinked integral pencils

Let γ₀ be a integral curve of an analytic vector field on a manifold of dimension 3. We suppose that γ₀ has oriented, iterated tangents. The integral pencil PI(γ₀) is the set of integral curves γ which have the same oriented, iterated tangent as γ₀. The curves of PI(γ₀) are either subanalitically separated or asymptotically linked. In this case PI(γ₀) has a formal axis which is divergent if and only if the curves of PI(γ₀) are not oscillating. To cite this article: F. Cano et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 855–858.     

More on the duality conjecture for entropy numbers

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Spectral theory of Sturm–Liouville difference operators

We present several classes of explicit self-adjoint Sturm–Liouville difference operators with either a non-Hermitian leading coefficient function, or a non-Hermitian potential function, or a non-definite weight function, or a non-self-adjoint boundary condition. These examples are obtained using a general procedure for constructing difference operators realizing discrete Sturm–Liouville problems, and the minimum conditions for such difference operators to be self-adjoint with respect to a natural quadratic form. It is shown that a discrete Sturm–Liouville problem admits a difference operator realization if and only if it does not have all complex numbers as eigenvalues. Spectral properties of self-adjoint Sturm–Liouville difference operators are studied. In particular, several eigenvalue comparison results are proved.     

Multi-poly-Bernoulli numbers and polynomials with a <Emphasis Type="Italic">q</Emphasis> parameter

We introduce themulti-poly-Bernoulli numbers and polynomialswith a q parameter, which are generalizations of the poly-Bernoulli numbers and polynomials with a q parameter, respectively.We give several combinatorial identities and properties of these new numbers and polynomials.     

Arbres de Markov couplePairwise Markov trees

The Hidden Markov Chain (HMC) models are widely applied in various problems. This succes is mainly due to the fact that the hidden model distribution conditional on observations remains a Markov chain distribution, and thus different processings, like Bayesian restorations, are handleable. These models have been recetly generalized to “Pairwise” Markov chains, which admit the same processing power and a better modeling one. The aim of this Note is to show that the Hidden Markov trees, which can be seen as extensions of the HMC models, can also be generalized to “Pairwise” Markov trees, which present the same processing advantages and better modelling power. To cite this article: W. Pieczynski, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 79–82.     

Sur le nombre de points algébriques où une fonction analytique transcendante prend des valeurs algébriquesOn the number of algebraic points where an analytic transcendental function takes algebraic values

We study the set of algebraic numbers of bounded height and bounded degree where an analytic transcendental function takes algebraic values. To cite this article: A. Surroca, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 721–725.     

Une remarque sur les ensembles hyperboliques localement maximauxA remark on locally maximal hyperbolic sets

On an example we show that there exist some hyperbolic sets which are not contained in any locally maximal hyperbolic set. To cite this article: S. Crovisier, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 401–404.