Décomposition spectrale dans des algèbres munies d'un star-produit

Summary The purpose of this paper is to study, in intrinsic way, the Moyal's product, defined in the flat space R ²ⁿ. This product is defined here with the twisted convolution and the Fourier transform. The S(R ²ⁿ) and L²(R ²ⁿ) spaces are_*5-algebras. Because of this definition, the_*V-product of some tempered distributions is defined. Let O _M ^v be the set of multiplication operators in S(R ²ⁿ). By transposition, the S(R ²ⁿ) space is a right-module on O _M ^v . The support of f_*v g is different from the support of f·g; under large enough hypotheses, there is a Taylor's formula for the star-product function of the v variable. The ^v space of the multiplication operators in L²(R ²ⁿ) is defined here as the space of tempered distributions, the image of which is the set of bounded operators in L²(R ²ⁿ) by the Weyl map. After the study of ^v space, it is possible to show the spectral resolution of the real elements of ^v or of O _M ^v , which satisfies a, probably superfluous, hypothesis.     

Windmill polynomials over fields of characteristic two

Chambers andSmeets [3] have designed a windmill arrangement of linear feedback shift registers (LFSRs) to generate pn-sequences overGF(2) with high speed. When the windmill hasv vanes, the associated minimal feedback polynomial (having degreen, relatively prime tov) can be taken to have the shapef ₁(x ^v )+x ⁿ f ₂(x ^–v ), where the polynomialsf ₁ andf ₂ have degree [n/v]. Their numerical evidence, whenv is divisible by 4, suggests that, surprisingly, there areno such windmill polynomials which are irreducible ifn±3 (mod 8), while about twice as many irreducible and primitive windmill polynomials as they expected occur ifn±1 (mod 8). A discussion of this behaviour is presented here with proofs. The brief explanation is that the Galois group of the underlying generic windmill polynomial overGF (4) is equal to the alternating groupA _n .     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

Convolutions of random measures on a compact topological semigroup

We investigate the asymptotic behavior of a sequence of convolutionsv ⁽ⁿ ₁() ** _n (), where { _n } _n=1 is some random process taking values in a semigroupM ¹(S) of probability Borel measures on a compact topological semigroupS.     

Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      6. A discrepancy theorem concerning polynomials of best approximation inL w p [−1, 1]      H. -P. Blatt  H. N. Mhaskar 《Monatshefte für Mathematik》1995,120(2):91-103 Letw be a suitable weight function,B n,p denote the polynomial of best approximation to a functionf inL w p [–1, 1],v n be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofB n+1,p–B n,p and be the arcsine measure defined by . We estimate the rate at which the sequencev n converges to in the weak-* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalL w p norm.This author gratefully acknowledges partial support from NSA contract #A4235802 during 1992, AFSOR Grant 226113 during 1993 and The Alexander von Humboldt Foundation during both of these years.      7. BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials      A. Okounkov 《Transformation Groups》1998,3(2):181-207 We consider 3-parametric polynomialsP * (x; q, t, s) which replace theA n-series interpolation Macdonald polynomialsP * (x; q, t) for theBC n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP * (x; q, t, s) becomeP * (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.      8. Nonparametric estimation of a regression function      Kuang-Fu Cheng  Pi-Erh Lin 《Probability Theory and Related Fields》1981,57(2):223-233 Summary Consider the regression model Y i * =g(x i * )+e i * , i=1,2,...,n, where x i * 's denote unordered design variables, and g is an unknown function defined on the interval [0,1]. Assume {e i * } are iid random variables with zero mean and finite variance. Priestley and Chao (1972) and Clark (1977) proposed estimators g 2n and g 3n , respectively for g. In this paper, the asymptotic behavior of g 2n and g 3n is studied utilizing the properties of the new estimator g 1n . It is shown that g 1n , g 2n , g 3n are asymptotically equivalent in various senses. Moreover, consistency results are established and rates of uniform convergence obtained. For example, if E¦e *¦3<, if g is Lipschitz of order 1, and if {n} is any sequence of constants tending to as n, then for all , as n. Finally, when g is monotone, a strong consistent isotonic estimator g n * is considered.      9. On unconditional bases in certain Banach function spaces      Kopaliani Tengiz 《Analysis Mathematica》2004,30(3):193-205 Let X be a Banach space, L ([0,1])XL 1([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f n} m=1 , where f n in C([0,1]), nN. In this paper, we introduce the notion that X *has the singularity property of X *at a point t 0[0,1]. It is proved that if X *has the singularity property at a point t 0 [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.      10. A constructive method of solving the Liapounov equation for complex matrices      Rita Meyer-Spasche 《Numerische Mathematik》1972,19(5):433-438 Summary This paper describes a method of solving the Liapounov equation (1)HM+M * H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M * A * =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH j andD j respectively for whichH j M+M * H j =2D j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.      11. ltivariate Hausdorff operators on the spaces H1(Rn) and BMO(Rn)      Ferenc Móricz 《Analysis Mathematica》2005,31(1):31-41 Summary A multivariate Hausdorff operator H = H(, c, A) is defined in terms of a -finite Borel measure on Rn, a Borel measurable function c on Rn, and an n × n matrix A whose entries are Borel measurable functions on rn and such that A is nonsingular -a.e. The operator H*:= H (, c | det A-1|, A-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H1(Rn) and BMO (Rn). Our main tool is proving commuting relations among H, H*, and the Riesz transforms Rj. We also prove commuting relations among H, H*, and the Fourier transform.      12. Weak limits of zeros of orthogonal polynomials      J. L. Ullman  M. F. Wyneken 《Constructive Approximation》1986,2(1):339-347 Letμ be a positive unit Borel measure with infinite support on the interval [?1, 1]. LetP n(x, μ) denote the monic orthogonal polynomial of degreen associated withμ, and letv n(μ) denote the unit measure with mass 1/n at each zero ofP n(x, μ). A carrier is a Borel subset of the support ofμ having unitμ-measure, and a measurev is carrier related toμ when it has the same carriers asμ. We demonstrate that for each carrierB of positive capacity there is a measurev, which is carrier related toμ, such that the equilibrium measure of the carrierB is the weak limit of the sequence {v n(v)} n =1/∞ .      13. Lower bounds for increasing complexity of derivations after cut elimination      V. P. Orevkov 《Journal of Mathematical Sciences》1982,20(4):2337-2350 We denote by C k * the formula. In this paper for all k there is constructed a derivation of C k * with cut, the number of sequents in which depends linearly on k. On the other hand, it is impossible to give an upper bound which is a Kalmar elementary function of k for the number of sequents in any derivation of the formula C k * without cuts, or for the number of disjunctions in a refutation by the method of resolutions of systems of disjunctions corresponding to the negation of the formula C k * . In particular, one can construct a derivation with cut of the formula C 6 * , in which there is contained no more than 253 sequents, but in seeking a derivation of C 6 * by the method of resolutions it is necessary to construct more than 1019200 disjunctions. With the help of Skolemization and taking out of quantifiers with respect to the formula C k * there is constructed a formula v0B k + (v0), which satisfies the following conditions: 1) one can construct a derivation with cuts of the formula v0B k + (v0) in the constructive predicate calculus, the number of sequents in which depends linearly on k; 2) it is impossible to give an upper bound which is a Kalmar elementary function of k of the length of a term t such that the formula B k + (t) is derivable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 137–161, 1979.      14. A family of (n−1)! Diagonal polynomial orders ofN n      Adolfo Sanchez-Flores 《Order》1995,12(2):173-187 Letn>0 be an element of the setN of nonnegative integers, and lets(x)=x 1+...+x n , forx=(x 1, ...,x n ) N n . Adiagonal polynomial order inN n is a bijective polynomialp:N n N (with real coefficients) such that, for allx,y N n ,p(x)<p(y) whenevers(x)<s(y). Two diagonal polynomial orders areequivalent if a relabeling of variables makes them identical. For eachn, Skolem (1937) found a diagonal polynomial order. Later, Morales and Lew (1992) generalized this polynomial order, obtaining a family of 2 n–2 (n>1) inequivalent diagonal polynomial orders. Here we present, for eachn>0, a family of (n – 1)! diagonal polynomial orders, up to equivalence, which contains the Morales and Lew diagonal orders.      15. Group connectivity of complementary graphs      Xinmin Hou  Hong‐Jian Lai  Ping Li  Cun‐Quan Zhang 《Journal of Graph Theory》2012,69(4):464-470 Let G be a 2‐edge‐connected undirected graph, A be an (additive) abelian group and A* = A?{0}. A graph G is A‐connected if G has an orientation D(G) such that for every function b: V(G)?A satisfying , there is a function f: E(G)?A* such that for each vertex v∈V(G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges directed into v equals b(v). For a 2‐edge‐connected graph G, define Λg(G) = min{k: for any abelian group A with |A|?k, G is A‐connected }. In this article, we prove the following Ramsey type results on group connectivity: Let G be a simple graph on n?6 vertices. If min{δ(G), δ(Gc)}?2, then either Λg(G)?4, or Λg(Gc)?4. Let Z3 denote the cyclic group of order 3, and G be a simple graph on n?44 vertices. If min{δ(G), δ(Gc)}?4, then either G is Z3‐connected, or Gc is Z3‐connected. © 2011 Wiley Periodicals, Inc. J Graph Theory       16. Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation      Johannes Giannoulis 《Mathematical Methods in the Applied Sciences》2005,28(5):607-629 We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?2 → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vn, we obtain a sequence of spatially highly oscillatory classical solutions un. By considering the Young measures (YMs) ν and µ generated by the sequences vn and un, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vn and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.      17. Temps d'arrÊt d'un système dynamique      Professor J. Neveu 《Probability Theory and Related Fields》1969,13(2):81-94 Summary The most general positive integer-valued random variable v such that for a given bijective measure preserving transformation , the transformation v is still bijective and measure preserving is shown to be a (generally infinite) superposition of return times. Equipe de Recherche n 1 «< Processus stochastiques et applications»> dépendant de la section n 2 «Théories Physiques et Probabilités» associée au C.N.R.S.      18. Analysis of Henrici's Transformation for Singular Problems      Mohammed Bellalij 《Numerical Algorithms》2003,33(1-4):65-82 Henrici's transformation is the underlying scheme that generates, by cycling, Steffensen's method for the approximation of the solution of a nonlinear equation in several variables. The aim of this paper is to analyze the asymptotic behavior of the obtained sequence (s n * ) by applying Henrici's transformation when the initial sequence (s n ) behaves sublinearly. We extend the work done in the regular case by Sadok [17] to vector sequences in the singular case. Under suitable conditions, we show that the slowest convergence rate of (s n * ) is to be expected in a certain subspace N of R p . More precisely, if we write s n * =s n * ,N+s n * ,N, the orthogonal decomposition into N and N , then the convergence is linear for (s n * ,N) but ( n * ,N) converges to the same limit faster than the initial one. In certain cases, we can have N=R p and the convergence is linear everywhere.      19. Graph C *-Algebras and Their Ideals Defined by Cuntz-Krieger Family of Possibly Row-Infinite Directed Graphs      Xiaochun Fang 《Integral Equations and Operator Theory》2006,54(3):301-316 Let E be a possibly row-infinite directed graph. In this paper, first we prove the existence of the universal C*-algebra C*(E) of E which is generated by a Cuntz-Krieger E-family {se, pv}, and the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the ideal of C*(E). Then we get our main results about the ideal structure of Finally the simplicity and the pure infiniteness of is discussed.      20. Bounds on the subdominant eigenvalue involving group inverse with applications to graphs      Stephen J. Kirkland  Michael Neumann  Bryan L. Shader 《Czechoslovak Mathematical Journal》1998,48(1):1-20 Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are 12 ... n. In this paper we derive several lower and upper bounds, in particular on 2 and n , but also, indirectly, on . The bounds are in terms of the diagonal entries of the group generalized inverse, Q #, of the singular and irreducible M-matrix Q = 1 I – A. Our starting point is a spectral resolution for Q #. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected graphs, where now Q becomes L, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of L # and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of L, by exploiting the diagonal entries of L #.

A discrepancy theorem concerning polynomials of best approximation inL w p [−1, 1]

Letw be a suitable weight function,B _n,p denote the polynomial of best approximation to a functionf inL _w ^p [–1, 1],v _n be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofB _n+1,p–B _n,p and be the arcsine measure defined by . We estimate the rate at which the sequencev _n converges to in the weak-^* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalL _w ^p norm.This author gratefully acknowledges partial support from NSA contract #A4235802 during 1992, AFSOR Grant 226113 during 1993 and The Alexander von Humboldt Foundation during both of these years.     

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP ^* (x; q, t, s) which replace theA _n-series interpolation Macdonald polynomialsP ^* (x; q, t) for theBC _n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP ^* (x; q, t, s) becomeP ^* (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.     

Nonparametric estimation of a regression function

Summary Consider the regression model Y _i ^* =g(x _i ^* )+e _i ^* , i=1,2,...,n, where x _i ^* 's denote unordered design variables, and g is an unknown function defined on the interval [0,1]. Assume {e _i ^* } are iid random variables with zero mean and finite variance. Priestley and Chao (1972) and Clark (1977) proposed estimators g _2n and g _3n , respectively for g. In this paper, the asymptotic behavior of g _2n and g _3n is studied utilizing the properties of the new estimator g _1n . It is shown that g _1n , g _2n , g _3n are asymptotically equivalent in various senses. Moreover, consistency results are established and rates of uniform convergence obtained. For example, if E¦e ^*¦³<, if g is Lipschitz of order 1, and if {_n} is any sequence of constants tending to as n, then for all , as n. Finally, when g is monotone, a strong consistent isotonic estimator g _n ^* is considered.     

On unconditional bases in certain Banach function spaces

Let X be a Banach space, L ([0,1])XL ¹([0,1]), with an unconditional basis. By the well-known stability property in X, there exists a unconditional basis {f _n} _m=1 , where f _n in C([0,1]), nN. In this paper, we introduce the notion that X ^*has the singularity property of X ^*at a point t ₀[0,1]. It is proved that if X ^*has the singularity property at a point t ₀ [0,1], then there exists no orthonormal, fundamental system in C([0,1]) which forms an unconditional basis in X.     

A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

ltivariate Hausdorff operators on the spaces H1(Rn) and BMO(Rn)

Summary A multivariate Hausdorff operator H = H(, c, A) is defined in terms of a -finite Borel measure on Rⁿ, a Borel measurable function c on Rⁿ, and an n × n matrix A whose entries are Borel measurable functions on rⁿ and such that A is nonsingular -a.e. The operator H*:= H (, c | det A^-1|, A^-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H¹(Rⁿ) and BMO (Rⁿ). Our main tool is proving commuting relations among H, H*, and the Riesz transforms R_j. We also prove commuting relations among H, H*, and the Fourier transform.     

Weak limits of zeros of orthogonal polynomials

Letμ be a positive unit Borel measure with infinite support on the interval [?1, 1]. LetP _n(x, μ) denote the monic orthogonal polynomial of degreen associated withμ, and letv _n(μ) denote the unit measure with mass 1/n at each zero ofP _n(x, μ). A carrier is a Borel subset of the support ofμ having unitμ-measure, and a measurev is carrier related toμ when it has the same carriers asμ. We demonstrate that for each carrierB of positive capacity there is a measurev, which is carrier related toμ, such that the equilibrium measure of the carrierB is the weak limit of the sequence {v _n(v)} _n ^=1/∞ .     

Lower bounds for increasing complexity of derivations after cut elimination

We denote by C _k ^* the formula. In this paper for all k there is constructed a derivation of C _k ^* with cut, the number of sequents in which depends linearly on k. On the other hand, it is impossible to give an upper bound which is a Kalmar elementary function of k for the number of sequents in any derivation of the formula C _k ^* without cuts, or for the number of disjunctions in a refutation by the method of resolutions of systems of disjunctions corresponding to the negation of the formula C _k ^* . In particular, one can construct a derivation with cut of the formula C ₆ ^* , in which there is contained no more than 253 sequents, but in seeking a derivation of C ₆ ^* by the method of resolutions it is necessary to construct more than 10¹⁹²⁰⁰ disjunctions. With the help of Skolemization and taking out of quantifiers with respect to the formula C _k ^* there is constructed a formula v₀B _k ⁺ (v₀), which satisfies the following conditions: 1) one can construct a derivation with cuts of the formula v₀B _k ⁺ (v₀) in the constructive predicate calculus, the number of sequents in which depends linearly on k; 2) it is impossible to give an upper bound which is a Kalmar elementary function of k of the length of a term t such that the formula B _k ⁺ (t) is derivable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 137–161, 1979.     

A family of (n−1)! Diagonal polynomial orders ofN n

Letn>0 be an element of the setN of nonnegative integers, and lets(x)=x ₁+...+x _n , forx=(x ₁, ...,x _n ) N _n . Adiagonal polynomial order inN _n is a bijective polynomialp:N _n N (with real coefficients) such that, for allx,y N _n ,p(x)<p(y) whenevers(x)<s(y). Two diagonal polynomial orders areequivalent if a relabeling of variables makes them identical. For eachn, Skolem (1937) found a diagonal polynomial order. Later, Morales and Lew (1992) generalized this polynomial order, obtaining a family of 2 ^n–2 (n>1) inequivalent diagonal polynomial orders. Here we present, for eachn>0, a family of (n – 1)! diagonal polynomial orders, up to equivalence, which contains the Morales and Lew diagonal orders.     

Group connectivity of complementary graphs

Let G be a 2‐edge‐connected undirected graph, A be an (additive) abelian group and A* = A?{0}. A graph G is A‐connected if G has an orientation D(G) such that for every function b: V(G)?A satisfying , there is a function f: E(G)?A* such that for each vertex v∈V(G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges directed into v equals b(v). For a 2‐edge‐connected graph G, define Λ_g(G) = min{k: for any abelian group A with |A|?k, G is A‐connected }. In this article, we prove the following Ramsey type results on group connectivity:

1. Let G be a simple graph on n?6 vertices. If min{δ(G), δ(G^c)}?2, then either Λ_g(G)?4, or Λ_g(G^c)?4.
2. Let Z₃ denote the cyclic group of order 3, and G be a simple graph on n?44 vertices. If min{δ(G), δ(G^c)}?4, then either G is Z₃‐connected, or G^c is Z₃‐connected. © 2011 Wiley Periodicals, Inc. J Graph Theory

     

Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation

We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?² → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vⁿ, we obtain a sequence of spatially highly oscillatory classical solutions uⁿ. By considering the Young measures (YMs) ν and µ generated by the sequences vⁿ and uⁿ, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vⁿ and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.     

Temps d'arrÊt d'un système dynamique

Summary The most general positive integer-valued random variable v such that for a given bijective measure preserving transformation , the transformation ^v is still bijective and measure preserving is shown to be a (generally infinite) superposition of return times.

---

---

Equipe de Recherche n 1 «< Processus stochastiques et applications»> dépendant de la section n 2 «Théories Physiques et Probabilités» associée au C.N.R.S.     

Analysis of Henrici's Transformation for Singular Problems

Henrici's transformation is the underlying scheme that generates, by cycling, Steffensen's method for the approximation of the solution of a nonlinear equation in several variables. The aim of this paper is to analyze the asymptotic behavior of the obtained sequence (s _n ^* ) by applying Henrici's transformation when the initial sequence (s _n ) behaves sublinearly. We extend the work done in the regular case by Sadok [17] to vector sequences in the singular case. Under suitable conditions, we show that the slowest convergence rate of (s _n ^* ) is to be expected in a certain subspace N of R ^p . More precisely, if we write s _n ^* =s _n ^* ,N+s _n ^* ,N, the orthogonal decomposition into N and N , then the convergence is linear for (s _n ^* ,N) but ( _n ^* ,N) converges to the same limit faster than the initial one. In certain cases, we can have N=R ^p and the convergence is linear everywhere.     

Graph C *-Algebras and Their Ideals Defined by Cuntz-Krieger Family of Possibly Row-Infinite Directed Graphs

Let E be a possibly row-infinite directed graph. In this paper, first we prove the existence of the universal C^*-algebra C^*(E) of E which is generated by a Cuntz-Krieger E-family {s_e, p_v}, and the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem for the ideal of C^*(E). Then we get our main results about the ideal structure of Finally the simplicity and the pure infiniteness of is discussed.     

Bounds on the subdominant eigenvalue involving group inverse with applications to graphs

Let A be an n × n symmetric, irreducible, and nonnegative matrix whose eigenvalues are ₁₂ ... _n. In this paper we derive several lower and upper bounds, in particular on ₂ and _n , but also, indirectly, on . The bounds are in terms of the diagonal entries of the group generalized inverse, Q ^#, of the singular and irreducible M-matrix Q = ₁ I – A. Our starting point is a spectral resolution for Q ^#. We consider the case of equality in some of these inequalities and we apply our results to the algebraic connectivity of undirected graphs, where now Q becomes L, the Laplacian of the graph. In case the graph is a tree we find a graph-theoretic interpretation for the entries of L ^# and we also sharpen an upper bound on the algebraic connectivity of a tree, which is due to Fiedler and which involves only the diagonal entries of L, by exploiting the diagonal entries of L ^#.