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In 1884, Konigs showed that when \(\varphi\)(z) is an analytic self-map of the unit disk fixing the origin, with 0 1. Under some natural hypotheses, they gave necessary and sufficient conditions for the existence of an analytic solution σ satisfying σ′(0) = I when \(\varphi\)′(0) is diagonalizable. In this paper, the problem when \(\varphi\)′(0) is not diagonalizable is considered. Both \(\varphi\)(z) and σ(z) will be regarded as vectors of purely formal power series, and it will not be assumed that \(\varphi\)(z) is analytic or that the series for \(\varphi\)(z) or σ(z) converge. Nevertheless, because of a process developed by Cowen and MacCluer, if the given \(\varphi\)(z) represents a map of the unit ball into itself of an appropriate form, then the results of this paper can be used to produce solutions of Schroder’s equation that are convergent power series, or (sometimes) to show that no such solution exists. A method of matrix completion is used.

Many papers study polynomial tractability for multivariate problems. Let n(@?,d) be the minimal number of information evaluations needed to reduce the initial error by a factor of @? for a multivariate problem defined on a space of d-variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(@?,d) is bounded by a polynomial in @?^-^1 and d and this holds for all (@?^-^1,d)@?[1,~)xN. In this paper we study generalized tractability by verifying when n(@?,d) can be bounded by a power of T(@?^-^1,d) for all (@?^-^1,d)@[email protected], where @W can be a proper subset of [1,~)xN. Here T is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set @W=[1,~)x{1,2,...,d^*}@?[1,@?"0^-^1)xN for some d^*>=1 and @?"[email protected]?(0,1). We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on T such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does.

Let ρ: G → GL(n, F) be a faithful representation of a finite group G. In this paper we proceed with the study of the image of the associated Noether map η G G : F[V(G)] G → F[V] G . In our 2005 paper it has been shown that the Noether map is surjective if V is a projective FG-module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for p-groups (where p is the characteristic of the ground field F) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of V.

This paper adresses the following problem: Given a closed orientable three-manifold M, are there at most finitely many closed orientable three-manifolds 1-dominated by M? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincare-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold M1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of M.

We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. There are two well-studied approximation methods: the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA). The WRGA is simpler than the WCGA in the sense of computational complexity. However, the WRGA has limited applicability. It converges only for elements of the closure of the convex hull of a dictionary. In this paper we study algorithms that combine good features of both algorithms, the WRGA and the WCGA. In the construction of such algorithms we use different forms of relaxation. First results on such algorithms have been obtained in a Hilbert space by A. Barron, A. Cohen, W. Dahmen, and R. DeVore. Their paper was a motivation for the research reported here.

This paper considers the Steiner Minimal Tree (SMT) problem in the rectilinear and octilinear planes. The study is motivated by the physical design of VLSI: The rectilinear case corresponds to the currently used M-architecture, which uses either horizontal or vertical routing, while the octilinear case corresponds to a new routing technique, X-architecture, that is based on the pervasive use of diagonal directions. The experimental studies show that the X-architecture demonstrates a length reduction of more than 10–20%. In this paper, we make a theoretical study on the lengths of SMTs in these two planes. Our mathematical analysis confirms that the length reduction is significant as the previous experimental studies claimed, but the reduction for three points is not as significant as for two points. We also obtain the lower and upper bounds on the expected lengths of SMTs in these two planes for arbitrary number of points.

For a set of primes $\mathcal{P}$, let $\Psi(x, \mathcal{P})$ be the number of positive integers $n \leq x$ all of whose prime factors lie in $\mathcal{P}$. In this paper we classify the sets of primes $\mathcal{P}$ such that $\Psi(x, \mathcal{P})$ is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matom\"aki and their main conjecture is proved in this paper. In particular our main theorem implies that, if not too many large primes are sieved out in the sense that \[ \sum_{\substack{p \in \mathcal{P} \\ x^{1/v} < p \leq x^{1/u}}} \frac{1}{p} \geq \frac{1 + \varepsilon}{u}, \] for some $\varepsilon > 0$ and $v \geq u \geq 1$, then \[ \Psi(x, \mathcal{P}) \gg_{\varepsilon, v} x \prod_{\substack{p \leq x\\ p \notin\mathcal{P}}} \left(1 - \frac{1}{p}\right). \], Comment: 23 pages

We study biased Maker-Breaker positional games between two players, one of whom is playing randomly against an opponent with an optimal strategy. In this paper we consider the scenario when Maker plays randomly and Breaker is "clever", and determine the sharp threshold bias of classical graph games, such as connectivity, Hamiltonicity, and minimum degree-$k$. We treat the other case, that is when Breaker plays randomly, in a separate paper. The traditional, deterministic version of these games, with two optimal players playing, are known to obey the so-called probabilistic intuition. That is, the threshold bias of these games is asymptotically equal to the threshold bias of their random counterpart, where players just take edges uniformly at random. We find, that despite this remarkably precise agreement of the results of the deterministic and the random games, playing randomly against an optimal opponent is not a good idea: the threshold bias becomes significantly more tilted towards the random player. An important qualitative aspect of the probabilistic intuition carries through nevertheless: the bottleneck for Maker to occupy a connected graph is still the ability to avoid isolated vertices in her graph., This paper has been split in two parts, the second part can be found at http://arxiv.org/abs/1602.04628 . A full version from before the split is submission v2 of this arXiv entry, available at http://arxiv.org/abs/1507.06688v2

Let $\mathbf{G}$ be one of the ind-groups $GL(\infty)$, $O(\infty)$, $Sp(\infty)$ and $\mathbf{P}\subset \mathbf{G}$ be a splitting parabolic ind-subgroup. The ind-variety $\mathbf{G}/\mathbf{P}$ has been identified with an ind-variety of generalized flags in the paper "Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups" (Int. Math. Res. Not. 2004, no. 55, 2935--2953) by I. Dimitrov and I. Penkov. In the present paper we define a Schubert cell on $\mathbf{G}/\mathbf{P}$ as a $\mathbf{B}$-orbit on $\mathbf{G}/\mathbf{P}$, where $\mathbf{B}$ is any Borel ind-subgroup of $\mathbf{G}$ which intersects $\mathbf{P}$ in a maximal ind-torus. A significant difference with the finite-dimensional case is that in general $\mathbf{B}$ is not conjugate to an ind-subgroup of $\mathbf{P}$, whence $\mathbf{G}/\mathbf{P}$ admits many non-conjugate Schubert decompositions. We study the basic properties of the Schubert cells, proving in particular that they are usual finite-dimensional cells or are isomorphic to affine ind-spaces. We then define Schubert ind-varieties as closures of Schubert cells and study the smoothness of Schubert ind-varieties. Our approach to Schubert ind-varieties differs from an earlier approach by H. Salmasian in "Direct limits of Schubert varieties and global sections of line bundles" (J. Algebra 320 (2008), 3187--3198)., Keywords: Classical ind-group, Bruhat decomposition, Schubert decomposition, generalized flag, homogeneous ind-variety. [26 pages]

All ten symmetry classes of plane partitions that fit in a given box are known to be enumerated by simple product formulas, but there is still no unified proof for all of them. Progress towards this goal can be made by establishing identities connecting the various symmetry classes. We present in this paper four such identities, involving all ten symmetry classes. We discuss their proofs and generalizations. The main result of this paper is to give a generalization of one of them, in the style of the identity presented in "A factorization theorem for rhombus tilings," M. Ciucu and C. Krattenthaler, arXiv:1403.3323., 14 pages

In this paper, we give a complete description of the associated primes of every power of the edge ideal in terms of generalized ear decompositions of the graph. This result establishes a surprising relationship between two seemingly unrelated notions of Commutative Algebra and Combinatorics. It covers all previous major results in this topic and has several interesting consequences., Comment: 28 pages, 7 figures. This revision contains more comments on the methods and the results of the paper. Important terminology and notations now appear in Definition environments for the convenience of the readers. That changes the numeration of the results considerably. This paper will appear in Trans. Amer. Math. Soc

For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $\ell(R/I) < \infty$, we introduce a new invariant $HKd(M, I)$ called the {\em Hilbert-Kunz density function}. In Theorem 1.1, we relate this to the Hilbert-Kunz multiplicity $e_{HK}(M,I)$ by an integral formula. We prove that the Hilbert-Kunz density function is additive. Moreover it satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of Projective curves is a rational number. As an another application we see that $e_{HK}(R, {\bf m}^k) - e(R, {\bf m}^k)/d!$ grows at least as a fixed positive multiple of $k^{d-1}$ as $k\to \infty$., This paper is one part of the earlier submission, which has been split in two parts. This part will introduce and develop the theory of Hilbert-Kunz Density. This will appear in Transactions of AMS. The second part (which will be another paper) will involve the study of asymptotic behaviour of Hilbert-Kunz multiplicities of powers of an ideal

In the first part of this paper, we consider nonlinear extension of frame theory by introducing bi-Lipschitz maps F between Banach spaces. Our linear model of bi-Lipschitz maps is the analysis operator associated with Hilbert frames, p-frames, Banach frames, g-frames and fusion frames. In general Banach space setting, stable algorithms to reconstruct a signal x from its noisy measurement $$F(x)+\epsilon $$ may not exist. In this paper, we establish exponential convergence of two iterative reconstruction algorithms when F is not too far from some bounded below linear operator with bounded pseudo-inverse, and when F is a well-localized map between two Banach spaces with dense Hilbert subspaces. The crucial step to prove the latter conclusion is a novel fixed point theorem for a well-localized map on a Banach space. In the second part of this paper, we consider stable reconstruction of sparse signals in a union $$\mathbf{A}$$ of closed linear subspaces of a Hilbert space $$\mathbf{H}$$ from their nonlinear measurements. We introduce an optimization framework called a sparse approximation triple $$(\mathbf{A}, \mathbf{M}, \mathbf{H})$$ , and show that the minimizer $$\begin{aligned} x^*=\mathrm{argmin}_{\hat{x}\in {\mathbf M}\ \mathrm{with} \ \Vert F(\hat{x})-F(x^0)\Vert \le \epsilon } \Vert \hat{x}\Vert _{\mathbf M} \end{aligned}$$ provides a suboptimal approximation to the original sparse signal $$x^0\in \mathbf{A}$$ when the measurement map F has the sparse Riesz property and the almost linear property on $${\mathbf A}$$ . The above two new properties are shown to be satisfied when F is not far away from a linear measurement operator T having the restricted isometry property.

In this paper, we study the following Neumann–Robin boundary value problem - ( ϕ p ( u ′ ( x ) ) ) ′ = λ f ( u ( x ) ) , x ∈ ( 0 , 1 ) , u ′ ( 0 ) = 0 , u ′ ( 1 ) + α u ( 1 ) = 0 , whereα ∈ R, λ > 0 are parameters and p > 1, and p ′ = p p - 1 is the conjugate exponent of p and ϕp(x): = ∣x∣p−2x for all x ∈ R where (ϕp(u′))′ is the one dimensional p-Laplacian and f ∈ C2[0, ∞) such that f(0) 0, and also f is increasing and concave up. We shall investigate the existence and multiplicity of nonnegative solutions. Note that in this paper, we shall establish our existence results by using the quadrature method.

∞n=0 ∞ pnF ∗n (x), where F ∗n (x) denotes the n-fold convolution of F (x) and where F ∗0 (x) denotes the unit mass at 0. The d.f. W (x )i s called subordinate to F (x) with subordinator {pn}. As in the univariate case in the paper, we shall assume that N satisfies condition (A): N has a generating function P (z )= E(z N ) that is analytic at z =1 . In the present paper, we discuss the relation between the asymptotic behavior of 1 − F (x) and that of 1 − F ∗n (x) and 1 − W (x). It turns out that, as in the univariate case, there are many cases in which 1 − F ∗n (x) asymptotically behaves as n(1 − F (x)) and 1 − W (x) behaves as E(N )(1 − F (x)). To specify the precise kind of asymptotic behavior, we present a form of multivariate subexponentiality. The paper is organized as follows. In Sec. 2, we briefly recall some basic properties and definitions concerning univariate subexponential d.f. In Sec. 3, we introduce and study multivariate subexponential d.f.’s. In Sec. 4, we discuss the relation with regular variation and, in Sec. 5, we provide some extensions. In our main results, we obtain first-order estimates for 1 − F ∗n (x) and 1 − W (x). In a forthcoming paper, we discuss second-order estimates. Without further comment, in the paper, we shall assume that all random vectors X, Y, Z, etc. are positive and have infinite support, i.e., the d.f. satisfies F (0+) = 0 and F (x) < 1, ∀x ∈ R d . We also use the notation F (x )=1 − F (x), and for vectors x and a ,w e setx ◦ = min(xi )a nda ∗ x =( a1x1 ,a 2x2 ,...,a dxd). 2. Univariate Subexponential Distributions In the one-dimensional case, many papers have been devoted to the tail behavior of subordinated d.f.’s. In doing so, the class of subexponential d.f.’s (notation: S) plays an important role. Extending the class S, Chover et al. [6, 7], introduced the class S(γ), where γ ≥ 0. To define these classes, let F (x) denote a d.f. in R such that F (0+) = 0 and F (x) < 1, ∀x ∈ R. Also, let f (s )= E(e −sX ) denote the generating function of X or F (x). The d.f. F (x) belongs to the subexponential class S (notation: F ∈ S) if it satisfies lim

The Brownian loop soup introduced by Lawler and Werner (2004) is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction property. In this paper, we define a random walk loop soup and show that it converges to the Brownian loop soup. In fact, we give a strong approximation result making use of the strong approximation result of Komlos, Major, and Tusnady. To make the paper self-contained, we include a proof of the approximation result that we need.

Total embedding distributions have been known for only a few classes of graphs. In this paper the total embedding distributions of the cacti and the necklaces are obtained. Furthermore we obtain the total embedding distributions of all graphs with maximum genus 1 by using the method of this paper.

results of Hagedorn gave me the courage to attempt such calculations for general groups and to obtain closed multiplicity formulas for orthogonal groups. It is a pleasure to thank Dick Gross for initiating the work in [12] which led to this paper, for helpful remarks on an earlier version, and for aquainting me with Hagedorn's thesis. The referee read the original version of this paper with care and insight, made valuable comments and simplified some of the arguments. In particular, the proof of Lemma 3.1 given below is due to the referee and is much shorter than the original one. Some general notation: The cardinality of a finite set X is denoted by \X\. Equivalence classes are generally denoted by [ ], sometimes with ornamentation. If g is an element of a group G, we write Ad(g) for the conjugation map Ad(g) : x i? gxg~x, and also write 9T := gTg~x for a subgroup T C G. The center of G is denoted Z(G) and the centralizer of g G G is denoted CG(g). We write ( , )# for the pairing on the space of class functions on a finite group H, for which the irreducible characters of H are an orthonormal basis. l?G,Gf D H are finite overgroups of H and i?, ip' are class functions on G, G' respectively, then (^^')h is understood to mean (^\h^'\h)h^ where \h denotes restriction to H. 2. Remarks on maximal tori Let G be a connected reductive algebraic J-group. We assume G is defined over f and has Frobenius F. If T is a maximal torus in G we denote its normalizer in G by NG(T) and write WG(T) = NG(T)/T for the Weyl group of T in G. If T is F-stable, we have W(T)F = NG(T)F/TF, by the Lang-Steinberg theorem. The reduction formula for Deligne-Lusztig characters (recalled in section 4 below) involves a sum over the following kind of subset of GF. Fix an F-stable maximal torus T C G, and let s be a semisimple element in GF. We must sum over the set NG(s,T)F:={j GF: *" T}. Note that NG(s, T)F, if nonempty, is a union of GF x NG(T)F double cosets, where Gs := CG(s)? is the identity component of the centralizer CG(s) of s in G. To say that s^ G T is to say that 7TcGs, so determining the GF x NG(T)F double cosets in NG(s,T)F amounts to determining the Gf-conjugacy classes of F-stable maximal tori in Gs which are contained in a given GF-conjugacy class. Such classes of tori are parameterized by twisted conjugacy classes in Weyl groups of Gs and G. The aim of this section is to parameterize the GF x NG(T)F double cosets in NG(s, T)F in terms of the fiber of a natural map between twisted conjugacy classes in the Weyl groups of Gs and G. This parameterization will be fundamental to our later calculations with Deligne-Lusztig characters. We begin by recalling the classification of F-stable maximal tori in G. See [5, chap. 3] for more details in what follows. Fix an F-stable maximal torus To in G contained in an F-stable Borel subgroup of G, and abbreviate NG = Ng(Tq), Wg = WG(T0). This content downloaded from 207.46.13.149 on Mon, 03 Oct 2016 06:07:19 UTC All use subject to http://about.jstor.org/terms ON THE RESTRICTION OF DELIGNE-LUSZTIG CHARACTERS 577 Let T(G) denote the set of all F-stable maximal tori in G. Then T(G) is a finite union of GF-orbits. For any T G T(G), let [T\g := {7T : 7 G GF} denote the GF-orbit of T. There is g G G such that T = 9T0. Since T is F-stable, we have g~1F(g) G AfcThis gives an element w := g~1F(g)T0 G WG. The map Ad(#)t =

In this paper we prove the existence and uniqueness of the solutions to the one-dimensional linear stochastic differential equation with Skorohod integral $$ X_{t} {\left( w \right)} = \eta {\left( w \right)} + {\int_0^t {a_{s} X_{s} {\left( w \right)}d} }{\text{\bf W}}_{s} + {\int_0^t {b_{s} X_{s} {\left( w \right)}ds,t \in {\left[ {0,1} \right]}} }, $$ where (Ws) is the canonical Wiener process defined on the standard Wiener space (\({\fancyscript W}\) ,ℋ , μ), a is non-smooth and adapted, but η and b may be anticipating to the filtration generated by (Ws). The intention of the paper is to eliminate the regularity of the diffusion coefficient a in the Malliavin sense, in the existing literature. The idea is to approach the non-smooth diffusion coefficient a by smooth ones.

In 1918 Pólya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Pólya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Pólya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters.

A new method, which enables us to compute rather efficiently the Galois group of a polynomial over ℚ or over ℤ, is presented. Reductions of this polynomial with respect to different prime modules are studied, and the information obtained is used for the calculation of the Galois group of the initial polynomial. This method uses an original modification of the Chebotarev density theorem, and it is in essence a probabilistic method. The irreducibility of the polynomial under consideration is not assumed. The appendix to this paper contains tables, which enable us to find the Galois group of polynomials of degree less than or equal to 10 as a subgroup of the symmetric group. Here the final part of the paper is published. The first part is contained in a previous issue (see Vol. 134, No. 6 (2006)). Bibliography: 10 titles.

Let ( R , m ) (R,\mathfrak {m}) be an n n -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an m \mathfrak {m} -primary ideal a \mathfrak {a} of R R , the relationship between the singularities of the scheme defined by a \mathfrak {a} and those defined by the multiplier ideals J ( a c ) \mathcal {J}(\mathfrak {a}^c) , with c c varying in Q + \mathbb {Q}_+ , are quantified in this paper by showing that the Samuel multiplicity of a \mathfrak {a} satisfies e ( a ) ≥ ( n + k ) n / c n e(\mathfrak {a}) \ge (n+k)^n/c^n whenever J ( a c ) ⊆ m k + 1 \mathcal {J}(\mathfrak {a}^c) \subseteq \mathfrak {m}^{k+1} . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustaţǎ and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

Every non-associative algebra Lcorresponds to its sym- metric semi-Lie algebra L(;) with respect to its commutator. It is an interesting problem whether the equality Autnon(L) = AutsemiiLie(L) holds or not (2), (13). We flnd the non-associative algebra au- tomorphism groups Autnon (WN0;0;1(0;1;r1;:::;rp) ) and AutsemiiLie (WN0;0;1(0;1;r1;:::;rp)), where every automorphism of the automor- phism groups is the composition of elementary maps (3), (4), (7), (8), (9), (10), (11). The results of the paper show that the F-algebra au- tomorphism groups of a polynomial ring and its Laurent extension make easy to flnd the automorphism groups of the algebras in the paper.

A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following: A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(log⁡n)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(log⁡n)1−o(1) such that not all graphs in the family have O(n) edges. Trevisan and Arora-Barak-Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges. Sudakov and the second author have recently shown that every graph with average degree d contains an n-vertex subgraph with average degree at least (1−o(1))d and vertex expansion 1/(log⁡n)1+o(1). We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1). The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super-linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.

By the classification theorem by F. Oort and J. Tate [6], any group scheme of prime order is isomorphic to a group scheme Ga,b under the suitable choice of a and b. We computed the torsors for some kinds of group schemes Ga,b in [8], which is a joint work with T. Sekiguchi, as in the following way: denote by p a prime number and by m = φ(p−1) the value of the Euler function φ. Suppose p is a prime ideal lying over p (which splits completely in Z[ζ]), where ζ is a primitive (p − 1)-st root of the unity. In case p is principal, the sequence 0 → μp,B → G m m,B p −→ Gm,B → 0 is exact, and the Galois descent of μp,B is isomorphic to Ga,b under the suitable choice of a and b, thus one can compute the torsors for this kinds of group schemes. The non-principal case is solved by Y. Koide [3] by using our method. The aim of this paper is to study some group schemes of order a power of a prime number. In section from 1 to 3, we would like to review the main result of the papers [6] by F. Oort and J. Tate, [4] by Y. Koide and T. Sekiguchi, and [8] by T. Sekiguchi and Y. Toda. In section 4, we give our main result, namely, the torsor for the Galois descent of μpn,B.

Assume that m ≥, p is a prime number, (m, p(p – 1)) = 1, −1 ∉ 〈p〉 ⊂ (ℤ/mℤ)* and [(ℤ/mℤ)* : 〈p〉] = 4. In this paper, we calculate the value of Gauss sum \( G{\left( \chi \right)} = {\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left( x \right)}\zeta ^{{T{\left( x \right)}}}_{p} } } \) over \({\mathbb F}\) q , where q = p f , \( f = \frac{{\varphi {\left( m \right)}}} {4} \), χ is a multiplicative character of \({\mathbb F}\) q and T is the trace map from \({\mathbb F}\) q to \({\mathbb F}\) p . Under our assumptions, G(χ) belongs to the decomposition field K of p in ℚ(ζ m ) and K is an imaginary quartic abelian number field. When the Galois group Gal(K/ℚ) is cyclic, we have studied this cyclic case in another paper: "Gauss sums of index four: (1) cyclic case" (accepted by Acta Mathematica Sinica, 2003). In this paper we deal with the non–cyclic case.

We introduce interpolation analogs of the Schur Q-functions — the multiparameter Schur Q-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and two-row functions, vanishing and characterization properties, a Pieri-type formula, a Nimmo-type formula (a quotient of two Pfaffians), a Giambelli-Schur-type Pfaffian formula, a determinantal formula for the transition coefficients between multiparameter Schur Q-functions with different parameters. We write an explicit Pfaffian expression for the dimension of a skew shifted Young diagram. This paper is a continuation of the author's paper math. CO/0303169 and a partial projective analog of the paper q-alg/9605042 by A. Okounkov and G. Olshanski and the paper math. CO/0110077 by G. Olshanski, A. Regev, and A. Vershik. Bibliography: 36 titles.

In this part of the paper, we investigate the structure of an arbitrary measure μ supported by a polyhedral cone C in R d in the case where the decumulative distribution function gμ of the measure μ satisfies certain continuity conditions. If a face Γ of the cone C satisfies appropriate conditions, the restriction μ|Γint of the measure μ to the interior part of Γ is proved to be absolutely continuous with respect to the Lebesgue measure λΓ on the face Γ. Besides, the density of the measure μ|Γint is expressed as the derivative of the function gμ multipied by a constant. This result was used in the first part of the paper to find the finite-dimensional distributions of a monotone random field on a poset. Bibliography: 6 titles.

Let p be a prime, m ≥ 2, and (m, p(p – 1)) = 1. In this paper, we will calculate explicitly the Gauss sum $$ G{\left( \chi \right)} = {\sum {_{{x \in \mathbb{F}^{ * }_{q} }} \chi {\left( x \right)}\zeta ^{{T{\left( x \right)}}}_{p} } } $$ in the case of [(ℤ/mℤ)* : 〈p〉] = 4, and −1 ∉ 〈p〉, where q = p f , $$ f = \frac{{\varphi {\left( m \right)}}} {4} $$ , χ is a multiplicative character of $${\Bbb F}$$ q with order m, and T is the trace map for $${\Bbb F}$$ q / $${\Bbb F}$$ p . Under the assumptions [(ℤ/mℤ)* : 〈p〉] = 4 and −1 ∉ 〈p〉, the decomposition field of p in the cyclotomic field ℚ(ζ m ) is an imaginary quartic (abelian) field. And G (χ) is an integer in K. We deal with the case where K is cyclic in this paper and leave the non–cyclic case to the next paper.

We will show in this paper that if λ is very close to 1, then $$ I(M,\lambda ,m) = {\mathop {\sup }\limits_{u \in H^{{1,n}}_{0} (M),\smallint _{M} {\left| {\nabla u} \right|}^{n} dV = 1} }{\int_\Omega {{\left( {e^{{\alpha _{n} {\left| u \right|}^{{\frac{n} {{n - 1}}}} }} - \lambda {\sum\limits_{k = 1}^m {\frac{{{\left| {\alpha _{n} u^{{\frac{n} {{n - 1}}}} } \right|}}} {{k!}}} }} \right)}dV} } $$ can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" (Comm. Pure. Appl. Math., 55, 135–152, 2002).

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 19, Number 1, Pages 171–180 S 0894-0347(05)00500-X Article electronically published on August 15, 2005 DOUBLE NODE NEIGHBORHOODS AND FAMILIES OF SIMPLY CONNECTED 4-MANIFOLDS WITH b + = 1 RONALD FINTUSHEL AND RONALD J. STERN 1. Introduction A basic question of 4-manifold topology is whether the complex projective plane, CP 2 , admits exotic smooth structures. Thus one is interested in knowing the smallest m for which CP 2 #m CP admits an exotic smooth structure. In the late 1980s, Dieter Kotschick [K] proved that the Barlow surface, which was known to be homeomorphic to CP 2 # 8CP , is not diffeomorphic to it. In the following years the subject of simply connected smooth 4-manifolds with b + = 1 languished because of a lack of suitable examples. However, largely due to a beautiful paper of Jongil Park [P], who found the first examples of exotic simply connected 4-manifolds with b + = 1 and b − = 7, the past year has found renewed interest in this subject. Peter Ozsvath and Zoltan Szabo [OS] proved that Park’s manifold is minimal, and Andras Stipsicz and Szabo [SS] used a technique similar to Park’s to construct an exotic manifold with b + = 1 and b − = 6. The rational elliptic surface E(1) ∼ CP 2 #9 CP admits infinitely many distinct smooth structures. Up to now it has not been known whether CP 2 #m CP can have an infinite family of smooth structures when m < 9. It is the purpose of this paper to introduce a new technique which we use to show that for m = 6, 7, 8, CP 2 #m CP does have an infinite family of smooth structures. The construction of these examples to some extent follows the ideas of Park [P] after performing suitable knot surgeries [FS3] on E(1). The essential new idea is that after certain knot surgeries on E(1), one is able to find an immersed sphere of self-intersection −1 representing the ‘pseudo-section’. This is accomplished by studying ‘double node neighborhoods’ as described below. Shortly after the appearance of a preliminary version of this article, Park, Stip- sicz, and Szabo [PSS] used the techniques described herein to give even better examples: an infinite family of pairwise nondiffeomorphic smooth 4-manifolds all homeomorphic to CP 2 #5 CP . In the final section of this paper we give a quick construction of such a family of examples. Received by the editors January 13, 2005. 2000 Mathematics Subject Classification. Primary 14J26, 53D05, 57R55, 57R57. Key words and phrases. 4-manifold, Seiberg-Witten invariant, knot surgery, rational blowdown. The first author was partially supported by NSF Grant DMS0305818 and the second author by NSF Grant DMS0204041. c 2005 American Mathematical Society Reverts to public domain 28 years from publication

The main aim of this paper is to use the continuation theorem of coincidence degree theory for studying the existence of periodic solutions to a kind of neutral functional differential equation as follows: $$ {\left( {x{\left( t \right)} - {\sum\limits_{i = 1}^n {c_{i} x{\left( {t - r_{i} } \right)}} }} \right)}^{{\prime \prime }} = f{\left( {x{\left( t \right)}} \right)}{x}\ifmmode{'}\else$'$\fi{\left( t \right)} + g{\left( {x{\left( {t - \tau } \right)}} \right)} + p{\left( t \right)}. $$ In order to do so, we analyze the structure of the linear difference operator A : C 2π →C 2π, \( {\left[ {Ax} \right]}{\left( t \right)} = x{\left( t \right)} - {\sum\nolimits_{i = 1}^n {c_{i} x{\left( {t - r_{i} } \right)}} } \) to determine some fundamental properties first, which we are going to use throughout this paper. Meanwhile, we also prove some new inequalities which are useful for estimating a priori bounds of periodic solutions.

This paper presents the working notes by S. V. Kerov (1946–2000) written in 1993. The author introduces a multidimensional analog of the classical hypergeometric distribution. This is a probability measure Mn on the set of Young diagrams contained in the rectangle with n rows and m columns. The fact that the expression for Mn defines a probability measure is a nontrivial combinatorial identity, which is proved in various ways. Another combinatorial identity analyzed in the paper expresses a certain coherence between the measures Mn and Mn+1. A connection with Selberg-type integrals is also pointed out. The work is motivated by harmonic analysis on the infinite-dimensional unitary group. Bibliography: 25 titles.

For A|F a central simple algebra over a p-adic local field the group of units A× ∼= GLm(Dd) is a general linear group over a central division algebra Dd|F of index d. The product n = dm being fixed, the Abstract Matching Theorem (AMT) implies the existence of bijective maps JL between the sets of discrete series representations of the groups A× such that a character relation is preserved. In this paper we construct maximal level zero extended type components for every level zero discrete series representation of A×. Its maximal level zero extended type determines the discrete series representation uniquely (without any twist ambiguities as for the usual types) and, conversely, every level zero discrete series representation Π contains a maximal level zero extended type component Σ(Π) which is unique up to conjugacy. In order to determine how JL matches the extended types we find certain regular elliptic elements where the characters of Σ(Π) and Π are the same and we compute the character values at these elements by using a version of Shintani descent which we develop in Appendix B. Surprisingly, we find that AMT also implies explicit Shintani descent for irreducible characters of finite general linear groups which have cuspidal descent. named author Let F be a locally compact p-adic field, let Dd be a central division algebra of index d over F , and let A = Mm(Dd), the algebra of all m × m matrices with entries in Dd. Then A is a central simple F algebra of reduced degree n := dm and the group of units A× of A is the group of F points of a connected reductive algebraic group defined over F . The group A× is separable, totally disconnected, and unimodular and the harmonic analysis of A×, as is well known, is linked to numerous problems in arithmetic. It is interesting, and perhaps may be regarded as a version of the Abstract Matching Theorem (AMT) (see §0.2), that the representation theory of A× depends upon the parameters d,m associated to A, but not more specifically on the isomorphism class of A. We follow the convention of writing Dd for any central F division algebra of index d; we also write Dn for any one of the φ(n) isomorphism class representatives of central F division algebras of index n. 1991 Mathematics Subject Classification. 22E50. 1The work reported in this paper grew out of a “research in pairs” sojourn at the Mathematisches Institut, Oberwolfach. The authors wish to express their appreciation to the Institut for its support of their research as well as for an enjoyable stay at the Institut.

This paper is an English translation (made by V. Gelfreich) of V. F. Lazutkin’s work that was published in 1984 by VINITI and thus was not easily available for readers. In the paper, a formula for an exponentially small angle of separatrix splitting of the Chirikov standard map was obtained for the first time. Bibliography: 16 titles and 17 titles added by the translator.

In this paper, we are concerned with the existence of solutions of the following multi-point boundary value problem consisting of the higher-order differential equations $ x^{(n)}(t)=f(t,x(t),x'(t),\cdots,x^{(n-1)}(t))+e(t),\;\;0 and the following multi-point boundary value conditions $ \begin{array}{ll} x^{(i)}(0)=0\;\;for\;i=0,1,\cdots,n-3,\\ x^{(n-2)}(0)=\alpha x^{(n-1)}(\xi),\;\;x^{(n-1)}(1)=\beta x^{(n-2)}(\eta),\end{array} \eqno{(\ast\ast)} $ Sufficient conditions for the existence of at least one solution of the BVP$ (\ast) $ and $ (\ast\ast) $ at resonance are established. This paper is directly motivated by Liu and Yu [India J. Pure Appl. Math., 33(4)(2002)475-494] and Qi [Acta Math. Appl. Sinica, 17(2)(2001)271-278].

In this paper, we will study the nonelementary groups of Mobius transformations in \( \overline{R} ^{n} \) and some properties are obtained. Also in this paper we will prove several theorems about discreteness criteria and group convergence of nonelementary groups of M(\( \overline{R} ^{n} \)).

When A ∈ B(H) and B ∈ B(K) are given, we denote by MC an operator acting on the Hilbert space H ⊕ K of the form \( M_{C} = {\left( {\begin{array}{*{20}c} {A} & {C} \\ {0} & {B} \\ \end{array} } \right)} .\)In this paper, first we give the necessary and sufficient condition for MC to be an upper semi-Fredholm (lower semi–Fredholm, or Fredholm) operator for some C ∈ B(K,H). In addition, let \( \sigma _{{SF_{ + } }} \) (A) ={λ ∈ ℂ : A − λI is not an upper semi-Fredholm operator} be the upper semi–Fredholm spectrum of A ∈ B(H) and let \( \sigma _{{SF_{ - } }} \) (A) = {λ ∈ ℂ : A − λI is not a lower semi–Fredholm operator} be the lower semi–Fredholm spectrum of A. We show that the passage from \( \sigma _{{SF_{ ±} }} {\left( A \right)} \cap \sigma _{{SF_{ ±} }} {\left( B \right)}\;{\text{to}}\;\sigma_{{SF_{±} }} {\left( {M_{C} } \right)} \) is accomplished by removing certain open subsets of \( \sigma _{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)} \) from the former, that is, there is an equality $$ \sigma_{{SF_{± } }} {\left( A \right)} \cup \sigma _{{SF_{ ±} }} {\left( B \right)} = \sigma_{{SF_{±} }} {\left( {M_{C} } \right)} \cup {\fancyscript G},$$ where \({\fancyscript G}\)is the union of certain of the holes in \( \sigma_{{SF_{± } }} {\left( {M_{C} } \right)} \) which happen to be subsets of \( \sigma_{{SF_{ - } }} {\left( A \right)} \cap \sigma_{{SF_{ + } }} {\left( B \right)} .\)Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl's theorem, Browder's theorem, a–Weyl's theorem and a–Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.

In this paper, we introduce the SP-integral and the SPfi-integral deflned on an interval in the n-dimensional Euclidean space R n . We also investigate the relationship between these two integrals. It is well known (3) that the Perron integral deflned on an interval of the real line R by major and minor functions which are not assumed to be continuous is equivalent to the one deflned by continuous major and minor functions and that the strong Perron integral deflned on an interval of R by strong major and minor functions is equivalent to the McShane integral. In this paper, we introduce Perron-type integrals deflned on an inter- val of the n-dimensional Euclidean spaceR n using the strong major and minor functions, and investigate the relationship between these integrals. We shall call it the strong Perron integral, or brie∞y SP-integral. For a subset E of the n-dimensional Euclidean spaceR n , the Lebesgue measure of E is denoted by jEj. For a point x = (x1;x2;¢¢¢ ;xn) 2R n , the norm of x is kxk = max1•in jxij and the --neighborhood U(x;-) of x is an open cube centered at x with sides equal to 2-. For an interval I = (a1;b1)£(a2;b2)£¢¢¢(an;bn) ofR n with ai fi(fi 2 (0;1)), then the interval I is said to be fi-regular.

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

The paper considers a relationship between the Chern numbers $K^2_X,\,c_2(X)$ of a smooth minimal surface $X$ of general type and the dimension of the space of infinitesimal deformations of $X$, i.e. $h^1(\Theta_X)$, where $\Theta_X$ is the holomorphic tangent bundle of $X$. We prove that if the ratio of the Chern numbers $\alpha(X) = \frac{c_2(X)}{K^2_X} \leq \frac38$ and $K_X$ is ample then $$h^1(\Theta_X) \leq 9(3c_2-K^2).$$ On the geometric side it is shown that a smooth surface of general type $X$ with $\alpha(X)\leq \frac38$ and $h^1(\Theta_X) \geq 3$ has two distinguished effective divisors $ F$ and $E$ such that $H^1(\Theta_X)$ admits a direct sum decomposition $H^1(\Theta_X) = V_1 \oplus V_0$, where $V_1$ is identified with a subspace of $H^0(\mathcal{O}_X (F))$ while $V_0$ is identified with a subspace of $H^0(\Theta_X \otimes \mathcal{O}_E (E))$. This gives a geometric interpretation of the cohomology classes in $H^1(\Theta_X)$ and allows to bound the dimension of $V_0$ (resp. $V_1$) in terms of geometry of the divisor $E$ (resp. $F$). ¶ The main idea of the paper is to use the natural identification $$H^1(\Theta_X) = Ext^1(\Omega_X, \mathcal{O}_X)$$ where $\Omega_X$ is the holomorphic cotangent bundle of $X$. Then the "universal" extension gives rise to a certain vector bundle whose study constitutes the essential part of the paper.

Let β > 1 \beta >1 be a Pisot unit. A family of sets { X i } 1 ≤ i ≤ q \{X_i\}_{1\leq i\leq q} defined by a β \beta -numeration system has been extensively studied as an atomic surface or Rauzy fractal. For the purpose of constructing a Markov partition, a domain X ^ = ⋃ i = 1 q X ^ i \hat X=\bigcup _{i=1}^q \hat X_i constructed by an atomic surface has appeared in several papers. In this paper we show that the domain X ^ \hat X completely characterizes the set of purely periodic β \beta -expansions.

An excellent introduction to the topic of poset matroids is due to M. Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rank axioms for poset matroids. In this paper, we study the special integral function f and obtain a new class of poset matroids from the old ones, and then we generalize this result according to the properties of f. Almost all of these results can be regarded as the application of global rank axioms for poset matroids. The main results in our paper have, indeed, investigated the restriction of the basis of the poset matroid, and we give them the corresponding geometric interpretation.

In the first paper of the series, we proved the standardness of a subgroup H containing a split maximal torus in the split spinor group Spin(n,R) over a field K of characteristic different from 2 containing at least 7 elements under one of the following additional assumptions: (1) H is reducible, (2) H is imprimitive, (3) H contains a nontrivial root element. In the present paper, we complete the proof of a result announced by the author in 1990 and prove the standardness of all intermediate subgroups, provided that n=2l and \(\left| K \right| \geqslant 9\). For an algebraically closed K, this follows from a classical result of Borel and Tits, and for a finite K this was proved by Seitz. Similar results for subgroups of the orthogonal groups SO(n,R) were previously obtained by the author not only for fields, but for any commutative semilocal rings R with residue fields large enough. Bibliography: 52 titles.

Suppose that X- = (X 0 , X 1 ) and Y-=(Y 0 ,Y 1 ) are Banach couples and suppose that T 0 : X 0 → Y 0 and T 1 : X 1 → Y 1 are bounded and linear. Also assume that Γ ∈ (Δ(X-))' and that T 0 and T 1 agree as maps from Δ(X-) ∩ ker Γ to Σ(Y-). If the maps do not agree as maps from all of Δ(X-) we cannot interpolate T 0 and T 1 to a map T : J 0,p (X- → J 0,p (Y-), where J 0,p denotes the classical J-method. This situation can for example be found in an article on interpolation of Hardy-type inequalities by Krugljak, Maligranda and Persson. We will in this paper define functors J 0,p;Γ such that T 0 and T 1 interpolate to a map T: J 0,p:Γ (X-) → J 0,p (Y-). The main purpose of this paper is to make the definition of the J 0,p;Γ (X-) spaces and build a theory for them. We will also do this for more general real parameters. If Γ is bounded on X 0 it holds that J 0,p:Γ (X-)=J 0,p (X 0 ∩ ker Γ, X 1 ). These spaces have been studied by Kalton, Ivanov and Lofstrom. Their results will follow as corollaries to the more general results of this article and our new theory can be thought of as a theory for generalized subcouples of codimension one. In the last section, we apply our theory to a situation considered by Krugljak, Maligranda and Persson in connection with Hardy-type inequalities. We prove new results and provide a new way of understanding that kind of problems.

In [Y], Yau proposed to study the uniformization of complete Kahler manifolds with nonnegative curvature. In particular, one wishes to determine whether or not a complete Kahler manifold M with positive bisectional curvature is biholomorphic to Cm. See also [GW], [Si]. For this sake, it was further asked in [Y] whether or not the ring of the holomorphic functions with polynomial growth, which we denote by Op(M), is finitely generated, and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on Cm. This paper addresses the latter questions. We denote by 0?(M) the space of holomorphic functions of polynomial growth with degree d. (See Section 3 for the precise definition.) Then Op(M) = \Jd>0 Od(M). In this paper, we show that

Loop groups are very important in differential geometry, algebraic topology, and theoretical physics [7, 9, 16, 37, 47], but nothing was known about Gaussian quasi-invariant differentiable measures on them. Only the simplest possible representations associated with path integrals were constructed for loop groups of the circle, i.e., for the manifold M = S1 and (real) Riemannian manifolds N [37]. On the other hand, quasi-invariant measures can be used for construction of regular unitary representations [30–32,35]. Moreover, quasi-invariant measures are helpful for investigation of a group itself. In the previous papers of the author [33, 34], loop groups of Riemannian manifolds M and N were investigated, where either M = S is an n-dimensional real sphere, n = 1, 2, . . . , or M = S∞ is the unit sphere in a real separable Hilbert space l2(R). This was progress in comparison with previous works of others authors, which considered only loop groups for the simplest case M = S1. This paper treats arbitrary complex separable connected metrizable manifolds M and N . For example, products of odd-dimensional real spheres S2n−1×S2m−1 can be endonved with structures of complex manifolds in different ways [29]. Other numerous examples of complex manifolds can be found in [25] and references therein; they are domains in Cn, the complex torus Cn/D, where D is a discrete additive subgroup of Cn generated by a basis τ1, . . . , τ2n of C n over R; the quotient space G/D of a complex Lie group G by a discrete subgroup D, submanifolds of the complex Grassmann manifold Gp,q(C), and also their different products and submanifolds. In general, there are complex compact manifolds which are not Kahler manifolds [28,36]. For construction of loop groups here, we use manifolds M with some mild additional conditions. When M is finite-dimensional over C, we assume that it is compact. This condition is not very restrictive, since each locally compact space admits the Alexandrov (one-point) compactification (see [13, Theorem 3.5.11]). When M is infinite-dimensional over C, it is assumed that M is embedded as a closed bounded subset in the corresponding Banach space XM over C. This is necessary for defining a group structure on the quotient space of a free loop space. The free loop space is considered as consisting of continuous functions f : M → N which are holomorphic on M \ M ′ and preserving distinguished points f(s0) = y0, where M ′ is a closed real submanifold depending on f of codimension codimRM ′ = 1, s0 ∈ M , and y0 ∈ N are distinguished points. There are two reasons to consider such a class of mappings. The first is the need to correctly define compositions of elements in the loop group (see below). The second is the isoperimetric inequality for holomorphic loops, which can impose the condition that the loop be constant on a sufficiently small neighborhood of s0 in M if this loop is in some small neighborhood of w0, where w0(M) := {y0} is a constant loop (see [23, Remark 3.2]). In this paper, loop groups of different classes are considered. Classes analogous to Gevrey classes of f : M \M ′ → N are considered for construction of dense loop subgroups and quasi-invariant measures. Henceforth, we consider only orientable manifolds M and N , since for a nonorientable manifold, there always exists its orientable double covering manifold (see [1, 6.5]). Loop commutative monoids with cancellation property are quotients of families of mappings f from M into a manifold N with f(s0) = y0

It seems that in Mane’s proof of the C 1Ω-stability conjecture containing in the famous paper which published in I. H. E. S. (1988), there exists a deficiency in the main lemma which says that for f∈ $${\fancyscript F}$$ 1(M) there exists a dominated splitting $$ TM_{{\left| {\overline{P} _{i} {\left( f \right)}} \right.}} = \widetilde{E}^{s}_{i} \oplus \widetilde{F}^{u}_{i} $$ (0 < i < dim M) such that if $$ \widetilde{E}^{s}_{i} $$ is contracting, then $$ \widetilde{F}^{u}_{i} $$ is expanding. In the first part of the paper, we give a proof to fill up this deficiency. In the last part of the paper, we, under a weak assumption, prove a result that seems to be useful in the study of dynamics in some other stability context.

Let $ R $ be an integral domain and $ \alpha $ an anti-integral element of degree $ d $ over $ R $. In the paper [3] we give a condition for $ \alpha^2-a$ to be a unit of $ R[\alpha] $. In this paper we will generalize the result to an arbitrary positive integer $n$ and give a condition, in terms of the ideal $ I_{[\alpha]}^{n}D(\sqrt[n]{a}) $ of $ R $, for $ \alpha^{n}-a$ to be a unit of $ R[\alpha] $.

Let Q: R → R be even, nonnegative and continuous, Q' be continuous, Q' > 0 in (0, ∞), and let Q'' be continuous in (0, ∞). Furthermore, Q satisfies further conditions. We consider a certain generalized Freud-type weight WrQ2(x) = |x|2r exp(-2Q(x)). In previous paper (J. Approx. Theory 121 (2003) 13) we studied the properties of orthonormal polynomials {Pn(WrQ2; x)}n=0x with the generalized Freud-type weight WrQ2(x) on R. In this paper we treat three themes. Firstly, we give an estimate of Pn(WrQ2; x) in the Lp-space, 0 < p ≤ ∞. Secondly, we obtain the Markov inequalities, and third we study the higher-order Hermite Fejer interpolation polynomials based at the zeros {xkn}k=1n of Pn(WrQ2; x). In Section 5 we show that our results are applicable to the study of approximation for continuous functions by the higher-order Hermite-Fejer interpolation polynomials.