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The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

The object of this note is to point out and correct an error in the paper [21 of Fox, purporting to prove Fenchel's conjecture that a finitely generated. infinite Fuchsian group has a torsion-free normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar. Given any three integers a > 1, b > 1 and c > 1, there exist permutations A of order a and B of order b, such that AB has order c. Fox divided the proof of this lemma into the four cases: (I) ab 0 (mod 2) and max{a, b} ? c < a + b, (II) ab 0 (mod 2) and c : a + b, (III) a b c 1 (mod 2), (IV) a b 1 and c 0 (mod 2), and subcases. In the subcase m(a + b 2) + 2 ? c < (m + l)(b + 1) 1 of Case (III) where m is the largest integer not exceeding (c l)/(b + 1), and it is assumed that a ? b < c, the number of distinct symbols in the permutations [2, p. 64] nl~~~W B ( Wv-sI .. WI2 I7 (vWb B = (vl * *t*v W_ ***W P2) * (V W2 .. * ]Pi+]1)'

In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z

This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement "that distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with G [p] the same subgroup of G" is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of G supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

F(k) ={xkj x F}; F,=F; Fk= (Fk_l,F). Let (u, v, 0)=u, (u, v, 1)=(u, v), (u, v, n)=((u, v, n-1), v). Then Sanov [3] proved that (1. 1) (u, v, apa _1) p-a E F(p#)Fapa+?, /3 a = 1, 2, where p is a prime. In this paper, (1.1) is proved for the cases a= 1, 2; 3 arbitrary. A slight generalization of these results is also proved. Sanov's proof involved an investigation of ideals in a Lie Ring. In this paper, Hall's Collection Process will be used. The method also yields other formulas, e.g. (1.2) (u, v, p2 1)P' (E F2p2_pF(p) (1.3) (u, v, pa+l 1)p~l C F2pa+l paF(pl), a = 1 2, and can be used to produce numerous formulas of a similar nature. The author hopes that some of these formulas and/or the method may be of use in solving other group-theoretic problems. The author was unable to use the method to prove (1.1) for a =3. Note that for a=f = 1, (1. 1) becomes (1.4) (U, v, p 1) E F(p)Fv+1. (1.4) plays an important role in the theory of the Restricted Burnside Problem.

If BP denotes the space of smooth alternating p-forms on the C' manifold M, we are interested in finding the spaces of R-linear maps from iPP to be that commute with the diffeomorphisms of M. For a compact manifold M these spaces were found by R. S. Palais. In this note we find them for noncompact M. Let V, W be the spaces of sections of two tensor bundles over a connected Coo manifold M of dimension n. In a paper [3] appearing in Trans. Amer. Math. Soc. 92 (1959), R. S. Palais studied a number of the spaces 4 (V, W) of the R-linear maps V->W natural with respect to the action on V and W of the group G of diffeomorphisms on M. We sketch the definition of this action: If g-G we define R,:F(TM) -4F(TM) to be the differential of g, and Rg:r(T*M)-->F(T*M) to be the dual of the differential of g-1. For the other spaces of tensor fields Rg is defined as the obvious tensor product of these two R-linear isomorphisms. A map c: V->W is then called natural if cRg =R,c for all g G. Palais was particularly interested in the case when V and W are the space IP of p-forms and the space bq of q-forms respectively, 0 < p, q

The first part of the theorem of the above paper states that if ois a closed convex set in real X1 ... Xn-space which is invariant under permutations of coordinates, and if C(o-) denotes the set of real symmetric nXn matrices whose eigenvalues X1, X. form the coordinates of points in a, then C(o) is convex. I am obliged to Professor R. T. Rockafellar for pointing out that this statement is essentially contained in a theorem of Chandler Davis.2 The statement also follows from an earlier result of V. B. Lidskii,3 which was not known to me at the time of publication.

C. Hobby and C. R. B. Wright [2] have just published the following Theorem A. However, their proof seems to contain an error.2 The notation of [2] is used except that G. is not reserved for the nth term of the lower central series of G: +/(G) denotes the Frattini subgroup of G; (G, H) means the group generated by the commutators g-lh-1gh where gEG, hEH; (A1, A2, * * *, A.+,) is defined inductively as ((A1, A2, . , A.), A.+,); HCG means that H is properly included in G.

PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence

In this short paper we establish a (non-trivial) lower bound on the degree two entry h 2 of a Gorenstein h-vector of any given socle degree e and any codimension r. In particular, when e = 4, that is, for Gorenstein h-vectors of the form h = (1, r, h 2 , r, 1), our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say f(r), that h 2 may assume. In fact, we show that limr→∞ f(r) r 2/3 =6 2/3 . In general, we wonder whether our lower bound is sharp for all integers e > 4 and r > 2.

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160

Let p p be an odd prime. h 1 ( p ) {h_1}(p) is the first factor of the class number of field Q ( ζ p ) Q({\zeta _p}) . We proved that \[ h 1 ( p ) ⩽ { 2 p ( p − 1 8 ( 2 l / 2 + 1 ) 4 / l ) ( p − 1 ) / 4 , if l is even, 2 p ( p − 1 8 ( 2 l − 1 ) 2 / l ) ( p − 1 ) / 4 , if l is odd . {h_1}(p) \leqslant \left \{ \begin {gathered} 2p{\left ( {\frac {{p - 1}} {{8{{({2^{l/2}} + 1)}^{4/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is even,}} \hfill \\ 2p{\left ( {\frac {{p - 1}} {{8{{({2^l} - 1)}^{2/l}}}}} \right )^{(p - 1)/4}},\quad {\text {if }}l\;{\text {is odd}}{\text {.}} \hfill \\ \end {gathered} \right . \] From that we obtain h 1 ( p ) ⩽ 2 p ( ( p − 1 ) / 31.997158 … ) ( p − 1 ) / 4 {h_1}(p) \leqslant 2p{((p - 1)/31.997158 \ldots )^{(p - 1)/4}} which is better than Carlitz’s and Metsänkyla’s results. For the fields Q ( ζ 2 n ) Q({\zeta _{{2^n}}}) and Q ( ζ p n ) ( n ⩾ 2 ) Q({\zeta _{{p^n}}})(n \geqslant 2) , we get the similar results.

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.

In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

In this paper we study the b b -ary expansions of the square roots of the function defined by the recurrence f b ( n ) = b f b ( n − 1 ) + n f_b(n)=b f_b(n-1)+n with initial value f ( 0 ) = 0 f(0)=0 taken at odd positive integers n n , of which the special case b = 10 b=10 is often referred to as the “schizophrenic” or “mock-rational” numbers. Defined by Darling in 2004 2004 and studied in more detail by Brown in 2009 2009 , these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 b\geq 2 by formally defining the schizophrenic pattern present in the b b -ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n (this hypothesis appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N and an explicit monic polynomial $f_0(x)\in \mathbb{Z}[x]$ of degree n, such that the Jacobian $J$ of every curve of the form $y^2=f(x)$ has $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and $Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, whenever $f(x)\in\mathbb{Z}[x]$ is monic with $f(x)\equiv f_0(x) \bmod{N}$ and with no roots of multiplicity greater than $2$ in $\overline{\mathbb{F}}_p$ for any p not dividing N., Comment: 24 pages, minor corrections

Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric., Comment: 50 pages, 14 figures. Final version

George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz, again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of Huh and Huh-Katz left open the more general Rota-Welsh conjecture where graphs are generalized to (not necessarily representable) matroids and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh-Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota-Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota-Welsh conjecture would follow from combinatorial analogues of the Hard Lefschetz Theorem and Hodge-Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial Hard Lefschetz Theorem and Hodge-Riemann relations using purely combinatorial arguments. We will survey these developments., Comment: 22 pages. This is an expository paper to accompany my lecture at the 2017 AMS Current Events Bulletin. v2: Numerous minor corrections

In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].

The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.

This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.

In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module V ♮ = ⨁ n V n ♮ V^\natural = \bigoplus _n V_n^\natural . Their results show that as n n tends to infinity, V n ♮ V_n^\natural is dominated by direct sums of copies of the regular representation. That is, if we view V n ♮ V_n^\natural as a module over the group ring Z [ M ] \mathbb {Z}[\mathbb {M}] , the free part dominates. A natural problem, posed at the end of the aforementioned paper, is to characterize the distribution of irreducible representations in the non-free part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson series to answer this question. We arrive at an ordering of the series by the magnitude of their coefficients, which corresponds to various contributions to the distribution. In particular, we show how the asymptotic distribution of the non-free part is dictated by the column for conjugacy class 2A in the monster’s character table. We find analogous results for the other monster modules V ( − m ) V^{(-m)} and W ♮ W^\natural studied by Duncan, Griffin, and Ono.

Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.

The aim of this paper is to define the notion of the Cohen- Macaulay cone of a Noetherian local domain R R and to present its applications to the theory of Hilbert-Kunz functions. It has been shown by the second author that with a mild condition on R R , the Grothendieck group G 0 ( R ) ¯ \overline {G_0(R)} of finitely generated R R -modules modulo numerical equivalence is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R R is the cone in G 0 ( R ) ¯ R \overline {G_0(R)}_{\mathbb R} spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the Cohen-Macaulay cone in this paper. As an application, various examples of Hilbert-Kunz functions in the polynomial type will be produced. Precisely, for any given integers ϵ i = 0 , ± 1 \epsilon _i = 0, \pm 1 ( d / 2 > i > d d/2 > i > d ), we shall construct a d d -dimensional Cohen-Macaulay local ring R R (of characteristic p p ) and a maximal primary ideal I I of R R such that the function ℓ R ( R / I [ p n ] ) \ell _R(R/I^{[p^n]}) is a polynomial in p n p^n of degree d d whose coefficient of ( p n ) i (p^n)^i is the product of ϵ i \epsilon _i and a positive rational number for d / 2 > i > d d/2 > i > d . The existence of such ring is proved by using Segre products to construct a Cohen-Macaulay ring such that the Chow group of the ring is of certain simplicity and that test modules exist for it.

In this paper we disprove a conjecture stated in (4) on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative to the following question, raised in the same paper. Given a compact family C of closed cones and a set S such that every blow-up of S at every point x 2 S is contained in some element of C, is it true that the dimension of S is smaller than or equal to the largest dimension of a vector space contained is some element of C?

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in the boundary of a sutured manifold., Paper completely rewritten. Main sutured manifold theory results have been moved to a separate paper

We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros. This paper concerns the question of how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑∞ n=1 1 ns as the prototype and are important objects in number theory. The Selberg class S of L-functions is the set of all Dirichlet series L(s) = ∑∞ n=1 a(n) ns of a complex variable s = σ + it with a(1) = 1, satisfying the following axioms (see [7]): (i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series. (ii) (Analytic continuation) There is a nonnegative integer k such that (s − 1)L(s) is an entire function of finite order. (iii) (Functional equation) L satisfies a functional equation of type ΛL(s) = ωΛL(1− s), where ΛL(s) = L(s)Q s ∏K j=1 Γ(λjs+μj) with positive real numbers Q, λj and with complex numbers μj , ω with Reμj ≥ 0 and |ω| = 1. (iv) (Ramanujan hypothesis) a(n) n for every e > 0; (v) (Euler product) logL(s) = ∑∞ n=1 b(n) ns , where b(n) = 0 unless n is a positive power of a prime and b(n) n for some θ < 12 . The degree dL of an L-function L is defined to be dL = 2 ∑K j=1 λj , where K, λj are the numbers in axiom (iii). The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis. At the same time, there are a whole host of interesting Dirichlet series not possessing a Euler product (see e.g. [3], [8]). Throughout the paper, all L-functions are assumed to be functions from the extended Selberg class of those only satisfying the axioms (i)-(iii) (see [3]). Thus, the results obtained in the present paper particularly apply to L-functions in the Selberg class. Received by the editors October 5, 2011 and, in revised form, February 12, 2012. 2010 Mathematics Subject Classification. Primary 11M36, 30D30.

In a paper by Z. Lu and J. Rowlett, it is shown that the eigenvalues of the weighted Laplacian can be approximated by eigenvalues of a naturally associated family of narrow graphs. In this paper, we generalize this result to the p p -Laplacian. Our approach features overcoming the nonlinearity of the p p -Laplacian when p ≠ 2 p\neq 2 , which is different from the Laplacian case.

The question of which functions acting entrywise preserve positive semidefiniteness has a long history, beginning with the Schur product theorem [Crelle 1911], which implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions. A famous result of Schoenberg and of Rudin [Duke Math. J. 1942, 1959] shows the converse: there are no other such functions. Motivated by modern applications, Guillot and Rajaratnam [Trans. Amer. Math. Soc. 2015] classified the entrywise positivity preservers in all dimensions, which act only on the off-diagonal entries. These two results are at "opposite ends", and in both cases the preservers have to be absolutely monotonic. We complete the classification of positivity preservers that act entrywise except on specified "diagonal/principal blocks", in every case other than the two above. (In fact we achieve this in a more general framework.) This yields the first examples of dimension-free entrywise positivity preservers - with certain forbidden principal blocks - that are not absolutely monotonic., Minor edits in exposition, 19 pages. The paper now uses the style file of Trans. AMS (to appear)

Let k be an infinite perfect field and let X = SpecR, where R is the localization of a k-algebra B of finite type with respect to a regular prime ideal p ∈ SpecB. LetD be a divisor onX with a decompositionD = ∪i=1Di into its irreducible components Di such that each Di is nonsingular and they intersect transversally. Consider Y = X − D, a deleted neighbourhood of the closed point x0 of X . Let E be a vector bundle on Y . In this set up, E is trivial in the following cases : (i) dim(X) ≤ 3 (Gabber [Ga1]) ; (ii) t = 1 (Bhatwadekar-Rao [BR]) ; (iii) rankE ≥ min{[dimR 2 ], t} (Rao [Ra1]). The above exposition is borrowed from a paper of Nisnevich [Ni], which contains some beautiful applications of these ‘purity theorems’. In the same set up, now let y ∈ Y be a closed point and let my be the corresponding maximal ideal of OY,y. It is natural to ask whether my is a complete intersection. Observing that the conormal bundle my/my is trivial, we consider a more general situation : Let I be the defining ideal of a closed subscheme Z of Y such that the conormal bundle I/I2 (defined on Z) is trivial. Is I a complete intersection? The questions we are addressing in this paper are a little stronger in nature : Can a given basis of I/I2 be lifted to a minimal set of generators of I? We remark that with the hypothesis on X , each component Di is defined by a single equation fi which vanishes at the closed point x0 of X . Let m be the maximal ideal at x0. The condition of transversality of intersection of Di’s is equivalent to saying that the images of f1, · · · , ft in the A/m-vector space m/m2 are linearly independent. Keeping this discussion in mind, we now give an excerpt of our main results in commutative algebraic terms (see 4.2, 4.3, 4.4).

The object of this short note is to give some observations on Bernoulli numbers and their function field analogs and point out ‘known’ counter-examples to a conjecture of Chowla. Bernoulli numbers Bn defined (for integer n > 1) by z/(e z − 1) = ∑ Bnz /n!, and their important cousins Bn/n, play interesting roles in many areas of mathematics. (Below we only restrict to these for n even, precisely the case when they are non-zero.) We mention some key words by which the reader can search: Power sums, Zeta special values, Eisenstein series, measures, p-adic L-functions, finite differences, combinatorics, Euler-Maclaurin formula, Todd classes in topology, Grothendieck-Hirzebruch-Riemann-Roch formula, K-theory of integers, Stable homotopy, Bhargava factorial associated to the set of primes, Kummer-HerbrandRibet theorems in cyclotomic theory, Kervaire-Milnor formula for diffeomorphism classes of exotic spheres. Their factorization is of interest, the denominators (which show up explicitly in the third-fourth items from the end) are well understood via theorems of von-Staudt, but the numerators (which show up explicitly in the last two items above) are mysterious and connected to many interesting phenomena. In one of the rare lapses, Ramanujan, in his very first paper [R1911, (14), (18) and Sec. 12], claimed to have proved (editors downgrade it to a conjecture) that the numerator Nn of Bn/n is always a prime, when it was already known since Kummer (in Fermat’s last theorem connection) that ‘irregular’ prime 37 is a proper divisor of N32, and even N20 is composite. In [C1930], Chowla showed that Ramanujan’s claim had infinity of counter-examples. Note that this also follows from one counterexample and the Kummer congruences (recalled below) for that prime! Interestingly, in his last paper [CC1986], Chowla (jointly with his daughter) asks as unsolved problem whether the numerator is always square-free. (This is also mentioned in the nice survey article by Murtys and Williams on Chowla’s work in Vol. 1 of [C1999], where the author learned about it.) Theorem 1. Chowla’s conjecture stated above has infinity of counter-examples. In fact, for any given irregular prime p less than 163 million, and given arbitrarily large k, there is n such that p divides Nn. Proof. Using the tables (or the reader can try to check directly!) giving factorizations of Bn/n, for example the table by Wagstaff at the Bernoulli web page www.bernoulli.org, we see that 37 divides N284. Now recall the well-known Kummer congruences that the value of (1 − p)Bn/n modulo p depends only on (even) n modulo pk−1(p − 1), for n not divisible by p − 1. The first claim follows by taking p = 37. Supported in part by NSA grant H98230-10-1-0200.

In this paper we investigate the Lipschitz equivalence of dust-like self-similar sets in Rd. One of the fundamental results by Falconer and Marsh [On the Lipschitz equivalence of Cantor sets, Mathematika, 39 (1992), 223– 233] establishes conditions for Lipschitz equivalence based on the algebraic properties of the contraction ratios of the self-similar sets. In this paper we extend the study by examining deeper such connections. A key ingredient of our study is the introduction of a new equivalent relation between two dust-like self-similar sets called matchable condition. Thanks to a certain measure-preserving property of bi-Lipschitz maps between dust-like self-similar sets, we show that the matchable condition is a necessary condition for Lipschitz equivalence. Using the matchable condition we prove several conditions on the Lipschitz equivalence of dust-like self-similar sets based on the algebraic properties of the contraction ratios, which include a complete characterization of Lipschitz equivalence when the multiplication groups generated by the contraction ratios have full rank. We also completely characterize the Lipschitz equivalence of dust-like self-similar sets with two branches (i.e. they are generated by IFS with two contractive similarities). Some other results are also presented, including a complete characterization of Lipschitz equivalence when one of the self-similar sets has uniform contraction ratio.

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

As a first goal, it is explained why Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P^m only for m < 16. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential, but critical, high-order obstructions in the corresponding Taylor towers. For m > 15, the relation TC^S(P^m) > n-1 is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an n-dimensional Euclidean embedding of P^m. A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis' BP-approach to the immersion problem of P^m. A form of the Euler class viewpoint is applied to show TC^S(P^3) = 5, as well as to suggest a few higher dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber's work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber's lead, this concept is connected to the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P^3., updated, final version to appear in Transactions of the American Mathematical Society

In this paper we present a probabilistic construction of countable abelian p p -groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian p p -groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin’s theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability 1 1 , produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. … It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian p p -groups. Valuated abelian p p -groups are natural generalizations of abelian p p -groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian p p -groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.

In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and Schutzenberger) could be used to dene a new basis for the ring of quasisymmetric functions we call \Quasisymmetric Schur functions" (QS functions for short). In this paper we develop the combinatorics of these polynomials futher, by showing that the product of a Schur function and a Demazure atom has a positive expansion in terms of Demazure atoms. We use these techniques, together with the fact that both a QS function and a Demazure character have explicit expressions as a positive sum of atoms, to obtain the expansion of a product of a Schur function with a QS function (Demazure character) as a positive sum of QS functions (Demazure characters). Our formula for the coecients in the expansion of a product of a Demazure character and a Schur function into Demazure characters is similar to known results and includes in particular the famous Littlewood-Richardson rule for the expansion of a product of Schur functions in terms of the Schur basis. A composition (weak composition) with n parts is a sequence of n positive (nonnegative) integers, respectively. A partition is a composition whose parts are monotone nonincreasing. If is a weak composition, composition, or partition, we let '( ) denote the number of parts of . Throughout this article is a weak composition with '( ) = n while and denote compositions and partitions, respectively, with '( ) n, '( ) n. The polynomials in this paper (Schur functions, Demazure atoms and characters, QS functions) depend on a

In an earlier paper we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, including all connected symmetric pairs over C. In this paper we show that the pairs (GL(V),O(V)), (GL(V),U(V)), (U(V), O(V)), (O(V ⊕ W), O(V) x O(W)), (U(V ⊕ W), U(V) × U(W)) are regular, where V and W are quadratic or Hermitian spaces over an arbitrary local field of characteristic zero. We deduce from this that the pairs (GL n (ℂ), O n (ℂ)) and (O n+m (ℂ),O n (ℂ) × O m (ℂ)) are Gelfand pairs.

Recently P. Hubert, C. Mauduit and A. Sarkozy introduced and studied the notion of pseudorandomness of binary lattices and gave a pseudorandom binary lattice. Later in other papers C. Mauduit and A. Sarkozy constructed some large families of "good" binary lattices. In this paper a large family of pseudorandom binary lattices is presented by using the multiplicative inverse and the quadratic character of finite fields.

In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of C cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let n be so large that v satisfies δn≤ 1. We show that any polynomial in has at least zeros in any disk with center on the unit circle and radius αn.

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H1 L(R n ) and a BMO L (R n ) space associated with the operator L were introduced and studied. In this paper we define a class of H p L (R n ) spaces associated with the operator L for a range of p < 1 acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical H p (R n ) spaces. We then establish a duality theorem between the H p L (R n ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on H p L (R n ) and give the inclusion between the classical H p (R n ) spaces and the H p L (R n ) spaces associated with operators.

This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely { x = -y + xy, {y = x+Dx 2 +Fy 2 . In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle K = (-7, 2) x (0,4). More concretely, we prove that if (D, F) ∉ K, then the period function of the center is monotonically increasing.

We prove that ΩSp(n) is stably indecomposable at the prime 2 if n ≥ 2 or n = ∞. Hopkins [2] proved that ΩSp(2) and ΩSp(3) are stably indecomposable. Later Hubbuck [3] added ΩG2 and ΩF4 to the list of such spaces. We [4] also proved that ΩE6 and ΩE7 are stably indecomposable. In this paper we will show that ΩSp(n) are stably indecomposable at the prime 2 for n ≥ 2, which was conjectured by Hubbuck. Theorem. ΩSp(n) is stably indecomposable at the prime 2 if n ≥ 2 or n = ∞. In contrast to our result ΩSU(n) is stably decomposable [1]. From now on until the end of this paper all spaces are assumed to be localized at the prime 2 and H∗(X) stands for H∗(X;F2). First we recall the ring structure of H∗(ΩSp(n)) and the dual action of the Steenrod algebra on them [5]: H∗(ΩSp(n)) ∼= F2[z1, z3, . . . , z2n−1], where z2i−1 ∈ H4i−2(ΩSp(n)). To state the action of the Steenrod algebra on z2i−1, for a positive integer i we define zi = (zb(i)) 2a(i) , where a(i) and b(i) are unique non-negative integers such that i = 2b(i) and b(i) is odd. Then we have Sqz2i−1 = ( 2i− 2− j j ) z2i−1−j. In particular, Sqz2i−1 = z2i−2. We also have Sqzi = ( i− 1− j j ) zi−j for any positive integer i. 2000 Mathematics Subject Classification. Primary 55P35, 55P42, 55S10.

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.

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.

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 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.

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.

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