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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.

A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = -" The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.

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.

Among other results, it is proved that if a sequence {t 1l) of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all a-compact sets or all open sets, then there exists a maximal open set U such that ft,1 ( U) O-* { f t,) being the associated submeasures. In [5], J. D. Stein, Jr. considers some versions of Phillip's lemma and the Vitali-Hahn-Saks theorem for sequences of regular scalar-valued Borel measures on Hausdorff spaces. The measures he considers are bounded, regular, and finitely additive Borel measures which are easily seen to be countably additive (to prove this, first note that I pi, the variation of such a measure /L, is regular, bounded, and finitely additive [5, Lemma 1]) and so for a sequence {Bi) of Borel sets, B1j0, and > 0, C a sequence of compact sets (Ki), Ki c Bi and vi(Bi \ K1) some no and so I til(BiWO), an observation which enables one to prove his results easily, in more general forms, and under weaker conditions. Theory of submeasures developed in [1] will be used. Let G be an Abelian Hausdorff topological group, X a Hausdorff topological space, and yt a countably additive, regular, G-valued measure on X (by regularity we mean that for any Borel set B in X and a 0-nbd U in G, there exists a compact set C c B such that tt(K) C U, VK c B \ C, K Borel). If G is normed [1, p. 270], we have an associated submeasure ft, AK(B) = sup{14(A)I:A c B,A Borel) = sup{ I(C)I: C c B, Ccompact), which is finite [1, p. 279, Corollary 4.1 1], exhaustive, a-subadditive, and order-continuous [1, II]. It is also regular in the sense that given E > 0 and a Borel set B, 3 a compact set K c B such that i(B \ K) < (proof by contradiction, using exhaustivity). For a collection of measures or subReceived by the editors August 25, 1976. AMS (MOS) subject classifications (1970). Primary 28A25; Secondary 46G10.

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.

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.

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 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].

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.

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?

Schoenberg (1938) identified the class of positive definite radial (or isotropic) functions φ{symbol}: ℝd → ℝ, φ{symbol}(0) = 1, as having a representation φ{symbol}(x) = ∫ℝ+ Ωd(tu)Gd(du), t = ||x||, for some uniquely identified probability measure Gd on ℝ+ and Ωd(t) = E(eit(e1,η)), where η is a vector uniformly distributed on the unit spherical shell Sd-1 ⊂ ℝd and e1 is a fixed unit vector. Call such Gd a d-Schoenberg measure, and let Φd denote the class of all functions f: ℝ+ → ℝ for which such a d-dimensional radial function φ{symbol} exists with f(t) = φ{symbol}(x) for t = ||x||. Matheron (1965) introduced operators Ĩ and D, called Montee and Descente, that map suitable f ∈ Φd into Φd' for some different dimension d': Wendland described such mappings as dimension walks. This paper characterizes Matheron's operators in terms of Schoenberg measures and describes functions, even in the class Φ∞ of completely monotone functions, for which neither Ĩf nor Df is well defined. Because f ∈ Φd implies f ∈ Φd' for d' < d, any f ∈ Φd has a d' Schoenberg measure Gd' for 1 ≤ d' < d and finite d ≥ 2. This paper identifies Gd' in terms of Gd via another 'dimension walk' relating the Fourier transforms Ωd' and Ωd that reflect projections on ℝd' within ℝd. A study of the Euclid hat function shows the indecomposability of Ωd. © 2014 American Mathematical Society.

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 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 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.

Let k, n ∈ N with k < n and let g k (R n ) denote the Grassmann manifold consisting of all k-dimensional linear subspaces in R n . In an earlier paper the authors showed that if the projections of a nonconvex closed set C C R n are convex and proper for projection directions from some nonempty open set P ⊂ g k (R n ), then C contains a closed copy of an (n―k―1)-manifold. In this paper we improve on that result by showing that that result remains valid under the weaker assumption that P is somewhere dense in g k (R n ).

The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.

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.

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.

A number of open problems in the field of several complex variables naturally lead to the study of bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) on C n + 1 \mathbb {C}^{n+1} . In particular, both the Ebenfelt sum of squares conjecture and the degree estimate conjecture for proper rational mappings between balls in complex Euclidean spaces lead to the study of the rank of the bihomogeneous polynomial r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z}\right \rVert ^2 under certain additional hypotheses. When r r has a diagonal coefficient matrix, these questions reduce to questions about real homogeneous polynomials. More specifically, we are led to study the rank of P = S Q P=SQ when Q Q is a homogeneous polynomial and S ( x ) = ∑ j = 0 n x j S(x) = \sum _{j=0}^n x_j . In this paper, we use techniques from commutative algebra to estimate the minimum rank of P = S Q P=SQ under the additional hypothesis that Q Q has maximum rank. The problem has already been solved for n + 1 ≤ 3 n+1 \leq 3 , and so we consider n + 1 ≥ 4 n+1\geq 4 . We obtain a minimum rank estimate that is sharp when n + 1 = 4 n+1=4 , and we exhibit a family of polynomials having this minimum rank. We also prove an estimate for n + 1 > 4 n+1>4 that, while not sharp, is non-trivial.

For any prime number p p , and integer k ⩾ 1 k\geqslant 1 , let F p k \mathbb {F}_{p^k} be the finite field of p k p^k elements. A famous problem in the theory of polynomials over finite fields is the characterization of all nonconstant polynomials F ∈ F p k [ x ] F\in \mathbb {F}_{p^k}[x] for which the value set { F ( α ) : α ∈ F p k } \{F(\alpha ): \alpha \in \mathbb {F}_{p^k}\} has the minimum possible size ⌊ ( p k − 1 ) / deg ⁡ F ⌋ + 1 \left \lfloor (p^k-1)/\deg F \right \rfloor +1 . For k ⩽ 2 k\leqslant 2 , the problem was solved in the early 1960s by Carlitz, Lewis, Mills, and Straus. This paper solves the problem for k = 3 k=3 .

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.

This paper answers a question of Jan J. Dijkstra by giving a proof that all three-point sets are zero dimensional. It is known that all two-point sets are zero dimensional, and it is known that for all n > 3, there are n-point sets which are not zero dimensional, so this paper answers the question for the last remaining case.

In this paper we shall deduce a Bailey type formula as a consequence of the residual identity of a q q -series transformation due to Gasper. Our formula leads to a variety of q q -series identities which are related to the arithmetic function counting integer representations of the form \[ n ( A n + B ) 2 + r ( C r + D ) 2 + E n r . \frac {n(An+B)}{2}+\frac {r(Cr+D)}{2} + Enr. \]

When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields., Fixed a typo with Theorem 1.3 that is present in the published version of the paper. The main results remain unchanged

Consider the affine space consisting of pairs of matrices ( A , B ) (A,B) of fixed size, and its closed subvariety given by the rank conditions rank ⁡ A ≤ a , rank ⁡ B ≤ b \operatorname {rank} A \leq a, \, \operatorname {rank} B \leq b , and rank ⁡ ( A ⋅ B ) ≤ c \operatorname {rank} (A\cdot B) \leq c , for three non-negative integers a , b , c a,b,c . These varieties are precisely the orbit closures of representations for the equioriented A 3 \mathbb {A}_3 quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest.

In a previous paper, the author proved that in characteristic zero the jacobian J(C) of a hyperelliptic curve C: y 2 = f(x) has only trivial endomorphisms over an algebraic closure K a of the ground field K if the Galois group Gal(f) of the irreducible polynomial f(x) ∈ K[x] is either the symmetric group S n or the alternating group An. Here n > 4 is the degree of f. In another paper by the author this result was extended to the case of certain smaller Galois groups. In particular, the infinite series n = 2 r + 1, Gal(f) = L 2 (2 r ):= PSL 2 (F 2 r) and n = 2 4r+2 +1, Gal(f) = Sz(2 2r+1 ) were treated. In this paper the case of Gal(f) = U 3 (2 m ):= PSU 3 (F 2 m) and n = 2 3m + I is treated.

Let G be a finite group of Lie type in characteristic p. This paper addresses the problem of describing the irreducible complex (or p-adic) representations of G that remain absolutely irreducible under the Brauer reduction modulo p. An efficient approach to solve this problem for p > 3 has been elaborated in earlier papers by the authors. In this paper, we use arithmetical properties of character degrees to solve this problem for the groups G ∈ { 2 B 2 (q), 2 G 2 (q), G 2 (q), 2 F 4 (q), F 4 (q), 3 D 4 (q)} provided that p < 3. We also prove an asymptotical result, which solves the problem for all finite groups of Lie type over F q with q large enough.

We give a reduction of the irregular case for the effective non vanishing conjecture by virtue of the Fourier-Mukai transform. As a conse quence, we reprove that the effective non-vanishing conjecture holds on alge braic surfaces. In this paper we consider the following so-called effective non-vanishing conjec ture, which has been put forward by Ambro and Kawamata [Am99, KaOO]. Conjecture 1 (EN,). Let X be a proper normal variety of dimension n, B an effective R-divisor on X such that the pair (X, B) is Kawamata log terminal, and D a Cartier divisor on X. Assume that D is nef and that D (Kx + B) is nef and big. Then Ho (X, D) 74 0. This conjecture is closely related to the minimal model program and plays an important role in the classification theory of Fano varieties. For a detailed intro duction to this conjecture, we refer the reader to [XieO6]. By the Kawamata-Viehweg vanishing theorem, we have HL(X, D) = 0 for any positive integer i. Thus H?(X, D) =& 0 is equivalent to X(X, D) /& 0. Under the same assumptions as in Conjecture 1, the Kawamata-Shokurov non-vanishing theorem says that Ho (X, mD) =A 0 for all m > 0. Thus the effective non-vanishing conjecture is an improvement of the non-vanishing theorem in some sense. Note that EN1 is trivial by the Riemann-Roch theorem and that EN2 was settled by Kawamata [KaOO, Theorem 3.1] by virtue of the logarithmic semipositivity theo rem. For n > 3, only a few results are known. For instance, ENn holds trivially for toric varieties [Mu02], EN3 (X, 0) holds for all canonical projective minimal three folds X [KaOO, Proposition 4.1], and EN3 (X, 0) also holds for almost all canonical projective threefolds X with -Kx nef [XieO5, Corollary 4.5]. In this paper, we shall prove that, in the irregular case, the effective non vanishing conjecture can be reduced to lower-dimensional cases by means of the Fourier-Mukai transform. As consequences, EN2 is reproved after Kawamata, and ENn holds for all varieties of maximal Albanese dimension. Throughout this paper, we work over the complex number field C. For the definition of the Kawamata log terminal (KLT, for short) and the other notions, we refer the reader to [KMM87, KM98]. For irregular varieties, the study of the Albanese map provides enough infor mation to understand their birational structure. Therefore, through the Albanese Received by the editors January 17, 2007, and, in revised form, October 18, 2007, November 16, 2007, and December 27, 2007. 2000 Mathematics Subject Classification. Primary 14E30. (@)2008 American Mathematical Society Reverts to public domain 28 years from publication 61 This content downloaded from 157.55.39.35 on Thu, 01 Sep 2016 04:38:29 UTC All use subject to http://about.jstor.org/terms

The Steenrod problem asks: given a G-module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a ZG-module is ZG-realizable if and only if it is ZHrealizable for all p-Sylow subgroups H, for all primes pl GI. Let G be a finite group. Let M be a finitely generated ZG-module. We say that M is a Steenrod representation if there exists a Moore space X with G-action such that the homology of X is isomorphic to M as a ZG-module. Recall that a Moore space is a topological space whose reduced homology vanishes in all dimensions except one. Without loss of generality, M will be assumed 2-free [1] from now on. In this paper, the following statement will be proved: M is a Steenrod representation as a ZG-module if and only if M is a Steenrod representation as a 7H-module for all p-Sylow subgroups H of G, for all primes pII GIC. This statement is inspired by papers by J. Arnold [1, 2] and P. Vogel [18]. Arnold [2] showed that if G is cyclic, then every finitely generated ZG-module is a Steenrod representation. And Vogel showed that if for any ZG-module M, we can always find a G-Moore space X realizes M, then G has only cyclic Sylow subgroups. Let us first consider a Moore space X whose homology is isomorphic to M as a 2-module. Let G(X) denote the space of self homotopy equivalences of X and let BG(X) [9] denote the classifying space of the H-space G(X) for a certain fibration defined by Dold and Lashof [10]; see also Stasheff [15]. Since a homotopy equivalence of X induces an automorphism on M, there is a map G(X) -* Aut(M). And the map G(X) -* Aut(M) induces another map a : BG(X) -' BAut(M). On the other hand, since M is a ZG-module, there is a map G -* Aut(M) (= GL(n, 2) if M = Zn), which induces .p : BG -) BAut(M). Following Cooke's theory [9], as pointed out in [12], M is a Steenrod representation as a ZG-module if and only if the lifting f exists in

We estimate double exponential sums of the form Sa(X, Y) = E E exp(27riaI0xY/p), xGE X yG Y where V is of multiplicative order t modulo the prime p and X and Y are arbitrary subsets of the residue ring modulo t. In the special case t = p 1, our bound is nontrivial for I XI > I Yl > p15/16+6 with any fixed 6 > 0, while if in addition we have Xl > p1-6/4 it is nontrivial for I YI ? p3/4+6. Let p be a prime and let Fp be a finite field of p elements. For an integer m > 1 we denote by Z,m = {0, . .. , m 1} the residue ring modulo m. We also identify Fp with the set {o,... ,p-1}. Finally we define e(z) = exp(27ri/p) and use log z for the natural logarithm of z. Throughout the paper the implied constants in symbols 'O', ' ' may occasionally, where obvious, depend on the small positive parameter E and are absolute otherwise (we recall that A A are equivalent to A 0 (B)). We fix an element ) E IFp of multiplicative order t, that is, 79s 7 1, 1 pl5/16+6 with any fixed 6 > 0. Further examples are given below. Our results rely on the following estimate for certain double exponential sums from [1]; see the proof of Theorem 8 of that paper. Let A E Fp be of multiplicative order T. For any a, b (E IF we have the bound 4 (1) E| e(a\v +bAuv) 1 satisfies (2) T(m) < m(1). 2 Our main estimate is the following. Theorem. For any sets X, Y C Zt, the bound max ISa( X, Y)| << I x 1/21 yI5/6tl/2pl/8+E aEF* p holds. Proof. For a divisor dlt we denote by Y(d) the subset of y Y Y with gcd(y, t) d. Then ISa(X, Y)I ? Z ud| dlt

We prove the following theorems: (1) Suppose that f; 2W 2W is a continuous function and X is a Sierpiiiski set. Then (A) for any strongly measure zero set Y, the image f[X + Y] is an so-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (see [N], [NSW], [S]) in which properties (mainly, the algebraic sum) of certain singular subsets of the real line R and of the Cantor set 2' were investigated. Throughout the paper, by a set of real numbers we mean a subset of 2' and by "+" we denote the standard modulo 2 coordinatewise addition in 2W. Let us also assume that a "measure zero" (or "negligible") set always denotes a Lebesgue measure zero set. We apply the following definition of sets of real numbers. Definition 1. An uncountable set X is said to be a Luzin (respectively, Sierpin'ski) set iff for each meager (respectively, measure zero) set Y, XnY is at most countable. We say that a set X is of strong measure zero (respectively, strongly meager) iff for each meager (respectively, measure zero) set Y, X + Y 7& 2'. Remark 1. It is well known (see [M] for example) that every Luzin set is strongly measure zero. Quite recently J. Pawlikowski proved that each Sierpin'ski set must be strongly meager as well (see [P]). Let us recall that a set X is called an so-set (or Marczewski set) iff for each perfect set P one can find a perfect set QCP that is disjoint from X. M. Scheepers showed in [S] that for a Sierpin'ski set X and a strong measure zero set Y, X + Y is an so-set. Later, in [NSW] it was proven that this also holds when X is strongly meager. We have the following functional version of the M. Scheepers' result. Theorem 1. Let X be a Sierpin'ski set and let Y be a strong measure zero set. Assume also that f: 2' -* 2W is a continuous function. Then the image f[X + Y] is an so-set. Received by the editors July 16, 1998 and, in revised form, September 9, 1998 and March 10, 1999. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15.

In this paper it is shown that if MC = ( iS a 2 x 2 upper triangular operator matrix acting on the Hilbert space 7H E IC and if W(.) denotes the "Weyl spectrum", then the passage from w(A) U w(B) to w(MC) is accomplished by removing certain open subsets of w(A) n w(B) from the former, that is, there is equality w(A) U w(B) = w(MC) U G, where G is the union of certain of the holes in w(Mc) which happen to be subsets of w (A) n w (B). Let At and IC be Hilbert spaces, let 1C(7-, 1C) denote the set of bounded linear operators from 7t to IC, and abbreviate LC(QH,7) to C(H). When A E LC(t) and B E L(/C) are given we denote by Mc an operator acting on 1t E 1C of the form M A C) Mc:=( 0 ), where C E iC(!, 7-). The invertibility and spectra of Mc were considered by Du and Jin [5]. In this paper we give some conditions for operators A and B to exist an operator C such that Mc is Weyl, and describe the Weyl spectra of Mc. Recall ([7], [8]) that an operator A E 12(X, Y) for Banach spaces X and Y is called regular if there is an operator A' E L(Y, X) for which A = AA'A; then A' is called a generalized inverse for A. In this case, X and Y can be decomposed as follows (cf. [8, Theorem 3.8.2]): A-1(0) e A'A(X) = X and A(X) e (AA')-'(0) = Y. It is familiar ([6], [8]) that A E LC((, IC) is regular if and only if A has closed range. An operator A E LC(7-, IC) is called relatively Weyl if there is an invertible operator A' E 12(IC,7t) for which A = AA'A. It is known ([8, Theorem 3.8.6]) that A is relatively Weyl if and only if A is regular and A-1(0) A(H)', where means a topological isomorphism between spaces. An operator A E LQ(H, IC) is called left-Fredholm if it is regular with finite dimensional null space and rightFredholm if it is regular with its range of finite co-dimension. If A is both leftand right-Fredholm, we call it Fredholm. The index, ind A, of a leftor right-Fredholm Received by the editors November 21, 1997 and, in revised form, May 1, 1998 and March 10, 1999. 1991 Mathematics Subject Classification. Primary 47A53, 47A55.

Necessary and sufficient conditions are presented to insure that the direct sum of two reflexive representations of a finite dimensional algebra is reflexive, and it is shown that for each such algebra, there is an integer k such that the direct sum of k copies of each of its representations is reflexive. Given a ring i\, our results are actually presented in the more general setting of A\-representations of a ring R. Over the last thirty years, a significant amount of attention has been given to the problem of determining when an algebra A of operators on a vector space V (often a Hilbert space) over a field K is reflexive in the sense that no larger algebra of operators on that space has the same lattice of invariant subspaces. (See, for example, [2], [3], [6], [7], [12], [13], [14].) Of course, A is an algebra of operators on a vector space V if and only if V is a faithful left A-module. Thus, to attack this problem from a more module theoretic point of view, based on notation of Halmos [14], the following notions were presented in [9]: For any R A-bimodule M= RMA one defines alglat(M) {a E End(MA\) I aL C L for all RL alglat(MA) denote the canonical ring homomorphism, M is called a reflexive bimodule (or A-representation of R) if A is surjective. Thus the K-algebra of operators A on V is reflexive if and only if A VK is a reflexive bimodule (and then we simply say that the A-module V is reflexive). The problems we shall deal with here have their roots in the papers [5] and [6] of Deddens and Fillmore and [2] of Azoff. Stated in module theoretic terms, in the first pair of papers the question was posed and answered (affirmatively) of whether a direct sum of two finitely generated reflexive modules over an algebra generated by a single complex matrix is again reflexive; and Azoff showed that for each finite dimensional module over a C-algebra, there is a positive integer k, depending on its dimension, such that the direct sum of k copies of that module is reflexive. Other results in this vein can be found in [4] where Brenner and Butler showed that the direct sum of two copies of the regular representation of a finite dimensional algebra is reflexive, and in Habibi and Gustafson's [12] from which the same result follows for any faithful representation of a split serial algebra. (See [9], [10] and [11] for related and more general results.) Our main objectives are to provide, in Theorem 1.2, a necessary and sufficient condition for a direct sum of two reflexive bimodules to be reflexive; and to show, in Received by the editors September 10, 1998 and, in revised form, November 10, 1998. 1991 Mathematics Subject Classification. Primary 16D20, 16G99, 16P10; Secondary 47A15. ?)2000 American Mathematical Society 2855 This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:10:44 UTC All use subject to http://about.jstor.org/terms 2856 V. P. CAMILLO AND K. R. FULLER Theorem 3.2, that whenever R is a left artinian ring with composition length c(RR), there is a positive integer k (am, fn). Thus if -y E alglat(M e N), then a = (-YIM, -YIN). A common subquotient of a pair of left modules RM and RN is a module RX that is isomorphic to a subquotient of both RM and RN. This notion together with the following lemma allows us to determine just when the direct sum of a pair of reflexive modules is reflexive. Lemma 1.1. If r, s E R, the ordered pair (r, s) E alglat(M D N) if and only if (r s)X = 0, for every (equivalently, every cyclic) common subquotient of RM and RN. Proof. Note that a E R annihilates every common subquotient of M and N if and only it annihilates each of their cyclic common subquotients. According to Goursats' Lemma, W is an R-submodule of M ED N, if and only if there are submodules M2 N,N2 such that W {(ml, ni) E Ml @ N, f(ml + M2) ni + N2}. (Given W and the orthogonal projections 7rM and 7rN for M ED N, one checks that M1 = rM(W), M2 = M n W, N1 = 7rN(W), and N2 = N n W.) (=>) If (ml,ni) E W and (rmi,sni) E W, then f(rmi + M2) = sn1 + N2 = sf(ml + M2) so, since f is an R-isomorphism, (r s)(Ml/M2)= 0. (^=) Suppose (r s)(Ml/M2) = 0, and f: M1/M2 -* N1/N2 is an isomorphism and (m1,ri) E W. Then f(rmi + M2) = rf(ml + M2) = rnm + N2 = sn, + N2 so (rm1,sn1) E W. Now we are able to provide the promised characterization in terms of annihilators of subquotients of M and N. The left annihilator of a module M is ?R(M) {r E RI rM= O}. This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:10:44 UTC All use subject to http://about.jstor.org/terms REFLEXIVITY OF DIRECT SUMS 2857 Theorem 1.2. Let M and N be reflexive, and let {Xi I i E I} represent one copy of each of the (cyclic) common subquotients of RM and RN. Then M ED N is reflexive if and only if ?R(M) + tR(N) = iR(D xi) iEl Proof. ( ) Let r E iR(GiE' Xi). Then by Lemma 1.1, (r, 0) E alglat(M ED N). Thus, assuming that M ED N is reflexive, (r,0) = (s,s) and r = (r-s) + s EE R(M) + eR(N). Since always ?R(M) + ?R(N) C iR(@iEI Xi), the proof is complete. F We note that the proof (=>) above shows that if M e N is reflexive (regardless of reflexivity of M and N), then ?R(M) + ?R(N) = ?R(@iCI Xi). From the inclusions 0 C iR(M) + eR(N) C ?R(ED Xi) C iR(Xi) C R iCI we easily obtain the following two corollaries: Corollary 1.3. If M and N are reflexive and have a common faithful subquotient, then M D N is reflexive. Corollary 1.4. If RZ is semiperfect and M and N are reflexive and have no common composition factor, then M ffl N is reflexive. Proof. Suppose that e1, ..., em, em?1, ..., en is a complete set of primitive idempotents such that Rei/Jej is not a composition factor of M for i = 1, ..., m, and Rei/Jej is not a composition factor of N for i = m+ 1, ..., n. Then ?R(M) + iR( N) R. (See [1, Section 27].) Regarding the Deddens-Fillmore result, the algebra generated by a matrix over a field K, being isomorphic to a proper factor of the polynomial ring K[x], is an example of a split commutative uniserial algebra, i.e., a direct product of local uniserial rings. Since a module over a direct product of rings is reflexive precisely when its corresponding components are reflexive, to show that direct sums of reflexive modules are reflexive over such an algebra R, we may assume that R is a local uniserial algebra, so that the ideals of R are linearly ordered and every R-module is a direct sum of factors of R (see [1, Section 32]). In this case, R/IR(M) embeds in M and every subquotient of RM is a factor of R/lR(M). In the presence of these facts Theorem 1.2 yields the following corollary almost at once. We note, however, that [10, Theorem 1] is more general. This content downloaded from 157.55.39.191 on Tue, 11 Oct 2016 05:10:44 UTC All use subject to http://about.jstor.org/terms 2858 V. P. CAMILLO AND K. R. FULLER Corollary 1.5. The direct sum of a finite number of reflexive modules over a split uniserial algebra is reflexive. Proof. Since we may assume that the ideals of R are linearly ordered, given modules M and N, we may also assume that ?R(M) C ?R(N) C ?R(X) for every common subquotient X of M and N. So since R/lR(N) embeds in N, we see that both sides of the desired equality are equal to ?R(N). Next, as further applications of this theorem, we shall see that it yields particularly nice proofs of some key results in [10]. In [10], in order to show that direct sums of reflexive modules may fail to be reflexive over a split K-algebra R with radical J whose quiver contains a triple arrow, a pair of reflexive modules was constructed with diagrams (as in [8]) u V w M: al b\ cl and N: a, c x y z with a, b, c linearly independent elements of J \ J2. (The diagram indicates that au = x, aw = cw = z, etc.) Here ?R(M) + ?R(N) = ?R(M) + K(a c) + Kb, but the only common subquotients of M and N are simple modules, so ?R(G X =) =R(M) + J R ?R(M) + tR(N). iEI Thus the theorem shows that M D N is not reflexive. In the positive vein we shall employ the following lemma and Theorem 1.2 to obtain a simple proof of a key part of [10, Proposition 4]. Lemma 1.6. Let R be a split local K-algebra of dim(KR) < 3. If M is a faithful left R-module, then every proper cyclic module is isomorphic to a subquotient of M. Proof. (This proof is essentially contained in the proof of [10, Proposition 4].) If R is uniserial, then RR embeds in M, so we may assume that J2 = 0 and that dim(KJ) = 2. Let u E M\SocM and suppose that WR(u) : 0. Then, since u ? SocM, there is a b C J with ?R(u) = Kb. But then there is a v E M with bv z 0, and there is an a E J with Ka =R(V). Thus it follows that J = Ka DKb.

We prove a strong version of a theorem of Balcerzak-RoslanowskiShelah by showing, in ZFC, that there exists a simply definable Borel CT-ideal for which both the ccc and property (M) fail. The proof involves Polish group actions. Definition. A Borel ideal is an ideal I on a Polish space X with the following property: for all A E I, there exists a Borel subset B of X such that A C B and B E I. We consider three types of properties for a Borel ideal I on a Polish space X. (1) I has a H' definition if the following set is Hl: C.1 = {c c 2': c is a Borel code for a subset B, of X and B, c I}. (2) For K a cardinal, I satisfies the /c-cc if there does not exist a family of i I-almost disjoint Borel sets that are not in I. The w1-cc is usually called the ccc. (3) I satisfies property (M) if there exists a Borel-measurable function f : X -? 2W with f-1(y) ? I for all y E 2W. Property (M) was introduced in Balcerzak [1]. Obviously, an ideal satisfying property (M) violates the ccc. Both the above paper and the later paper of Balcerzak-Roslanowski-Shelah [2] are concerned with circumstances in which it is possible for both the ccc and property (M) to fail. We refer the reader to these two references for information on these properties-and several other properties of ideals. The latter paper contains the following result. Theorem 1 (Balcerzak-Roslanowski-Shelah [2, 5.6]). Assume that either CH fails or every Al set of reals has the Baire property. Then there exists a Borel a-ideal I*, containing all singletons, such that: (a) I* has a H1 definition; (b) the ccc fails for I*; (c) property (M) fails for I*; (d) I* satisfies the w2-cc. Andrzej Roslanowski presented this result in a seminar talk at Ohio State University in 1993, and asked whether or not it is provable in ZFC. I thank him for bringing this question to my attention, and I thank Ohio State for their support. Received by the editors October 23, 1998 and, in revised form, December 3, 1998. 2000 Mathematics Subject Classification. Primary 03E15.

Consider an independent random cascade acting on the positive Borel measures defined on the boundary of a Galton-Watson tree. Assuming an offspring distribution with finite moments of all orders, J. Peyriere computed the fine scale structure of an independent random cascade on Galton-Watson trees. In this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger class of initial measures is covered and, as a result, it is shown that it is the Holder exponent of the initial measure which is the critical parameter in the Peyriere theory. We start by defining the space of Galton-Watson trees and introducing some related terminology. 1. GALTON-WATSON TREES Let Tr be the space of labelled tree graphs rooted at 4. An element r of ?7 is a set of finite sequences of positive integers (v1, v2,... , Vn) E r such that: (i) X5E r is coded as the empty sequence. (ii) If (Vl,...,Vk) ET, then (vl,...,vi) ETV 1 1. The countable dense subset To of finite labelled tree graphs rooted at 0 makes Tr a Polish space. An important class of probability distributions on the Borel sigma field B(T) of Tr for this paper is the Galton-Watson distribution with single progenitor and offspring distribution Pk, k = 0, ,.. ., for which the probability Received by the editors May 14, 1998 and, in revised form, October 8, 1998. 2000 Mathematics Subject Classification. Primary 60G57, 60G30, 60G42; Secondary 60K35. @)2000 American Mathematical Society

In this paper we show that given any compact set $E \subset \hat{\mathbb{C}}$, we can always find a conformally removable subset with the same Hausdorff dimension as $E$.

In the space I r I_r of invariant r r -dimensional subspaces of a null system in ( 2 r + 1 ) (2r+1) -dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections φ : I r → I r \varphi :I_r\to I_r , for which both φ \varphi and φ − 1 \varphi ^{-1} preserve adjacency. In the present paper we show that the two conditions φ : I r → I r \varphi :I_r\to I_r is a surjection and φ \varphi preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.