On Sets Where Iterates of a Meromorphic Function Zip Towards Infinity

For a transcendental meromorphic function f, various propertiesof the set [formula] were obtained in [8] and [9]. Here we establish analogous propertiesfor the smaller sets [formula] introduced in [5], and [formula] We deduce a symmetry result for Julia sets J(f), and also indicatesome techniques for showing that certain invariant curves liein I'(f), Z(f) and J(f). 2000 Mathematics Subject Classification30D05, 37F10, 37F50.     

Corrigendum

SOLUTIONS OF p-SUBLINEAR p-LAPLACIAN EQUATION VIA MORSE THEORY (J. London Math. Soc. (2) 72 (2005) 632–644) YUXIA GUO AND JIAQUAN LIU The above-mentioned paper was influenced by the work of V. Moroz[1] and Z. Q. Wang [2], which had been inadvertently omittedfrom the bibliography.     

On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Optimality conditions for n x n infinite-order parabolic coupled systems with control constraints and general performance index

W. Kotarski Institute of Informatics, Silesian University, Bedzinska 60, 41-200 Sosnowiec, Poland Email: bahaa_gm{at}hotmail.com Email: kotarski{at}gate.math.us.edu.pl Received on March 14, 2006; Accepted on December 20, 2006 A distributed control problem for n x n parabolic coupled systemsinvolving operators with infinite order is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem, the necessary and sufficientconditions of optimality are derived for the Dirichlet problem.Yet, the problem considered here is more general than the problemsin El-Saify & Bahaa (2002, Optimal control for n x n hyperbolicsystems involving operators of infinite order. Math. Slovaca,52, 409–424), El-Zahaby (2002, Optimal control of systemsgoverned by infinite order operators. Proceeding (Abstracts)of the International Conference of Mathematics (Trends and Developments)of the Egyptian Mathematical Society, Cairo, Egypt, 28–31December 2002. J. Egypt. Math. Soc. (submitted)), Gali &El-Saify (1983, Control of system governed by infinite orderequation of hyperbolic type. Proceeding of the InternationalConference on Functional-Differential Systems and Related Topics,vol. III. Poland, pp. 99–103), Gali et al. (1983, Distributedcontrol of a system governed by Dirichlet and Neumann problemsfor elliptic equations of infinite order. Proceeding of theInternational Conference on Functional-Differential Systemsand Related Topics, vol. III. Poland, pp. 83–87) and Kotarskiet al. (200b, Optimal control problem for a hyperbolic systemwith mixed control-state constraints involving operator of infiniteorder. Int. J. Pure Appl. Math., 1, 241–254).     

Errata to ``Classification of three-dimensional flips'

This erratum makes two corrections to J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), 533-703.

     

The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval

In the 1960s, Richard J. Thompson introduced a triple of groupsF T G which, among them, supplied the first examples of infinite,finitely presented, simple groups [14] (see [6] for publisheddetails), a technique for constructing an elementary exampleof a finitely presented group with an unsolvable word problem[12], the universal obstruction to a problem in homotopy theory[8], and the first examples of torsion free groups of type FPand not of type FP [5]. In abstract measure theory, it has beensuggested by Geoghegan (see [3] or [9, Question 13]) that Fmight be a counterexample to the conjecture that any finitelypresented group with no non-cyclic free subgroup is amenable(admits a bounded, non-trivial, finitely additive measure onall subsets that is invariant under left multiplication). Recently,F has arisen in the theory of groups of diagrams over semigrouppresentations [10], and as the object of questions in the algebraof string rewriting systems [7]. For more extensive bibliographiesand more results on Thompson's groups and their generalizationssee [1, 4, 6]. A persistent peculiarity of Thompson's groups is their abilityto pop up in diverse areas of mathematics. This suggests thatthere might be something very natural about Thompson's groups.We support this idea by showing (Theorem 1.1 below) that PL_o(I),the group of piecewise linear (finitely many changes of slope),orientation-preserving, self-homeomorphisms of the unit interval,is riddled with copies of F: a very weak criterion implies thata subgroup of PL_o(I) must contain an isomorphic copy of F.     

Semisimple Modules for Finite Groups of Lie Type

Let k be an algebraically closed field of characteristic p >0, and let G be a connected, reductive algebraic group overk. In [8] and [11], conditions on the dimension of rationalG modules were seen to imply semisimplicity of these modules.In [8], certain of these conditions were extended to cover thefinite groups of Lie type. In this paper, we extend some ofthe results of [11] to cover these finite Lie type groups. Themain such extension is the following result.     

Optimal control for cooperative parabolic systems governed by Schrodinger operator with control constraints

^** Email: Bahaa_gm{at}hotmail.com A distributed control problem for cooperative parabolic systemsgoverned by Schrödinger operator is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem given by Walczak (1984, Onesome control problems. Acta Univ. Lod. Folia Math., 1, 187–196),the optimality conditions are derived for the Dirichlet problem.     

On the Representations of a Number as the Sum of Four Fifth Powers

It is known from Vaughan and Wooley's work on Waring's problemthat every sufficiently large natural number is the sum of atmost 17 fifth powers [13]. It is also known that at least sixfifth powers are required to be able to express every sufficientlylarge natural number as a sum of fifth powers (see, for instance,[5, Theorem 394]). The techniques of [13] allow one to showthat almost all natural numbers are the sum of nine fifth powers.A problem of related interest is to obtain an upper bound forthe number of representations of a number as a sum of a fixednumber of powers. Let R(n) denote the number of representationsof the natural number n as a sum of four fifth powers. In thispaper, we establish a non-trivial upper bound for R(n), whichis expressed in the following theorem.     

One-Dimensional p-Adic Subanalytic Sets

In this paper we extend two theorems from [2] on p-adic subanalyticsets, where p is a fixed prime number, Q_p is the field of p-adicnumbers and Z_p is the ring of p-adic integers. One of thesetheorems [2, 3.32] says that each subanalytic subset of Z_p issemialgebraic. This is extended here as follows.     

Stability of uniformly Morse-Smale gradient-like numerical methods for flows

In Garay (1996, Numer. Math., 72, 449–479) and Li (1997b,SIAM J. Math. Anal., 28, 381–388), it was shown that thequalitative properties of a Morse–Smale gradient-likeflow are preserved by its numerical approximations. In thispaper, we show that the qualitative properties of a family ofuniformly Morse–Smale gradient-like numerical methodsare preserved by the approximated flow. The techniques usedin the study of the structural stability theorem for diffeomorphismsare the main tools for this work.     

Harmonic Analogues of G. R. Maclane's Universal Functions

Let E denote the space of all entire functions, equipped withthe topology of local uniform convergence (the compact-opentopology). MacLane [15] constructed an entire function f whosesequence of derivatives (f, f', f', ...) is dense in E; hisconstruction is succinctly presented in a much later note byBlair and Rubel [2], who unwittingly rederived it (see also[3]). We shall call such a function f a universal entire function.In this note we show that analogous universal functions existin the space H_N of functions harmonic on R^N, where N2. We alsostudy the permissible growth rates of universal functions inH_N and show that the set of all such functions is very large. For purposes of comparison, we first review relevant facts aboutuniversal entire functions. The function constructed by MacLaneis of exponential type 1. Duyos Ruiz [7] observed that a universalentire function cannot be of exponential type less than 1. G.Herzog [11] refined MacLane's growth estimate by proving theexistence of a universal entire function f such that |f(z)|=O(re^r)as |z|=r. Finally, Grosse–Erdmann [10] proved the followingsharp result.     

Weak Cayley Tables

In [1] Brauer puts forward a series of questions on group representationtheory in order to point out areas which were not well understood.One of these, which we denote by (B1), is the following: whatinformation in addition to the character table determines a(finite) group? In previous papers [5, 7–13], the originalwork of Frobenius on group characters has been re-examined andhas shed light on some of Brauer's questions, in particularan answer to (B1) has been given as follows. Frobenius defined for each character of a group G functions^(k):G^(k) C for k = 1, ..., deg with ⁽¹⁾ = . These functionsare called the k-characters (see [10] or [11] for their definition).The 1-, 2- and 3-characters of the irreducible representationsdetermine a group [7, 8] but the 1- and 2-characters do not[12]. Summaries of this work are given in [11] and [13].     

Linear Spaces on Cubic Hypersurfaces, and Pairs of Homogeneous Cubic Equations

A remarkable theorem of Birch [2] shows that a system of homogeneouspolynomials with rational coefficients has a non-trivial zero,provided only that these polynomials are of odd degree, andthe system has sufficiently many variables in terms of the numberand degrees of these polynomials. Despite four decades of effort,the problem of obtaining a reasonable bound for the latter numberof variables has proved to be one of great difficulty. Whenthe system consists of a single cubic form, Davenport [4] hassucceeded in showing that 16 variables suffice, and Schmidt[17, 18, 19, 20] has devoted a series of papers to systems ofcubic forms, showing in particular that 5140 variables sufficefor pairs of cubic forms, and that (10r)⁵ variables sufficefor systems of r cubic forms. The current state of knowledgefor forms of higher degree is, by comparison, extremely weak(but see [21, 22]), and so it seems worthwhile expending furthereffort on the case of systems of cubic forms. In this paperwe improve on Schmidt's result for pairs of cubic forms. Incontrast with the sophisticated versions of the Hardy–Littlewoodmethod employed by Davenport and Schmidt, our approach is basedon an elementary idea of Lewis [12], and is applicable in arbitrarynumber fields. This method also has consequences for the existenceof linear spaces of rational solutions on cubic hypersurfaces,thereby improving on work of Lewis and Schulze-Pillot [14] onthis topic. 1991 Mathematics Subject Classification 11D72, 11E76.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.     

On Irregularities of Distribution II

Let a=(a₁, a₂, a₃, ...) be an arbitrary infinite sequence inU=[0, 1). Let Van der Corput [5] conjectured that d(a, n) (n=1, 2, ...) isunbounded, and this was proved in 1945 by van Aardenne-Ehrenfest[1]. Later she refined this [2], obtaining for infinitely many n. Here and later c₁, c₂, ... denote positiveabsolute constants. In 1954, Roth [8] showed that the quantity is closely related to the discrepancy of a suitable point setin U².     

Confined Subgroups of Simple Locally Finite Groups and Ideals of their Group Rings

We are concerned in this paper with the ideal structure of grouprings of infinite simple locally finite groups over fields ofcharacteristic zero, and its relation with certain subgroupsof the groups, called confined subgroups. The systematic studyof the ideals in these group rings was initiated by the secondauthor in[15], although some results had been obtained previously(see [3, 1]). Let G be an infinite simple locally finite groupand K a field of characteristic zero. It is expected that inmost cases, the group ring KG will have the smallest possiblenumber of ideals, namely three, (KG itself, {0} and the augmentationideal), and this has been verified in some cases. In some interestingcases, however, the situation is different, and there are moreideals. We mention in particular the infinite alternating groups[3] and the stable special linear groups [9], in which the ideallattice has been completely determined. The second author hasconjectured that the presence of ideals in KG, other than thethree unavoidable ones, is synonymous with the presence in thegroup of proper confined subgroups. Here a subgroup H of a locallyfinite group G is called confined, if there exists a finitesubgroup F of G such that H^gF1 for all gG. This amounts to sayingthat F has no regular orbit in the permutation representationof G on the cosets of H.     

On a theorem of supersoluble automorphism groups

The maximal nilpotent and supersoluble automorphism groups of Riemann surfaces were given in earlier papers by this author. In this note the author wishes to correct the necessity of the condition given in Theorem (4.3) of Bounds for the order of supersoluble automorphism groups of Riemann surfaces (Proc. Amer. Math. Soc. 108 (1990), 587-600), which was left out at the time of writing the paper. The author also wishes to apologize to the readers for that.

     

Correction to ``Formal degrees and adjoint -factors'

This is a correction to the paper Formal degrees and adjoint -factors, J. Amer. Math. Soc. 21 (2008), 283-304.

     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.