On the Optimum Criterion of Polynomial Stability

The purpose of this note is to answer the question raised byNie & Xie (1987). Let f(x)=a₀xⁿ+a₁x^n–1+...+a_n be apositive-coefficient polynomial. The numbers ₁=a_i-1a_i+2/a_ia_i+1(i=1, ..., n–2) are called determining coefficients. Theoptimum criterion problem was posed as follows: for n3, findthe maximal number (n) such that the polynomial f(x) is stableif _i < (n) (1in–2). For n6, we show that (n)=ß,where ß is the unique real root of the equation x(x+1)²=1.     

Inner Derivations and Primal Ideals of C*-Algebras

Let A be a C*-algebra. For a A let D(a, A) denote the innerderivation induced by a, regarded as a bounded operator on A,and let d(a, Z(A)) denote the distance of a from Z(A), the centreof A. Let K(A) be the smallest number in [0, ] such that d(a,Z(A)) K(A)||D(a, A)|| for all a A. It is shown that if A isnon-commutative and has an identity then either K(A) = , or K(A) = 1 / 3, or K(A) 1. Necessaryand sufficient conditions for these three possibilities aregiven in terms of the primitive and primal ideals of A. If Ais a quotient of an AW*-algebra then K(A) . Helly's Theorem is used to show that if A is aweakly central C*-algebra then K(A) 1.     

Differential Simplicity in Polynomial Rings and Algebraic Independence of Power Series

Let k be a field of characteristic zero, f(X,Y), g(X,Y)k[X,Y],g(X,Y) (X,Y) and d:=g(X,Y)/X + f(X,Y)/Y. A connection is establishedbetween the d-simplicity of the local ring k[X,Y]_(X,Y) and thetranscendency of the solution in tk[[t]] of the algebraic differentialequation g(t,y(t))·(/t)y(t)+f(t,y(t)). This connectionis used to obtain some interesting results in the theory ofthe formal power series and to construct new examples of differentiallysimple rings.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

A Remark on The Essential Spectra of Dirac Systems

A potential for the one-dimensional Dirac operator is constructedsuch that its essential spectrum does not cover the whole realline, whereas the potential q(x) tends to as |x| . Furthermore,a criterion by Hartman and Wintner for points of the essentialspectrum of Sturm–Liouville operators is generalised toa purely operator-theoretical setting, and a simplified proofis given. 1991 Mathematics Subject Classification 34L40, 47A10,81Q10.     

Embedded Thick Finite Generalized Hexagons in Projective Space

We show that every embedded finite thick generalized hexagon of order (s, t) in PG(n,q) which satisfies the conditions

1. s = q
2. the set of all points of generates PG(n, q)
3. for anypoint x of , the set of all points collinear in withx is containedin a plane of PG(n, q)
4. for any point x of , the set of allpoints of not oppositex in is contained in a hyperplane ofPG,(n, q)

is necessarily the standard representation of H(q) in PG(6,q) (on the quadric Q(6, q)), the standard representation ofH(q) for q even in PG(5, q) (inside a symplectic space), orthe standard representation of H(q, ) in PG(7, q) (where the lines of are the lines fixed by a trialityon the quadric Q⁺(7, q)). This generalizes a result by Cameronand Kantor [3], which is used in our proof.     

Spectral Multipliers for Hardy Spaces Associated with Schrodinger Operators with Polynomial Potentials

Let T_t be the semigroup of linear operators generated by a Schrödingeroperator – A = – V, where V is a non-negative polynomial,and let be the spectral resolution of A. We say that f is an element of if the maximal function Mf(x) = sup_t>0|T_tf(x)| belongs toL_p. We prove a criterion of Mihlin type on functions F whichimplies boundedness of the operators on , 0 < p 1. 1991 MathematicsSubject Classification 42B30, 35J10.     

The Series of Norms in a Soluble p-Group

The norm of a group G is the subgroup of elements of G whichnormalise every subgroup of G. We shall denote it (G). An ascendingseries of subgroups _i(G) in G may be defined recursively by:₀(G) = 1 and, for i 0, _i+1(G)/_i(G) = (G/_i(G)). For each i,the section _i+1(G)/i(G) clearly contains the centre of the groupG/_i(G). A result of Schenkman [8] gives a very close connectionbetween this norm series and the upper central series: _i(G) _i(G) _2i(G). 1991 Mathematics Subject Classification 20E15.     

Uncertainty Inequalities for Fourier Series of Pairs of Reciprocal Positive Functions

In the early 1930s, Wiener proved that if f(x) is a strictlypositive periodic function whose Fourier series is absolutelyconvergent, then the Fourier series of g(x)=1/f(x) is also absolutelyconvergent [8, pp. 10–14]. This phenomenon can be easilyunderstood nowadays using Banach algebra techniques (see, forexample, [4, pp. 202–203]). In fact, these techniquesallow us to study the absolute convergence of g(x)=F(f(x)),where F is holomorphic in an open subset of C that containsthe range of f(x) (for xR). In this context, Wiener's originalproblem corresponds to the choice F(z)=1/z. In this work we want to analyse the constraints on the simultaneousrate of vanishing of the Fourier coefficients f(n) and (n) asn. We shall focus on g=1/f, but we shall also study the generalcase g=F(f). In either case, there are obviously no constraintswhen f is a constant function. Although this problem does not seem to be directly related touncertainty inequalities for the Fourier Transform, we observethat there are some analogies, both in the nature of the resultsand in the proof techniques. The general fact with which weare dealing is that f(n) and (n) cannot vanish too quickly atthe same time as n, unless f(x) is constant. The general factthat underlies uncertainty inequalities is that a non-periodicfunction (x) and its Fourier Transform circ;(u) cannot vanishtoo quickly at the same time as x and u, unless (x) is zero(almost everywhere). For a simple introduction to some aspectsof uncertainty inequalities, see [5]; for a thorough and recentintroduction to this vast subject, see [3]. 1991 MathematicsSubject Classification 42A05, 42A16, 42A99.     

Specht Filtrations for Hecke Algebras of Type A

Let H_q(d) be the Iwahori–Hecke algebra of the symmetricgroup, where q is a primitive 1th root of unity. Using resultsfrom the cohomology of quantum groups and recent results aboutthe Schur functor and adjoint Schur functor, it is proved that,contrary to expectations, for l 4 the multiplicities in a Spechtor dual Specht module filtration of an H_q(d)-module are welldefined. A cohomological criterion is given for when an H_q(d)-modulehas such a filtration. Finally, these results are used to givea new construction of Young modules that is analogous to theDonkin–Ringel construction of tilting modules. As a corollary,certain decomposition numbers can be equated with extensionsbetween Specht modules. Setting q = 1, results are obtainedfor the symmetric group in characteristic p 5. These resultsare false in general for p = 2 or 3.     

Root-Clustering Criteria (I): The Composite-Matrix Approach

In a series of articles we investigate different approachesto root clustering criteria. In particular, given A C^nxn andan algebraic region A C, we are looking for a criterion (asingle set of inequalities) ensuring the inclusion (A) A. Here,in the first article, we collect previous ideas on transformableregions and composite matrices, state new results, and correctand refine others.     

The Tracial Topological Rank of C*-Algebras

We introduce the notion of tracial topological rank for C*-algebras.In the commutative case, this notion coincides with the coveringdimension. Inductive limits of C*-algebrasof the form PM_n(C(X))P,where X is a compact metric space with dim X k, and P is aprojection in M_n(C(X)), have tracial topological rank no morethan k. Non-nuclear C*-algebras can have small tracial topologicalrank. It is shown that if A is a simple unital C*-algebra withtracial topological rank k (< ), then

(i) A is quasidiagonal,

(ii) A has stable rank 1,

(iii) A has weakly unperforatedK₀(A),

(iv) A has the following Fundamental Comparabilityof Blackadar:if p, q A are two projections with (p) < (q)for all tracialstates on A, then p q

. 2000 MathematicsSubject Classification: 46L05, 46L35.     

On the Compactness of the Nonlinear Evolution Operator in a Banach Space

Let X be a real Banach space and let A(t): X 2^x be dissipativefor all t(0, T). Assume that {A(t)} generates an evolution operatorU(t, s) of type (D, , f). Necessary and sufficient conditionsare given for the compactness of U(t, s) for 0 s < t <T.     

The Second Dual Algebra of the Measure Algebra of a Compact Group

It is shown that for every compact group G, L¹(G)^ is uniqueand minimal among all the closed subsets I of M(G)** such thatI is a proper (0, M(G)**) algebraic ideal, and such that I issolid with respect to absolute continuity; that is, n L¹(G)^whenever n M(G)** and n << µ L¹(G)^. 1991 MathematicsSubject Classification 43A20, 43A22.     

Normal Families and Shared Values

For f a meromorphic function on the plane domain D and a C,let _f(a) = {z D: f(z) = a}. Let F be a family of meromorphicfunctions on D, all of whose zeros are of multiplicity at leastk. If there exist b 0 and h > 0 such that for every f F,_f(0) = _f^(k)(b) and 0 < |f^(k+1)(z)| h whenever z _f(0), thenF is a normal family on D. The case _f(0) = Ø is a celebratedresult of Gu [5]. 1991 Mathematics Subject Classification 30D45,30D35.     

Some Properties of Free Groups of Some Soluble Varieties of Groups

Let F be a free group, and let _n(F) be the nth term of the lowercentral series of F. It is proved that F/[_j(F), _i(F), _k(F)]and F/[_j(F), _i(F), _k(F), _l(F)] are torsion free and residuallynilpotent for certain values of i, j, k and i, j, k, l, respectively.In the process of proving this, it is proved that the analogousLie rings are torsion free.     

Integer Points of Entire Functions

A result is proved which implies the following conjecture ofOsgood and Yang from 1976: if f and g are non-constant entirefunctions, such that T(r, f) = O(T(r,g)) as r and such thatg(z Z implies that f(z) Z, then there exists a polynomialG with coefficients in Q, such that G(Z) Z and f = G g. 2000Mathematics Subject Classification 30D20, 30D35.     

Finiteness of integrals of functions of Levy processes

We prove necessary and sufficient conditions for the almostsure convergence of the integrals

and thus of ,where M_t = sup{|X_s|: s t} is the two-sided maximum processcorresponding to a Lévy process (X_t)_{t 0}, a(·)is a non-decreasing function on [0, ) with a(0) = 0, g(·)is a positive non-increasing function on (0, ), possibly withg(0 + ) = , and f(·) is a positive non-decreasing functionon [0, ) with f(0) = 0. The conditions are expressed in termsof the canonical measure, (·), of the process X_t. Thespecial case when a(x) = 0, f(x) = x and g(·) is equivalentto the tail of (at zero or infinity) leads to an interestingcomparison of M_t with the largest jump of X_t in (0, t]. Some results concerning the convergence at zero and infinityof integrals like t g(a(t) + |X_t|) dt, t g(S_t) dt,and t g(R_t) dt, where S_t is the supremum process and R_t= S_t – X_t is the process reflected in its supremum, arealso given. We also consider the convergence of integrals suchas , etc.     

Periodic solutions and oscillation of discrete non-linear delay population dynamics model with external force

^** Email: emelabbasy{at}mans.edu.eg^*** Email: shsaker{at}mans.edu.eg In this paper, we consider the discrete non-linear delay populationdynamics model [graphic: see PDF] where m is a positive integer, p(n), Q(n) and (n) are positiveperiodic sequences of period . By the method that involves theapplication of the Gaines and Mawhins coincidence degree theory,we prove that there exists a positive -periodic solution (n). We prove that every positive solutionof (*) which does not oscillate about (n)satisfies lim_t[y(n)–(n)]=0.We establish some necessary and sufficient conditions for theoscillation of every positive solution about (n), and finally, we establish the lower and upperbounds of the oscillatory solutions.     

Julia Sets of Polynomials are Uniformly Perfect

Let f be a polynomial of degree at least two. We shall showthat the Julia set J(f) of f is uniformly perfect. This meansthat there is a constant c(0,1) depending on f only such thatwhenever zJ(f) and 0 < r < diam J(f) then J(f) intersectsthe annulus {w:cr |w — z| r}.