Link Polynomials of Higher Order

In this paper, we study certain polynomial invariants of links(singular or non-singular) that are related to the Homfly polynomialand Vassiliev's invariants. The Homfly polynomial H_L [3] (alsoknown as the Flypmoth polynomial) satisfies the well-known skeinrelation The Vassiliev invariants [1, 2] (of order 1) satisfy the relations and The invariants that we study satisfy the skein relations      

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Strong Weighted Mean Summability and Kuttner's Theorem

In this paper we study sequence spaces that arise from the conceptof strong weighted mean summability. Let q = (q_n) be a sequenceof positive terms and set Q_n = ⁿ_k=1q_k. Then the weighted meanmatrix M_q = (a_nk) is defined by if kn, a_nk=0 if k>n. It is well known that M_q defines a regular summability methodif and only if Q_n. Passing to strong summability, we let 0<p<.Then , are the spaces of all sequences that are strongly M_q-summablewith index p to 0, strongly M_q-summable with index p and stronglyM_q-bounded with index p, respectively. The most important specialcase is obtained by taking M_q = C₁, the Cesàro matrix,which leads to the familiar sequence spaces w₀(p), w(p) and w(p), respectively, see [4, 21]. We remark that strong summabilitywas first studied by Hardy and Littlewood [8] in 1913 when theyapplied strong Cesàro summability of index 1 and 2 toFourier series; orthogonal series have remained the main areaof application for strong summability. See [32, 6] for furtherreferences. When we abstract from the needs of summability theory certainfeatures of the above sequence spaces become irrelevant; forinstance, the q_k simply constitute a diagonal transform. Hence,from a sequence space theoretic point of view we are led tostudy the spaces     

Tangential Boundary Behaviour of Harmonic and Holomorphic Functions

Let K be a kernel on Rⁿ, that is, K is a non-negative, unboundedL¹ function that is radially symmetric and decreasing. We definethe convolution K * F by and note from L^p-capacity theory [11, Theorem 3] that, if F L^p, p > 1, then K * F exists as a finite Lebesgue integraloutside a set A Rⁿ with C_K,p(A) = 0. For a Borel set A, where We define the Poisson kernel for = {(x, y) : x Rⁿ, y > 0} by and set Thus u is the Poisson integral of the potential f = K * F, andwe write u=P_y*(K*F)=P_y*f=P[f]. We are concerned here with the limiting behaviour of such harmonicfunctions at boundary points of , and in particular with the tangential boundary behaviour ofthese functions, outside exceptional sets of capacity zero orHausdorff content zero.     

On the Index of Vector Fields Tangent to Hypersurfaces with Non-Isolated Singularities

Let F be a germ of a holomorphic function at 0 in Cⁿ⁺¹, having0 as a critical point not necessarily isolated, and let be a germ of a holomorphic vectorfield at 0 in Cⁿ⁺¹ with an isolated zero at 0, and tangent toV := F^–1(0). Consider the O_V,0-complex obtained by contractingthe germs of Kähler differential forms of V at 0 (0.1) with the vector field X:=|_Von V: (0.2)     

The Cauchy-Schwarz Norm Inequality for Elementary Operators in Schatten Ideals

For bounded Hilbert space operators X, A_n and B_n, n = 1, 2,..., for all p < and These inequalities involve some estimates for the norm of elementaryoperators with the range contained in the Schatten p-ideals.     

The Singular Homology of the Hawaiian Earring

The singular homology groups of compact CW-complexes are finitelygenerated, but the groups of compact metric spaces in generalare very easy to generate infinitely and our understanding ofthese groups is far from complete even for the following compactsubset of the plane, called the Hawaiian earring: Griffiths [11] gave a presentation of the fundamental groupof H and the proof was completed by Morgan and Morrison [15].The same group is presented as the free -product of integers Z in [4, Appendix]. Hence the firstintegral singular homology group H₁(H) is the abelianizationof the group . These results have been generalized to non-metrizable counterparts H_I of H(see Section 3). In Section 2 we prove that H₁(X) is torsion-free and H_i(X) =0 for each one-dimensional normal space X and for each i 2.The result for i 2 is a slight generalization of [2, Theorem5]. In Section 3 we provide an explicit presentation of H₁(H)and also H₁(H_I) by using results of [4]. Throughout this paper, a continuum means a compact connectedmetric space and all maps are assumed to be continuous. Allhomology groups have the integers Z as the coefficients. Thebouquet with n circles is denoted by B_n. The base point (0, 0) of B_n is denoted by o forsimplicity.     

Weighted Integral Inequalities for the Hardy Type Operator and the FractionalMaximal Operator

Let w(x), u(x) and (x) be weight functions. In this paper, underappropriate conditions on Young's functions ₁, ₂ we characterizethe inequality for the Hardy-typeoperator T defined in [1] and the inequality for the fractional maximal operator M_{, ;} definedin [8], as well as the corresponding weak-type inequalities.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Asymptotic Expansions of Multiple Zeta Functions and Power Mean Values of Hurwitz Zeta Functions

Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where _r(s₁,...,s_r;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.     

Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(–(py')'+qy), , where det = > 0, c 0, det > 0, t 0 and (cs + dr –au – tb)² < 4(cr – ta)(ds – ub). Denoteby (l; ; ) the eigenvalue problem ly = y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define (; ; ) as above but with l replacedby . Let w_n denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions w_n(0)= sin and pw'_n(0) = –cos and let _n = –aw_n(1)+cpw'_n(1).Define _n and _n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;; ) and (; ; ) have the samespectrum or for all n, thenq/r = /.     

Recurrence, Dimension and Entropy

Let (_A, T) be a topologically mixing subshift of finite typeon an alphabet consisting of m symbols and let :_A R^d be a continuousfunction. Denote by (x) the ergodic limit when the limit exists. Possible ergodic limits arejust mean values dµ for all T-invariant measures. Forany possible ergodic limit , the following variational formulais proved: where h_µ denotes the entropy of µ and h_top denotestopological entropy. It is also proved that unless all pointshave the same ergodic limit, then the set of points whose ergodiclimit does not exist has the same topological entropy as thewhole space _A     

Interpolation of Vector-Valued Real Analytic Functions

Let R^d be an open domain. The sequentially complete DF-spacesE are characterized such that for each (some) discrete sequence(z_n) , a sequence of natural numbers (k_n) and any family the infinite system of equations has an E-valued real analytic solution f.     

On the Zeros of the Riemann Zeta-Function

It is well known that the multiplicity of a complex zero =ß+iof the zeta-function is O(log||). This may be proved by meansof Jensen's formula, as in Titchmarsh [7, Chapter 9]. It mayalso be seen from the formula for the number N(T) of zeros suchthat 0<<T, (1) due to Backlund [1], in which E(T) is a continuous functionsatisfying E(T)=O(1/T) and (2) We assume here that T is not the ordinate of a zero; with appropriatedefinitions of N(T) and S(T) the formula is valid for all T.We have S(T)=O(logT). On the Lindelöf Hypothesis S(T)=o(logT),(Cramér [2]), and on the Riemann Hypothesis (Littlewood [5]). These results are over 70 years old. Because the multiplicity problem is hard, it seems worthwhileto see what can be said about the number of distinct zeros ina short T-interval. We obtain the following result, which isindependent of any unproved hypothesis.     

On the Compactness and Approximation Numbers of Hardy-Type Integral Operators in Lorentz Spaces

We characterize the mapping properties such as the boundedness,compactness and measure of noncompactness for those real weightfunctions , , u0, v0, for which the Hardy-type integral operatorof the form acts from to , when the parameters are restricted to the range 1 < max (r,s) min (p, q) < and the kernel k(x, y) 0 satisfies theOinarov condition (see (2) below). For the case k(x, y) = 1,we obtain lower and upper estimates of the approximation numbers,extending the result of [5].     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

Periodic Solutions of Lienard Equations with Asymmetric Nonlinearities at Resonance

The existence of 2-periodic solutions of the second-order differentialequation where a, b satisfy and p(t)=p(t+2),t R, is examined. Assume that limits lim_x±F(x)=F(±)(F(x)=) and lim_x±g(x)=g(±)exist and are finite. It is proved that the equation has atleast one 2-periodic solution provided that the zeros of thefunction ₁ are simple and the zeros of the functions ₁, ₂ aredifferent and the signs of ₂ at the zeros of ₁ in [0,2/n) donot change or change more than two times, where ₁ and ₂ aredefined as follows: Moreover, it is also proved that the given equation has at leastone 2-periodic solution provided that the following conditionshold: with 1 p < q 2.