The parametrized family of metric Mahler measures

Let M(α) denote the (logarithmic) Mahler measure of the algebraic number α. Dubickas and Smyth, and later Fili and the author, examined metric versions of M. The author generalized these constructions in order to associate, to each point in t∈(0,∞], a metric version Mt of the Mahler measure, each having a triangle inequality of a different strength. We further examine the functions Mt, using them to present an equivalent form of Lehmer?s conjecture. We show that the function t?Mtt(α) is constructed piecewise from certain sums of exponential functions. We pose a conjecture that, if true, enables us to graph t?Mt(α) for rational α.     

A description of continued fraction expansions of quadratic surds represented by polynomials

The principal thrust of this investigation is to provide families of quadratic polynomials , where e_k²−f_k²C=n (for any given nonzero integer n) satisfying the property that for any , the period length of the simple continued fraction expansion of is constant for fixed k and lim_k→∞?_k=∞. This generalizes, and completes, numerous results in the literature, where the primary focus was upon |n|=1, including the work of this author, and coauthors, in Mollin (Far East J. Math. Sci. Special Vol. 1998, Part III, 257-293; Serdica Math. J. 27 (2001) 317) Mollin and Cheng (Math. Rep. Acad. Sci. Canada 24 (2002) 102; Internat Math J 2 (2002) 951) and Mollin et al. (JP J. Algebra Number Theory Appl. 2 (2002) 47).     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

Hilbert-Speiser number fields for a prime p inside the p-cyclotomic field

Let p be a prime number. We say that a number field F satisfies the condition when for any cyclic extension N/F of degree p, the ring of p-integers of N has a normal integral basis over . It is known that F=Q satisfies for any p. It is also known that when p?19, any subfield F of Q(ζp) satisfies . In this paper, we prove that when p?23, an imaginary subfield F of Q(ζp) satisfies if and only if and p=43, 67 or 163 (under GRH). For a real subfield F of Q(ζp) with F≠Q, we give a corresponding but weaker assertion to the effect that it quite rarely satisfies .     

A Chebysheff recursion formula for Coxeter polynomials

A polynomial f(T)∈Z[T] is represented by q(T)∈Z[T] if ; f(T) is graphically represented if for χ_M(T) the characteristic polynomial of a symmetric matrix M. Many instances of Coxeter polynomialsf_A(T), for A a finite dimensional algebra, are (graphically) representable. We study the case of extended canonical algebras A, see [H. Lenzing, J.A. de la Peña, Extended canonical algebras and Fuchsian singularities, in press], show that the corresponding polynomials f_A(T) are representable and satisfy a Chebysheff type recursion formula. We get consequences for the eigenvalues of the Coxeter transformation of A showing, for instance, that at most four eigenvalues may lie outside the unit circle.     

The space of adding machines generated by continuous self maps of manifolds

Let M be the Cantor space or an n-dimensional manifold with C(M,M) the set of continuous self-maps of M, and . We prove the following:

(1)

If α≠∞, then Sα(M) is a nowhere dense subset of M×C(M,M) that contains no isolated points.

(2)

If α?β, then .

     

Existence of fast positive wavefronts for a non-local delayed reaction-diffusion equation

We establish the existence of a continuous family of fast positive wavefronts u(t,x)=?(x+ct), ?(−∞)=0, ?(+∞)=κ, for the non-local delayed reaction-diffusion equation . Here 0 and κ>0 are fixed points of g∈C²(R₊,R₊) and the non-negative K is such that is finite for every real λ. We also prove that the fast wavefronts are non-monotone if .     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

On locally compact quantum groups whose algebras are factors

In this paper we are interested in examples of locally compact quantum groups (M,Δ) such that both von Neumann algebras, M and the dual , are factors. There is a lot of known examples such that are respectively of type (I_∞,I_∞) but there is no example with factors of other types. We construct new examples of type (I_∞,II_∞), (II_∞,II_∞) and (IIIλ,IIIλ) for each λ∈[0,1]. We also show that there is no such example with M or a finite factor.     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

Haj?asz-Sobolev imbedding and extension

The author establishes some geometric criteria for a Haj?asz-Sobolev -extension (resp. -imbedding) domain of Rn with n?2, s∈(0,1] and p∈[n/s,∞] (resp. p∈(n/s,∞]). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α-cigar domain with α∈(0,1) if and only if for some/all s∈[α,1) and p=(2−α)/(s−α), where denotes the restriction of the Triebel-Lizorkin space on Ω.     

Congruences involving Bernoulli polynomials

Let {B_n(x)} be the Bernoulli polynomials. In the paper we establish some congruences for , where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for and the sum , where h(d) is the class number of the quadratic field of discriminant d and p-regular functions are those functions f such that are rational p-integers and for n=1,2,3,… . We also establish many congruences for Euler numbers.     

Spectral radius and infinity norm of matrices

Let Mn(R) be the linear space of all n×n matrices over the real field R. For any A∈Mn(R), let ρ(A) and ‖A_‖∞ denote the spectral radius and the infinity norm of A, respectively. By introducing a class of transformations φa on Mn(R), we show that, for any A∈Mn(R), ρ(A)<‖A_‖∞ if . If A∈Mn(R) is nonnegative, we prove that ρ(A)<‖A_‖∞ if and only if , and ρ(A)=‖A_‖∞ if and only if the transformation φ‖A_‖∞ preserves the spectral radius and the infinity norm of A. As an application, we investigate a class of linear discrete dynamic systems in the form of X(k+1)=AX(k). The asymptotical stability of the zero solution of the system is established by a simple algebraic method.     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

A subadditive property of the gamma function

Let Δ=min_x?0Γ(2x)/Γ(x) and . We prove that the function x?(Γ(x))^α is subadditive on (0,∞) if and only if α∗?α?0.     

Dixmier traces as singular symmetric functionals and applications to measurable operators

We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the operator ideal L^(1,∞) (which is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L^(1,∞), i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L^(1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L^(1,∞) if and only if the sequence of singular numbers {s_n(x)}_n?1 (in the descending order and counting the multiplicities) satisfies . In this case, our characterization amounts to saying that a positive element x∈L^(1,∞) is measurable if and only if exists; (ii) the set of Dixmier traces and the set of Connes-Dixmier traces are norming sets (up to equivalence) for the space , where the space is the closure of all finite rank operators in L^(1,∞) in the norm ∥.∥_(1,∞).     

Sequences and filters of characters characterizing subgroups of compact Abelian groups

Let H be a countable subgroup of the metrizable compact Abelian group G and a (not necessarily continuous) character of H. Then there exists a sequence of (continuous) characters of G such that limn→∞χn(α)=f(α) for all α∈H and does not converge whenever α∈G?H. If one drops the countability and metrizability requirement one can obtain similar results by using filters of characters instead of sequences. Furthermore the introduced methods allow to answer questions of Dikranjan et al.     

On the value distribution and moments of the Epstein zeta function to the right of the critical strip

We study the Epstein zeta function En(L,s) for and a random lattice L of large dimension n. For any fixed we determine the value distribution and moments of En(⋅,cn) (suitably normalized) as n→∞. We further discuss the random function c?En(⋅,cn) for c∈[A,B] with and determine its limit distribution as n→∞.     

Computation of the Iwasawa invariants of certain real abelian fields

Let p be a prime number and k a finite extension of . It is conjectured that the Iwasawa invariants λ_p(k) and μ_p(k) vanish for all p and totally real number fields k. Some methods to verify the conjecture for each real abelian field k are known, in which cyclotomic units and a set of auxiliary prime numbers are used. We give an effective method, based on the previous one, to compute the exact value of the other Iwasawa invariant ν_p(k) by using Gauss sums and another set of auxiliary prime numbers. As numerical examples, we compute the Iwasawa invariants associated to in the range 1<f<200 and 5?p<10000.