Asymptotic Formula for the Coordinates of the Zeros of Sections of the Zeta Function, zeta(N)(s), Near s = 1

The sum, Σ1^Nn^-8 = ζN(s), s = σ + it, is called a section of the Riemann zeta function, ζ(s). Here an asymptotic formula, for large N, giving the location of the zeros of ζN(s) near s = 1 will be demonstrated. In particular these zeros will lie in σ < 1. Relationships between the location of the zeros of ζN(s) and the zeros of ζ(s) were discovered by Turán. The location in σ < 1 will also be demonstrated directly, without the asymptotic formula, together with results valid also when s - 1 is not small.     

Subordinate quadratic forms and their complementary forms

Theorem 1. For α, β on the range 1,..., μ, let Q(z) = ^*aαβzαzβ be a real valued, nonsingular, symmetric quadratic form. For positive integers r and s such that μ = r + s set (z₁,..., z_μ) = (u₁,..., u_r:S₁,..., S_n), Q(z) = P(u, s) and [Formula: see text] Let B = (z⁽¹⁾,..., z^(r)) be a base “over R” for points z ε π_r. For an arbitrary r-tuple ω₁,..., ω_r set [Formula: see text] index H_B(ω) = κ and nullity H_B(ω) = ν. Then [Formula: see text]     

omega-Theorems for Quotients of Zeta-Functions at Combinations of Points

Theorem A. Let q ≥ O and r ≥ O be integers. Let s = σ + it, let ζ(s) be the Riemann zeta-function, let G_o(s) = 1, and [Formula: see text] and let F(s) = G_q(s)/H_r(s). Then as t → ∞ lim sup [unk]F(1 + it)[unk]/(log log t)^q+r+1 ≥ (6/π)²)^r+1 exp {(q + r + 1)γ}, where γ is Euler's constant.     

Characters of irreducible Harish-Chandra modules and Goldie ranks of their annihilators

Assume that G_C is a simply connected complex semi-simple Lie group with Lie algebra [unk]. Let G G_C be a real form, H G be a maximally split Cartan subgroup with Lie algebra [unk]₀, [unk] = [unk]₀ [unk] C, and P [unk]^* be the weight lattice. If X is an irreducible Harish-Chandra module with infinitesimal character λ [unk]^*, one can associate to X a family {θ(μ): μ λ + P} of Z-linear combinations of distribution characters of G, so that θ(λ) = X. θ(μ) is irreducible when μ lies in C_λ, a certain positive “Weyl” chamber containing λ. In this case let Ann θ(μ) be its annihilator in U([unk]) and set p(μ) = Goldie rank of U([unk])/ Ann θ(μ). Let d = Gelfand-Kirillov dimension of X. For most x [unk]₀ if exp tx is regular for small t > 0 then (i) c(μ) = lim_t→0⁺t^d θ(μ) (exp tx) exists for all μ λ + P; (ii) c(μ) extends to a homogeneous Weyl group harmonic polynomial on [unk]^* of degree ½(dim G - dim H) - d; (iii) up to a constant, c = the polynomial extending p to [unk]^*. c is said to be the character polynomial of Ann X.     

Lipoarabinomannan of Mycobacterium: Mannose capping by a multifunctional terminal mannosyltransferase

Biosynthesis of phosphatidylinositol (PI)-containing lipoarabinomannan (LAM) and lipomannan (LM) of Mycobacterium spp. follows a conserved pathway involving multiple membrane-associated, substrate-specific mannosyltransferases (ManTs) responsible for the sequential addition of α-mannopyranosyl (Manp) units donated by decaprenyl-P-Manp on the periplasmic side of the plasma membrane. Because of their receptor-binding and immunomodulatory properties, the α(1→2)-linked di- and tri-Manp motifs that functionalize the nonreducing arabinan termini of LAM (ManLAM) in Mycobacterium tuberculosis are of crucial importance. We now show that the M. tuberculosis ManT, Rv2181, is required for the addition of these α(1→2)-linked Manp residues but also at other locations of the LAM molecule. Structural analyses of the LM and LAM variants produced by a M. tuberculosis Rv2181 knockout mutant revealed the presence of but a single Manp residue on the nonreducing arabinan termini of LAM and also a complete absence of α(1→2)-linked Man branching on the mannan backbones of LM and LAM. A recombinant strain was constructed in ManLAM-deficient Mycobacterium smegmatis that coexpressed Rv2181 and Rv1635c—the ManT responsible for the addition of the first Manp capping residue of ManLAM. Analysis revealed LAM termini fully capped with di- and tri-Manp motifs in addition to α(1→2)Man branching on the mannan backbones of LM and LAM, confirming the involvement of the α(1→2)ManT Rv2181 in the dual role of Man capping and mannan-core branching, and in the process generated a rapidly growing, ManLAM-containing strain, a tool for the study of the role of ManLAM in the pathogenesis of tuberculosis.     

Ultrametric skeletons

We prove that for every ε∈(0,1) there exists C_ε∈(0,∞) with the following property. If (X,d) is a compact metric space and μ is a Borel probability measure on X then there exists a compact subset S⊆X that embeds into an ultrametric space with distortion O(1/ε), and a probability measure ν supported on S satisfying ν(B_d(x,r))⩽(μ(B_d(x,C_εr))^1-ε for all x∈X and r∈(0,∞). The dependence of the distortion on ε is sharp. We discuss an extension of this statement to multiple measures, as well as how it implies Talagrand’s majorizing measure theorem.     

Existence of nonarithmetic monodromy groups

In an 1885 paper, E. Picard defined a subgroup Τ(Λ) of PU(2,1) generated by monodromies and depending on parameters Λ = (λ₁,λ₂,λ₃,λ₄), 0 < λ_i < 1, < λ_i < 3, λ_i + λ_j ≥ 1, 1 ≤ i < j ≤ 4. The family Τ(Λ) resembles the family of groups Τ([unk]) defined in 1978 but is a different family. In common with the groups Τ([unk]), (i) Τ(Λ) is discrete for a finite number of Λ, (ii) Τ(Λ) is a nonarithmetic lattice for some Λ, and (iii) for all Λ [unk]⁴, there is a compact complex surface S(Λ) with π₁ [S(Λ)] of finite index in Τ(Λ).     

Structural effects in solvolytic reactions; carbon-13 NMR studies of carbocations: Effect of increasing electron demand on the carbon-13 NMR shifts in substituted tert-cumyl and 1-aryl-1-cyclopentyl carbocations-correlation of the data by a new set of substituent constants, sigma

The cationic carbon substituent chemical shifts (ΔδC⁺) for nine representative meta-substituted tert-cumyl carbocations are correlated satisfactorily by the σ_m⁺ substituent constants (slope ρ+ = -18.18, correlation coefficient r = 0.990). However, the substituent chemical shifts (ΔδC⁺) for the corresponding para derivatives are not correlated by the σ_p⁺ substituent constants. The possibility of developing a set of substituent constants capable of correlating such ¹³C NMR shifts was examined. The slope of the line defined by the meta substituents (ρ+ = -18.18) was utilized to calculate σ^C+ constants for both meta and para substituents. The utility of these constants was then tested by their ability to correlate the ¹³C NMR shifts in the cations for a different system, the 1-aryl-1-cyclopentyl cations. Indeed, these σ^C+ values correlate very well with the ΔδC⁺ values, yielding ρ^C+ = -16.84, r = 0.999.     

Isotropic probability measures in infinite-dimensional spaces

Every isotropic probability measure on the space R^∞ of real sequences x = (x₁, x₂,...) is a convex combination of the measure concentrated at 0 and a member of I₀(R^∞), the set of all isotropic probability measures p_∞ on R^∞ with p_∞({0}) = 0. Each p_∞ [unk] I₀(R^∞) is completely determined by any one of its finite-dimensional marginal distributions p_n. Each p_n has a density function f_n with dp_n(x₁,..., x_n) = dx₁... dx_nf_n(x₁² +... + x_n²). Each f_n is completely monotone in 0 < ξ < ∞ (hence analytic in the right complex ξ half-plane), and π^n/2Γ(n/2)^-1 ʃ₀^∞dξ ξ^n/2-1f_n(ξ) = 1. Every f that satisfies these two conditions is f_n for a unique p_∞ [unk] I₀(R^∞). Hence the equation πʃ_ξ^∞dζ f₂(ζ) = ʃ₀^∞dμ (t)e^-tξ defines a bijection between I₀(R^∞) and the set of all probability measures μ on 0 ≤ t < ∞. If p_∞ [unk] I₀(R^∞) then p_∞({x: Σ_i=1^∞x_i² < ∞}) = 0, so p_∞ is not a “softened” or “fuzzy” version of the inequality Σ_i=1^∞x_i² ≤ 1. If the prior information in a linear inverse problem consists of this inequality and nothing else, stochastic inversion and Bayesian inference are both unsuitable inversion techniques.     

omega-Theorems for Riemann's Zeta-Function at Harmonic Combinations of Points

For integer r ≥ 2 there are arbitrarily large T for which [Formula: see text] With s = σ + it, corresponding results hold for σ = c, 1/2 ≤ c < 1, and for 1/ζ(1 + it). A similar result holds for [Formula: see text]     

Analytic construction of Hartree-Fock density matrices

A method is presented for variational calculation of the energy and the spin densities derived from a single-determinant wavefunction. Sum and difference coordinates [unk]R = ½([unk]r₁ + [unk]r₁) and [unk]r = [unk]r₁ - [unk]r₁ are introduced, and the density matrix P([unk]r₁,[unk]r₁) is expanded in partial waves in the new coordinate frame: [Formula: see text] The functions h_L(ε,r) are bound or continuum hydrogenic functions with energy ε.     

Lower bounds for the life-span of solutions of nonlinear wave equations in three dimensions

The paper deals with strict solutions u(x,t) = u(x₁,x₂,x₃,t) of an equation [Formula: see text] where Du is the set of four first derivatives of u. For given initial values u(x,0) = εF(x), u_t(x,0) = εG(x), the life span T(ε) is defined as the supremum of all t to which the local solution can be extended for all x. Blowup in finite time corresponds to T(ε) < ∞. Examples show that this can occur for arbitrarily small ε. On the other hand, T(ε) must at least be very large for small ε. By assuming that a_ik,F,G [unk] C^∞, that a_ik(0) = 0, and that F,G have compact support, it is shown that [Formula: see text] for every N. This result had been established previously only for N < 4.     

Electric and Dielectric Properties in Low-Frequency Fields of Composites Consisting of Silicone Rubber and Al Particles for Flexible Electronic Devices

Understanding the electrical conduction and dielectric polarization properties of elastomer-based composites is important for the design of flexible and elastic electronic devices and circuits. Five samples were manufactured by mixing silicone rubber (RTV-530) with Al particles in different volume fractions, x equal to 0%, 0.5%, 1%, 2.5% and 5.1%. Using the complex impedance measurements, the electric modulus, M, the electrical conductivity, σ, and the dielectric permittivity, ε, over the frequency range 100 Hz–200 kHz were analyzed. The electrical conductivity spectrum, σ(f), follows the Jonscher universal law and the DC conductivity of the samples, σ_DC, increases from 2.637·10⁻⁸ S/m to 5.725·10⁻⁸ S/m, with increasing x from, 0 to 5.1%. The conduction process was analyzed in terms of Mott’s variable-range-hopping (VRH) model. The hopping distance of the charge carriers, R_h decreases with increasing x, from 7.30 nm (for x = 0) to 5.92 nm (for x = 5.1%). The frequency dependence of permittivity, ε(f) = ε′(f) − iε″(f), reveals a relaxation process with the maximum of ε″(f) shifting from 301 Hz to 385 Hz and values of ε′(f) increasing with the increase of x.     

Conductimetric determination of thermodynamic pairing constants for symmetrical electrolytes

Earlier theories of electrolytic conductance are reviewed; all of these, with the exception of the Arrhenius-Ostwald theory, are based on physical models. Their theory failed to describe the conductance of strong electrolytes because it did not include the effects (then unsuspected) of long-range forces on mobility. Thermodynamic derivations are independent of model; applied to the postulated equilibrium A⁺ + B^- A⁺B^- between free ions and nonconducting paired ions, the thermodynamic pairing constant K_a equals a_p/(a±)², and ΔG, the difference in free energy between paired ions (activity = a_p) and free ions (activity = a_±), equals (-RT ln K_a). Converting to the molarity scale, K_a = (1000 ρ/M)[1 - γ)/cy²(y_±)²]. Here ρ is the density of the solvent of molecular weight M, c is stoichiometric concentration of electrolyte (mol/liter), γ is the fraction of solute present as unpaired ions, and y_± is their activity coefficient. The corresponding conductance function Λ = Λ(c;Λ₀,R,G)involves three parameters: Λ₀, the limiting equivalent conductance; R, the sum of the radii of the cospheres of the ions; and ΔG. Conductance data for cesium bromide and for lithium chloride in water/dioxane mixtures and for the alkali halides in water are analyzed to determine these parameters. Correlations between the values found for R and ΔG and properties characteristic of salt and solvent are then discussed.     

Differential contribution of HLA-DR,DQ, and TAP2 alleles to systemic lupus erythematosus susceptibility in Spanish patients: role of TAP2*01 alleles in Ro autoantibody production

OBJECTIVE—To study the influence MHC class II and TAP2 alleles exert on systemic lupus erythematosus (SLE) susceptibility and on the clinical and serological manifestations of the disease, in a cohort of Spanish patients.
METHODS—HLA-DR serological typing and HLA-DQA, DQB, and TAP2 DNA sequence specific oligotyping, were carried out in 85 unrelated Spanish SLE patients and 186 healthy controls. Autoantibodies detection was carried out by indirect immunofluorescence and counter immunoelectrophoresis.
RESULTS—Total SLE group: the frequency of HLA-DR3 and HLA-DQA1*0501 is significantly increased in this group (p_c<0.005, δ=0.34 and p_c<0.005, δ= 0.45, respectively) although the highest δ value (δ=0.87) is obtained when the TAP2*01 alleles are considered. No DQB allele shows significant deviation from the control group. Renal damage: it mainly occurs in HLA-DR3 patients (p_c<0.0005 and δ=0.72). HLA-DQA1*0501 (p_c<0.05, δ=0.57) and DQB1*0201 (p_c NS, δ=0.56) are weaker susceptibility factors. Ro+ (but not La) group: this autoantibody response is associated with TAP2*01 alleles in homozygosity (p<0.05, δ=0.81). Ro/La+ group: it has a different genetic background as HLA-DQA1*0501 (δ=1) and HLA-DQB1*0201 (δ=1) are the main susceptibility factors.
CONCLUSIONS—A differential association between HLA-DR, DQA1, and DQB1 alleles and SLE or its clinical and serological manifestations are found. Furthermore, the associations are different to the ones reported in other ethnic groups. Finally, TAP2*01 group of alleles are associated with the highest susceptibility to SLE (higher than HLA-DR3) and may influence Ro (but not La) autoantibodies production, whereas HLA-DQA1*0501 and DQB1*0201 mediates concomitant Ro and La production.

     

Low-Temperature Dielectric Response of Strontium Titanate Thin Films Manipulated by Zn Doping

The voltage dependence of the dielectric permittivity ε′ and the low dielectric loss tanδ of incipient ferroelectrics have drawn vast attention to the use of these materials for the development of tuning elements in electronics and telecommunications. Here, we study the DC electric field dependence of low-temperature ε′ in ~320 nm thick sol-gel-derived SrTi_1−xZn_xO_3−δ thin films with x = 0.01 and 0.05, deposited on Pt/TiO₂/SiO₂/Si substrates. Incorporation of Zn onto Ti sites is found to decrease ε′ compared to undoped SrTiO₃ films, while increasing the relative tunability n_r up to ~32.9% under a DC electric field of 125 kV/cm at low temperatures. The hysteresis-free variation in ε′ with electric field and tanδ values below 0.6% observed for SrTi_1−xZn_xO_3−δ film with x = 0.01 make this compound more attractive for tunable device applications.     

Maximal functions associated to smooth curves

Let t → γ(t), 0 ≤ t ≤ 1, be a smooth curve in IRⁿ. Define the maximal function [unk](f) by [unk](f)(x) = sup_0<h≤1 (1/h) ʃ₀^h | f(x - γ(t))| dt. We state conditions under which [unk](f)_p ≤ A_p f_p, for 1 < p ≤ ∞.     

Problems on periodic simple continued fractions

Let N be a positive non-square integer and a₁,a₂,...,a₃ be the partial denominators in the period of length s = s(N) of the continued fraction for √N. Also let Σ_N = a₃ - a_3-1 + -... ± a₁, and let h(d) be the class-number of Q(√d). Hirzebruch (unpublished) recently found the surprising theorem (which is a special case of more general results): If p is a prime3(4) and p > 3, then h(p) = 1 implies that Σ_p = 3h(-p).     

Synthesis,Crystal Structure and Luminescent Property of Mg(II) Complex with N-Benzenesulphonyl-L-Leucine and 1,10-Phenanthroline

A new complex [Mg(L)₂(phen)(H₂O)₂](phen)(H₂O)₂ [L= N-benzenesulphonyl-L-leucine] was synthesized by the reaction of magnesium chloride hexahydrate with N-benzenesulphonyl-L-leucine and 1,10-phenanthroline in the CH₃CH₂OH/H₂O (v:v = 5:1). It was characterized by elemental analysis, IR and X-ray single crystal diffraction analysis. The crystal of the title complex [Mg(L)₂(phen)(H₂O)₂](phen)(H₂O)₂ belongs to triclinic, space group P-1 with a = 0.72772(15) nm, b = 1.4279(3) nm, c = 1.4418(3) nm, α = 63.53(3)°, β = 79.75(3)°, γ = 81.83(3)°, V = 1.3163(5) nm³, Z =1, D_c= 1.258 μg·m⁻³, μ = 0.177 mm⁻¹, F(000) = 526, and final R₁ = 0.0506, ωR₂ = 0.1328. The complex comprises a six-coordinated magnesium(II) center, with a N₂O₄ distorted octahedron coordination environment. The molecules are connected by hydrogen bonds and π-π stacking to form one dimensional chain structure. The luminescent property of the Mg(II) complex has been investigated in solid.     

Transformation of Tri-Titanium(IV)-Substituted α-Keggin Polyoxometalate (POM) into Tetra-Titanium(IV)-Substituted POMs : Reaction Products of Titanium(IV) Sulfate with the Dimeric Keggin POM Precursor under Acidic Conditions

Reaction products of titanium(IV) sulfate in HCl-acidic aqueous solution with the dimeric species linked through three intermolecular Ti-O-Ti bonds of the two tri-titanium(IV)-substituted α-Keggin polyoxometalate (POM) subunits are described. Two novel titanium(IV)-containing α-Keggin POMs were obtained under different conditions. One product was a dimeric species through two intermolecular Ti-O-Ti bonds of the two tetra-titanium(IV)-substituted α-Keggin POM subunits, i.e., [[{Ti(H₂O)₃}₂(μ-O)](α-PW₉Ti₂O₃₈)]₂^6- (1). The other product was a monomeric α-Keggin species containing the tetra-titanium(IV) oxide cluster and two coordinated sulfate ions, i.e., [{Ti₄(μ-O)₃(SO₄)₂(H₂O)₈}(α-PW₉O₃₄)]^3- (2). Molecular structures of 1 and 2 were also discussed based on host (lacunary site)-guest (titanium atom) chemistry.