Sharp Weak Type Inequalities for Fractional Integral Operators

Suppose that d≥1 is an integer, α∈(0,d) is a fixed parameter and let I _α be the fractional integral operator associated with d-dimensional Walsh-Fourier series on (0,1] ^d . Let p, q be arbitrary numbers satisfying the conditions 1≤p<d/α and 1/q=1/p?α/d. We determine the optimal constant K, which depends on α, d and p, such that for any f∈L ^p ((0,1] ^d ) we have

$$ ||I_{\alpha } f||_{L^{q,\infty }((0,1]^{d})}\leq K||f||_{L^{p}((0,1]^{d})}. $$

In fact, we shall prove this inequality in the more general context of probability spaces equipped with a regular tree-like structures. This allows us to obtain this result also for non-integer dimension. The proof exploits a certain modification of the so-called Bellman function method and appropriate interpolation-type arguments. We also present a sharp weighted weak-type bound for I _α , which can be regarded as a version of the Muckenhoupt-Wheeden conjecture for fractional integral operators.

     

Insight into hydrogen bonding of terephthalamides with amino acids: Synthesis,structural and spectroscopic investigations

Several bis-terephthalamides based on methyl esters of amino acids including glycine (1), β-alanine (2), γ-aminobutyric acid (3) and ε-aminocaproic acid (4), X(CH₂)_nHNOCC₆H₄CONH(CH₂)_nX (XCO₂CH_3,n = 1, 2, 3 and 5), have been synthesized and characterized by single-crystal X-ray diffraction and spectroscopic methods: FT-IR, polarized FT-IR, Raman, ¹H NMR and ¹³C NMR. The four structures assemble via classical NH?O hydrogen bonds between amide functionalities linking the molecules into chains parallel to the short axis. The analysis of polarized IR spectra of pure and deuterated compounds reveals that a weak interchain (“through-space”) exciton coupling involves two closely-spaced hydrogen bonds belonging to two different adjacent chains. The exciton coupling magnitude decreases with the addition of methylene groups to the terephthalamide system. Isotope effects in terephthalamides show that the distribution of protons and deuterons in the crystalline lattice depends on the strength of the exciton couplings involving hydrogen bonds.     

On a class of equations stemming from various quadrature rules

We deal with the functional equation

$F(y)-F(x)=(y-x)\sum_{i=1}^n a_if(\lambda_ix+(1-\lambda_i)y),\quad x,y\in \mathbb {R}$

motivated by quadrature rules of approximate integration. In previous results the solutions of this equation were found only in some particular cases. For example the coefficients λ _i were supposed to be rational or the equation in question was solved only for n=2. In the current paper we do not assume any particular form of coefficients occurring in this equation and we allow n to be any positive integer. Moreover, we obtain a solution of our equation without any regularity assumptions concerning the functions f and F.

     

Quantification of methyl thiocyanate in the headspace of Pseudomonas aeruginosa cultures and in the breath of cystic fibrosis patients by selected ion flow tube mass spectrometry

Infection by Pseudomonas aeruginosa (PA) is a major cause of morbidity and mortality in patients with cystic fibrosis (CF). Breath analysis could potentially be a useful diagnostic of such infection, and analyses of volatile organic compounds (VOCs) emitted from PA cultures are an important part of the search for volatile breath markers of PA lung infection. Our pilot experiments using solid‐phase microextraction, SPME and gas chromatography/mass spectrometric (GC/MS) analyses of volatile compounds produced by PA strains indicated a clear presence of methyl thiocyanate. This provided a motivation to develop a method for real‐time online quantification of this compound by selected ion flow tube mass spectrometry, SIFT‐MS. The kinetics of reactions of H₃O⁺, NO⁺ and O₂^+? with methyl thiocyanate at 300 K were characterized and the characteristic product ions determined (proton transfer for H₃O⁺, rate constant 4.6 × 10^–9 cm³ s^–1; association for NO⁺, 1.7 × 10^–9 cm³ s^–1 and nondissociative charge transfer for O₂^+?_, 4.3 × 10^–9 cm³ s^–1). The kinetics library was extended by a new entry for methyl thiocyanate accounting for overlaps with isotopologues of hydrated hydronium ions. Solubility of methyl thiocyanate in water (Henry's law constant) was determined using standard reference solutions and the linearity and limits of detection of both SIFT‐MS and SPME‐GC/MS methods were characterized. Thirty‐six strains of PA with distinct genotype were cultivated under identical conditions and 28 of them (all also producing HCN) were found to release methyl thiocyanate in headspace concentrations greater than 6 parts per billion by volume (ppbv). SIFT‐MS was also used to analyze the breath of 28 children with CF and the concentrations of methyl thiocyanate were found to be in the range 2–21 ppbv (median 7 ppbv). Copyright © 2011 John Wiley & Sons, Ltd.     

Unprecedented Architecture of the Linear (μ-Cl)-Tetranuclear Cobalt Complex with Quaterpyridine Ligand

Abstract  

The cationic tetranuclear cobalt complex of the geometry Cl-CoL-Cl-CoL-Cl-CoL-Cl-CoL (where L is 6,6′′′-dimethyl-2,2′:6′,2′′:6′′,2′′′-quaterpyridine) has been synthesized and structurally characterized. The structural unit contains one cationic ligand, two Eu(NO₃)₅ dianions, two solvent acetonitrile molecules and some disordered water molecules. The salt crystallizes in the monoclinic space group C2/c with a = 17.405 (3) ?, b = 28.700 (5) ? and c = 23.350 (5) ?, β = 95.14 (2)o. The cation is C ₂ -symmetrical, the central Cl atom lies on the twofold axis. This structure is another example of centro-noncentro ambiguity: in the centrosymmetric model, the terminal Cl in the cation is disordered between the two ends of the chain. The Eu ion is coordinated by five nitro groups, the coordination number is 10–two oxygen atoms of every nitro group are involved in the coordination with the mean Eu–O distance of 2.52(3) ?. The quaterpyridine ligands are folded, the dihedral angle between the terminal ring planes is 29.3 (2) and 32.8 (3)o in the two symmetry independent ligands. In the crystal structure the coulombic attraction between the cations and anions seems to be a driving force; the ionic species make a chessboard-like pattern, the voids in such created structure are filled with the solvent-acetonitrile molecules as well as by some disordered waters.     

On relations between gradient and classical equivariant homotopy groups of spheres

We investigate relations between stable equivariant homotopy groups of spheres in classical and gradient categories. To this end, the auxiliary category of orthogonal equivariant maps, a natural enlargement of the category of gradient maps, is used. Our result allows for describing stable equivariant homotopy groups of spheres in the category of orthogonal maps in terms of classical stable equivariant groups of spheres with shifted stems. We conjecture that stable equivariant homotopy groups of spheres for orthogonal maps and for gradient maps are isomorphic.     

A Padé family of iterations for the matrix sign function and related problems

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p‐sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p‐sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1?z)^?σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log‐concavity property. Copyright © 2011 John Wiley & Sons, Ltd.     

Long Time,Large Scale Limit of the Wigner Transform for a System of Linear Oscillators in One Dimension

We consider the long time, large scale behavior of the Wigner transform W _? (t,x,k) of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. This model has been introduced in Basile et al. in Phys. Rev. Lett. 96 (2006) to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile et al. in Arch. Rat. Mech. 195(1):171–203 (2009). In the present paper we prove that in the unpinned case there exists γ ₀>0 such that for any γ∈(0,γ ₀] the weak limit of W _? (t/? ^3/2γ ,x/? ^γ ,k), as ??1, satisfies a one dimensional fractional heat equation \(\partial_{t} W(t,x)=-\hat{c}(-\partial_{x}^{2})^{3/4}W(t,x)\) with \(\hat{c}>0\). In the pinned case an analogous result can be claimed for W _? (t/? ^2γ ,x/? ^γ ,k) but the limit satisfies then the usual heat equation.