Hyperharmonic series involving Hurwitz zeta function

We show that the sum of the series formed by the so-called hyperharmonic numbers can be expressed in terms of the Riemann zeta function. These results enable us to reformulate Euler's formula involving the Hurwitz zeta function. In additon, we improve Conway and Guy's formula for hyperharmonic numbers.     

Fekete-like polynomials

In 2001, Borwein, Choi, and Yazdani looked at an extremal property of a class of polynomial with ±1 coefficients. Their key result was:

Theorem. (See Borwein, Choi, Yazdani, 2001.) Letf(z)=±z±z²±?±zN−1, and ζ a primitive Nth root of unity. If N is an odd positive integer then

     

Unprovability threshold for the planar graph minor theorem

This note is part of the implementation of a programme in foundations of mathematics to find exact threshold versions of all mathematical unprovability results known so far, a programme initiated by Weiermann. Here we find the exact versions of unprovability of the finite graph minor theorem with growth rate condition restricted to planar graphs, connected planar graphs and graphs embeddable into a given surface, assuming an unproved conjecture (*): ‘there is a number a>0 such that for all k≥3, and all n≥1, the proportion of connected graphs among unlabelled planar graphs of size n omitting the k-element circle as minor is greater than a’. Let γ be the unlabelled planar growth constant (27.2269≤γ<30.061). Let P(c) be the following first-order arithmetical statement with real parameter c: “for every K there is N such that whenever G₁,G₂,…,G_N are unlabelled planar graphs with |G_i|<K+c⋅log₂i then for some i<j≤N, G_i is isomorphic to a minor of G_j”. Then

1.

for every , P(c) is provable in IΔ₀+exp;

2.

for every , P(c) is unprovable in .

We also give proofs of some upper and lower bounds for unprovability thresholds in the general case of the finite graph minor theorem.     

Diophantine approximation and badly approximable sets

Let (X,d) be a metric space and (Ω,d) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).     

Parametric duality on minimax programming involving generalized convexity in complex space

In this paper we employ generalized convexity of complex functions to establish several sufficient optimality conditions for minimax programming in complex spaces. Using such criteria, we constitute a parametrical dual, and establish the weak, strong, and strict converse duality theorems in the framework.     

Catalyst-Substrate Effects on Biocompatible SABRE Hyperpolarization

The hyperpolarization technique, Signal Amplification by Reversible Exchange (SABRE), has the potential to improve clinical diagnosis by making molecular magnetic resonance imaging in vivo a reality. Essential to this goal is the ability to produce a biocompatible bolus for administration. We seek here to determine how the identity of the catalyst and substrate affects the cytotoxicity by in vitro study, in addition to reporting how the use of biocompatible solvent mixtures influence the polarization transfer efficiency. By illustrating this across five catalysts and 8 substrates, we are able to identify routes to produce a bolus with minimal cytotoxic effects.     

Heterogeneous Parahydrogen-Induced Polarization of Diethyl Ether for Magnetic Resonance Imaging Applications

Magnetic resonance imaging (MRI) with the use of hyperpolarized gases as contrast agents provides valuable information on lungs structure and function. While the technology of ¹²⁹Xe hyperpolarization for clinical MRI research is well developed, it requires the expensive equipment for production and detection of hyperpolarized ¹²⁹Xe. Herein we present the ¹H hyperpolarization of diethyl ether vapor that can be imaged on any clinical MRI scanner. ¹H nuclear spin polarization of up to 1.3 % was achieved using heterogeneous hydrogenation of ethyl vinyl ether with parahydrogen over Rh/TiO₂ catalyst. Liquefaction of diethyl ether vapor proceeds with partial preservation of hyperpolarization and prolongs its lifetime by ≈10 times. The proof-of-principle 2D ¹H MRI of hyperpolarized diethyl ether was demonstrated with 0.1×1.1 mm² spatial and 120 ms temporal resolution. The long history of use of diethyl ether for anesthesia is expected to facilitate the clinical translation of the presented approach.     

Natural Abundance 15N NMR by Dynamic Nuclear Polarization: Fast Analysis of Binding Sites of a Novel Amine‐Carboxyl‐Linked Immobilized Dirhodium Catalyst

A novel heterogeneous dirhodium catalyst has been synthesized. This stable catalyst is constructed from dirhodium acetate dimer (Rh₂(OAc)₄) units, which are covalently linked to amine‐ and carboxyl‐bifunctionalized mesoporous silica (SBA‐15?NH₂?COOH). It shows good efficiency in catalyzing the cyclopropanation reaction of styrene and ethyl diazoacetate (EDA) forming cis‐ and trans‐1‐ethoxycarbonyl‐2‐phenylcyclopropane. To characterize the structure of this catalyst and to confirm the successful immobilization, heteronuclear solid‐state NMR experiments have been performed. The high application potential of dynamic nuclear polarization (DNP) NMR for the analysis of binding sites in this novel catalyst is demonstrated. Signal‐enhanced ¹³C CP MAS and ¹⁵N CP MAS techniques have been employed to detect different carboxyl and amine binding sites in natural abundance on a fast time scale. The interpretation of the experimental chemical shift values for different binding sites has been corroborated by quantum chemical calculations on dirhodium model complexes.