Differentiability of arithmetic Fourier series arising from Eisenstein series

Let \(k\in \mathbb {N}^*\) be even. We consider two trigonometric series \( F_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \sin (2\pi n x)\) and \(G_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k-1}(n)}{n^{k+1}} \cos (2\pi n x),\) where \(\sigma _{k-1}\) is the divisor function. They converge on \(\mathbb {R}\) to continuous functions. In this paper, we examine the differentiability of \(F_k\) and \(G_k\). These functions are related to Eisenstein series, and their (quasi-)modular properties allow us to apply the method proposed by Itatsu in 1981 in the study of the Riemann series. We focus on the case \(k=2\) and we show that the sine series exhibits a different behaviour with respect to differentiability than the cosine series. We prove that the differentiability of \(F_2\) at an irrational x is related to the continued fraction expansion of x. We estimate the modulus of continuity of \(F_2\). We formulate a conjecture concerning differentiability of \(F_k\) and \(G_k\) for any k even.     

Measuring the price of anarchy in critical care unit interactions

Hospital throughput is often studied and optimised in isolation, ignoring the interactions between hospitals. In this paper, critical care unit (CCU) interaction is placed within a game theoretic framework. The methodology involves the use of a normal form game underpinned by a two-dimensional continuous Markov chain. A theorem is given that proves that a Nash Equilibrium exists in pure strategies for the games considered. In the United Kingdom, a variety of utilisation targets are often discussed: aiming to ensure that wards/hospitals operate at a given utilisation value. The effect of these target policies is investigated justifying their use to align the interests of individual hospitals and social welfare. In particular, we identify the lowest value of a utilisation target that aligns these.     

Metals and Metal-Nanoparticles in Human Pathologies: From Exposure to Therapy

An increasing number of pathologies correlates with both toxic and essential metal ions dyshomeostasis. Next to known genetic disorders (e.g., Wilson’s Disease and β-Thalassemia) other pathological states such as neurodegeneration and diabetes are characterized by an imbalance of essential metal ions. Metal ions can enter the human body from the surrounding environment in the form of free metal ions or metal-nanoparticles, and successively translocate to different tissues, where they are accumulated and develop distinct pathologies. There are no characteristic symptoms of metal intoxication, and the exact diagnosis is still difficult. In this review, we present metal-related pathologies with the most common onsets, biomarkers of metal intoxication, and proper techniques of metal qualitative and quantitative analysis. We discuss the possible role of drugs with metal-chelating ability in metal dyshomeostasis, and present recent advances in therapies of metal-related diseases.     

Cationic Emulsion Polymerization of Octamethylcyclotetrasiloxane (D4) in Mixtures with Alkoxysilanes

The cationic emulsion polymerization of octamethylcyclotetrasiloxane (D4) in mixtures with methyltriethoxysilane (MTES) and vinyltriethoxysilane (VTES) was studied by FTIR ATR, GC, the development of a toluene insoluble fraction of the polymer and a gravimetric analysis. The polymerization of D4 alone was also conducted for comparison and, additionally, the development of molecular weight of polydimethylsiloxane (PDMS) obtained in that process was studied by GPC. Dodecylbenzenesulphonic acid (DBSA) was used as a surfactant and catalyst. The process was carried out in a “starved feed” mode by adding dropwise the monomer mixture to the aqueous solution of DBSA. FTIR ATR spectra were recorded by the sensor placed in the probe tip of a ReactIR 15^TM apparatus. It was found that the silicone polymer formation proceeded faster when D4 was polymerized in the mixture with alkoxysilanes, especially in the beginning of the process, and that already at the beginning of the process, the partly crosslinked polymer was formed. The induction period of ca. 30 min was observed and the concentration of cyclic siloxanes (D4 and decamethylcyclopentasiloxane—D5) remained at a very low level in the course of the reaction and only traces were detected in the final product. The particle size development in the course of the reaction was also studied and it was found that the particle size distribution was bimodal and was broadening as the reaction proceeded, though this phenomenon was less distinct when D4 was polymerized in the mixtures with alkoxysilanes. The structure of the reaction product was confirmed by ²⁹Si NMR.     

A Comprehensive Investigation of the Structural,Thermal, and Biological Properties of Fully Randomized Biomedical Polyesters Synthesized with a Nontoxic Bismuth(III) Catalyst

Aliphatic polyesters are the most common type of biodegradable synthetic polymer used in many pharmaceutical applications nowadays. This report describes the ring-opening polymerization (ROP) of l-lactide (L-LA), ε-caprolactone (CL) and glycolide (Gly) in the presence of a simple, inexpensive and convenient PEG200-BiOct₃ catalytic system. The chemical structures of the obtained copolymers were characterized by ¹H- or ¹³C-NMR. GPC was used to estimate the average molecular weight of the resulting polyesters, whereas TGA and DSC were employed to determine the thermal properties of polymeric products. The effects of temperature, reaction time, and catalyst content on the polymerization process were investigated. Importantly, the obtained polyesters were not cyto- or genotoxic, which is significant in terms of the potential for medical applications (e.g., for drug delivery systems). As a result of transesterification, the copolymers obtained had a random distribution of comonomer units along the polymer chain. The thermal analysis indicated an amorphous nature of poly(l-lactide-co-ε-caprolactone) (PLACL) and a low degree of crystallinity of poly(ε-caprolactone-co-glycolide) (PCLGA, X_c = 15.1%), in accordance with the microstructures with random distributions and short sequences of comonomer units (l = 1.02–2.82). Significant differences in reactivity were observed among comonomers, confirming preferential ring opening of L-LA during the copolymerization process.     

Enhanced and Heteromolecular Guest Encapsulation in Nonporous Crystals of a Perfluorinated Triketonato Dinuclear Copper Complex

The flexible host framework of a perfluorinated mononuclear copper complex, [Cu(L¹)₂] ( 1 , HL¹=3-hydroxy-1,3-bis(pentafluorophenyl)-2-propen-1-one), with a CuO₄ core reversibly encapsulated several organic guest molecules through electrostatic interactions in its crystals. Hence, the corresponding dinuclear complex, [Cu₂(L²)₂] ( 2 , H₂L²=1,5-dihydroxy-1,5-bis(pentafluorophenyl)-1,4-pentadien-3-one), was prepared to enhance guest recognition and the ability to separate molecular mixtures. Complex 2 comprises a Cu₂O₆ core and four pentafluorophenyl groups. In crystal 2 , cavities are formed on the axial sites of the metal core that are surrounded by pentafluorophenyl groups. The crystal of 2 encapsulates various guest molecules, that is, benzene ( 3 ), toluene ( 4 ), xylene ( 5 ), mesitylene ( 6 ), durene ( 7 ), and anisole ( 8 ). X-ray crystallographic and thermogravimetric (TG) studies show that three guest molecules are present in the crystal cavities. The number of guest molecules found in complex 2 was higher than that in complex 1 , for example, ( 2 )₃ ⋅ ( 6 )₁₀> 1⋅ ( 6 )₂, ( 2 )₂ ⋅ ( 7 )₇> 1⋅7 , or 2⋅ ( 8 )₃> 1⋅ ( 8 )₂. Naphthalene ( 9 ), was encapsulated in 2 to give 2⋅ ( 9 )₃, but not in 1 . In the crystal of complex 2 , heteromolecular guest encapsulation was confirmed, designated as 2⋅ ( 3 )₂ ⋅9 . TG analysis indicates that the thermal stability of the guest-included crystals of 2 is higher than that of 1 .     

Polymerization of hydroxyacetylenes by ruthenium alkylidene complexes

Polymerization of a series of acetylenes with a hydroxy functional group was investigated by using modified 2nd generation Grubbs (A) and Grubbs–Hoveyda (B) initiators. Owing to excellent tolerance for polar functional groups, catalysts A and B polymerized 3-butyn-2-ol (1), 2-methyl-3-butyn-2-ol (2) and 3-butyn-1-ol (3). The catalytic activities of catalyst B were greater than those of initiator A for these polymerizations. The steric bulk and the position of hydroxyl group of the monomer had an influence on the rate of polymerization. In order to investigate the role of hydroxyl group of monomers in the polymerization, the reaction between hydroxyacetylenes and the ruthenium complexes were monitored by ¹H NMR spectroscopy. The results revealed the formation of new alkylidene species via α-insertion. The calculated relative energies of propagating species formed in the reaction of A with monomer 1 suggested the formation of oxygen-chelated species. The structures of resulting polymers were characterized by various methods such as NMR, IR and UV–Vis spectroscopies. The ruthenium initiators gave polymers with different geometric structure of main chain than conventional catalysts.