On local convexity of nonlinear mappings between Banach spaces

We find conditions for a smooth nonlinear map f: U → V between open subsets of Hilbert or Banach spaces to be locally convex in the sense that for some c and each positive ? < c the image f(B _?(x)) of each ?-ball B _?(x) ? U is convex. We give a lower bound on c via the second order Lipschitz constant Lip₂(f), the Lipschitz-open constant Lip_o(f) of f, and the 2-convexity number conv₂(X) of the Banach space X.     

Metrizability of Clifford topological semigroups

We prove that a countably compact Clifford topological semigroup S is metrizable if and only if the set E={e∈S:ee=e} of idempotents of S is a metrizable G _δ -set in S.     

A topological characterization of LF-spaces

We present a topological characterization of LF-spaces and detect small box-products that are (locally) homeomorphic to LF-spaces.     

A New Quantum Chemical Approach to the Magnetic Properties of Oligonuclear Transition‐Metal Complexes: Application to a Model for the Tetranuclear Manganese Cluster of Photosystem II

Broken‐symmetry DFT calculations on transition‐metal clusters with more than two centers allow the hyperfine coupling constants to be extracted. Application of the proposed theoretical scheme to a tetranuclear manganese complex that models the S₂ state of the oxygen‐evolving complex of photosystem II yields hyperfine parameters that can be directly compared with experimental data. The picture shows the metal–oxo core of the model and the following parameters; exchange coupling constant J_ij, the expectation value of the site‐spin operator , and the isotropic hyperfine coupling parameters.

     

High resolution Fourier transform infrared spectra and analysis of the ν14, ν15 and ν16 bands of azetidine

Rotationally resolved vibrational spectra of the three lowest frequency bands of the four-membered heterocycle azetidine (c-C₃H₆NH) have been collected with a resolution of 0.00096 cm⁻¹ using the far infrared beamline at the Canadian Light Source synchrotron. The modes observed correspond principally to motions best described as: β-CH₂ rock (ν₁₄) at 736.701310(7) cm⁻¹, ring deformation (ν₁₅) at 648.116041(8) cm⁻¹, and the ring puckering mode (ν₁₆) at 207.727053(9) cm⁻¹. A global fit of 14 276 rovibrational transitions from the three bands provided an accurate set of ground state spectroscopic constants as well as excited state parameters for each of the three vibrational modes. The ground state structure was determined to be that of the puckered conformer having the NH bond in an equatorial arrangement.     

The Role of Astaxanthin as a Nutraceutical in Health and Age-Related Conditions

The current review provides an up-to-date analysis of scientific data on astaxanthin (ASX) sources and experimental studies on its health benefits as a potent antioxidant in the aging process. ASX is a liposoluble carotenoid nutrient and reddish-orange pigment, naturally synthesized by numerous microalgae, yeasts, and bacteria as secondary metabolites. Provides a reddish hue to redfish and shellfish flesh that feed on ASX-producing microorganisms. The microalga Haematococcus pluvialis is the most important source for its industrial bioproduction. Due to its strong antioxidant properties, numerous investigations reported that natural ASX is a more significant antioxidant agent than other antioxidants, such as vitamin C, vitamin E, and β-carotene. Furthermore, several data show that ASX possesses important nutraceutical applications and health benefits, especially in healthy aging processes. However, further studies are needed for a deeper understanding of the potential mechanisms through which ASX could lead to its effective role in the healthy aging process, such as supporting brain health and skin homeostasis. This review highlights the current investigations on the effective role of ASX in oxidative stress, aging mechanisms, skin physiology, and central nervous system functioning, and shows the potential clinical implications related to its consumption.     

On index of total boundedness of (strictly) o-bounded groups

Let G be a metrizable topological group. Denote by itb(G) the smallest cardinality of a cover of G by totally bounded subsets of G. A group G is defined to be σ-bounded if itb(G)₀. The group G is called o-bounded if for every sequence (U_n)_nω of neighborhoods of the identity in G there exists a sequence (F_n)_nω of finite subsets in G such that G=_nωF_n·U_n; G is called strictly o-bounded (respectively OF-determined) if the second player (respectively one of the players) has a winning strategy in the following game OF: two players, I and II, choose at every step n an open neighborhood U_n of the identity in G and a finite subset F_n of G, respectively. The player II wins if G=_nωF_n·U_n.

For a second countable group G the following results are proven. . If G is strictly o-bounded, then itb(G)₁ and G is σ-bounded or meager. If the space G is analytic, then the group is OF-determined and satisfies . G is σ-bounded if it is strictly o-bounded and one of the following conditions holds: (i) G is analytic; (ii) ; (iii) (MA+¬CH) holds; (iv) analytic games are determined; (v) there exists a measurable cardinal. Also we show that under (MA) every non-locally compact Polish Abelian divisible group contains a Baire o-bounded OF-undetermined subgroup.     

Free energies of adsorption of amino acids, short linear peptides and 2,5-piperazinediones on silica from water as estimated from high-performance liquid-chromatographic retention data

Equilibrium constants (K) and free energies (— G) of adsorption of amino acids, short linear peptides and cyclic dipeptides (2,5-piperazinediones) on silica from neutral aqueous solutions were calculated from the retention values measured by means of high-performance liquid chromatography on a silica gel column. For most amino acids and linear peptides — G values were negative and K < 1, thus showing very weak adsorption. 2,5-Piperazinediones exhibited higher adsorbability (for most of them, — G > 0 and K > 1) as compared to related dipeptides. Influence of the structure of a-substituent on the adsorbability is analyzed. A linear dependence of — G on the number of aliphatic carbon atoms in a sorbate molecule was found for the series of aliphatic bifunctional amino acids, related dipeptides and 2,5-piperazinediones, as well as for the group from glycine to triglycyl glycine.     

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.     

All-optical multichannel 2R regeneration in a fiber-based device

We propose the design of an all-optical 2R regenerator capable of handling multiple wavelength-division-multiplexed channels simultaneously. It extends the known concept of off-center filtering of self-phase-modulation-broadened signal spectra. The novel feature of the proposed device is a dispersion map that strongly suppresses interchannel impairments. The map employs several sections of nonlinear fiber with high normal dispersion, separated by dispersion compensators with spectrally periodic group delay. The results of our numerical simulations indicate the feasibility of such a multichannel regenerator.