Advancing Mathematical Activity: A Practice-Oriented View of Advanced Mathematical Thinking

The purpose of this article is to contribute to the dialogue about the notion of advanced mathematical thinking by offering an alternative characterization for this idea, namely advancing mathematical activity. We use the term advancing (versus advanced) because we emphasize the progression and evolution of students' reasoning in relation to their previous activity. We also use the term activity, rather than thinking. This shift in language reflects our characterization of progression in mathematical thinking as acts of participation in a variety of different socially or culturally situated mathematical practices. For these practices, we emphasize the changing nature of students' mathematical activity and frame the process of progression in terms of multiple layers of horizontal and vertical mathematizing.     

Probabilities of the deviation of the sample mean from its median

One investigates the asymptotic behavior of as n, where is the sample mean constructed from the independent, identically distributed random variables X₁...,X_n, >0. One considers the powerlike and the exponential rates of decrease of this probability.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 130–136, 1983.     

Finite-temperature phase diagram of a polarized fermi gas in an optical lattice

We present phase diagrams for a polarized Fermi gas in an optical lattice as a function of temperature, polarization, and lattice filling factor. We consider the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO), Sarma or breached pair, and BCS phases, and the normal state and phase separation. We show that the FFLO phase appears in a considerable portion of the phase diagram. The diagrams have two critical points of different nature. We show how various phases leave clear signatures to momentum distributions of the atoms which can be observed after time of flight expansion.     

Creation of a monopole in a spinor condensate

We propose a method to create a monopole structure in a multicomponent condensate by applying the basic methods used to create vortices and solitons experimentally in single-component condensates. We also show that by using a two-component structure for a monopole, we can avoid many problems related to the previously suggested three-component monopole. We discuss the observation and dynamics of such a monopole structure, and note that the dynamics of the two-component monopole differs from the dynamics of the three-component monopole.     

Quasi-two-dimensional superfluid fermionic gases

We study a quasi-two-dimensional superfluid Fermi gas where the confinement in the third direction is due to a strong harmonic trapping. We investigate the behavior of such a system when the chemical potential is varied and find strong modifications of the superfluid properties due to the discrete harmonic oscillator states. We show that such quasi-two-dimensional behavior can be created and observed with current experimental capabilities.     

Boundedness of a class of bi-parameter square functions in the upper half-space

We consider a class of bi-parameter kernels and related square functions in the upper half-space, and give an efficient proof of a boundedness criterion for them. The proof uses modern probabilistic averaging methods and is based on controlling double Whitney averages over good cubes.     

A fast direct solver for elliptic problems with a divergence constraint

A fast direct solution method for a discretized vector‐valued elliptic partial differential equation with a divergence constraint is considered. Such problems are typical in many disciplines such as fluid dynamics, elasticity and electromagnetics. The method requires the problem to be posed in a rectangle and boundary conditions to be either periodic boundary conditions or the so‐called slip boundary conditions in one co‐ordinate direction. The arising saddle‐point matrix has a separable form when bilinear finite elements are used in the discretization. Based on a result for so‐called p‐circulant matrices, the saddle‐point matrix can be transformed into a block‐diagonal form by fast Fourier transformations. Thus, the fast direct solver has the same structure as methods for scalar‐valued problems which are based on Fourier analysis and, therefore, it has the same computational cost ??(N log N). Numerical experiments demonstrate the good efficiency and accuracy of the proposed method. Copyright © 2002 John Wiley & Sons, Ltd.     

Chlovalicin B,a Chlorinated Sesquiterpene Isolated from the Marine Mushroom Digitatispora marina

As part of our search for bioactive metabolites from understudied marine microorganisms, the new chlorinated metabolite chlovalicin B (1) was isolated from liquid cultures of the marine basidiomycete Digitatispora marina, which was collected and isolated from driftwood found at Vannøya, Norway. The structure of the novel compound was elucidated by spectroscopic methods including 1D and 2D NMR and analysis of HRMS data, revealing that 1 shares its molecular scaffold with a previously isolated compound, chlovalicin. This represents the first compound isolated from the Digitatispora genus, and the first reported fumagillin/ovalicin-like compound isolated from Basidiomycota. Compound 1 was evaluated for antibacterial activities against a panel of five bacteria, its ability to inhibit bacterial biofilm formation, for antifungal activity against Candida albicans, and for cytotoxic activities against malignant and non-malignant human cell lines. Compound 1 displayed weak cytotoxic activity against the human melanoma cell line A2058 (~50% survival at 50 µM). No activity was detected against biofilm formation or C. albicans at 50 µM, or against bacterial growth at 100 µM nor against the production of cytokines by the human acute monocytic leukemia cell line THP-1 at 50 µM.     

Optimal payout policy in presence of downside risk

We analyze the determination of a value maximizing dividend payout policy for a broad class of cash reserve processes modeled as spectrally negative jump diffusions. We extend previous results based on continuous diffusion models and characterize the value of the optimal dividend distribution strategy explicitly. We also characterize explicitly the values as well as the optimal dividend thresholds for a class of associated optimal liquidation and sequential lump sum dividend control problems. Our results indicate that both the value as well as the marginal value of the optimal policies are increasing functions of policy flexibility in the discontinuous setting as well.      

Two Novel Lyso-Ornithine Lipids Isolated from an Arctic Marine Lacinutrix sp. Bacterium

The Lacinutrix genus was discovered in 2005 and includes 12 Gram-negative bacterial species. To the best of our knowledge, the secondary metabolite production potential of this genus has not been explored before, and examination of Lacinutrix species may reveal novel chemistry. As part of a screening project of Arctic marine bacteria, the Lacinutrix sp. strain M09B143 was cultivated, extracted, fractionated and tested for antibacterial and cytotoxic activities. One fraction had antibacterial activity and was subjected to mass spectrometry analysis, which revealed two compounds with elemental composition that did not match any known compounds in databases. This resulted in the identification and isolation of two novel isobranched lyso-ornithine lipids, whose structures were elucidated by mass spectrometry and NMR spectroscopy. Lyso-ornithine lipids consist of a 3-hydroxy fatty acid linked to the alpha amino group of an ornithine amino acid through an amide bond. The fatty acid chains were determined to be iso-C15:0 (1) and iso-C16:0 (2). Compound 1 was active against the Gram-positive S. agalactiae, while 2 showed cytotoxic activity against A2058 human melanoma cells.