Boundaries of weak peak points in noncommutative algebras of Lipschitz functions

It has been shown that any Banach algebra satisfying ‖f ²‖ = ‖f‖² has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip(X, \mathbbF\mathbb{F}) the algebra over R of F-valued Lipschitz functions on a compact metric space (X, d), where \mathbbF\mathbb{F} = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip(X, \mathbbF\mathbb{F}) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary.     

A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Network Dynamics in Elemental Assimilation and Metabolism

Metabolism and physiology frequently follow non-linear rhythmic patterns which are reflected in concepts of homeostasis and circadian rhythms, yet few biomarkers are studied as dynamical systems. For instance, healthy human development depends on the assimilation and metabolism of essential elements, often accompanied by exposures to non-essential elements which may be toxic. In this study, we applied laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS) to reconstruct longitudinal exposure profiles of essential and non-essential elements throughout prenatal and early post-natal development. We applied cross-recurrence quantification analysis (CRQA) to characterize dynamics involved in elemental integration, and to construct a graph-theory based analysis of elemental metabolism. Our findings show how exposure to lead, a well-characterized toxicant, perturbs the metabolism of essential elements. In particular, our findings indicate that high levels of lead exposure dysregulate global aspects of metabolic network connectivity. For example, the magnitude of each element’s degree was increased in children exposed to high lead levels. Similarly, high lead exposure yielded discrete effects on specific essential elements, particularly zinc and magnesium, which showed reduced network metrics compared to other elements. In sum, this approach presents a new, systems-based perspective on the dynamics involved in elemental metabolism during critical periods of human development.     

On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

Consider the homogeneous equation $$u'(t) = l(u)(t){\rm{ for a}}{\rm{.e}}{\rm{. }}t \in [a,b]$$ where ?: C([a, b];?) → L([a, b];?) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.     

Middle‐School Mathematics Teachers' Beliefs in NCTM's Vision

This study examined the extent to which seventh‐ and eighth‐grade mathematics teachers are aware of National Council of Teachers of Mathematics (NCTM) standards documents, Curriculum and Evaluation Standards for School Mathematics and Principles and Standards for School Mathematics and agree with NCTM's vision of school mathematics as expressed in these documents. Quantitative data were collected through the Mathematics Standards Belief Survey (MSBS), a survey specifically designed to measure teachers' overall belief in NCTM's vision as well as in certain philosophical tenets of NCTM. Of the 82 seventh‐ and eighth‐grade mathematics teachers in the identified school district of Nevada, 73 (89.0%) participated in this study. The data revealed that, among seventh‐ and eighth‐grade mathematics teachers, secondary‐certified teachers had significantly higher MSBS scores than elementary‐certified teachers. A number of other findings were made, including significant differences among mean belief scores in the philosophical tenets of NCTM.     

Convective heat transfer in a conducting fluid over a permeable stretching surface with suction and internal heat generation/absorption

We consider a convective flow in a porous medium of an incompressible viscous conducting fluid impinging on a permeable stretching surface with suction, and internal heat generation/absorption. Using a similarity transformation the governing equations of the problem are reduced to a coupled third-order nonlinear ordinary differential equations. We first examine a number of special cases for which we may obtain exact solutions. We then obtain analytical solutions (by the Homotopy Analysis Method) and numerical solutions (by a boundary value problem solver), in order to further study the behavior of the nonlinear differential equations, for various values of the physical parameters. Our numerical solutions are shown to agree with the available results in the literature. We then employ the numerical results to bring out the effects of the suction parameter, heat source/sink parameter, stretching parameter, porosity parameter, the Prandtl number and the free convection parameter on the flow and heat transfer characteristics. In the absence of suction and free convection, our findings are in agreement with the corresponding numerical results of Attia [H.A. Attia, On the effectiveness of porosity on stagnation point flow towards a stretching surface with heat generation, Comput. Mater. Sci. 38 (2007) 741-745].     

Topological defects and cosmology

Many particle physics models of matter admit solutions corresponding to stable or long-lived topological defects. In the context of standard cosmology it is then unavoidable that such defects will form during phase transitions in the very early Universe. Certain types of defects lead to disastrous consequences for cosmology, others may play a useful role, as possible seeds for the formation of structure in the Universe, or in mediating baryon number violating processes. In all cases, topological defects lead to a fruitful interplay between particle physics and cosmology.     

The spin density distribution in some phenyl substituted imidazyl and pyrryl radicals

ENDOR spectroscopy has been used to interpret the E.S.R. spectra of the 2,4,5-triphenylimidazyl, tetraphenylpyrryl, tetrakis (p-tolyl) pyrryl, tetrakis (p-anisyl) pyrryl radicals. Using the hyperfine coupling constants thus obtained in conjunction with a McLachlan-type HMO calculation the spin density distribution in these radicals is calculated. The configuration of the phenyl rings and the deviation of their bond lengths from hexagonal symmetry is inferred from the MO parameters required to fit the experimental couplings.     

Studies of Hindered Rotation and Magnetic Anisotropy by 1H, 13C and 19F NMR. The Diels-Alder Adduct of N- Pentafluorophenylmaleimide and Phencyclone: A Model for Drugs.

The potent Diels-Alder diene, phencyclone, 1, reacts with N-pentafluorophenylmaleimide, 2, to form an adduct, 3, characterized by ¹H, ¹³C, and ¹⁹F NMR at 300, 75 and 282 MHz, respectively. The one-dimensional (1D) and two-dimensional (2D) ¹H and ¹³C NMR spectra of 3 at ambient temperatures imply a slow exchange limit (SEL) regime with respect to rotation of the unsubstituted bridgehead phenyl groups about severely hindered C(sp²)-C(sp³) bonds. Major non-bonded interactions are expected between the ortho protons of the C₆H₅ groups and H-1, 8 of the phenanthrenoid moiety of 3. ¹⁹F 1D and 2D (COSY) NMR spectra show that the SEL regime also obtains for rotation about the N-C₆F₅ bond of 3, with five separate fluorine signals seen, consistent with a preferred conformation in which the C₆F₅ may lie roughly perpendicular to the plane of the pyrrolidinedione moiety, and may be in the mirror symmetry plane of 3. The results are considered relevant to hindered aryl rotations in numerous Pharmaceuticals. Selected spectral data for 2 and precursors are also presented.