The noncommutative values of quantum observables

We discuss the notion of representing the values of physical quantities by the real numbers, and its limits to describe the nature to be understood in the relation to our appreciation that the quantum theory is a better theory of natural phenomena than its classical analog. Getting from the algebra of physical observables to their values for a fixed state is, at least for classical physics, really a homomorphic map from the algebra into the real number algebra. The limitation of the latter to represent the values of quantum observables with noncommutative algebraic relation is obvious. We introduce and discuss the idea of the noncommutative values of quantum observables and its feasibility, arguing that at least in terms of the representation of such a value as an infinite set of complex numbers, the idea makes reasonable sense theoretically as well as practically.     

Full cross-diffusion limit in the stationary Shigesada-Kawasaki-Teramoto model

This paper studies the asymptotic behavior of coexistence steady-states of the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. In the case when either one of two cross-diffusion coefficients tends to infinity, Lou and Ni [18] derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady-states. Recently, a formal observation by Kan-on [10] implied the existence of a limiting system including the nonstationary problem as both cross-diffusion coefficients tend to infinity at the same rate. This paper gives a rigorous proof of his observation as far as the stationary problem. As a key ingredient of the proof, we establish a uniform $L^{\infty }$ estimate for all steady-states. Thanks to this a priori estimate, we show that the asymptotic profile of coexistence steady-states can be characterized by a solution of the limiting system.     

Weak dissipation of electrostatic solitary structures in warm collisional pair-ion plasmas with non-Maxwellian electron population

The localized electrostatic structures with dissipation due to ion-neutral collisions in a symmetric warm pair-ion plasma in the presence of non-Maxwellian population of electrons are studied. The analytical model for ion dynamics is based on fluid equations and the evolution equation is derived by using the reductive perturbation scheme in the form of a damped Korteweg-de Vries equation. The parameter regime relevant to space-based observations and laboratory plasmas is considered and time evolution of the propagating ion-acoustic soliton is discussed. The energetic-particles-driven properties of soliton for various spectral indices, dissipation, ion temperature, and density are illustrated with comparison to the thermal mode for Boltzmann distribution of electrons.     

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length ≤3 and the diagonal lengths ≤3; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group G and the Iwahori-Hecke algebra of type A on the permutation module of G by its Borel subgroup.     

The 6j-Symbols for the SL(2, ℂ) Group

Theoretical and Mathematical Physics - We study 6j-symbols or Racah coefficients for the tensor products of infinite-dimensional unitary principal series representations of the group SL(2,...     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.     

Intracranial vessel wall imaging framework – Data acquisition,processing, and visualization

ObjectiveAssessment of vessel walls is an integral part in diagnosis and disease monitoring of vascular diseases such as vasculitis. Vessel wall imaging (VWI), in particular of intracranial arteries, is the domain of Magnetic Resonance Imaging (MRI) – but still remains a challenge. The tortuous anatomy of intracranial arteries and the need for high resolution within clinically acceptable scan times require special technical conditions regarding the hardware and software environments.Materials and methodsIn this work a dedicated framework for intracranial VWI is presented offering an optimized, black-blood 3D T1-weighted post-contrast Compressed Sensing (CS)-accelerated MRI sequence prototype combined with dedicated 3D-GUI supported post-processing tool for the CPR visualization of tortuous arbitrary vessel structures.ResultsUsing CS accelerated MRI sequence, the scanning time for high-resolution 3D black-blood CS-space data could be reduced to under 10 min. These data are adequate for a further processing to extract straightened visualizations (curved planar reformats – CPR). First patient data sets could be acquired in clinical environment.ConclusionA highly versatile framework for VWI visualization was demonstrated utilizing a post-processing tool to extract CPR reformats from high-resolution 3D black-blood CS-SPACE data, enabling simplified and optimized assessment of intracranial arteries in intracranial vascular disorders, especially in suspected intracranial vasculitis, by stretching their tortuous course. The processing time from about 15–20 min per patient (data acquisition and further processing) allows the integration into clinical routine.