The NMR line shape of a system of nuclear spins with equal spin-spin coupling constants

We obtain an expression for the NMR line at low temperatures for a system of nuclear spins described by a Hamiltonian with equal spin-spin coupling constants. We show that in the case of “easy axis” anisotropy, the line has a logarithmic low-frequency singularity and an exponentially decreasing high-frequency asymptotic behavior at the temperature of an anomalous peak of heat capacity. In the case of “easy plane” anisotropy, the line has the traditional Gaussian form. We discuss the possibility of using NMR data to discover specific thermodynamic and magnetic properties of the considered model system.     

Fast reaction limit of a three-component reaction-diffusion system

We consider a three-component reaction-diffusion system with a reaction rate parameter, and investigate its singular limit as the reaction rate tends to infinity. The limit problem is given by a free boundary problem which possesses three regions separated by the free boundaries. One component vanishes and the other two components remain positive in each region. Therefore, the dynamics is governed by a system of two equations.     

Integrable models of the dynamics of longitudinal-transverse acoustic pulses in a paramagnetic crystal

We study the interaction between longitudinal-transverse acoustic pulses and a system of paramagnetic impurities with the effective spin S = 1 in a statically deformed crystal. We show that the dynamics of a pulse propagating at an arbitrary angle to the static-deformation direction and of the effective spins satisfy the modified reduced Maxwell-Bloch equations and, if the spectrum of the acoustic pulse overlaps the quantum transitions between spin sublevels, the modified sine-Gordon equation. These equations generalize the well-known models in the theory of the inverse scattering method and in the theory of self-induced transparency and also belong to the class of integrable equations. Analyzing soliton solutions shows that the pulse-medium interaction reveals some qualitatively new features in these models compared with the cases of purely transverse or purely longitudinal acoustic fields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 228–247, May, 2007.     

Nonlinear analysis of flow pulses and shock waves in arteries

Zusammenfassung In jeder Gruppe von Menschen variieren die Kreislaufparameterwerte von Individuum zu Individuum ziemlich stark. Um die Effekte realistischer Schwankungen dieser Parameter auf die Strom-und Druckpulse in arteriellen Leitungen zu untersuchen und um mögliche Manifestationen gewisser pathologischer Zustände wie Arteriosklerosis und die Insuffizienz der Aortenklappen zu identifizieren, werden hier einige der Resultate präsentiert, die sich auf Grund des im ersten Teil beschriebenen mathematischen Modells ergeben. Insuffizienz der Aortenklappen ist nachgebildet durch einen Ausströmungspuls mit aussergewöhnlich grossem Rückstrom. Der zugehörige Druckpuls zeigt die bekannten Eigenschaften von Stosswellen, die anscheinend verantwortlich sind für die pistol shot sounds, die man über der arteria femoralis und der arteria radialis von Patienten mit Aorteninsuffizienz hört.

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For Part I see Z. ang. Math. Phys.22, 217 (1971).     

Nonlinear analysis of flow pulses and shock waves in arteries

Zusammenfassung Druck- und Strompulse mit grosser Amplitude erzeugt in der Aorta und in andern Hauptarterien des Hundes werden theoretisch berechnet für vorgeschriebene Ausströmungspulse von der linken Herzkammer und für gegebene physikalische und geometrische Eigenschaften des Kreislauf-systems. Der Blutausfluss durch die Äste und Verzweigungen der uns interessierenden arteriellen Leitung ist durch ein kontinuierlich verteiltes, von Druck und Ort abhängiges Ausflussmodell nachgeahmt. Am herzfernen Ende der Leitung ist als Randbedingung entweder der periphere Widerstand oder ein konstanter druck vorgeschrieben. Die Geometrie der Leitung ist durch ihren kreisförmingen Querschnitt und den mit Herzdistanz exponentiell abnehmenden Radius definiert. Die elastischen Eigenschaften der Gefässwand sind durch die von Ort und Druck abhängige Geschwindigkeit kleiner Druckwellen gegeben. Durch Integration der Beziehung zwischen Wellen-geschwindigkeit und Querschnittsdehnung ist damit auch der Querschnitt als Funktion des Druckes und der Herzdistanz vorgeschrieben. Die nichtlinearen Gleichungen für eindimensionale Strömung einer inkompressibler Flüssigkeit werden mit Hilfe der Charakteristikenmethode integriert für Kreislaufparameterwerte die einem hypothetischen Hund von 30 kg Gewicht entsprechen. Das verwendete mathematische Modell für die arterielle Leitung wiedergibt manche der bekannten Eigenschaften des vom Herzen erzeugten Pulses, einschliesslich die Klappenincisur, das Ansteigen und Abfallen der Höhe des systolischen Druckgipfels mit wachsender Entfernung vom Herzen. Während der Fortpflanzung der Pulswelle zeichnet sich eine zunehmende Steilheit der Wellenfront ab, die jedoch nicht merkbar ist wenn man die grundlegenden Gleichungen linearisiert. Die numerischen Ergebnisse weisen darauf hin, dass die sekundäre (dicrotic) Welle durch Reflexionen erzeugt wird, und als solche von der Verjüngung des Querschnittes und vom Blutausfluss abhängt.

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Notation A cross-sectional area of artery at the reference intraluminal pressurep ₀ - c local wave speed - c ₀,c ₁ parameters in wave speed expression - c _L local wave speed for linearized analysis - C, C designations for general curves inz, t plane - D ₀ diameter at aortic valve whenp=p ₀ - D _t diameter at distal end of artery whenp=p ₀ - E circumferential Young's modulus - f axial frictional force per unit mass of fluid - h wall thickness - L distance from aortic valve to distal end of artery - n parameter in wave speed expression - p intraluminal pressure - calculated mean pressure - p ₀ reference pressure - p _c capillary pressure - p _L pressure at distal end of artery - q S v=local volume flow rate - q ₀ q ₀(t)=volume flow rate ejected by heart - r radial coordinate - R _e Reynolds number for steady flow - R _L peripheral resistance - s curvilinear coordinate - S cross-sectional area of artery - S _L cross-sectional area for linearized analysis - t time - v axial fluid velocity averaged over cross section - z axial distance coordinate - z ^* distance from aortic valve to femoral artery - exponent in cross-sectional area expression - outflow parameter - undetermined multiplier - blood viscosity coefficient - blood density - outflow function simulating effect of side branches     

Computational modeling of the propagation of acoustic pulses in hemodynamics

The present paper deals with the numerical simulation of the propagation of pulses of blood pressure and velocity in a blood vessel. The numerical solution of the system of linear hemodynamic equations is formed as a superposition of progressing waves (Riemann invariants) satisfying the transport equations. Considerable attention is paid to the construction of a difference scheme for the linear and quasilinear transport equations. Examples of computations are presented. The suggested algorithm can be generalized to the case of a quasilinear system of equations.     

Symmetry and regularity of solutions to a system with three-component integral equations

Consider the system with three-component integral equations u(x) = Rn |x y|α nw(y)rv(y)q dy,v(x) = Rn |x y|α nu(y)pw(y)rdy,w(x) = Rn |x y|α nv(y)q u(y)pdy,where 0 < α < n,n is a positive constant,p,q and r satisfy some suitable conditions.It is shown that every positive regular solution(u(x),v(x),w(x)) is radially symmetric and monotonic about some point by developing the moving plane method in an integral form.In addition,the regularity of the solutions is also proved by the contraction mapping principle.The conformal invariant property of the system is also investigated.     

Nonlinear analysis in a Lorenz-like system

In this paper we study the nonlinear dynamics of a Lorenz-like system. More precisely, we study the stability and bifurcations which occur in a new three parameter quadratic chaotic system. We also study the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters. As a consequence we show the existence of chaotic attractors when these cycles disappear.     

Nonlinear dynamics and chaos in a fractional-order financial system

This study examines the two most attractive characteristics, memory and chaos, in simulations of financial systems. A fractional-order financial system is proposed in this study. It is a generalization of a dynamic financial model recently reported in the literature. The fractional-order financial system displays many interesting dynamic behaviors, such as fixed points, periodic motions, and chaotic motions. It has been found that chaos exists in fractional-order financial systems with orders less than 3. In this study, the lowest order at which this system yielded chaos was 2.35. Period doubling and intermittency routes to chaos in the fractional-order financial system were found.     

Nonlinear dynamics of a simplified engine-propeller system

This paper presents a procedure for studying dynamical behaviors of a simplified engine-propeller dynamical system consisting of a number of bodies of plane motions. The equation of motion of the complex system is obtained using the Lagrange equation and solved numerically using the 4th order Runge–Kutta method. Various simulations were performed to investigate the transient and steady state behaviors of the multiple body system while taking into consideration the engine pressure pulsations, nonlinear inertia of moving bodies, and nonlinear aerodynamic load. Sub-harmonics and super harmonics in the steady state responses for different power and propeller pitch settings are obtained using the fast Fourier transform. Numerical simulations indicate that the 1.5 order is the dominant order of harmonics in the steady state oscillatory motion of the crankshaft. The findings and procedure presented in the paper are useful to the aerospace industry in certifying reciprocating engines and propellers. The crankshaft oscillatory velocities obtained from the simplified rigid body model are in good agreement with the experimental data for a SAITO-450 engine and a SOLO propeller at a 6″ pitch setting.     

A Comparison Theorem for a Nonlinear system Arising in Porous Medium Combustion

Comparison theorems for the initial value finite domain one dimensional heat equation with a discontinuous forcing term are extended to a coupled system of a heat equation and an ordinary differential equation in space, rather than the usual ordinary differential equation in time, that arises in combustion theory.     

Nonlinear modelling and control of a common rail injection system for diesel engines

Modern trends in designing mechatronic systems call for a synergic design of the separated subsystems (mechanic, electronic parts, control modules, etc.) concurring to the overall performance. Following this point of view, this paper presents a control oriented model and a nonlinear control design for a Common Rail injection system. First a model is developed, which is tuned in a virtual simulation environment, representing the injection system in details in a reliable replication of reality. Then a sliding mode control is developed. Both the model of the injection process and of the control law are validated by a virtual detailed simulation environment. The prediction capability of the model and the control efficiency are clearly shown.     

Gumbel statistics for the longest interval of identical spins in a one-dimensional Gibbs measure

We consider one-dimensional Gibbs measures on spin configurations σ ∈ {–1,+1}^ℤ. For N ∈ ℕ let l _N denote the length of the longest interval of consecutive spins of the same kind in the interval [0,N]. We show that the distribution of a suitable continuous modification l _c (N) of l _N converges to the Gumbel distribution, i.e., for some α, β ∈ (0, ∞) and γ ∈ ℝ, lim _{N →∞} ℙ(l _c (N) ≤ α log N + βx + γ) = e ^{–e –x} . Received: 2 September 2002     

Nonlinear stability of the equilibria in a double-bar rotating system

We study the nonlinear stability of the equilibria corresponding to the motion of a particle orbiting around a two finite orthogonal straight segment. The potential is a logarithmic function and may be considered as an approximation to the one generated by irregular celestial bodies. Using Arnold’s theorem for non-definite quadratic forms we determine the nonlinear stability of the equilibria, for all values of the parameter of the problem. Moreover, the resonant cases are determined and the stability is investigated.