An analytic approach to stability

The stability method is very useful for obtaining exact solutions of many extremal graph problems. Its key step is to establish the stability property which, roughly speaking, states that any two almost optimal graphs of the same order n can be made isomorphic by changing o(n²) edges.Here we show how the recently developed theory of graph limits can be used to give an analytic approach to stability. As an application, we present a new proof of the Erd?s-Simonovits stability theorem.Also, we investigate various properties of the edit distance. In particular, we show that the combinatorial and fractional versions are within a constant factor from each other, thus answering a question of Goldreich, Krivelevich, Newman, and Rozenberg.     

An analytic approach to stochastic calculus

An Itô type stochastic differential calculus is constructed in an analytic way, without use of the notion of filtration, for processes that satisfy certain homogeneity and smoothness conditions. This calculus is developed in the framework of Lie groups.     

An analytic approach to the ergodic theory of a stochastic variational inequality

In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality.     

An analytic approach to the normalized Ricci flow-like equation

This paper is concerned with a parabolic equation with a non-local term defined on a compact two-dimensional Riemannian surface Ω. If the total mass of the solution, λ, is equal to 8π and Ω is the standard sphere S², it is a Hamilton’s normalized Ricci flow. We obtain the global in time existence of the solution to this problem for 0<λ≤8π. If 0<λ<8π, the orbit is compact while for λ=8π, there is a time sequence along which the solution converges to a stationary solution.     

An approach to generate deterministic Brownian motion

We propose an approach for generation of deterministic Brownian motion. By adding an additional degree of freedom to the Langevin equation and transforming it into a system of three linear differential equations, we determine the position of switching surfaces, which act as a multi-well potential with a short fluctuation escape time. Although the model is based on the Langevin equation, the final system does not contain a stochastic term, and therefore the obtained motion is deterministic. Nevertheless, the system behavior exhibits important characteristic properties of Brownian motion, namely, a linear growth in time of the mean square displacement, a Gaussian distribution, and a −2 power law of the frequency spectrum. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.     

On the motion of a projectile in the atmosphere

The equation of rectilinear motion of a projectile which moves in an atmosphere, of which the density decreases exponentially with the altitude, is solved. It is found that the velocity of the projectile can be expressed explicitly in terms of confluent hypergeometric functions. This theory is applied to treat two specific problems: (1) the flight analysis of a sounding rocket during the freeflight period and (2) the calculation of ambient temperature from the trajectory of a spherical projectile.

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Zusammenfassung Die Bewegungsgleichung eines vertikal ansteigenden Geschosses wird gelöst für den Fall, dass die Dichte der Atmosphäre exponentiell mit der Höhe abnimmt. Die Geschwindigkeit des Projektils kann dabei in konfluenten hypergeometrischen Funktionen ausgedrückt werden. Die Theorie wird hierauf auf zwei spezifische Probleme angewendet: 1. Fluganalyse einer Vertikalrakete während der Zeit des schubfreien Fluges und 2. Berechnung der Umgebungstemperatur der Bahn eines sphärischen Projektils.

     

Auto-oscillations during deceleration of a rigid body in a resisting medium

Some qualitative analysis is carried out of the rectilinear and spatial problems concerning the motion of a rigid body in a resisting medium.Anonlinearmodel is constructed of impact of the mediumon the rigid body, which takes into account the dependence of the arm of force on the reduced angular velocity of the body. Moreover, the moment of this force itself is also a function of the angle of attack. As was shown by the processing the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of the plane and spatial models of the interaction of a rigid body with a medium reveal the sufficient conditions of stability of the key regime of motion, i.e., the translational rectilinear deceleration. It is also shown that, under certain conditions, both stable or unstable auto-oscillating regimes can be presented in the system.     

An algebraic analysis approach to linear time-varying systems

^** Email: zerz{at}mathematik.uni-kl.de This paper introduces an algebraic analysis approach to lineartime-varying systems. The analysis is carried out in an ‘almosteverywhere’ setting, i.e. the considered signals are smoothexcept for a set of measure zero, and the coefficients of thelinear ordinary differential equations are supposed to be rationalor meromorphic functions. The methodology is based on a normalform for matrices over the resulting ring of differential operators,which is a non-commutative analogue of the Smith form. Thisis used to establish a duality between linear time-varying systemson the one hand and modules over the ring of differential operatorson the other. This correspondence is based on the fact thatthe signal space is an injective cogenerator when consideredas a module over this ring of differential operators.     

An alternative approach to super-Brownian motion with a locally infinite branching mass

Fleischmann and Mueller (Probab. Theory Related Fields 107 (1997) 325) constructed a super-Brownian motion in R¹ with a locally infinite branching rate function, and they showed that this super-Brownian motion has a strong killing property in the critical case. In this paper, we first construct, via a limiting procedure, a super-Brownian motion which is equivalent to the super-Brownian motion with a locally infinite branching. From this construction, one can easily see the connection between the superprocess and a killed Brownian motion. Next, by taking advantage of the new construction, we give a new proof of the strong killing property of the process.     

An analytic network process approach to operationalization of five forces model

The five forces model has been one of the most influential frameworks for strategic management. In contrast to its importance as a centerpiece of textbooks, however, it has attracted less attention from both academic researchers and practicing managers. This is due to its innate weakness, difficulty in operationalization. The vital requisites for operationalizing the five forces model are to deal with it as a complex system composed of interrelated forces and their sub-forces, and to prioritize them with consideration of their interdependency. The tenet of this study is the requisites can be achieved through the analytic network process (ANP). The ANP, which is a generalization of the analytic hierarchy process (AHP), produces priorities of elements in a complex network model with consideration of interdependency among elements. The five forces model is transformed into a network model of the ANP. The ANP procedure is then carried out to obtain the priority weights of the forces. Combining the derived weights and ratings on the forces produces the state-of-industry-competition index (SICI) values that represent the overall competitive condition of a given industry. The working of the proposed approach is provided with the help of a case study example of the Web portal Industry of Korea. The proposed ANP approach is expected to expand the five forces model into a workable system of analysis by improving its analytical power.     

An analytic approach to stochastic Volterra equations with completely monotone kernels

We apply the semigroup setting of Desch and Miller to a class of stochastic integral equations of Volterra type with completely monotone kernels with a multiplicative noise term; the corresponding equation is an infinite dimensional stochastic equation with unbounded diffusion operator that we solve with the semigroup approach of Da Prato and Zabczyk. As a motivation of our results, we study an optimal control problem when the control enters the system together with the noise.      

An elastic beam approach to predictive vehicle motion planning

Vehicle guidance systems are among the latest automotive developments. Examples are A daptive C ruise C ontrol (ACC), lane departure warning and active lane keeping. However, the rapid development of environmental sensor systems such as radar, lidar and video technologies in combination with data fusion of the single sensor signals enables further applications as for example lane changing assistance, collision avoidance or even autonomous driving. For all of these functions, trajectories have to be planned and, depending on the application, to be communicated to the driver. Recently, methods originally developed in robotics as for example so-called elastic bands are adapted for road vehicle applications. Here an virtual elastic beam is proposed to guide the vehicle. The beam is deflected by virtual potential fields generated by obstacles in the traffic space. In doing so, the elastic beam avoids obstacles and provides collision-free trajectories, while its bending stiffness counteracts the curvature of the trajectory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An analytic approach to existence and uniqueness for martingale problems in infinite dimensions

We prove existence and uniqueness for a class of martingale problems in a Hilbert space. We solve the associated Kolmogorov equation and prove that the corresponding semigroup is determined by a kernel of measures if a Schauder-type regularity is satisfied. Received: 18 May 1998 / Revised version: 27 September 1999 / Published online: 5 September 2000     

An inexact approach to solving linear semi-infinite programming problems

In this paper, we propose an “inexact solution” approach to deal with linear semi-infinite programming problems with finitely many variables and infinitely many constraints over a compact metric space. A general convergence proof with some numerical examples are given and the advantages of using this approach are discussed     

An MC2 linear programming approach to combined forecasting

Combined forecasting is a well-known forecasting technique of handling multiple forecasting methods. However, the previous combined forecasting approaches lack of incorporating the various possible opinions from several experts on an given forecasting problem. This paper proposes an MC² linear programming approach to determining weighted coefficients of combined forecasting that involves multiple experts. The numerical example of the paper shows that the proposed approach likely outperforms the current techniques of combined forecasting in dealing with the case of multiple experts.     

General theory of optimal trajectory for rocket flight in a resisting medium

This paper considers the problem of optimizing the flight trajectory of a rocket vehicle moving in a resisting medium and in a general gravitational force field. General control laws for the lift, the bank angle, and the thrusting program are obtained in terms of the primer vector, the adjoint vector associated to the velocity vector. Additional relations for the case of variable thrusting and integrals of motion for flight at maximum lift-to-drag ratio and flight in a constant gravitational field are obtained.This work was supported by Air Force Grant No. AFOSR-71-2129.     

An analytic solution of an oscillatory flow through a porous medium with radiation effect

Analytical solutions for two-dimensional oscillatory flow on free convective-radiation of an incompressible viscous fluid, through a highly porous medium bounded by an infinite vertical plate are reported. The Rosseland diffusion approximation is used to describe the radiation heat flux in the energy equation. The resulting non-linear partial differential equations were transformed into a set of ordinary differential equations using two-term series. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The free-stream velocity of the fluid vibrates about a mean constant value and the surface absorbs the fluid with constant velocity. Expressions for the velocity and the temperature are obtained. To know the physics of the problem analytical results are discussed with the help of graph.