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Principle of symmetric criticality and evolution equations
Goro Akagi, Jun Kobayashi and Mitsuharu Ôtani
2003, 2003(Special): 1-10 doi: 10.3934/proc.2003.2003.1 +[Abstract](2845) +[PDF](216.0KB)
   Let X be a Banach space on which a symmetry group G linearly acts and let J be a G-invariant functional defined on X. In 1979, R. Palais [6] gave some sufficient conditions to guarantee the so-called "Principle of Symmetric Criticality": every critical point of J restricted on the subspace of symmetric points becomes also a critical point of J on the whole space X. In [5], this principle was generalized to the case where J is non-smooth and the setting does not require the full variational structure when G is compact or isometric.
   The purpose of this paper is to combine this result with the abstract theory developed in [1] and [2] concerning the evolution equation: $du(t)$/$dt + \partial\upsilon^1(u(t))- \partial\upsilon^2(u(t))$ ∋ $f(t)$ in V*, where $\partial\upsilon^i$ is the so-called subdifferential operator from a Banach space X into its dual V*. It is assumed that there exists a Hilbert space H satisfying $V \subset H \subset V $ and that G acts on these spaces as isometries. In this setting, the existence of G-symmetric solution for above equation can be discussed.
   As an application, a parabolic problem with the p-Laplacian in unbounded domains is discussed.
Evolution equations and subdifferentials in Banach spaces
Goro Akagi and Mitsuharu Ôtani
2003, 2003(Special): 11-20 doi: 10.3934/proc.2003.2003.11 +[Abstract](2720) +[PDF](220.2KB)
Sufficient conditions for the existence of strong solutions to the Cauchy problem are given for the evolution equation $du(t)$/$dt + \partial\upsilon^1(u(t))- \partial\upsilon^2(u(t)) \in f(t)$ in V*, where $\partial\upsilon^1$ is the so-called subdifferential operator from a Banach space V into its dual space V* ($i$ = 1,2).
   Studies for this equation in the Hilbert space framework has been done by several authors. However the study in the V -V* setting is not pursued yet.
   Our method of proof relies on some approximation arguments in a Hilbert space. To carry out this procedure, it is assumed that there exists a Hilbert space H satisfying $V \subset H \-= H$* $\subset V$* with densely defined continuous injections.
   As an application of our abstract theory, the initial-boundary value problem is discussed for the nonlinear heat equation: $ut(x, t)-\Delta_p u(x, t)-|u|^(q-2) u(x, t) = f(x, t), x \in \Omega, u|_(\partial\Omega) = 0, t \>= 0$, where ­$\Omega$ is a bounded domain in $\mathbb(R)^N$. In particular, the local existence of solutions is assured under the so-called subcritical condition, i.e., $q < p$*, where $p$* denotes Sobolev’s critical exponent, provided that the initial data $u_0$ belongs to $W_0^(1,p)(\­Omega)$.
Quasi Normal modes in stochastic domains
Carlo Alabiso and Mario Casartelli
2003, 2003(Special): 21-29 doi: 10.3934/proc.2003.2003.21 +[Abstract](2329) +[PDF](566.3KB)
In anharmonic chains with even potentials, including classical Fermi- Pasta-Ulam model, we show how ordered structures can coexist with high degree stochasticity.
Constrained envelope for a general class of design problems
Ernesto Aranda and Pablo Pedregal
2003, 2003(Special): 30-41 doi: 10.3934/proc.2003.2003.30 +[Abstract](2407) +[PDF](203.6KB)
We analyze the relaxation and computation of the relaxed density when we reformulate a typical optimal design problem with volume constraint in two dimension as a fully vector variational problem. Our aim is to examine a general cost functional depending explicitly on all variables and in particular in the gradient variable, and see how far computations and properties of the relaxed integrand can be pushed.
Properties of solutions for nonlinear Volterra integral equations
M. R. Arias and R. Benítez
2003, 2003(Special): 42-47 doi: 10.3934/proc.2003.2003.42 +[Abstract](2546) +[PDF](155.8KB)
Some properties of non-locally bounded solutions for Abel integral equations are given. The case in which there exists two non-trivial solutions for such equations is also studied. Besides, some known results about existence, uniqueness and attractiveness of solutions for some Volterra equations are improved.
Constant J-unitary factor and operator-valued transfer functions
Yury Arlinskiĭ and Eduard Tsekanovskiĭ
2003, 2003(Special): 48-56 doi: 10.3934/proc.2003.2003.48 +[Abstract](2737) +[PDF](192.2KB)
We study linear, conservative, stationary dynamical systems (l.c.s.d.s) with unbounded main operator and their transfer operator-valued functions. A new class of transfer operator-valued functions is introduced and shown that this class is invariant with respect to multiplication (from the left, from the right or both) by a constant J-unitary operator. The problem how to realize the product W(z)B as a transfer function of some l.c.s.d.s with an arbitrary J-unitary operator B and given realization of W(z) is considered.
Controllability of a string under tension
Sergei A. Avdonin and Boris P. Belinskiy
2003, 2003(Special): 57-67 doi: 10.3934/proc.2003.2003.57 +[Abstract](2457) +[PDF](187.8KB)
We study controllability for a string under an axial stretching tension. The tension is a sum of a constant positive term and a small, slowly variable, load. We are looking for an exterior force $g(x)f(t)$ that drives the state solution to rest. The controllability problem is reduced to a moment problem for the control $f(t)$: We describe the set of initial data which may be driven to rest by a control $f(t) \in L^2(0, T)$: The description is obtained in terms of the Fourier coefficients of the initial data. The proof is based on an auxiliary basis property result.
A reducible representation of the generalized symmetry group of a quasiperiodic flow
L. Bakker
2003, 2003(Special): 68-77 doi: 10.3934/proc.2003.2003.68 +[Abstract](2641) +[PDF](171.9KB)
The generalized symmetry group of a quasiperiodic flow on a $n$-torus is the group theoretic normalizer, within the group of diffeomorphisms of the $n$-torus, of the one parameter abelian group of diffeomorphisms generated by the flow. Up to conjugacy, the generalized symmetry group of a quasiperiodic flow is determined by a system of uncoupled first order partial differential equations. New types of symmetries (other than the classical types of symmetries or time-reversing symmetries) may exist depending on certain algebraic relationships being satisfied by pair wise ratios of the frequencies of the quasiperiodic flow. These new symmetries, when they exist, are a dominant feature of a reducible linear representation of the generalized symmetry group in the de Rham cohomology of the $n$-torus.
Ramanujan type graphs and bigraphs
Cristina M. Ballantine
2003, 2003(Special): 78-82 doi: 10.3934/proc.2003.2003.78 +[Abstract](2323) +[PDF](205.2KB)
We show that quotients of the Bruhat-Tits building of S$L_3(mathbb(Q)_p)$ form an infinite family of graphs which are Ramanujan type. We investigate the Bruhat-Tits tree associated with U$_3(mathbb(Q)_p)$ and show how its relation to the building of S$L_3(mathbb(Q)_p)$should lead to an estimation of its spectrum.
Nonlinear boundary value problems with multiple positive solutions
John V. Baxley and Philip T. Carroll
2003, 2003(Special): 83-90 doi: 10.3934/proc.2003.2003.83 +[Abstract](2707) +[PDF](185.7KB)
We give conditions on $f(y)$ which guarantee that the boundary value problem $T_ny(x) = f(y(x))$, 0 < $x$ < $A$, $y^(2j)$(0) = 0 = $y^(2j+1)(A)$, $j$ = 0, 1,... , $n$-1, where $T_n$ is the $n$th iterate of the operator $Ty(x)$ = -$(1)/( w(x))$ $(p(x)y'(x))'$, have a prescribed number of multiple positive solutions. Our main tool is a fixed point theorem of Krasnosel’skiĭ.
Stochastic stability of some mechanical systems with a multiplicative white noise
Boris P. Belinskiy and Peter Caithamer
2003, 2003(Special): 91-99 doi: 10.3934/proc.2003.2003.91 +[Abstract](2826) +[PDF](175.3KB)
We discuss the behavior, for large values of time, of a class of linear mechanical systems with a multiplicative white noise in its parameters. The initial conditions may be random as well but are independent of white noise. It is well known that a deterministic linear mechanical system with viscous damping is stable, i.e., its energy approaches zero as time increases. We calculate the expected energy and check that this behavior takes place in the case when the initial conditions are random but the parameters are not. When the parameters contain a random noise the expected energy may be infinite, approach zero, remain bounded, or increase with no bound. We give necessary and sufficient conditions for stability of the systems considered in terms of the roots of an auxiliary equation.
Asymptotic constancy for nonhomogeneous linear differential equations with unbounded delays
Hüseyin Bereketoğlu and Mihály Pituk
2003, 2003(Special): 100-107 doi: 10.3934/proc.2003.2003.100 +[Abstract](2375) +[PDF](162.5KB)
Sufficient conditions are given for the asymptotic constancy of the solutions of a nonhomogeneous linear delay differential equation with unbounded delays. Moreover, it is shown that the limits of the solutions, as $t \to \infty$, can be computed in terms of the initial function and a special matrix solution of the corresponding adjoint equation.
Oscillation of second order difference equations with advanced argument
Bi Ping and Maoan Han
2003, 2003(Special): 108-112 doi: 10.3934/proc.2003.2003.108 +[Abstract](2545) +[PDF](131.3KB)
In this paper, we are mainly concerned with the second order difference equations with advanced argument and give sufficient conditions for their solutions to be oscillatory.
Stages of energy transfer in the FPU model
Joseph A. Biello, Peter R. Kramer and Yuri Lvov
2003, 2003(Special): 113-122 doi: 10.3934/proc.2003.2003.113 +[Abstract](2674) +[PDF](219.3KB)
The (alpha) version of the Fermi-Pasta-Ulam is revisited through direct numerical simulations and an application of weak turbulence theory. The energy spectrum, initialized with a large scale excitation, is traced through a series of distinct qualitative phases en route to eventual equipartition. Weak turbulence theory is applied in an attempt to provide an effective quantitative description of the evolution of the energy spectrum. Some scaling predictions are well-confirmed by the numerical simulations.
Stochastic global bifurcation in perturbed Hamiltonian systems
Lora Billings, Erik M. Bollt, David Morgan and Ira B. Schwartz
2003, 2003(Special): 123-132 doi: 10.3934/proc.2003.2003.123 +[Abstract](2996) +[PDF](787.2KB)
We study two perturbed Hamiltonian systems in which chaos-like dynamics can be induced by stochastic perturbations. We show the similarities of a class of population and laser models, analytically and topologically. Both systems have similar manifold structure that includes bi-instability and partially formed heteroclinic connections. Noise takes advantage of this structure, inducing a global bifurcation and chaotic-like dynamics which exhibits mixed mode behavior of the original bi-stable solutions. We support these claims with numerical approximations of the transport between basins.
Numerical approximation of normally hyperbolic invariant manifolds
Henk Broer, Aaron Hagen and Gert Vegter
2003, 2003(Special): 133-140 doi: 10.3934/proc.2003.2003.133 +[Abstract](3091) +[PDF](339.7KB)
This paper deals with the numerical continuation of invariant manifolds, regardless of the restricted dynamics. Typically, invariant manifolds make up the skeleton of the dynamics of phase space. Examples include limit sets, co-dimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/center manifolds, nested hierarchies of attracting manifolds in dissipative systems and manifolds in phase plus parameter space on which bifurcations occur. These manifolds are for the most part invisible to current numerical methods. The approach is based on the general principle of normal hyperbolicity, where the graph transform leads to the numerical algorithms. This gives a highly multiple purpose method. The key issue is the discretization of the differential geometric components of the graph transform, and its consequences. Examples of computations will be given, with and without non-uniform adaptive refinement.
On positive solutions of nonlinear fractional differential equations
Daria Bugajewska and Mirosława Zima
2003, 2003(Special): 141-146 doi: 10.3934/proc.2003.2003.141 +[Abstract](2426) +[PDF](152.9KB)
The aim of this note is to prove some existence and uniqueness theorems on positive solutions of some nonlinear fractional equations. Classical as well as Carathéodory’s solutions are under our considerations.
On the spectral radius of linearly bounded operators and existence results for functional-differential equations
Daria Bugajewska and Mirosława Zima
2003, 2003(Special): 147-155 doi: 10.3934/proc.2003.2003.147 +[Abstract](2407) +[PDF](168.9KB)
In this paper we formulate conditions, different from the global commutativity, under which one can estimate the spectral radius of the composition of linearly bounded operators. We apply this estimation to prove the existence of global solutions for some functional differential equations and systems of such equations. All our results are illustrated by suitable examples.
A learning theory approach to the construction of predictor models
G. Calafiore and M.C. Campi
2003, 2003(Special): 156-166 doi: 10.3934/proc.2003.2003.156 +[Abstract](2593) +[PDF](208.6KB)
This paper presents new results for the identification of predictive models for unknown dynamical systems. The three key elements of the proposed approach are: i) an unknown mechanism that generates the observed data; ii) a family of models, among which we select our predictor, on the basis of past observations; iii) an optimality criterion that we want to minimize. A major departure from standard identification theory is taken in that we consider interval models for prediction, that is models that return output intervals, as opposed to output values. Moreover, we introduce a consistency criterion (the model is required to be consistent with observations) which act as a constraint in the optimization procedure. In this framework, the model has not to be interpreted as a faithful description of reality, but rather as an instrument to perform prediction. To the optimal model, we attach a certificate of reliability, that is a statement of the probability that the computed model will actually be consistent with future unknown data.
Oscillation of mixed neutral differential equations with forcing term
T. Candan and R.S. Dahiya
2003, 2003(Special): 167-172 doi: 10.3934/proc.2003.2003.167 +[Abstract](2454) +[PDF](170.3KB)
In this paper, we provide sufficient conditions for the equation

$[x(t) + ax(t + \alpha h) + bx(t + \beta g)]^(n)$ - $cx(t + \mu H)$ - $dx(t + G) = r(t)$

to be oscillatory.

A quadratic Bolza-type problem in a non-complete Riemannian manifold
Anna Maria Candela, J.L. Flores and M. Sánchez
2003, 2003(Special): 173-181 doi: 10.3934/proc.2003.2003.173 +[Abstract](2524) +[PDF](197.4KB)
Let the nonlinear equation $D_s(dotx) + \lambda \nabla_x V (x, s) = 0$ be defined in a non–complete Riemannian manifold $M$ and consider those ones of its solutions which join any couple of fixed points in $M$ in a fixed arrival time $T > 0$. If $V$ has a quadratic growth with respect to $x$ and if $M$ has a convex boundary, then a "best constant" $bar(\lambda)(T)>$ 0 exists such that if $0 \<= \lambda \< bar(\lambda)(T)$ the problem admits at least one solution while infinitely many ones exist if the topology of $M$ is not trivial.
Blow-up estimates of positive solutions of a reaction-diffusion system
Hongwei Chen
2003, 2003(Special): 182-188 doi: 10.3934/proc.2003.2003.182 +[Abstract](2767) +[PDF](160.9KB)
This paper is concerned with positive solutions of the reaction-diffusion system

$u_t - \Delta u = u^(m_1)v^(n_1)$ ,
$v_t - \Delta v = u^(m_2)v^(n_2)$ ,

which blow up at $t = T$. We obtain the following estimates on the blow-up rates:

$c(T - t)^(-(n_1-n_2+1)/\gamma) <= max_(x\in\Omega) u(x, t) <= C(T - t)^(-(n_1-n_2+1)/\gamma)$,
$c(T - t)^(-(m_2-m_1+1)/\gamma) <= max_(x\in\Omega) v(x, t) <= C(T - t)^(-(m_2-m_1+1)/\gamma)$,

for some positive constants $c,C$ and $\gamma = m_2n_1 - (1 - m_1)(1 - n_2)$.

Energy conserving nonholonomic integrators
Jorge Cortés
2003, 2003(Special): 189-199 doi: 10.3934/proc.2003.2003.189 +[Abstract](2423) +[PDF](204.3KB)
We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on previous work on time-dependent discrete mechanics, our approach is based on a discrete version of the Lagrange-d’Alembert principle for nonautonomous systems.
Existence theorems for weakly inward semilinear operators
C. T. Cremins
2003, 2003(Special): 200-205 doi: 10.3934/proc.2003.2003.200 +[Abstract](2056) +[PDF](158.6KB)
We obtain existence theorems for semilinear equations of the form Lx = Nx, where the operators L and N satisfy a weakly inward condition and are such that L - N is A-proper. In particular, results involving positive and multiple solutions are proved.
Integrability in 1D quantum chaos
Yu. Dabaghian, R. V. Jensen and R. Blümel
2003, 2003(Special): 206-212 doi: 10.3934/proc.2003.2003.206 +[Abstract](2120) +[PDF](159.9KB)
Explicit, exact periodic orbit expansions for individual eigenvalues exist for a subclass of quantum networks called regular quantum graphs. We prove that all linear chain graphs have a regular regime.
Spectra of Heisenberg graphs over finite rings
M. DeDeo, M. Martínez, A. Medrano, M. Minei, H. Stark and A. Terras
2003, 2003(Special): 213-222 doi: 10.3934/proc.2003.2003.213 +[Abstract](2393) +[PDF](252.2KB)
We investigate spectra of Cayley graphs for the Heisenberg group over finite rings $\mathbb(Z)$/$p^n\mathbb(Z)$, where $p$ is a prime. Emphasis is on graphs of degree four. We show that for odd $p$ there is only one such connected graph up to isomorphism. When $p = 2$, there are at most two isomorphism classes. We study the spectra using representations of the Heisenberg group. This allows us to produce histograms and butterfly diagrams of the spectra.
On the geometry and topology of singular optimal control problems and their solutions
M. Delgado-Téllez and Alberto Ibort
2003, 2003(Special): 223-233 doi: 10.3934/proc.2003.2003.223 +[Abstract](2812) +[PDF](194.9KB)
The existence of singular arcs for optimal control problems is studied by using a geometric recursive algorithm inspired in Dirac’s theory of constraints. It is shown that singular arcs must lie in the singular locus of a projection map into the coestate space. After applying the geometrical recursive constraints algorithm, we arrive to a reduced set of hamiltonian equations that replace Pontriaguine’s maximum principle. Finally, a global singular perturbation theory is used to obtain nearly optimal solutions.
Kelvin-Helmholtz instability waves of supersonic multiple jets
Joshua Du
2003, 2003(Special): 234-245 doi: 10.3934/proc.2003.2003.234 +[Abstract](1934) +[PDF](154.9KB)
Abstract not present.
Geometric equivalence on nonholonomic three-manifolds
Kurt Ehlers
2003, 2003(Special): 246-255 doi: 10.3934/proc.2003.2003.246 +[Abstract](2422) +[PDF](194.1KB)
We apply Cartan’s method of equivalence to the case of nonholonomic geometry on three-dimensional contact manifolds. Our main result is to derive the differential invariants for these structures and give geometric interpretations. We show that the symmetry group of such a structure has dimension at most four. Our motivation is to study the geometry associated with classical mechanical systems with nonholonomic constraints.
Making the numerical abscissa negative for a class of neutral equations
R.H. Fabiano and J. Turi
2003, 2003(Special): 256-262 doi: 10.3934/proc.2003.2003.256 +[Abstract](2440) +[PDF](160.9KB)
We consider the question of exponential stability of the solution semi- group for a class of scalar differential-difference equations of neutral type. Under very weak assumptions on the coefficients in the equation we show how to construct an appropriate inner product on the underlying state space, which guarantees that the numerical abscissa of the infinitesimal generator is negative.
Solutions and positive solutions for some three-point boundary value problems
Wenying Feng
2003, 2003(Special): 263-272 doi: 10.3934/proc.2003.2003.263 +[Abstract](2455) +[PDF](169.7KB)
We will prove existence results for some three-point boundary value problems by applying a Gronwall inequality. Then, we establish the existence of at least one positive solution for a three-point boundary value problem where the nonlinearity involves the first derivative.
Existence of solutions to second order problems with nonlinear boundary conditions
Daniel Franco and Donal O'Regan
2003, 2003(Special): 273-280 doi: 10.3934/proc.2003.2003.273 +[Abstract](2506) +[PDF](169.6KB)
We prove several new existence results for second–order differential equations with nonlinear boundary value conditions including periodic, antiperiodic and Dirichlet data among others. New definitions of upper and lower solutions for our problems are presented and existence will be established via fixed point methods.
Fractal dimension of attractors for a Stefan problem
Michael L. Frankel and Victor Roytburd
2003, 2003(Special): 281-287 doi: 10.3934/proc.2003.2003.281 +[Abstract](2577) +[PDF](290.5KB)
For a one-phase free-boundary problem with kinetics, which is known to generate a rich dynamics, we present results of a numerical study of the correlation dimension of the attractor.
Semiclassical and large quantum number limits of the Schrödinger equation
Harald Friedrich
2003, 2003(Special): 288-294 doi: 10.3934/proc.2003.2003.288 +[Abstract](2810) +[PDF](161.8KB)
For bound one-dimensional systems, the semiclassical limit h $\to 0$ of the Schrödinger equation generally corresponds to the limit of infinite quantum numbers, and conventional WKB quantization becomes increasingly accurate in this limit. A potential well with a sufficiently strong attractive inverse-square tail supports an infinite dipole series of bound states, but the limit of infinite quantum numbers is not the semiclassical limit in this case. Semiclassical eigenvalues derived via conventional WKB quantization tend to a constant relative error in the large-quantum-number limit when the Langer modification is used. Without the Langer modification the relative error grows exponentially in the limit of large quantum numbers.
Systems with coupling in $mathbb(R)^N$ class of noncoercive potentials
Marcelo F. Furtado, Liliane A. Maia and Elves A. B. Silva
2003, 2003(Special): 295-304 doi: 10.3934/proc.2003.2003.295 +[Abstract](2245) +[PDF](212.9KB)
This paper deals with the existence and multiplicity of solutions to a class of resonant semilinear elliptic system in RN. The main goal is to consider systems with coupling where none of the potentials are coercive. The existence of solution is proved under a critical growth condition on the nonlinearity.
Uniform Gevrey regularity for the attractor of a damped wave equation
Cedric Galusinski and Serguei Zelik
2003, 2003(Special): 305-312 doi: 10.3934/proc.2003.2003.305 +[Abstract](3074) +[PDF](184.1KB)
The goal of this paper is to prove time-asymptotic regularity in Gevrey spaces of the solution of a singularly perturbed damped wave equation and to obtain the uniform (with respect to the perturbation parameter) bounds for the associated global and exponential attractors in the appropriate Gervey spaces.
A three point boundary value problem containing the operator
Marta García-Huidobro and Raul Manásevich
2003, 2003(Special): 313-319 doi: 10.3934/proc.2003.2003.313 +[Abstract](2864) +[PDF](166.8KB)
We consider problems of the form

$(\phi(u'))' = f(t, u, u'), t \in (0, 1)$;

under the three point boundary condition

$u'(0) = 0, u(n) = u(1);$

where $n \in$ (0, 1) is given. This problem is at resonance. Three-point boundary value problems at resonance have been studied in several papers, we present here some new result as well as generalizations of some results valid for particular forms of the operator -$(\phi(u'))'.

Optimal control problem of Bolza-type for evolution hemivariational inequality
Leszek Gasiński
2003, 2003(Special): 320-326 doi: 10.3934/proc.2003.2003.320 +[Abstract](2628) +[PDF](182.2KB)
In this paper we study an optimal control problem of Bolza-type described by evolution hemivariational inequality of second order. Sufficient conditions for obtaining an existence result for such problem are given.
Critical exponents which relate embedding inequalities with quasilinear elliptic problems
Filippo Gazzola
2003, 2003(Special): 327-335 doi: 10.3934/proc.2003.2003.327 +[Abstract](2243) +[PDF](228.1KB)
We show that three numbers which are critical for suitable embedding inequalities are also critical for existence results for some m-Laplace quasilinear elliptic problems with polynomial reaction term.
Remarks on quasilinear elliptic equations as models for elementary particles
Filippo Gazzola and Lorenzo Pisani
2003, 2003(Special): 336-341 doi: 10.3934/proc.2003.2003.336 +[Abstract](2206) +[PDF](176.7KB)
We study a class of quasilinear elliptic equations suggested by C.H. Derrick in 1964 as models for elementary particles. For scalar fields we prove some new nonexistence results. For vector-valued fields the situation is different as shown by recent results concerning the existence of solitary waves with a topological constraint.
Oscillatory properties of third order neutral delay differential equations
John R. Graef, R. Savithri and E. Thandapani
2003, 2003(Special): 342-350 doi: 10.3934/proc.2003.2003.342 +[Abstract](2822) +[PDF](186.2KB)
The authors consider the third order neutral delay differential equation

$a(t) b(t) (y(t) + py(t - \tau))'^'^' + q(t)f(y(t - \sigma)) = 0$

where $a(t) > 0, b(t) > 0, q(t) >= 0, 0 <= p < 1, \tau > 0$, and $\sigma > 0$. Criteria for the oscillation of all solutions of (*) are obtained. Examples illustrating the results are included.

On the damped semilinear wave equation with critical exponent
Maurizio Grasselli and Vittorino Pata
2003, 2003(Special): 351-358 doi: 10.3934/proc.2003.2003.351 +[Abstract](2752) +[PDF](177.2KB)
We provide an optimal regularity result for the universal attractor of the weakly damped semilinear wave equation, when the nonlinearity satisfies the critical growth condition. This allows us to prove an upper semicontinuity result as well as the existence of an exponential attractor.
A nonlinear controlled system of differential equations describing the process of production and sales of a consumer good
Ellina Grigorieva and Evgenii Khailov
2003, 2003(Special): 359-364 doi: 10.3934/proc.2003.2003.359 +[Abstract](2258) +[PDF](269.7KB)
A nonlinear controlled system of differential equations has been constructed to describe the process of production and sales of a consumer good. This model can be controlled either by the rate of production or by the price of the good. The attainable sets of corresponding controlled systems are studied. It is shown that in both cases the boundaries of these sets are the unions of two two-parameter surfaces. It is proved that every point on the boundaries of the attainable sets is a result of piecewise constant controls with at most two switchings. Attainable sets for different values of parameters of the model will be demonstrated using MAPLE.
Modified Chebyshev rational spectral method for the whole line
Guo Ben-Yu and Wang Zhong-Qing
2003, 2003(Special): 365-374 doi: 10.3934/proc.2003.2003.365 +[Abstract](2282) +[PDF](199.6KB)
A modified Chebyshev rational orthogonal system on the whole line is introduced. A rational spectral scheme for the Korteweg de Vries equation on the whole line is constructed. The convergence is proved. The numerical results show its efficiency.
A global semi-Lagrangian spectral model for the reformulated shallow water equations
Daniel Guo and John Drake
2003, 2003(Special): 375-385 doi: 10.3934/proc.2003.2003.375 +[Abstract](2613) +[PDF](406.0KB)
In this paper, we study the semi-Lagrangian spectral method for the shallow-water equations in a rotating, spherical geometry. With the reformulation of a vector calculus identity for spherical geometries, we are able to write the vorticity and divergence equations in advective form and directly apply the semi-Lagrangian, spectral method. The scalar vorticity and divergence equations are used to avoid the pole problems. Shape preserving interpolation is used for the calculation of departure point values for all fields. The results of the standard test set are presented showing accuracy, stability and regularity properties of the method for atmospheric flows.
Lorentz geometry technique in nonimaging optics
Manuel Gutiérrez
2003, 2003(Special): 386-392 doi: 10.3934/proc.2003.2003.386 +[Abstract](2218) +[PDF](175.8KB)
Nonimaging optics is a field that studies optimal concentration of light from a source distribution to a receiver. The relevant information is codified by a field of cones at each point of the concentrator, formed by those rays that we want to reach the receiver (perhaps after some reflections on the wall of the concentrator). This suggests that we can use Lorentz geometry to analyze the problem. We will establish a technique to design three dimensional ideal concentrators with arbitrary media which generalizes a previous one for the homogeneous case.
Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations
Takahiro Hashimoto
2003, 2003(Special): 393-402 doi: 10.3934/proc.2003.2003.393 +[Abstract](2567) +[PDF](206.1KB)
In this paper, we are concerned with the existence and nonexistence of nontrivial solutions for nonlinear elliptic equations involving a biharmonic operator. Concerning the second order equations, a complementary result was obtained for the problem of interior, exterior and whole space. The main purpose of this paper is to discuss whether the complementary result mentioned above is still valid for the nonlinear fourth order equations. We introduce "Kelvin type transformation" for a biharmonic operator to convert an exterior problem to an interior problem. The existence results in case of super-critical exterior problem are shown by introducing a weighted version of Sobolev-Poincaré type inequality, and the nonexistence results are shown by giving a Pohozaev-type identity for fourth order equations.
On continuity in parameters of integrated semigroups
Min He
2003, 2003(Special): 403-412 doi: 10.3934/proc.2003.2003.403 +[Abstract](2630) +[PDF](159.2KB)
Considering a family of non-densely defined Hille-Yosida operators A($\epsilon$), we discuss continuity in a multi-parameter $\epsilon$ of integrated semigroup S(t, $\epsilon$) generated by A($\epsilon$). Recent developed theorems are given for determining continuity in parameters of integrated semigroup on the entire space. The obtained results are effectively and conveniently employed for some hyperbolic and parabolic types of equations.
Remarks on the zeta function of a graph
J. William Hoffman
2003, 2003(Special): 413-422 doi: 10.3934/proc.2003.2003.413 +[Abstract](2924) +[PDF](205.4KB)
We make two observations about the zeta function of a graph. First we show how Bass’s proof of Ihara’s formula fits into the framework of torsion of complexes. Second, we show how in the special case of those graphs that are quotients of the Bruhat-Tits tree for SL(2, $K$) for a local nonarchimedean field $K$, the zeta function has a natural expression in terms of the $L$-functions of Coexter systems.
Oscillation of an Euler-Cauchy dynamic equation
S. Huff, G. Olumolode, N. Pennington and A. Peterson
2003, 2003(Special): 423-431 doi: 10.3934/proc.2003.2003.423 +[Abstract](2464) +[PDF](169.8KB)
The Euler-Cauchy differential equation and difference equation are well known. Here we study a more general Euler-Cauchy dynamic equation. For this more general equation when we have complex roots of the corresponding characteristic equation we for the first time write solutions of this dynamic equation in terms of a generalized exponential function and generalized sine and cosine functions. This result is even new in the difference equation case. We then spend most of our time studying the oscillation properties of the Euler-Cauchy dynamic equation. Several oscillation results are given and an open problem is posed.
Positive solutions of differential equations with nonlinear boundary conditions
Gennaro Infante
2003, 2003(Special): 432-438 doi: 10.3934/proc.2003.2003.432 +[Abstract](2525) +[PDF](160.3KB)
Using the theory of fixed point index, we establish new results for some differential equations subject to nonlinear boundary conditions. We obtain existence of at least one or of multiple positive solutions.
Strong solutions of magneto-micropolar fluid equation
Hiroshi Inoue, Kei Matsuura and Mitsuharu Ôtani
2003, 2003(Special): 439-448 doi: 10.3934/proc.2003.2003.439 +[Abstract](2625) +[PDF](196.1KB)
We show the existence and uniqueness of a strong solution for the system of magneto-micropolar fluid motions under some assumptions on the regularity of given data similar to those of Fujita-Kato [4]. The method of our proof relies on the abstract nonmonotone perturbation theory developed in ˆ Otani [10].
Transfer function approach to output specification in certain linear distributed parameter systems
Harry L. Johnson and David Russell
2003, 2003(Special): 449-458 doi: 10.3934/proc.2003.2003.449 +[Abstract](2489) +[PDF](198.1KB)
In this paper we study, using Laplace transform methods, some questions of output determination for the wave equation and the Euler–Bernoulli beam equation. Specifically we study the problem of determining, over a specified time interval, the displacement at a particular spatial point, via control exercised by means of an external force applied at another spatial point. Some indications of a more general theory are given.
Some recent findings concerning unsteady dipolar fluid flows
P. M. Jordan and P. Puri
2003, 2003(Special): 459-468 doi: 10.3934/proc.2003.2003.459 +[Abstract](2276) +[PDF](210.6KB)
Stokes' second problem for dipolar fluids is solved and analyzed under boundary conditions (BC)s involving the usual no-slip condition in conjunction with the specification of the vorticity. A comparison of results obtained with those given in other work where a different set of BCs was used is presented. In addition, special/limiting cases of the solution, including those which correspond to other fluid models, are examined and analytical and numerical results are given.
Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane
Shuichi Kawashima, Shinya Nishibata and Masataka Nishikawa
2003, 2003(Special): 469-476 doi: 10.3934/proc.2003.2003.469 +[Abstract](2539) +[PDF](183.9KB)
We investigate the asymptotic stability of a stationary solution to an initial boundary value problem for a 2-dimensional viscous conservation law in half plane. Precisely, we show that under suitable boundary and spatial asymptotic conditions, a solution converges to the corresponding stationary solution as time tends to infinity. The proof is based on an a priori estimate in the $H^2$-Sobolev space, which is obtained by a standard energy method. In this computation, we utilize the Poincaré type inequality. In addition, we obtain a convergence rate under the assumption that the initial data converges to a spatial asymptotic state algebraically fast. This result is obtained by a weighted energy estimate.
Conservation laws and invariant solutions for soil water equations
C. M. Khalique and G. S. Pai
2003, 2003(Special): 477-481 doi: 10.3934/proc.2003.2003.477 +[Abstract](2366) +[PDF](137.9KB)
We determine conservation laws for a class of soil water equations and associate these, where possible, with Lie symmetry generators. One cannot invoke Noether’s theorem here as there is no Lagrangian for these equations. We also obtain exact solutions for such a class of equations. These solutions are invariant under a three-dimensional subalgebra of the symmetry Lie algebra.
Application of weak turbulence theory to FPU model
Peter R. Kramer, Joseph A. Biello and Yuri Lvov
2003, 2003(Special): 482-491 doi: 10.3934/proc.2003.2003.482 +[Abstract](2360) +[PDF](190.8KB)
The foundations of weak turbulence theory is explored through its application to the (alpha) Fermi-Pasta-Ulam (FPU) model, a simple weakly nonlinear dispersive system. A direct application of the standard kinetic equations would miss interesting dynamics of the energy transfer process starting from a large-scale excitation. This failure is traced to an enforcement of the exact resonance condition, whereas mathematically the resonance should be broadened due to the energy transfer happening on large but finite time scales. By allowing for the broadened resonance, a modified three-wave kinetic equation is derived for the FPU model. This kinetic equation produces some correct scaling predictions about the statistical dynamics of the FPU model, but does not model accurately the detailed evolution of the energy spectrum. The reason for the failure seems not to be one of the previously clarified reasons for breakdown in the weak turbulence theory.
From quantum action to quantum chaos
Helmut Kröger
2003, 2003(Special): 492-500 doi: 10.3934/proc.2003.2003.492 +[Abstract](2569) +[PDF](176.8KB)
We discuss the concept of the quantum action with the purpose to characterize and quantitatively compute quantum chaos. As an example we consider in quantum mechanics a 2-D Hamiltonian system - harmonic oscillators with anharmonic coupling - which is classically a chaotic system. We compare Poincaré sections obtained from the quantum action with those from the classical action.
Multiple positive eigenvalues of conjugate boundary value problems with singularities
K. Q. Lan
2003, 2003(Special): 501-506 doi: 10.3934/proc.2003.2003.501 +[Abstract](2274) +[PDF](161.4KB)
This paper treats the existence of one or several positive eigenvalues for some nth order differential equations with conjugate boundary conditions. The approach is to employ a well-known result on the existence of positive eigenvalues for Hammerstein integral equations obtained recently by the author. Closed intervals which the eigenvalues belong to are studied and applied to obtain new results on the existence of multiple positive eigenvalues.
Optimal control problems with variable endpoints
Bavo Langerock
2003, 2003(Special): 507-516 doi: 10.3934/proc.2003.2003.507 +[Abstract](2449) +[PDF](193.3KB)
In a previous paper [5] we have proven a geometric formulation of the maximum principle for non-autonomous optimal control problems with fixed endpoint conditions. In this paper we shall reconsider and extend some results from [5] in order to obtain the maximum principle for optimal control problems with variable endpoint conditions. We only consider the case where one of the endpoints may vary, whereas the other is kept fixed.
Consistency of the KP approximation
D. Lannes
2003, 2003(Special): 517-525 doi: 10.3934/proc.2003.2003.517 +[Abstract](2169) +[PDF](194.6KB)
We consider here the consistency of the KP approximation for a Boussinesq system. We consider the general case of two counterpropagating waves, which do not have to satisfy strong zero mass assumptions. We show that without such strong assumption, the KP approximation is not consistent with the Boussinesq system, but that it is close to a consistent approximation. We give precise consistency results, and also consider the case were no zero mass assumption at all is made.
Existence and multiplicity results for a class of nonlinear elliptic problems in $\mathbb(R)^N$
Monica Lazzo
2003, 2003(Special): 526-535 doi: 10.3934/proc.2003.2003.526 +[Abstract](2198) +[PDF](223.0KB)
We study the effect of non constant coefficients $a$ and $b$ on the existence and multiplicity of positive solutions for the equation -div $a(x)\nabla u + \lambda u = b|x|^(p-2)$ in $\mathbb(R)^N.$
Exponential attractors for a chemotaxis growth system on domains of arbitrary dimension
Dung Le
2003, 2003(Special): 536-543 doi: 10.3934/proc.2003.2003.536 +[Abstract](2350) +[PDF](181.0KB)
It is proven that there exist exponential attractors for dynamical systems defined by a general chemotaxis system defined on a domain of arbitrary dimension $n$.
Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective
Wei Feng, Shuhua Hu and Xin Lu
2003, 2003(Special): 544-553 doi: 10.3934/proc.2003.2003.544 +[Abstract](2393) +[PDF](180.3KB)
Phase-locking in electrically coupled non-leaky integrate-and-fire neurons
Timothy J. Lewis
2003, 2003(Special): 554-562 doi: 10.3934/proc.2003.2003.554 +[Abstract](2512) +[PDF](180.3KB)
The synchronization properties of a pair of oscillating spiking neurons connected by electrical (diffusive) coupling is considered. The intrinsic behavior of the cells is described by the non-leaky integrate-and-fire model with a simple modification to account for the effect of spikes. Dynamics of the two-cell system are reduced to the consideration of a one-dimensional map. Periodic orbits of the map correspond to phase-locked solutions of the paired-cell system. The bifurcation structure of the map and therefore the phase-locked states in the two-cell system are analyzed. It is shown that increasing the effect of the spikes and increasing the intrinsic frequency of the cells promote synchronous activity. However, in some conditions, increasing the strength of electrical coupling can counter-intuitively lead to the destabilization of synchronous activity and the stabilization of the anti-phase state.
Nonlocal problems for quasilinear parabolic equations in divergence form
Gary Lieberman
2003, 2003(Special): 563-570 doi: 10.3934/proc.2003.2003.563 +[Abstract](2395) +[PDF](167.4KB)
We study a class of quasilinear parabolic equations with nonlocal initial conditions. The initial conditions are a generalization of periodicity with respect to time and include conditions studied by other authors, which can be used to study inverse problems and problems arising in reactor theory.
Discrete models and Fisher's maximum principle in ecology
Torsten Lindström
2003, 2003(Special): 571-579 doi: 10.3934/proc.2003.2003.571 +[Abstract](2937) +[PDF](164.5KB)
Fisher's (1930) maximum principle in ecology states that \Any net advantage gained by an organism will be conserved in the form of an increase in population, rather than in an increase in the average Malthusian parameter, which is kept by this adjustment always near zero." We know today that we cannot make such general statements. Nevertheless, several ecologists, including Nicholson (1960), have stressed this principle as a general ecological principle. Based on a number of theoretical counterexamples, we cannot conclude that this principle is not supported by any essential biological facts. This paper examines simple examples that illustrate when the principle is valid. We use a discrete modeling approach to account for the fact that several boreal populations are constrained to reproduce at well-defined discrete moments. Several authors have pointed out that the above maximum principle ceases to be valid when predation is present. With reference to the Ricker competition case, we suggest how the principle could be reformulated so as to cover that case.
Yorke and Wright 3/2-stability theorems from a unified point of view
Eduardo Liz, Victor Tkachenko and Sergei Trofimchuk
2003, 2003(Special): 580-589 doi: 10.3934/proc.2003.2003.580 +[Abstract](2145) +[PDF](204.6KB)
We consider a family of scalar delay differential equations $x'(t) = f(t, x_t)$, with a nonlinearity $f$ satisfying a negative feedback condition combined with a boundedness condition. We present a global stability criterion for this family, which in particular unifies the celebrated 3/2-conditions given for the Yorke and the Wright type equations. We illustrate our results with some applications.
Asymptotic solutions of a nonlinear equation
Chunqing Lu
2003, 2003(Special): 590-595 doi: 10.3934/proc.2003.2003.590 +[Abstract](2301) +[PDF](168.8KB)
This paper presents rigorous proofs of the asymptotic solutions of a nonlinear ordinary equation, $\epsilon n f^(iv) = (f-2\epsilon) f''' - f' f''$ subject to boundary conditions: $f(0) = 0, f(1) = 1, f'(1) = 0, lim_(n \to 0^+) sqrtn f''(n)=0.$
Classical and quantum chaos in fundamental field theories
Harald Markum and Rainer Pullirsch
2003, 2003(Special): 596-603 doi: 10.3934/proc.2003.2003.596 +[Abstract](2403) +[PDF](206.6KB)
An investigation of classical chaos and quantum chaos in gauge fields and fermion fields, respectively, is presented for (quantum) electrodynamics. We analyze the leading Lyapunov exponents of U(1) gauge field configurations on a $12^3$ lattice which are initialized by Monte Carlo simulations. We find that configurations in the strong coupling phase are substantially more chaotic than in the deconfinement phase. Considering the quantum case, complete eigenvalue spectra of the Dirac operator in quenched $4d$ compact QED are studied on $8^3 \times 4$ and $8^3 \times 6$ lattices. We investigate the behavior of the nearest-neighbor spacing distribution $P(s)$ as a measure of the fluctuation properties of the eigenvalues in the strong coupling and the Coulomb phase. In both phases we find agreement with the Wigner surmise of the unitary ensemble of random-matrix theory indicating quantum chaos.
An averaging method for the Helmholtz equation
S. L. Ma'u and P. Ramankutty
2003, 2003(Special): 604-609 doi: 10.3934/proc.2003.2003.604 +[Abstract](2792) +[PDF](108.7KB)
The well-known J. Schauder result on the existence of Lip$_\alpha (bar(\Omega))$ solutions of the Dirichlet problem for bounded domains with smooth boundaries is true for the Helmholtz equation $\Delta u + \lambda u = 0$ for $\lambda =< 0$. We suggest a method of constructing the solution based on an averaging procedure and mean-value theorem. We show some conditions under which, for $0 < \alpha < 1$, and $\lambda =< 0$, a sequence of iterated averages of an initial approximation converges geometrically to the solution.
Dynamics of torque-speed profiles for electric vehicles and nonlinear models based on differential-algebraic equations
Roderick V.N. Melnik, Ningning Song and Per Sandholdt
2003, 2003(Special): 610-617 doi: 10.3934/proc.2003.2003.610 +[Abstract](2778) +[PDF](2264.7KB)
The so-called $\mu - \lambda$ curves, where $\lambda$ is the slip ratio and $\mu$ is the normalised traction force or the friction index, are nonlinear functions of the velocity of the vehicle and the wheel rotational velocity. Despite their predominant use in the literature, linear approximations of such curves may fail to predict correctly key characteristics of vehicle performance efficiency such as torque-speed profiles. Although attempts to model these characteristics in the context of slip phenomena have been made before, to our best knowledge a general model with respect to the vehicle velocity, the wheel rotating velocity, the slip ratio, the traction force, and the torque, has never been formulated and solved as a coupled nonlinear problem based on a system of differential-algebraic equations arising naturally in this context. In this paper, such a model is formulated, solved numerically, and some results of numerical simulation of driving an electric vehicle on different surface conditions are presented.
On solutions to stochastic differential inclusions
Mariusz Michta
2003, 2003(Special): 618-622 doi: 10.3934/proc.2003.2003.618 +[Abstract](2589) +[PDF](162.6KB)
In the paper a martingale problem approach is used to analyze the problem of existence and topological properties of optimal weak solutions to stochastic differential inclusions of Ito type with convex integrands.
Positivity preserving discrete model for the coupled ODE's modeling glycolysis
Ronald E. Mickens
2003, 2003(Special): 623-629 doi: 10.3934/proc.2003.2003.623 +[Abstract](2553) +[PDF](162.3KB)
We construct a nonstandard finite difference scheme for the two coupled ODE's that model glycolysis. The primary emphasis is having the scheme satisfy a positivity condition and also retain the limit-cycle behavior for certain values of the parameters. We show that this is possible and give a full discussion of the scheme along with some of its numerical properties.
Existence of solutions for Cahn-Hilliard type equations
Alain Miranville
2003, 2003(Special): 630-637 doi: 10.3934/proc.2003.2003.630 +[Abstract](2605) +[PDF](162.9KB)
Our aim in this article is to study the existence of solutions for a system of equations which arises in the context of phase transitions in binary alloys.
L-functions and the Selberg trace formulas for semiregular bipartite graphs
Hirobumi Mizuno and Iwao Sato
2003, 2003(Special): 638-646 doi: 10.3934/proc.2003.2003.638 +[Abstract](2133) +[PDF](178.5KB)
We give a decomposition formula for the L-function of a semiregular bipartite graph G. Furthermore, we present the Selberg trace formula for the above L-function of G.
Existence of square integrable solutions of perturbed nonlinear differential equations
Octavian G. Mustafa and Yuri V. Rogovchenko
2003, 2003(Special): 647-655 doi: 10.3934/proc.2003.2003.647 +[Abstract](2267) +[PDF](181.8KB)
We give constructive proof of the existence of square integrable solutions for a class of $n$-th order nonlinear differential equations and discuss properties of square integrable solutions, obtaining as by-products an efficient estimate for the rate of decay of the $L^2$ norm of the solution and the nonexistence result due to Grammatikopoulos and Kulenovic [On the nonexistence of $L^2$-solutions of $n$-th order differential equations, Proc. Edinburgh Math. Soc. (2) 24 (1981), 131–136].
On stochastic stability of dynamic neural models in presence of noise
K Najarian
2003, 2003(Special): 656-663 doi: 10.3934/proc.2003.2003.656 +[Abstract](2223) +[PDF](161.5KB)
Dynamic feedback neural networks are known to present powerful tools in modeling of complex dynamic models. Since in many real applications, the stability of such models (specially in presence of noise) is of great importance, it is essential to address stochastic stability of such models. In this paper, sufficient conditions for stochastic stability of two families of feedback sigmoid neural networks are presented. These conditions are set on the weights of the networks and can be easily tested.
Dynamics of principal configurations near umbilics for surfaces in $mathbb(R)^4$
Matías Navarro and Federico Sánchez-Bringas
2003, 2003(Special): 664-671 doi: 10.3934/proc.2003.2003.664 +[Abstract](2263) +[PDF](178.1KB)
The $v$-principal configuration of an immersed surface $M$ in $\mathbb(R)^4$ is the set formed by the umbilical points and the lines of principal curvatures with respect to an unitary smooth vector field $v$ normal to $M$. In this article we describe the bifurcation set of $v$-principal configurations of a local surface $M$ depending on two parameters of the surface and depending also on the 1-jet of the vector field $v$ normal to $M$ which defines an isolated simple umbilical point of $M$.
On weak-almost periodic mild solutions of some linear abstract differential equations
Gaston N'Guerekata
2003, 2003(Special): 672-677 doi: 10.3934/proc.2003.2003.672 +[Abstract](2347) +[PDF](168.1KB)
We are concerned with the differential equation $x'(t) = Ax(t) + f(t)$ with a linear operator $A$ acting in a Banach space $X$ and $f : \mathbb(R) \to X$ a almost periodic function (in Bochner’s sense). We give necessary conditions to ensure that the so-called optimal mild solutions are also weakly almost periodic.
Quasi-regular graphs, cogrowth, and amenability
Sam Northshield
2003, 2003(Special): 678-687 doi: 10.3934/proc.2003.2003.678 +[Abstract](1967) +[PDF](183.6KB)
We extend Grigrochuk’s cogrowth criterion for amenability of groups to the case of non-regular graphs for which a certain regularity condition is satisfied. The proof involves generalized Laplacians which are inverses of growth series and whose determinants are closely related to zeta functions of graphs.
A semilinear elliptic system with vanishing nonlinearities
Rafael Ortega and James R. Ward Jr
2003, 2003(Special): 688-693 doi: 10.3934/proc.2003.2003.688 +[Abstract](2049) +[PDF](176.4KB)
The Neumann boundary value problem is examined for systems of elliptic equations of the form $\Delta u + g(u) = f(x), x \in \omega.$ It is assumed that $g \in 2 C(\mathbb(R)^N,\mathbb(R)^N)$ is a bounded function which may vanish at infinity. Leray-Schauder degree methods are used.
Stability of a linear functional equation in Banach modules
Chun-Gil Park
2003, 2003(Special): 694-700 doi: 10.3934/proc.2003.2003.694 +[Abstract](2216) +[PDF](159.7KB)
We prove the generalized Hyers-Ulam-Rassias stability of a linear functional equation in Banach modules over a unital C*-algebra.
Topological-numerical approach to the existence of periodic trajectories in ODE's
Paweł Pilarczyk
2003, 2003(Special): 701-708 doi: 10.3934/proc.2003.2003.701 +[Abstract](2395) +[PDF](202.5KB)
We discuss an application of a topological-numerical method for proving the existence of a periodic trajectory in a smooth dynamical system in $\mathbb(R)^n$ where a periodic trajectory is numerically observed. The method is based on the Conley index theory and rigorous numerics for ODEs and it is a generalization of the method introduced in [13]. We apply this method to the Rössler equations.
Analytic continuation into the future
David W. Pravica and Michael J. Spurr
2003, 2003(Special): 709-716 doi: 10.3934/proc.2003.2003.709 +[Abstract](2187) +[PDF](184.8KB)
A class of analytic advanced and delayed differential equations, which are defined in a neighborhood of an initial point, and which are assumed to have formal solutions in terms of power series, is studied. We provide growth conditions whereby the (perhaps non-convergent) formal series solutions can be extended to analytic solutions defined on a sectorial domain with vertex at the initial point. By introducing a new Laplace-Borel kernel, and obtaining estimates on its decay rate, the concept of a Gevrey series is generalized. The class of equations studied includes advanced and delayed initial value problems with polynomial coefficients. Key estimates are shown and an example of a new application is given.
Exponential attractor for the delayed logistic equation with a nonlinear diffusion
Dalibor Pražák
2003, 2003(Special): 717-726 doi: 10.3934/proc.2003.2003.717 +[Abstract](2737) +[PDF](193.9KB)
We study the generalized logistic equation where the feedback is captured by the time convolution with a nonnegative measure and the diffusion is the laplacian plus the p-laplacian with $p >= 2$. We prove that the equation has an exponential attractor provided that the solutions are asymptotically bounded.
Parabolic systems with non continuous coefficients
Maria Alessandra Ragusa
2003, 2003(Special): 727-733 doi: 10.3934/proc.2003.2003.727 +[Abstract](2426) +[PDF](195.6KB)
In this note we are interested in the local regularity of the highest order derivatives of the solutions of the system

$T u = fi(y)$      $i = 1,...,N

where the known terms $f_i$ are in Lebesgue spaces and the differential the parabolic operator $T$ has the form

$ut - \sum_{j=1}^{N}\sum_{|\alpha|=2s} a^(\alpha)_(ij) (y)D^(\alpha) u_j (y) + \sum_{j=1}^{N}\sum_{|\alpha|<=2s-1} b^(\alpha)_(ij) (y)D^(\alpha) u_j (y)$.

have discontinuous coefficients.

On the central stability zone for linear discrete-time Hamiltonian systems
Vladimir Răsvan
2003, 2003(Special): 734-741 doi: 10.3934/proc.2003.2003.734 +[Abstract](2451) +[PDF](173.5KB)
In this paper we start from the discrete version of linear Hamiltonian systems with periodic coefficients

$_y_(k+1) - _y(_k) = \lambda B(__k)_y(_k) + \lambda D_(k^z_(k+1))$
$_z_(k+1) - _z(_k) = -\lambda A(__k)_y(_k) - \lambda B*_(k^z_(k+1))$

where $A_k$ and $D_k$ are Hermitian matrices, $A_k$, $B_k$, $D_k$ define $N$-periodic sequences, and $\lambda$ is a complex parameter. For this system a Krein-type theory of the $\lambda$-zones of strong (robust) stability may be constructed. Within this theory the side $\lambda$-zones’ width may be estimated using the multipliers’ “traffic rules” of Krein while the central stability zone (centered around $\lambda$ = 0) is estimated using the eigenvalues of a certain boundary value problem which is self-adjoint. In the discrete-time there occur some specific differences with respect to the continuous time case due to the fact that the transition matrix (hence the monodromy matrix also) is not entire with respect to $\lambda$ but rational. During the paper we consider some specific cases (the matrix analogue of the discretized Hill equation, the J-unitary and symplectic systems, real scalar systems) for which the results on the eigenvalues are complete and obtain some simplified estimates of the central stability zones.

Minimization of the base transit time in semiconductor devices using optimal control
Paolo Rinaldi and Heinz Schättler
2003, 2003(Special): 742-751 doi: 10.3934/proc.2003.2003.742 +[Abstract](2573) +[PDF](184.7KB)
We consider the problem of determining the optimal profile of doping concentration that minimizes the base transit time in homojunction bipolar transistors. This is a well-studied problem in the electronics literature, but typically only numerical optimization is used to find solutions. In this paper we give an explicit analytic solution to the problem using the Pontryagin Maximum Principle with state-space constraints and prove its optimality using synthesis type arguments.
On certain elliptic systems with nonlinear self-cross diffusions
Kimun Ryu and Inkyung Ahn
2003, 2003(Special): 752-759 doi: 10.3934/proc.2003.2003.752 +[Abstract](2377) +[PDF](179.6KB)
We investigate the positive coexistence to certain strongly-coupled nonlinear elliptic systems with self-cross diffusions under homogeneous Robin boundary conditions. Competing interactions between two species are considered. Conditions of the positive coexistence to self-cross diffusive systems can be expressed in terms of the spectral property of differential operators of nonlinear Schrödinger type which reflect the influence of the domain and nonlinearity in the system. Decoupling method and nonlinear fixed point theorem are employed.
Nonlinear boundary value problems of the calculus of variations
Felix Sadyrbaev
2003, 2003(Special): 760-770 doi: 10.3934/proc.2003.2003.760 +[Abstract](3036) +[PDF](175.9KB)
We consider nonlinear boundary value problems arising in the classical one-dimensional calculus of variations for scalar-valued unknown functions. Conditions for the existence of extremals (solutions of the Euler equation subject to related boundary conditions) are obtained and properties of extremals are discussed. The method of upper and lower solutions (functions) is our main tool. Several Bernstein - Nagumo type conditions are derived directly in terms of the Lagrangian. Both coercive and non-coercive (slow-growth) variational problems are considered.
A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays
Yasuhisa Saito
2003, 2003(Special): 771-777 doi: 10.3934/proc.2003.2003.771 +[Abstract](2625) +[PDF](153.9KB)
This paper presents a sufficient condition for the global stability of an n-species Lotka-Volterra food chain system with distributed time delays. The result is sharp in the sense that it coincides with a necessary and sufficient condition for global stability when $n = 2$.
Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems
Addolorata Salvatore
2003, 2003(Special): 778-787 doi: 10.3934/proc.2003.2003.778 +[Abstract](2581) +[PDF](194.1KB)
We look for homoclinic orbits of the system of differential equations

$- dot(q) + L(t)q = V_q(t, q) + g(t)$

where $V : R \times R^N (\to) R$ is superquadratic and even in $q$ and $L(t)$ is a a symmetric, positive definite matrix. If $g(t) != 0$, in spite of the loss of symmetry a suitable perturbative method allows one to state the existence of infinitely many solutions of the problem.

Compactified isospectral sets of complex tridiagonal Hessenberg matrices
Barbara A. Shipman
2003, 2003(Special): 788-797 doi: 10.3934/proc.2003.2003.788 +[Abstract](2257) +[PDF](193.3KB)
The completed flows of the complex Toda lattice hierarchy are used to compactify an arbitrary isospectral set $J$^ of complex tridiagonal Hessenberg matrices. When the eigenvalues are distinct, this compactification, as found by other authors, is a toric variety; it is the closure of a generic orbit of a complex maximal torus inside a flag manifold. This torus becomes a direct product, A, of a nonmaximal diagonal subgroup and a unipotent group when eigenvalues coincide. We describe the compactification of $J$^ in this case as the closure of a generic orbit of A. We are interested mainly in the structure of its boundary, which is a union of nonmaximal orbits of A. There is a one-to-one correspondence between the connected components of the intersections of $J$^ with the lower-dimensional symplectic leaves and the faces of the moment polytope where at least one vertex is a minimal coset representative of a certain quotient of the Weyl group.
Asymptotic stability for dynamical systems associated with the one-dimensional Frémond model of shape memory alloys
Ken Shirakawa
2003, 2003(Special): 798-808 doi: 10.3934/proc.2003.2003.798 +[Abstract](2311) +[PDF](195.1KB)
In this paper, an evolution dynamical system, which is generated by one-dimensional Frémond models of shape memory alloys, is considered. Assuming that forcing terms converge to some time-independent terms in appropriate senses as time goes to infinity, we shall characterize the asymptotic stability for our dynamical system by the global attractor for the limiting autonomous dynamical system.
Exponential decay of oscillations in a multidimensional delay differential system
Eugenii Shustin
2003, 2003(Special): 809-816 doi: 10.3934/proc.2003.2003.809 +[Abstract](2425) +[PDF](171.7KB)
We consider a multi-dimensional dynamical system with a discontinuous delayed part, which is a model of a relay type negative feedback with an uncertain time delay. Our main result consists of a sufficient criterion for an exponential decay of oscillations even when the non-delayed part of the system has no stable solutions.
Numerical simulation of capillary formation during the onset of tumor angiogenesis
Michael W. Smiley, Howard A. Levine and Marit Nilsen Hamilton
2003, 2003(Special): 817-826 doi: 10.3934/proc.2003.2003.817 +[Abstract](2580) +[PDF](180.7KB)
A recently proposed mathematical model for turor angiogenesis consists of a coupled system of ordinary and partial differential equations, some of which are strongly convection dominated diffusion equations. A numerical method based on the use of characterics, which is mass conserving, can be used to effectively handle this feature. The model and the numerical method are presented.
Integral curves of derivations on locally semi-algebraic differential spaces
Jędrzej Śniatycki
2003, 2003(Special): 827-833 doi: 10.3934/proc.2003.2003.827 +[Abstract](2063) +[PDF](163.6KB)
We identify a class of differential spaces which contains orbit spaces of proper actions of Lie groups on smooth manifolds. In this class, we prove an existence and uniqueness theorem for integral curves of derivations of smooth functions. We give a necessary and sufficient condition for a derivation to generate a local one parameter group of local diffeomorphisms.
True laminations for complex Hènon maps
Meiyu Su
2003, 2003(Special): 834-841 doi: 10.3934/proc.2003.2003.834 +[Abstract](2155) +[PDF](212.1KB)
In this paper, we are specially interested in the lamination structure for a polynomial diffeomorphism $f$ of $\mathbb(C)^2$ that are conjugate to a finite decomposition of generalized complex Hènon maps on $\mathbb(C)$. We prove that there are true $f$-invariant contracting and expanding measured Riemann surface laminations–injected into the stable and unstable partitions $W^(s/u)$. Leaves of the laminations are conformally isomorphic to the complex plane $\mathbb(C)$. The new ingredients here are the countable collection of the Pesin boxes and a $\sigma$-finite topology, the ‘entropy topology’ on the transversals, defined by the logarithm of the measures obtained by conditioning the unique ergodic measure of maximal entropy $\mu$.
Optimal control of a space shuttle, and numerical simulations
Emmanuel Trélat
2003, 2003(Special): 842-851 doi: 10.3934/proc.2003.2003.842 +[Abstract](2656) +[PDF](217.3KB)
We study the Earth re-entry problem of a space shuttle where the control is the angle of bank, the cost is the total amount of thermal flux, and the system is subject to state constraints on the thermal flux, the normal acceleration and the dynamic pressure. The optimal solution is approximated by a concatenation of bang and boundary arcs, and is numerically computed using a multiple-shooting code.
Global existence and asymptotic analysis of solutions to a model for the dynamic combustion of compressible fluids
Konstantina Trivisa
2003, 2003(Special): 852-863 doi: 10.3934/proc.2003.2003.852 +[Abstract](2505) +[PDF](197.2KB)
A model is introduced for the dynamic combustion of exothermically reacting, compressible fluids formulated by the Navier-Stokes equations expressing the conservation of mass, the balance of momentum and energy and the two species chemical kinetics. The global existence of the discontinuous solutions to the Navier- Stokes equations with large discontinuous initial data is established by combining the difference approximations techniques and energy estimates. The asymptotic analysis of solutions is also discussed.
Cellular neural networks: asymmetric templates and spatial chaos
Larry Turyn
2003, 2003(Special): 864-871 doi: 10.3934/proc.2003.2003.864 +[Abstract](2184) +[PDF](187.0KB)
We consider a Cellular Neural Network (CNN), with a bias term, on the integer lattice $Z^2$ in the plane $R^2$. A space-dependent, asymmetric coupling (template) appropriate for CNN on the hexagonal lattice on $R^2$ is studied. We characterize the mosaic patterns and study their spatial entropy. Asymmetry of the template has a decisive effect on spatial entropy for all known results.
Travelling waves for solid-gas reaction-diffusion systems
C. van der Mee and Stella Vernier Piro
2003, 2003(Special): 872-879 doi: 10.3934/proc.2003.2003.872 +[Abstract](2344) +[PDF](172.9KB)
We study a two-species reaction-diffusion problem described by a system consisting of a semilinear parabolic equation and a first order ordinary differential equation, endowed with suitable conditions. We prove the existing of a unique traveling wave profile and give necessary conditions and sufficient conditions for the occurrence of penetration and conversion fronts.
The primitive equations formulated in mean vorticity
Cheng Wang
2003, 2003(Special): 880-887 doi: 10.3934/proc.2003.2003.880 +[Abstract](2255) +[PDF](153.4KB)
The primitive equations (PEs) of large-scale oceanic flow formulated in mean vorticity is proposed. In the reformulation of the PEs, the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity. The total velocity field (both horizontal and vertical) is statically determined by differential equations at each fixed horizontal point. Its equivalence to the original formulation is also presented.
Global existence and dynamical properties of large solutions for combustion flows
Dehua Wang
2003, 2003(Special): 888-897 doi: 10.3934/proc.2003.2003.888 +[Abstract](2045) +[PDF](184.9KB)
A mathematical model for viscous compressible realistic reactive flows without species diffusion in dynamic combustion is investigated. The initial-boundary value problem with Dirichlet-Neumann mixed boundaries in a finite domain is studied. The existence, uniqueness, and regularity of global solutions are established with general large initial data in $H^1$. It is proved that, although the solutions have large oscillations and the chemical reaction generates heat, there is no shock wave, turbulence, vacuum, mass or heat concentration developed in a finite time.
Learning theory applied to Sigmoid network classification of protein biological function using primary protein structure
D. Warren and K Najarian
2003, 2003(Special): 898-904 doi: 10.3934/proc.2003.2003.898 +[Abstract](3031) +[PDF](174.7KB)
Recently, Valiant’s Probably Approximately Correct (PAC) learning theory has been extended to learning m-dependent data. With this extension, training data set size for sigmoid neural networks have been bounded without underlying assumptions for the distribution of the training data. These extensions allow learning theory to be applied to training sets which are definitely not independent samples of a complete input space. In our work, we are developing length independent measures as training data for protein classification. This paper applies these learning theory methods to the problem of training a sigmoid neural network to recognize protein biological activity classes as a function of protein primary structure. Specifically, we explore the theoretical training set sizes for classifiers using the full amino acid sequence of the protein as the training data and using length independent measures as the training data. Results show bounds for training set sizes given protein size limits for the full sequence input compared to bounds for input that is sequence length independent.
Remarks on positive solutions of some three point boundary value problems
J. R. L. Webb
2003, 2003(Special): 905-915 doi: 10.3934/proc.2003.2003.905 +[Abstract](2302) +[PDF](174.9KB)
We investigate constants that appear in the study of positive solutions of some nonlocal boundary value problems with a view to obtaining optimal values. We succeed for each of two problems and see that the 'natural' choice is not optimal.
Boundary conditions for multi-dimensional hyperbolic relaxation problems
Wen-Qing Xu
2003, 2003(Special): 916-925 doi: 10.3934/proc.2003.2003.916 +[Abstract](2132) +[PDF](186.1KB)
We study the IBVP for a class of linear relaxation systems in a half space with arbitrary space dimensions. The goal is to determine the appropriate structural stability conditions, particularly, the formulation of boundary conditions such that the relaxation IBVP is stiffly well-posed or uniformly well-posed independent of the relaxation parameter. Our main contribution is the derivation, in an explicit and easily checkable form, of a stiff version of the classical Uniform Kreiss Condition (and hence referred to as Stiff Kreiss Condition). The Stiff Kreiss Condition is shown to be necessary and su±cient for the stiff well-posedness of the relaxation IBVP and its asymptotic convergence to the underlying equilibrium system in the zero relaxation limit.
Multiple solutions of super-quadratic second order dynamical systems
Xiangjin Xu
2003, 2003(Special): 926-934 doi: 10.3934/proc.2003.2003.926 +[Abstract](2217) +[PDF](184.3KB)
In this paper the existence of periodic solutions of large norm for the super-quadratic second order dynamical systems $A dot(x) = - \nabla V (x)$ is proved. And some results for forced systems are also gained.
Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems
Noriaki Yamazaki
2003, 2003(Special): 935-944 doi: 10.3934/proc.2003.2003.935 +[Abstract](2299) +[PDF](180.6KB)
In this paper we consider double obstacle problems including regional economic growth models. Unfortunately, by prescribed double obstacles, our problems lose the uniqueness of solutions. So, our problems have multiple solutions for a given initial value. Hence, the associated dynamical systems are multivalued. In this paper we shall consider the large-time behaviour of multiple solutions from the viewpoint of attractors. Namely, the main object of this paper is to construct the global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems.
Valuing investment project in competitive environment
Ming Yang and Chulin Li
2003, 2003(Special): 945-950 doi: 10.3934/proc.2003.2003.945 +[Abstract](2427) +[PDF](151.2KB)
This paper provides a model of investment project in a dynamic competitive environment and values the system. The option value of the investment is modelled as a solution of a stochastic differential equation with free boundary through a combination of real option approach as well as dynamic investing analysis. The optimal investment rule is the free boundary of the equation, $p$*, such that the firm invests at once when $p \>= p$* and waits when $p < p$*. The result shows how uncertainty in project and competitive pressure of a rival affect the value of the project and firm’s investment behavior. The results imply that the investment option value of following investor will be valueless when the research effort intensity of the leader investor is larger enough. The results demonstrate that the model can be applied as an extension from those discussing value of real options for a non -competitive environment.
Solution filtering technique for solving Burgers' equation
Tianliang Yang and J. M. McDonough
2003, 2003(Special): 951-959 doi: 10.3934/proc.2003.2003.951 +[Abstract](2440) +[PDF](697.3KB)
In this paper we focus on using a solution filtering technique as an alternative to the conventional large-eddy simulation approach of filtering the governing equations. Our research shows that the solution filtering technique works quite well when applied to Burgers' equation, and since this equation embodies many important mathematical features of the Navier-Stokes equations, the solution filtering technique possesses significant potential for solving practical turbulence problems governed by these equations.
Formally gradient reaction-diffusion systems in Rn have zero spatio-temporal topological
Zelik S.
2003, 2003(Special): 960-966 doi: 10.3934/proc.2003.2003.960 +[Abstract](2175) +[PDF](186.4KB)
We prove that the spatio-temporal topological entropy (= the topological entropy per unit volume) is equal to zero for formally gradient reaction-diffusion systems in $\mathbb(R)^n$. This result generalizes the well-known fact that gradient ODEs have zero topological entropy.
Linear conservation laws of nonholonomic systems with symmetry
Dmitry V. Zenkov
2003, 2003(Special): 967-976 doi: 10.3934/proc.2003.2003.967 +[Abstract](2631) +[PDF](195.9KB)
In nonholonomic dynamics, symmetries do not always lead to conservation laws. In this paper we study conditions for a nonholonomic system with symmetry to have conservation laws linear in momentum.

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