Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis

We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient. We first provide with a complete answer in the definitively autonomous case, when a(t) takes a constant value l outside a bounded interval. Then we consider the case in which a(t) is definitively monotone, converges from above, as t → ±∞, to two positive limits l _* and l ^*, and never goes below min(l _*, l ^*). Furthermore, the convergence to max(l _*, l ^*) is supposed to be not too fast (slower than a suitable exponential term).     

Exact propagators for some degenerate hyperbolic operators

Exact propagators are obtained for the degenerate second order hyperbolic operators ∂² _t -t ^2l Δ _x , l=1,2,..., by analytic continuation from the degenerate elliptic operators ∂² _t +t ^2l Δ _x . The partial Fourier transforms are also obtained in closed form, leading to integral transform formulas for certain combinations of Bessel functions and modified Bessel functions.     

Nonsingular Equations over Groups II

Let G be a group, t an element distinct from G, and r(t) = g ₁ t ^{l ₁} …g _k t ^{l _k}  ∈ G* ?t?, where each g _i is an element of G order greater than 2, and the l _i are nonzero integers such that l ₁ + l ₂+…+l _k  ≠ 0 and |l _i | ≠ |l _j | for i ≠ j. We prove that if k = 4, then the natural map from G to the one-relator product ?G*t | r(t)? is injective. This together with previous results show that the natural map from G is injective for k ≥ 1.     

A generalization of an example by V. Ya. Ivrii

A well-known example by Ivrii concerning the operatorP=D _t ² −t^2lD _x ² +at^kD_x (a≠0), shows that there exists a delicate relation amongl, k and the Gevrey index of well-posedness of the Cauchy problem. In this paper we give a generalization to a class of pseudo-differential operators includingP.

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Sunto Un famoso esempio di Ivrii riguardante l'operatoreP=D _t ² −t^2lD _x ² +at^kD_x (a≠0), mostra che c'è una relazione sottile tral, k e l'indice di Gevrey di buona positura del problema di Cauchy. In questo articolo viene data una generalizzazione ad una classe di operatori pseudodifferenziali che comprendeP.

     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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ADDITIVE FAMILIES OF INVARIANTS OF SKEWSYMMETRIC TENSORS

Abstract

Let d be a positive integer and F be a field of characteristic 0. Suppose that for each positive integer n, I _n is a polynomial invariant of the usual action of GL_n (F) on Λ^d(Fⁿ), such that for t ? Λ^d(F ^k) and s ? Λ^d(F ^l), I _{k + l} (t l s) = I _k(t)I _t (s), where t ⊥ s is defined in §1.4. Then we say that {I_n} is an additive family of invariants of the skewsymmetric tensors of degree d, or, briefly, an additive family of invariants. If not all the I_n are constant we say that the family is non-trivial. We show that in each even degree d there is a non-trivial additive family of invariants, but that this is not so for any odd d. These results are analogous to those in our paper [3] for symmetric tensors. Our proofs rely on the symbolic method for representing invariants of skewsymmetric tensors. To keep this paper self-contained we expound some of that theory, but for the proofs we refer to the book [2] of Grosshans, Rota and Stein.     

On the structure of linear graphs

Denote byG(n; m) a graph ofn vertices andm edges. We prove that everyG(n; [n ²/4]+1) contains a circuit ofl edges for every 3 ≦l<c ₂ n, also that everyG(n; [n ²/4]+1) contains ak _e(u _n, u_n) withu _n=[c ₁ logn] (for the definition ofk _e(u _n, u_n) see the introduction). Finally fort>t ₀ everyG(n; [tn ^3/2]) contains a circuit of 2l edges for 2≦l<c ₃ t ². This work was done while the author received support from the National Science Foundation, N.S.F. G.88.     

First non-zero terms for the Taylor expansion at 1 of the Conway potential function

The Conway potential function ∇ _L (t ₁,...,t _l ) of an ordered oriented link L = L ₁ ∪ L ₂ ∪ ... ∪ L _l ⊂ S ³ is considered. In general, this function is not determined by the linking numbers and the Conway potential functions of the components. However, the first two nonzero terms of the Taylor expansion at the point 1 of the function ∇ _L are determined by the linking numbers only. We give the explicit formulas for these terms using summation over trees with l vertices.     

Some remarks on hilbert functions of veronese algebras

We study the Hilbert polynomials of finitely generated graded algebras R, with generators not all of degree one (i.e. non-standard). Given an expression P(R,t)=a(t)/(1-t^l ) ⁿ for the Poincare series of R as a rational function, we study for 0 ≤ i ≤ l the graded subspaces ? _kR_kl+i (which we denote R[l;i]) of R, in particular their Poincaré series and Hilbert functions. We prove, for example, that if R is Cohen-Macaulay then the Hilbert polynomials of all non-zeroR[l;i] share a common degree. Furthermore, if R is also a domain then these Hilbert polynomials have the same leading coefficient.     

On the Propagation of Regularity and Decay of Solutions to the k-Generalized Korteweg-de Vries Equation

We study special regularity and decay properties of solutions to the IVP associated to the k-generalized KdV equations. In particular, for datum u ₀ ∈ H ^3/4+ (?) whose restriction belongs to H ^l ((b, ∞)) for some l ∈ ?⁺ and b ∈ ? we prove that the restriction of the corresponding solution u(·, t) belongs to H ^l ((β, ∞)) for any β ∈ ? and any t ∈ (0, T). Thus, this type of regularity propagates with infinite speed to its left as time evolves.     

The Banach spaceT and the fast growing hierarchy from logic

Necessary and sufficient conditions for (t _k) to be equivalent to (t _f(k)) are given where (t _k) is the natural basis of the Banach spaceT, the so-called “Tsirelson space”. The condition is in terms of the growth rate off(k) relative to the fast growing hierarchy. Roughly speaking, the ω-th level is needed to obtain nonequivalence. In particular, primitive recursivef(k) yield equivalent basic sequences. The proof is obtained via some asymptotically sharp estimates on the divergence of finite-dimensional subspaces ofT froml ₁ ^d of the same dimension.     

Direct decompositions of torsion-free Abelian groups of finite rank

It is proved that ifr ₁ ,r ₂ , ...,r _s ;l ₁ ,l ₂ , ...,l _t are the ranks of the indecomposable summands of two direct decompositions of a torsion-free Abelian group of finite rank and if s₀ is the number of units among the numbers r_i, while t₀ is the number of units among the numbers l_j, thenr _i ⩽n - t ₀ ,l _j ⩽n−s ₀ for all i, j. Moreover, if for some i we have r_i=n−t₀, then among the l_j's only one term is different from 1 and it is equal to n−t₀; similarly if l_j=n−s₀ for some j. In addition, a construction is presented, allowing to form, from several indecomposable groups, a new group, called a flower group, and it is proved that a flower group is indecomposable under natural restrictions on its defining parameters. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 272–285, 1987.     

The subindependence of coordinate slabs inl p n balls

It is proved that if the probabilityP is normalised Lebesgue measure on one of thel _p ⁿ balls in R ⁿ , then for any sequencet ₁ , t ₂ , …, t _n of positive numbers, the coordinate slabs {|x _i |≤t _i } are subindependent, namely, . A consequence of this result is that the proportion of the volume of thel ₁ ⁿ ball which is inside the cube[−1, t] ⁿ is less than or equal tof _n (t)=(1−(1−t) ⁿ ) ⁿ . It turns out that this estimate is remarkably accurate over most of the range of values oft. A reverse inequality, demonstrating this, is the second major result of the article. Supported in part by NSF DMS-9257020. Supported by a grant from Public Benefit Foundation Alexander S. Onassis. This work will form part of a Ph.D. thesis written by the second-named author.     

Control of stochastic distributed-parameter systems

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.Notation a(t) l-vector noisy dynamic input to system - A(t, a) l-vector function - A frequency factor for first-order rate law (5.68×10⁶ sec^–1) - b distance to the centerline between two coil banks in the reactor (4.7 cm) - B k-vector function defining the control action - c(, ) concentration of styrene monomer, molel ^–1 - C _p heat capacity (0.43 cal · g^–1 · K^–1) - C _ij constants in approximate filter, Eqs. (49)–(52) - E activation energy (20330 cal · mole^–1) - expectation operator - f(t,...) n-vector function - g ₀,g ₁(t,...) n-vector functions - h(t, u) m-vector function relating observations to states - H(t) function defined in Eq. (36) - k dimensionality of control vectorv(x, t) - k _i constants in approximate filter, Eqs. (49)–(52) - K dimensionless proportional gain - l dimensionality of dynamic inputa(t) - m dimensionality of observation vectory(t) - n dimensionality of state vectoru(x, t) - P ^(vv)(x, s, t) n×n matrix governed by Eq. (9) - P ^(va)(x, t) n×l matrix governed by Eq. (10) - P ^(aa)(t) l×l matrix governed by Eq. (11) - q _i (t) diagonal elements ofm×m matrixQ(x, s, t) - Q(x, s, t) m×m weighting matrix - R universal gas constant (1.987 cal · mole^–1 · K^–1) - R(x, s, t) n×n weighting matrix - R _i (t) n×n weighting matrix - s dimensionless spatial variable - S(x, s, t) matrix defined in Eq. (11) - t dimensionless time variable - T(, ) temperature, K - u(x, t) n-dimensional state vector - u _c (t) wall temperature - u _d desired value ofu ₁(1,t) - u _c * reference control value ofu _c - u _c ^max maximum value ofu _c - u _c ^min minimum value of _c - v(x, t) k-dimensional control vector - W(t) l×l weighting matrix - x dimensionless spatial variable - y(t) m-dimensional observation vector - _i constants in approximate filter, Eqs. (49)–(52) - dimensionless parameter defined in Eq. (29) - H heat of reaction (17500 cal · mole^–1) - dimensionless activation energy, defined in Eq. (29) - (x) Dirac delta function - (x, t) m-dimensional observation noise - thermal conductivity (0.43×10^–3 cal · cm^–1 · sec^–1 · K^–1) - density (1 g · cm^–3) - time, sec - dimensionless parameter defined in Eq. (29) - spatial variable, cm - * reference value - ^ estimated value     

HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

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Precise Asymptotic Formulas for Semilinear Eigenvalue Problems

. We consider the nonlinear Sturm-Liouville problem¶¶-u"(t) = | u(t) | ^p-1u(t) - lu(t), t ? I :=(0,1), u(0) = u(1) = 0 -u'(t) = \mid u(t)\mid^{p-1}u(t) - \lambda u(t), t \in I :=(0,1), u(0) = u(1) = 0 ,¶¶ where p > 1 and l ? R \lambda \in {\bf R} is an eigenvalue parameter. To investigate the global L²-bifurcation phenomena, we establish asymptotic formulas for the n-th bifurcation branch l = l_n (a) \lambda = \lambda_n (\alpha) with precise remainder term, where a \alpha is the L² norm of the eigenfunction associated with l \lambda .     

Elliptic operators with general Wentzell boundary conditions,analytic semigroups and the angle concavity theorem

We prove a very general form of the Angle Concavity Theorem, which says that if (T (t)) defines a one parameter semigroup acting over various L^p spaces (over a fixed measure space), which is analytic in a sector of opening angle θ_p, then the maximal choice for θ_p is a concave function of 1 – 1/p. This and related results are applied to give improved estimates on the optimal L^p angle of ellipticity for a parabolic equation of the form ?u /?t = Au, where A is a uniformly elliptic second order partial differential operator with Wentzell or dynamic boundary conditions. Similar results are obtained for the higher order equation ?u /?t = (–1)^{m +l}A^mu, for all positive integers m.     

Homoclinic Orbits of Nonperiodic Super Quadratic Hamiltonian System

This paper is concerned with solutions of the Hamiltonian system: [(u)\dot]=JH_u(t,u)\dot{u}=\mathcal{J}H_{u}(t,u) , where H(t,u)=\frac12u·Lu+W(t,u)H(t,u)=\frac{1}{2}u\cdot Lu+W(t,u) with L being a 2N×2N symmetric matrix and W∈C ¹(ℝ×ℝ^2N ,ℝ) being super quadratic at infinity in u∈ℝ^2N . We use variational methods to study this problem. By virtue of some auxiliary system related to the “limit equation” of the Hamiltonian system, we constructed linking levels of the variational functional Φ such that the minimax value E _l based on the linking structure of Φ satisfies 0 < E_l < E_l00E_l0E_{l_{0}} is the least action of the “limit equation”. Thus we can show the (PS) _c -condition holds true for all c < E_l0c相似文献