Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

A Bound for Solutions of a Fourth-order Dynamical System

We consider the homogeneous linearized equation of yawing motionof a spinning projectile ⁿ+A)s)' + B(s) = 0, where A and B are complex functions of the real independentvariable s. It is shown that where K is a positive constant and v(s) is a real function ofs.     

Bounds for the Singular Values of a Matrix

A lower-bound theorem is developed for the singular values ofa matrix A (and therefore for the eigenvalues of B = A^HA). Itis found that there is always a unique column scaling of A whichproduces the optimum bound. However, sharper bounds still maysometimes be obtained by taking advantage of matrix partitioning.It is shown that the resulting bounds may often (but not always)be better than those obtained by applying Gerschgorin's theoremto B. The equivalent upper-bound theorem is found to be weak.     

Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k 3 be an integer. For 0<s<1, let D_s R² be the setthat is constructed iteratively as follows. Take a regular openk-gon with sides of unit length, attach regular open k-gonswith sides of length s to the middles of the edges, and so on.At each stage of the iteration the k-gons that are added area factor s smaller than the previous generation and are attachedto the outer edges of the family grown so far. The set D_s isdefined to be the interior of the closure of the union of allthe k-gons. It is easy to see that there must exist some s_k> 0 such that no k-gons overlap if and only if 0 < s s_k. We derive an explicit formula for s_k. The set D_s is open, bounded, connected and has a fractal polygonalboundary. Let denote the heat content of D_s at time t when D_s initially has temperature 0and D_s is kept at temperature 1. We derive the complete short-timeexpansion of up to terms that are exponentially small in 1/t. It turns out that there arethree regimes, corresponding to 0<s<1/(k–1), s=1/(k–1),and 1/(k–1)<s s_k. For s 1/(k–1) the expansionhas the form where p_s is a log (1/s²)-periodic function, d_s=log (k–1)/log(1/s) is a similarity dimension, A_s and B are constants relatedto the edges and vertices, respectively, of D_s, and r_s is anerror exponent. For s=1/(k–1), the t^1/2-term carries anadditional log t. 1991 Mathematics Subject Classification: 11D25,11G05, 14G05.     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

Base sizes for simple groups and a conjecture of Cameron

Let G be a permutation group on a finite set . A base for Gis a subset B with pointwise stabilizer in G that is trivial;we write b(G) for the smallest size of a base for G. In thispaper we prove that b(G) 6 if G is an almost simple group ofexceptional Lie type and is a primitive faithful G-set. Animportant consequence of this result, when combined with otherrecent work, is that b(G) 7 for any almost simple group G ina non-standard action, proving a conjecture of Cameron. Theproof is probabilistic and uses bounds on fixed point ratios.     

Holder Continuity of the Dirichlet Solution for a General Domain

Let D be a bounded domain in Rⁿ. For a function f on the boundaryD, the Dirichlet solution of f over D is denoted by H_Df, providedthat such a solution exists. Conditions on D for H_D to transforma Hölder continuous function on D to a Hölder continuousfunction on D with the same Hölder exponent are studied.In particular, it is demonstrated here that there is no boundeddomain that preserves the Hölder continuity with exponent1. It is also also proved that a bounded regular domain D preservesthe Hölder continuity with some exponent , 0<<1,if and only if D satisfies the capacity density condition, whichis equivalent to the uniform perfectness of D if n = 2. 2000Mathematics Subject Classification 31A05, 31A20, 31B05, 31B25.     

Superconvergence of a Collocation-type Method for Hummerstein Equations

This paper considers the numerical solution of Hammerstein equationsof the form by a collocation method applied not to this equation, but ratherto an equivalent equation for z(t) :=g(t, y(t)). The desiredapproximation to y is then obtained by use of the (exact) equation In an earlier paper, questions of existence and optimal convergenceof the respective approximations to z and y were examined. Inthis sequel, collocation approximations to z are sought in certainpiecewise polynomial function spaces, and analogous of knownsuperconvergence results for the iterated collocation solutionof (linear) second-kind Fredhoim integral equations are statedand proved for the approximation to y.     

A Comparison Theorem for a Second Order Linear Quasi-Differential Equation

We are concerned with the non-oscillatory behaviour of a secondorder linear quasi-differential equation defined a.e. over ahalf-line. More particularly given that (py')' + qy = 0 is non-oscillatorywe show that by writing q in a form which is amenable to furtheranalysis we can generalise Sturm's comparison theorem.     

A multigrid-type method for thin plate spline interpolation on a circle

We consider the problem of thin plate spline interpolation ton equally spaced points on a circle, where the number of datapoints is sufficiently large for work of O(n³ to be unacceptable.We develop an iterative multigrid-type method, each iterationcomprising ngrid stages, and n being an integer multiple of2^ngrid–1. We let the first grid, V₁ be the full set ofdata points, V say, and each subsequent (coarser) grid, V_k,k=2, 3,...,ngrid, contain exactly half of the data points ofthe preceding (finer) grid, these data points being equallyspaced. At each stage of the iteration, we correct our current approximationto the thin plate spline interpolant by an estimate of the interpolantto the current residuals on V_k, where the correction is constructedfrom Lagrange functions of interpolation on small local subsetsof p data points in V_k. When the coarsest grid is reached, however,then the interpolation problem is solved exactly on its q=n/2^ngrid–1points. The iterative process continues until the maximum residualdoes not exceed a specified tolerance. Each iteration has the effect of premultiplying the vector ofresiduals by an n x n matrix R, and thus convergence will dependupon the spectral radius, (R), of this matrix. We investigatethe dependence of the spectral radius on the values of n, p,and q. In all the cases we have considered, we find (R) <<1, and thus rapid convergence is assured.     

A Necessary Condition for Transitivity of a Finite Permutation Group

Suppose the group G is generated by permutations g₁, g₂, ...,g₈ acting on a set of size n, such that g₁g₂...g₈ is the identitypermutation. If the generator g_i has exactly c_i cycles (for1 i s), and G is transitive on , then n(s–2)– is a non-negative even integer. Thisis proved using an elementary graph-theoretic argument.     

Viscoelastic boundary stabilization for a transmission problem in elasticity

We consider an anisotropic body which is constituted of twodifferent types of materials supporting a memory boundary conditionand we show that its energy decays uniformly as time goes toinfinity with the same rate as the relaxation function g, thatis, the energy decays exponentially when g decays exponentially,and polynomially when g decays polynomially.     

A root-distribution criterion for an interval polynomial in a sector

A simple test is proposed for checking whether a given intervalpolynomial of degree n has robust root distribution with respectto a given symmetric sector in the complex plane. The test amountsto checking the real roots of at most 2n real polynomials withfixed coefficients, of degree n. The test is based on a detailedanalysis of the value set (s) as s travels along the sectorboundary. A simple algorithm for implementing this test, andan example, are also provided.     

An Optimal Krasnosel'skii-Type Theorem for the Dimension of the Kernel of a Starshaped Set

It is shown for a closed connected nonconvex and locally compactsubset S of a real normed linear space that ker S = {conv S_z: z D reg S}, where reg S denotes the set of regular pointsof S, D is a relatively open subset of S containing the setlnc S of local nonconvexity points of S, and S_z = {s S : zis visible from s via S}. An analogous intersection formula,with the set sph S of spherical points of S in place of regS, is shown to hold for a closed connected nonconvex set S ina real Banach space which is uniformly convex and uniformlysmooth. A routine procedure then leads to a Krasnosel'skii-typecharacterization of the dimension of the kernel of a closedconnected nonconvex subset S of R^d with lnc S bounded. Thisimproves a recent result by removing the hypothesis of compactnessof S and settles an open problem.     

Exponential stability for a contact problem in thermoelasticity

We consider the unilateral problem for the thermoelastic equationand we show that the solution decays exponentially to zero astime goes to infinity; that is, denoting by E(t) the first-orderenergy of the system, we show that positive constants C and exist which satisfy E(t)CE(0)e^–$$$.     

A Simple Proof for a Spectral Factorization Theorem

It is shown using simple methods that the transform Z(z), z C, of the coefficient sequence of the Wold decomposition ofany full-rank wide-sense stationary purely non-deterministicstochastic process satisfies (i) Z(z) H² (D) and (ii) Z^–1(z) H(D). Further it is shown that all spectral factors satisfying(i) and (ii) are equal up to right multiplication by orthogonalmatrices, and that among these the normalized (Z(0) =I) spectralfactors are equal to the transform of the Wold decomposition.An elementary proof of Youla's Theorem is then given togetherwith a simple proof that the rows of a Cholesky factor of abanded block Toeplitz matrix converge to the coefficients ofa stable matrix polynomial. Computer and Automation Institute of the Hungarian Academyof Sciences, Budapes, Hungary.     

On a Product Formula for Unitary Groups

For any nonnegative self-adjoint operators A and B in a separableHilbert space, the Trotter-type formula is shown to converge strongly in the norm closureof dom (A^1/2) dom (B^1/2 for some subsequence and for almostevery t R. This result extends to the degenerate case, andto Kato-functions following the method of T. Kato (see ‘Trotter'sproduct formula for an arbitrary pair of self-adjoint contractionsemigroup’, Topics in functional analysis (ed. M. Kac,Academic Press, New York, 1978) 185–195). Moreover, therestrictions on the convergence can be removed by consideringfunctions other than the exponential. 2000 Mathematics SubjectClassification 47D03 (primary), 47B25 (secondary).     

On a question of Sarkozy and Sos for bilinear forms

We prove that, if 2 k₁ k₂, then there is no infinite sequence of positive integers such that the representation functionr(n) = #{(a, a'): n = k₁a + k₂a', a, a' } is constant for nlarge enough. This result completes the previous work of Diracand Moser for the special case k₁ = 1 and answers a questionposed by Sárkozy and Sós.     

Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis

^** Email: saito{at}infsup.jp Finite-element approximation for a non-linear parabolic–ellipticsystem is considered. The system describes the aggregation ofslime moulds resulting from their chemotactic features and iscalled a simplified Keller–Segel system. Applying an upwindtechnique, first we present a finite-element scheme that satisfiesboth positivity and mass conservation properties. Consequently,if the triangulation is of acute type, our finite-element approximationpreserves the L¹ norm, which is an important property of theoriginal system. Then, under some assumptions on the regularityof a solution and on the triangulation, we establish error estimatesin L^p x W^1, with a suitable p > d, where d is the dimensionof a spatial domain. Our scheme is well suited for practicalcomputations. Some numerical examples that validate our theoreticalresults are also presented.