Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Detailed Error Analysis for a Fractional Adams Method

We investigate a method for the numerical solution of the nonlinear fractional differential equation D _* y(t)=f(t,y(t)), equipped with initial conditions y ^(k)(0)=y ₀ ^(k), k=0,1,...,–1. Here may be an arbitrary positive real number, and the differential operator is the Caputo derivative. The numerical method can be seen as a generalization of the classical one-step Adams–Bashforth–Moulton scheme for first-order equations. We give a detailed error analysis for this algorithm. This includes, in particular, error bounds under various types of assumptions on the equation. Asymptotic expansions for the error are also mentioned briefly. The latter may be used in connection with Richardson's extrapolation principle to obtain modified versions of the algorithm that exhibit faster convergence behaviour.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

Idempotent medialn-groupoids defined on fields

Fajtlowicz and Mycielski ([2]) showed that if we define an operationx·y=px+(1-p)x on the real numbersR for somep R, then the idempotent and medial laws form a basis for the equational theory of the groupoid (R,) if and only ifp is a transcendental number. In this paper, we generalize this ton-groupoids. Namely, if we define ann-ary operation [x ₁x₂h.x_n]=₁x₁ + ₂x₂+h.+(1-₁-h._n–1) onR for some ₁, ₂, h., _n–1 in R, then the idempotent and medial laws form a basis for the equational theory of then-groupoid (R, [ ]) if and only if ₁, ₂,..., _n–1 are algebraically independent.This is a part of Ph.D. dissertation developed under the direction of professor Trevor Evans while the author was at Emory University.     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

A generalization of the results of Pillai

In a recent article Pillai (1990,Ann. Inst. Statist. Math.,42, 157–161) showed that the distribution 1–E (–x ), 0<1; 0x, whereE (x) is the Mittag-Leffler function, is infinitely divisible and geometrically infinitely divisible. He also clarified the relation between this distribution and a stable distribution. In the present paper, we generalize his results by using Bernstein functions. In statistics, this generalization is important, because it gives a new characterization of geometrically infinitely divisible distributions with support in (0, ).     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Quasi-Symmetric Designs with Good Blocks and Intersection Number One

We show that a quasi-symmetric design with intersection numbers 1 and y > 1 and a good block belongs to one of three types: (a) it has the same parameters as PG ₂(4, q), the design of points and planes in projective 4-space; (b) it is the 2-(23, 7, 21) Witt design; (c) its parameters may be written v = 1 + (( – 1) + 1)(y – 1) and k = 1 + (y – 1), where is an integer and > y 5, and the design induced on a good block is a 2-(k, y, 1) design. No design of type (c) is known; moreover, for large ranges of the parameters, it cannot exist for arithmetic reasons concerning the parameters. We show also that PG ₂(4, q) is the only design of type (a) in which all blocks are good.     

Oscillation and non-oscillation theorems for superlinear Emden–Fowler equations of the fourth order

We study the oscillatory behavior of solutions of the fourth-order Emden–Fowler equation: (E) y^(iv)+q(t)|y|sgny=0, where >1 and q(t) is a positive continuous function on [t₀,), t₀>0. Our main results Theorem 2 – if (q(t)t^(3+5)/2)0, then equation (E) has oscillatory solutions; Theorem 3 – if lim_tq(t)t^4+(-1)=0, >0, then every solution y(t) of equation (E) is either non-oscillatory or satisfies limsup_tt^-+i|y⁽ⁱ⁾(t)|= for < and i=0,1,2,3,4. These results complement those given by Kura for equation (E) when q(t)<0 and provide analogues to the results of the second-order equation, y+q(t)|y|sgny=0,>1. Mathematics Subject Classification (2000) 34C10, 34C15     

An estimate for the region of regularity of solutions of certain nonlinear differential inequalities

In the diskx ²+y ²R ² of thex, y-plane we consider the differential inequalityz _xxz_yy–z _xy ² –(1+z _x ^/2 +z _y ^/2 )^k, where the constants >0 andk>1. In the case =1 andk=2 this inequality means that the surfacez(x, y) has Gaussian curvatureK1. Efimov has shown that in this case the radius of the disk has an upper bound. In the present article we establish an analogous upper bound for the radiusR of the disk in which the functionz(x, y) satisfies the differential inequality above.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 19–21.     

Extensions of Laguerre operators in indefinite inner product spaces

The Laguerre-Sonin polynomialsL _n ⁽⁾ are orthogonal in linear spaces with indefinite inner product if<–1. We construct the completion () of this space and describe self-adjoint extensions of the Laguerre operatorl(y)=xy+(1+–x)y,<–1, in the space (). In particular, we write out the self-adjoint extension of the Laguerre operator whose eigenfunctions coincide with the Laguerre-Sonin polynomials and form an orthogonal basis in ().Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 509–521, April, 1998.This research was partially supported by the INTAS foundation under grant No. 93-02449.     

Some extensions of the principles of idealization transfer and choice in the relative internal set theory

The results established in this paper are in connection with the Relative Internal Set Theory (R.I.S.T.). The main result is the general principle of choice: Let be a level and let (x, y) be anexternalbounded formula of the language of R.I.S.T.. Suppose that to each elementx, dominated by , corresponds an elementy _x such that (x, y _x ) holds, then there exists a function of choice such that, which is a very general principle of choice, for everyx dominated by , (x, (x)) holds. More than that, we establish that if all the elementsy _x are uniformly dominated by a level then we can prescribe that the function of choice is also dominated by .     

A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15     

A note on the small-time development of the solution to a scalar, non-linear, singular reaction-diffusion equation

In this note, we consider a class of scalar, non-linear, singular (in the sense that the reaction terms in the equation are not Lipschitz continuous) reaction-diffusion equations with positive initial data being of (a) O(x^–) or (b) O(x^–e^{– x}) at large x (dimensionless distance), where , > 0 and are constants. We establish, by developing the small–t (dimensionless time) asymptotic structure of the solution, that the support of the solution becomes finite in infinitesimal time in both cases (a) and (b) above. The asymptotic form for the location of the edge of the support as t 0 is given in both cases.Received: June 6, 2002; revised: May 6 and June 4, 2003     

On a Modification of the Jacobi Linear Functional: Asymptotic Properties and Zeros of the Corresponding Orthogonal Polynomials

The paper deals with orthogonal polynomials in the case where the orthogonality condition is related to semiclassical functionals. The polynomials that we discuss are a generalization of Jacobi polynomials and Jacobi-type polynomials. More precisely, we study some algebraic properties as well as the asymptotic behaviour of polynomials orthogonal with respect to the linear functional U U=J _,+A ₁(x–1)+B ₁(x+1)–A ₂(x–1)–B ₂(x+1), where J _, is the Jacobi linear functional, i.e. J _,,p›=_–1 ¹ p(x)(1–x)(1+x)dx,,>–1, pP, and P is the linear space of polynomials with complex coefficients. The asymptotic properties are analyzed in (–1,1) (inner asymptotics) and C[–1,1] (outer asymptotics) with respect to the behaviour of Jacobi polynomials. In a second step, we use the above results in order to obtain the location of zeros of such orthogonal polynomials. Notice that the linear functional U is a generalization of one studied by T. H. Koornwinder when A ₂=B ₂=0. From the point of view of rational approximation, the corresponding Markov function is a perturbation of the Jacobi–Markov function by a rational function with two double poles at ±1. The denominators of the [n–1/n] Padé approximants are our orthogonal polynomials.     

Lp-convergence for expansions in terms of the eigenfunctions of a Sturm — Liouville problem

For the operator L_v=–(x^2ay). x [0, 1], y(0)=y(1)=0 with 0 < 1/2, or ¦y¦ < , y(1)=0 with 1/2 <1, we investigate the effect which the singularity of the Sturm-Liouville operator derived from this self-adjoint expression has on L_p-convergence of expansions in terms of the eigenfunctions of this operator. We will prove that the orthonormalized system of eigenfunctions forms a basis in L_p [0, 1] for 2/(2–) < p < 2/.Translated from Matematicheskii Zametki, Vol. 3, No. 6, pp. 683–692, June, 1968.The author is grateful to V. M. Tikhomirov for his many valuable remarks and his constant attention to this work.     

Weak growth conditions for ODEs in pre-ordered Banach spaces

Let E be a pre-ordered real Banach space and f:[0,T)×EE a quasimontone increasing function. We prove one-sided estimates of the form ₊q[y–x,f(t,y)–f(t,x)](t,q(y–x)) with respect to seminorms q generated by a single positive linear functional. Such estimates lead to growth conditions, for example for the total variation of the solution of u=f(t,u) in function spaces.Mathematics Subject Classification (2000): 34C11, 34C12, 34G20     

A nonlocal boundary-value problem for the Euler-Darboux equation

We consider the boundary-value problem for the equation (x–y)U_xy-U_x+U_y=0,>0, >0, + <1, in the characteristic square and investigate its unique solvability when the boundary conditions contain generalized fractional integrodifferentiation operators.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 26–35, 1985.