Recurrence-Transience for Self-similar Additive Processes Associated with Stable Distributions

For any stable distribution on the line, recurrence-transience of the selfsimilar additive process {X _t ,t0} with (X ₁)= is determined. Comparison with the stable Lévy process {Y _t ,t0} with (Y ₁)= is made: if is not strictly stable, then {Y _t } is transient but {X _t } is recurrent except the obviously transient case of monotone sample functions.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Isotone solutions of parametric linear complementarity problems

The parametric linear complementarity problem is given by the conditions:q + p + Mz 0, 0,z 0,z ^T (q + p + Mz) = 0. Under the assumption thatM is a P-matrix, Cottle proved that the solution mapz() of the above problem is montonically nondecreasing in the parameter for every nonnegativeq and everyp if and only ifM is a Minkowski matrix. This paper examines whether a similar result holds in various other settings including a nonlinear case.     

An n-Dimensional Hahn-Banach Extension Theorem and Minimal Projections

Let T~=_i=1 ⁿ _irv_i:V V=[v₁,. . . .,v_n] X, where _i V* and X is a Banach space. Let T= _i=1 ⁿu_iv_i: X V be an extension of T~ to all of X (i.e., u_i X*) such that T has minimal (operator) norm. (E.g., if T~=I, T is a minimal projection from X onto V.) Then it is necessary and sufficient that u:=u_1,. . . ,u_n is given by (v:=v₁,. . . ,v_n)ext_v(u) Vⁿ,where the notion of a v-extremal (ext_v) of u is properly defined.The condition above leads in many important cases to a simple geometric interpretation of minimal projections. Furthermore, by applying this formula to the case X=L^p, we obtain a linear n-dimensional analog of the Hölder equality condition (M is given by ext_v(u)=Mv)1/p u · Mv = 1/q u · Mv,wherever v is differentiable.We point out several applications, including the determination of the absolute projection constant of _n ^p      

On well-ordered mono-unary algebras

Given a linearly ordered set (A, _R ) and an R-monotone function f: AA, we give a necessary and sufficient condition on A, f, _R , involving generating sets and forbidden subalgebras, for _R to be a well-ordering.Partially supported by Hungarian National Foundation for Scientific Research Grant nr. 1813.     

Asymptotic stability at infinity of planar vector fields

Let >0 andX be aC ¹ vector field on the plane such that: (i) for allq², Det(DX(q))>0; and (ii) for allp², with p, Trace(D(X(p))<0. IfX has a singularity and ² Trace(DX)dxdy is less than 0 (resp. greater or equal than 0), then the point at infinity of the Riemann sphere ²{} is a repellor (resp. an attractor) ofX.     

D λ -cycles in 3-cyclable graphs

Letk and be positive integers, andG a 2-connected graph of ordern with minimum degree and independence number. A cycleC ofG is called aD -cycle if every component ofG – V(C) has order smaller than. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that if k 2, G isk-connected and every connected subgraphH ofG of order has at leastn +k ² + 1/k + 1 – vertices outsideH adjacent to at least one vertex ofH, thenG contains aD -cycle. Here it is conjectured that k-connected can be replaced by k-cyclable, and this is proved fork = 3. As a consequence it is shown that ifn 4 – 6, or ifG is triangle-free andn 8 – 10, thenG contains aD ₃-cycle orG , where denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn 4 – 6 and – 1, thenG is hamiltonian orG . The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are close to claw-free.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Finite Representability of ℓp in Quotients of Banach Spaces

A typical result of the paper states that if X is a Banach space with a basis and for some 1pq, the spaces _p and _q are finitely block representable in every block subspace of X, then every block subspace of X admits a block quotient Z such that for every r[p,q], the space _r is finitely block representable in Z. Results of a similar nature are also established for ^N _p-block-sequences and asymptotic spaces.     

On the radicals of structural matrix rings

The relationship between the radical of a ringR and a structural matrix ring overR has been determined for some radicals. We continue these investigations, amongst others, determining exactly which radicals have the property (M(,R))=M( _s ,(R))+M( _a ,⁺(R))for any structural matrix ringM(,R) and finding (M(,R)) for any hereditary subidempotent radical .     

Parametrices for a class of differential operators with multiple characteristics

Summary In this paper we give the construction of a parametrix for a class of differential operators of the type ²/t²–t^2k+1 + t^q(/x_i), k N, q N, qk.     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n ) of the best L ₁-approximations of classes of periodic functions by splines s S _{2n, r – 1} and s S _{2n, r + k – 1} (S _{2n, r} is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

Compact groups on compact projective planes

LetP=(P, L) be a compact projective plane with 0P< and let be a compact connected subgroup of Aut(P). If dim dimE – dimP, whereE is the elliptic motion group of the corresponding classical plane, then E or is isomorphic to a point stabilizerE ₀ inE, cf. [31]. Here we consider the case E ₀. It is shown that the action of on the point spaceP is equivalent to the classical action ofE ₀. For dimP {8, 16} the planeP is uniquely determined by a 2-dimensional subplane with SO₂ Aut().Für H. Reiner Salzmann zum 65. Geburtstag     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

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Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.

     

Regular (2, 2)-systems

Let (E,I) be an independence system over the finite setE = {e ₁, ,e _n }, whose elements are orderede ₁ e _n . (E,I) is called regular, if the independence of {e _l , ,e _{l k} },l ₁ < <l _k , implies that of {e _{m l} , ,e _{m k} }, wherem _l < ··· <m _k andl ₁ m ₁, ,l _k m _k . (E,I) is called a 2-system, if for anyI I,e E I the setI {e } contains at most 2 distinct circuitsC, C I and the number 2 is minimal with respect to this property. If, in addition, for any two independent setsI andJ the family (C J, C C (J, I)), whereC(J, I) denotes {C C:e J I C {e}}, can be partitioned into 2 subfamilies each of which possesses a transversal, then (E,I) is called a (2, 2)-system. In this paper we characterize regular 2-systems and we show that the classes of regular 2-systems resp. regular (2, 2)-systems are identical.     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.