Extending n times differentiable functions of several variables

It is shown that n times Peano differentiable functions defined on a closed subset of and satisfying a certain condition on that set can be extended to n times Peano differentiable functions defined on if and only if the nth order Peano derivatives are Baire class one functions.     

Zur Struktur von Quadraturformeln

Summary Usually the errorR _n (j) of a quadrature formula is estimated with the aid of theL ₁-norm of the Peano kernel. It is shown that this term may be estimated rather sharp using the norm Q _n of the quadrature rule. Then it follows that formulas with non-negative weights are favourable also in the sense of minimizing theL ₁-norm of the kernel. A remainder term of the typeR _n (f)=cf⁽ⁿ⁺¹⁾ () is possible iff the kernel is definite. In the case of an interpolatory formula this definiteness is usually shown by an application of the so-called V-method. We determine the optimal formulas in the sense of this method. Then we analyse the influence of the structure of the mesh on the norm of a formula. We find that on an equidistant mesh withm nodes there exists a rule with a small norm if the order is not greater than .     

A nonlinear equation for linear programming

We present a characterization of the normal optimal solution of the linear program given in canonical form max{c ^tx: Ax = b, x 0}. (P) We show thatx ^* is the optimal solution of (P), of minimal norm, if and only if there exists anR > 0 such that, for eachr R, we havex ^* = (rc – A^t_r)₊. Thus, we can findx ^* by solving the following equation for _r A(rc – A^t_r)₊ = b. Moreover,(1/r) _r then converges to a solution of the dual program.On leave from The University of Alberta, Edmonton, Canada. Research partially supported by the National Science and Engineering Research Council of Canada.     

Definable sets and expansions of models of Peano arithmetic

We consider expansions of models of Peano arithmetic to models ofA ₂ ^s -¦ ₁ ¹ + ₁ ¹ –AC which consist of families of sets definable by nonstandard formulas.     

The Resolution Method for One Reducible Class of Formulas of the First-Order Modal Logic S4

In the paper, we describe a resolution method for the family of the formulas of the form *ⁱ**(L ₁ L _s ), where i = 0 1 and L _j are modal literals. Negations here stand directly before classical atomic formulas, the formulas may contain constants. We also present the absorption tactics for a set of such formulas.__________Translated from Lietuvos Matematikos Rinkinys, Vol. 44, No. 4, pp. 481–492, October–December, 2004.     

Absolutely continuous invariant measures forC 2 unimodal maps satisfying the Collet-Eckmann conditions

Summary In this paper we show that unimodal mappingsf[0, 1][0, 1] have absolutely continuous measures of positive entropy if these maps areC ² and satisfy the so-called Collet-Eckmann conditions. No conditions on the Schwarzian derivative off are assumed.     

Ergodic properties of semi-Markovian operators on the Z1-part

Summary In this note we consider a semi-Markovian operator, that is a positive linear mapping T: L ¹ L ¹ such that sup T ⁿ <. We study the behavior of T ⁿ on the Z ¹-part of the space (the disappearing part in Sucheston's terminology). We show in particular, that if the operator T has a non-trivial conservative part in Z ¹, then the ratio theorem must fail.Research supported by the U.S.Army Research Office (Durham) under contract DA-31-124-ARO(D)-288.     

On Localization and Stabilization for Factorization Systems

If (, M)is a factorization system on a category C, we define new classes of maps as follows: a map f:AB is in if each of its pullbacks lies in (that is, if it is stably in ), and is in M ^* if some pullback of it along an effective descent map lies in M(that is, if it is locally in M). We find necessary and sufficient conditions for (, M ^*) to be another factorization system, and show that a number of interesting factorization systems arise in this way. We further make the connexion with Galois theory, where M ^*is the class of coverings; and include self-contained modern accounts of factorization systems, descent theory, and Galois theory.     

Peano Corto and Peano Basso: A Study of Local Induction in the Context of Weak Theories

In this paper we study local induction w.r.t. Σ₁‐formulas over the weak arithmetic . The local induction scheme, which was introduced in 7 , says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non‐empty virtual class that is definable by a Σ₁‐formula without parameters, the intersection of and is non‐empty. In other words, we have, for all Σ₁‐sentences S, that S implies , whenever is progressive. Since, in the weak context, we have (at least) two definitions of Σ₁, we obtain two minimal theories of local induction w.r.t. Σ₁‐formulas, which we call Peano Corto and Peano Basso. In the paper we give careful definitions of Peano Corto and Peano Basso. We establish their naturalness both by giving a model theoretic characterization and by providing an equivalent formulation in terms of a sentential reflection scheme. The theories Peano Corto and Peano Basso occupy a salient place among the sequential theories on the boundary between weak and strong theories. They bring together a powerful collection of principles that is locally interpretable in . Moreover, they have an important role as examples of various phenomena in the metamathematics of arithmetical (and, more generally, sequential) theories. We illustrate this by studying their behavior w.r.t. interpretability, model interpretability and local interpretability. In many ways the theories are more like Peano arithmetic or Zermelo Fraenkel set theory, than like finitely axiomatized theories as Elementary Arithmetic, and . On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut‐interpretable in . On the other hand, they behave as if they were strong: they are not contained in any consistent finitely axiomatized arithmetical theory, however strong. Moreover, they extend , the theory of parameter‐free Π₁‐induction.     

More about signatures and approximation

Consider a non-singular real algebraic varietyM together with a codimension 1 real algebraic setY M. SupposeY=^–1(0) for a smooth function :M and denote by the signature induced by onM–Y. The following results are proved.For compactM, is induced by a regular functionf R(M) if and only if the setY ^c, where changes sign, is the union of the (d–1)-dimensional parts of some irreducible components ofY if and only if can be approximated by regular functions with the same zero-set. For non-compactM this is true only ifR(M) is a factorial ring. Similar results are proved whenM andY are real analytic instead of algebraic.Dedicated to the memory of our friend Mario RaimondoThe authors are members of GNSAGA of CNR. This work is partially supported by MURST.     

Ramsey families which exclude a graph

For graphsA andB the relationA(B) _r ¹ means that for everyr-coloring of the vertices ofA there is a monochromatic copy ofB inA. Forb (G) is the family of graphs which do not embedG. A familyof graphs is Ramsey if for all graphsBthere is a graphAsuch thatA(B) _r ¹ . The only graphsG for which it is not known whether Forb (G) is Ramsey are graphs which have a cutpoint adjacent to every other vertex except one. In this paper we prove for a large subclass of those graphsG, that Forb (G) does not have the Ramsey property.This research has been supported in part by NSERC grant 69-1325.     

The Rate of Convergence for Subexponential Distributions and Densities

A distribution function F on the nonnegative real line is called subexponential if lim_x(1-F ^*n (x)/(1 - F(x)) = n for all n 2, where F ^*n denotes the nfold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term R _n (x) defined by R _n (x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue r_n (x) = -(R_n (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form _n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ₀ ^x (1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form R _n(x)= ₂ ⁿ R₂(x) + O(1)(-f'(x))R²(x) where f' is the derivative of f.     

Saturation of Kantorovich type operators

It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL ¹ metric in the optimal order {n ^–1} if and only if ² f is of bounded variation where and , respectively.     

Some Explicit Distributions Related to the First Exit Time from a Bounded Interval for Certain Functionals of Brownian Motion

Let (B _t ) _{t≥ 0} be standard Brownian motion starting at y and set X _t = for , with V(y) = y ^γ if y≥ 0, V(y) = −K(−y)^γ if y≤ 0, where γ and K are some given positive constants. Set . In this paper, we provide some formulas for the probability distribution of the random variable as well as for the probability (or b)}. The formulas corresponding to the particular cases x = a or b are explicitly expressed by means of hypergeometric functions.      

Quadrature formulas with positive weights

Recent developments in the theory of stability or contractivity of numerical methods for solving ordinary differential equations (see for instance [4], [5], [8]) have renewed the interest for the study of quadrature formulas with positive weights. Nørsett-Wanner [8] and Burrage [2], [3] have given characterisation of such quadrature formulas of order 2m–2 or 2m–3. In this paper we extend these investigations to the case of formulas of order 2m–4 and then to the case where the order is 2m–7. Finally we use these results to characterise the algebraically stable methods out of a 12-parameter family of implicit Runge-Kutta methods of order 2m–4.     

Automatic continuity over Moore groups

LetG be a Moore group, letB be a Banach algebra, and let :L ¹(G)B be a homomorphism. We show that is continuous if and only if its restriction to the center ofL ¹(G) is continuous. As a consequence, we obtain that (i) every homomorphism fromL ¹(G) orC ^*(G) onto a dense subalgebra of a semisimple Banach algebra, and (ii) every epimorphism fromC ^*(G) onto a Banach algebra is automatically continuous.     

Optimal control problems that test for envelope contacts

Systems are studied whose state vector x is governed by the usual set of first-order differential equations. When the extremals x(t) that originate at a fixed point inn-dimensional state-variable space are stopped at a fixed final time, the locus of their endpoints defines a hypersurface called the wavefront. The well-known adjoint vector is normal to the wavefront. The principal point of this paper is that a second wavefront normal can be constructed from the x(t) vectors that are available if the test for Jacobi conjugate points is performed. Verifying that the two normals are almost collinear shows that the errors due to computer truncation and numerical integration are negligible. This check is particularly useful when using the finite-difference approximation x(t) x ⁱ (t) – x ^j (t), where x ⁱ (t) and x ^j (t) are close but nonneighboring extremals. This approximation can simplify considerably the analysis and computation required for a conjugate-point test, particularly if the extremals have corners.     

Empirical Bayes Procedures for Selecting the Best Population with Multiple Criteria

Consider k (k 2) populations whose mean _i and variance _i ² are all unknown. For given control values ₀ and ₀ ² , we are interested in selecting some population whose mean is the largest in the qualified subset in which each mean is larger than or equal to ₀ and whose variance is less than or equal to ₀ ² . In this paper we focus on the normal populations in details. However, the analogous method can be applied for the cases other than normal in some situations. A Bayes approach is set up and an empirical Bayes procedure is proposed which has been shown to be asymptotically optimal with convergence rate of order O(ln² n/n). A simulation study is carried out for the performance of the proposed procedure and it is found satisfactory.     

On some additive mappings in rings with involution

Summary LetR be a *-ring. We study an additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) for allx R.It is shown that, in caseR contains the unit element, the element 1/2, and an invertible skew-hermitian element which lies in the center ofR, then there exists ana R such thatD(x) = ax ^* – xa for allx R. IfR is a noncommutative prime real algebra, thenD is linear. In our main result we prove that a noncommutative prime ring of characteristic different from 2 is normal (i.e.xx ^* =x ^* x for allx R) if and only if there exists a nonzero additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) and [D(x), x] = 0 for allx R. This result is in the spirit of the well-known theorem of E. Posner, which states that the existence of a nonzero derivationD on a prime ringR, such that [D(x), x] lies in the center ofR for allx R, forcesR to be commutative.