The problem of conformal transformations of a circle into nonoverlapping regions

Let a, a0, a, be a fixed point in the z-plane, (a, 0, ), the class of all systemsf _k()_l ³ of functions z=f ^k(), k=1, 2, 3, of which the first two map conformally and in a s ingle-sheeted manner the circle ¦¦<1, and the third maps in a similar manner the region ¦¦>1, into pair-wise nonintersecting regions B_k, k=1, 2, 3, containing the points a, 0, and , respectively, so thatf ₁(0)=a,f ₂(0)=0 andf ₃()=. The region of values (a, 0, ) of the system M(¦f ₁'(0)¦, ¦f ₂'(0)¦, 1/¦f ₃'()¦) in the class (a, 0, ) is determined.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 417–424, October, 1969.     

Minimization of the maximal penalty in the case of preemption of jobs

For a class of structural sets of penalty functions={_i} _i=1 ⁿ with lower quasiconvex functions _i defined for sets of jobs={_i} _i=1 ⁿ , one gives an algorithm for solving the problem n /1/ preemp ¦ max, having order 0(np), where n is the number of jobs _i and p is the total length of the completion of all jobs of the set.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 102, pp. 61–67, 1980.In conclusion, the author expresses her gratitude to K. V. Shakhbazyan for his interest in this paper.     

Inequalities for the distribution of a sum of functions of independent random variables

Let where ₁,..., n are independent random variables and the are functions (e.g., taking the values 0 and 1). For cases when almost all the summands forming are equal to 0 with a probability close to 1, estimates from above and below are obtained for the quantity P{=0}, as well as upper estimates for the distance in variation between the distribution , and the distribution of the approximating sum of independent random variables.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 745–758, November, 1977.The author is grateful to V. G. Mikhailov for numerous discussions of the results of this paper and for his help in carrying out the tedious auxiliary calculations.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

Arithmetic properties of the Markov function for the Jacobi weight

. . .     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.     

Pointwise convergence of multiple Fourier series using different convergence notions

, . , L ^p (²) p>1 . , C(T³), - , . - .

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This work was completed while the first named author was a visiting professor at Indiana University, Bloomington, Indiana; and the second named author was a visiting professor at Ohio State University, Columbus, Ohio, U.S.A.     

On the strongly spherical Sasakian metric of a spherical tangent bundle

Let Mⁿ denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mⁿ there exists a -dimensional subspace _Q T_QMⁿ such that the curvature operator of the metric of Mⁿ satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y _Q , X, Z #x2208; T_QMⁿ. The number is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T₁Mⁿ be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M² has constant Gaussian curvature K 1 and k = K²/4; b) = 3 if and only if M² has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T₁Mⁿ (n Mⁿ) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 0. Let us denote by (Mⁿ, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T₁(Mⁿ, K) (n 3) be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.     

Limit distribution of the last exit time for stationary random sequences

Summary For a strictly stationary random sequence (X _i) _i0 we find sufficient conditions such that the distribution of the last exit time t = max{i X _i>i} (>0) tends weakly to a nondegenerate limit distribution as 0.     

An algorithm for constructing an optimal structural schedule

The article is devoted to the problem of finding an optimal schedule for a class of functionals ƒ which allows for the existence of a structural set of activities. The functionalƒ(R), where, is defined in the following way: where {_i(t)} is a structural set of functions, and the function F is defined on any finite set of arguments and satisfies the following conditions: 1)F(x)=(x); 2) F(x₁,x₂)=(x₁,x₂), F(x₁,x₂,...x₃)= (x₁, F(x₂,...,x_s)), S2; 3) and do not decrease in each of their arguments, and moreover, 3a) strictly increases with the increase of both arguments, 3b) if (x₁,x₂)>(x₁, x₂ (x₂, x₃)> (x₂,x₃), then F(x₁,x₂,x₃)>F(x₁,x₂,x₃).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 124, pp. 5–20, 1983.     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

Sugli insiemi di permutazioni strettamente k-transitivi (k ≥ 3)

Summary Let G be a sharply 3-transitive permutation set on a finite set E of even cardinality and let 1 be in G. The following theorems are proved. G is one of the known examples if and only if there exists a non-identity normal subgroup N of G and an element of E such that NG G.G is a group if and only if G for every G and for every G and for every G .By using the classification of finite single groups a result concerning sharply k-transitive permutation sets k>3 is also proved.

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Dedicato a Guido Zappa in occasione del suo 70° compleanno

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Lavoro eseguito nell'ambito dei progetti finanziati dal Ministero della Pubblica Istruzione.     

Quasifractional Approximants in the Theory of Composite Materials

We study the effective heat conductivity of regular arrays of perfectly conducting spheres embedded in a matrix with unit conductivity. Quasifractional approximants allow us to derive an approximate analytical solution, valid for all values of the spheres volume fraction [0, _max] (_max is the maximum volume fraction of a spheres). As a starting point we use a perturbation approach for 0 and an asymptotic solution for _max. Three different spatial arrangements of the spheres, simple cubic, body centred and face centred cubic arrays, are considered. Results obtained give a good agreement with numerical data.     

Infinitesimal,first order bendings of smooth,convex surfaces of revolution subject to conic,sleeve-like constraints along the boundary

We prove that on a closed, smooth, convex surface of revolution , whose poles are not flattening points, there exists only a countable set of parallels _n. Each of these parallels cuts surface into two parts so that one of the parts, , admits nontrivial, infinitesimal bendings in the process of which all the points of its boundary _n are displaced on a preassigned, conic sleeve K that is coaxial with the surface. The sequence of such parallels _n converges to parallel ^*, which has the following properties: 1) the tangent cone to surface along ^* is orthogonal to sleeve K; 2) surface , cut off from surface by parallel ^*, has rigidity of first order in the indicated class of bendings.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 3–8, 1990.     

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

Summary We describe a large class of one-parameter families , {}, , of two-dimensional diffeomorphisms which arestable for <0, exhibit acycle for =0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals [0, ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.The second author acknowledges hospitality and financial support from IMPA/CNPq during the period this paper was prepared     

Stabilization of ill-posed Cauchy problems for parabolic equations

Summary In this paper we study the noncharacteristic Cauchy problem, u_t–(a(x)u_x)_x=0, x(0, l), t.(0, T], u(0, t)=(t), u_x(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.

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Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto u_t= (a(x)u_x)_x, x(0, l), t(0, T), u(0, t)=(t), u_x(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

On the limit superior of analytic sets

. , A ₀,A ₁,— - lim supA _j - H, . , - - . , , ; , , . - . - .     

A note on rational approximation rate for Müntz system {x λn } with λ n ↘ 0

[Zho2] {x ⁿ } , _n 0 n .

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Supported in part by an NSERC Postdoctoral Fellowship and a CRF grant of University of Alberta.     

Oscillating processes with independent increments and nondegenerate Wiener component

For an oscillating process _z(t) (_z(0)=2,t0), which is defined with the help of two homogeneous processes ₁(t) and ₂(t) with independent increments and nondegemerate Wiener components, under certain restrictions we establish a relation of the form and find the characteristic function of the ergodic distribution of the process considered.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1415–1421, October, 1990.