Quasimonotonicity,regularity and duality for nonlinear systems of partial differential equations

Summary We prove partial regularity for the vector-valued differential forms solving the system (A(x, ))=0, d=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) (1+ + ¦¦²)^{(p – 2)/2} (p2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, )=Df(x, ) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1相似文献   

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

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Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

The embedding of linearly ordered sets

It is shown that if a linearly ordered set B does not contain as subsets sets of order type and ^* then B can be embedded in 2 . We construct an example of a set satisfying the above conditions which cannot be embedded in any 2 if < . Simultaneously we show that for any ordinal, 2 ⁺¹ cannot be embedded in 2 and that there exists at least ₊₁ distinct dense order types of cardinality 2 .Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 83–88, January, 1972.In conclusion, I wish to take the opportunity to thank Yu. L. Ershov for kindness and assistance in this work.     

Ornstein–Uhlenbeck Path Integral and Its Application

Introducing a path integral for the Ornstein–Uhlenbeck process distorted by a potential V(x), we find out the T limit of the probability distributions of X[]:=1/T ₀ ^T V((t))dt for Ornstein–Uhlenbeck process (t), with appropriate values of the exponent that depend on V. The results are compared with those for the Wiener process.     

1-formes fermées singulières et groupe fondamental

Summary We study the influence of the ₁ of a closed manifoldM ⁿ (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If ₁ M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on ₁ M and is related to finitely generated normal subgroups of ₁ M with quotient . If ₁ M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

A comparison theorem for the mean exit time from a domain in a Kähler manifold

Let M be a Kähler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary . The main result of this paper is a comparison theorem between the Mean Exit Time function defined on and the Mean Exit Time from a geodesic ball of the complex projective space ⁿ () which involves a characterization of the geodesic balls among the domain . In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of , using the Index Lemma for Submanifolds.Work partially supported by a DGICYT Grant No. PS87-0115-C03-01.     

Classifying countable Boolean terms

We deal with the Borel and difference hierarchies in the space P of all subsets of endowed with the Scott topology. (The spaces P and 2 coincide set-theoretically but differ topologically.) We look at the Wadge reducibility in P. The results obtained are applied to the problem of characterizing ₁-terms t which satisfy C = t( ₁ ⁰ ) for a given Borel-Wadge class C. We give its solution for some levels of the Wadge hierarchy, in particular, all levels of the Hausdorff difference hierarchy. Finally, we come up with a discussion of some relevant facts and open questions.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 173–197, March–April, 2005.     

On meromorphic solutions of algebraic differential equations in angular domains

We obtain asymptotic estimates of meromorphic solutions to the differential equationP _n (z, , )=P _n–1 (z, , ,..., ^(m) ) in the angular domain P={z: arg z · }. Here P_n(z, w, w) is a polynomial in all variables, and of degree n with respect to w and w; P_n–1(z, w, w, ..., w^(m)) is a polynomial in all variables, and of degree n –1 with respect to w, w, ..., w^(m) In the particular case, when the solutions are entire functions, these estimates are more precise than the known estimates that are obtained by using the method of Wiman-Valiron, which cannot be applied to meromorphic solutions in the domain P.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 514–523, April, 1992.     

Computation of efficient solutions of discretely distributed stochastic optimization problems

In engineering and economics often a certain vectorx of inputs or decisions must be chosen, subject to some constraints, such that the expected costs (or loss) arising from the deviation between the outputA() x of a stochastic linear systemxA()x and a desired stochastic target vectorb() are minimal. Hence, one has the following stochastic linear optimization problem minimizeF(x)=Eu(A()x b()) s.t.xD, (1) whereu is a convex loss function on ^m , (A(), b()) is a random (m,n + 1)-matrix, E denotes the expectation operator andD is a convex subset of ⁿ . Concrete problems of this type are e.g. stochastic linear programs with recourse, error minimization and optimal design problems, acid rain abatement methods, problems in scenario analysis and non-least square regression analysis.Solving (1), the loss functionu should be exactly known. However, in practice mostly there is some uncertainty in assigning appropriate penalty costs to the deviation between the outputA ()x and the targetb(). For finding in this situation solutions hedging against uncertainty a set of so-called efficient points of (1) is defined and a numerical procedure for determining these compromise solutions is derived. Several applications are discussed.     

On some variational problems for 2-dimensional Hermitian metrics

Summary Let (M, J, g) be a compact complex 2-dimensional Hermitian manifold with the Kähler form , and the torsion 1-form defined by d = . In this note we obtain the Euler-Lagrange equations for the variational functionals defined by ² and d², whereg runs in the space of all the Hermitian metrics onM. In the first case, the extremals are precisely the Kähler metrics [Gd]. In the second case, we also write down a formula for the second variation.Communicated by J. Szenthe     

Estimates for a Group of Deviations in Generalized Hölder Metric

We present order relations for a group of deviations of a function f(·) H in terms of partial Fourier sums of this function in a generalized Hölder metric defined in a generalized Hölder space H _* H .     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

Definability properties and the congruence closure

We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL (Th) or countably compact regular sublogic ofL (Th), properly extendingL , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies either-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic ofL (Th) in which Chang's quantifier or some cardinality quantifierQ , with 1, is definable.     

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Criteria of algebraic independence of complex numbers over a field of finite transcendence type (II)

In the present paper we establish a criterion of algebraic independence of complex numbers ₁, ..., _n over a field of finite transcendence type using a sequence of nonzero polynomials in several variables with integral coefficients, which satisfy simultaneously certain upper and lower estimates in different orders of magnitude at the point ( ₁, ..., _q , ₁, ..., _n ), where { ₁, ..., _q } is a transcendence basis of over .The Project Supported by the National Natural Science Foundation of China     

Some topological properties of ω-covering sets

We prove the following theorems:1. There exists an -covering with the property s ₀.2. Under cov there exists X such that is not an -covering orX \ B is not an -covering].3. Also we characterize the property of being an -covering.     

On local automorphisms of certain quadrics of codimension 2

The article considers nondegenerate quadrics in Cⁿ⁺¹ with codimension 2 that are of the form M={zCⁿ, C²: Im_j=z, z_j; j=1, 2}, where are Hermitian forms, and thje stability groups Aut_x M that preserve the point x. It is proved that if the matrix ¹ is stable and the matrix (¹)^–12 has more than two different eigenvalues, all automorphisms of Aut_x M are linear transformations.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 9–14, July, 1992.     

Calculation of Some Integrals Describing Wave Fields

Two methods of calculating the scattering amplitude f(, ₀) of a wave scattered by the vertex of an arbitrarily shaped cone are justified. It is shown that the approximation f _d (, ₀,t) obtained by a method similar to the AbelPoisson method of summation converges uniformly in the regularity region for f. Also, the possibility of calculating f(, ₀) for N ₁( ₀) with the help of rapidly convergent integrals is proved. Bibliography: 7 titles.     

Sufficient Conditions for the Existence of Disintegrations

Let be a probability space and a partition of . A necessary and sufficient condition is given for the existence of a -additive and measurable disintegration of P on . It is also shown that P admits a -additive (but not measurable) disintegration on whenever is a standard space and the set (₁, ₂):₁ and ₂ are in the same element of } is coanalytic in ×. Finally, sufficient statistics (in the classical Fisherian sense) are investigated by using -additive disintegrations as conditional probabilities.