Exact solutions of the d'Alembert and Liouville equations in the pseudo-Euclidean space R2, 2. II

Invariants of the maximal rank-3 and rank-4 subalgebras of the Poincaré algebras AP(2,2), which is the maximal invariance algebra of the d'Alembert equation u+u^k=0,k=1, and the Liouville equation u+ expu=0 are constructed. The reduction of the given equations with respect to the maximal rank-3 subalgebra is carried out and some exact solutions of these equations are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 9, pp. 1237–1244, September, 1990.     

Nonoscillatory solutions of linear differential equations with deviating arguments

Summary The equation to be considered is of the form (1) x(ⁿ)(t)+p(t)x(g(t))=0 (t>a), where =±1, p(t) > 0 for ta and g(t) as t. It is well- known that a nonoscillatory solution x(t) of (1) satisfies (2) x(t)x(ⁱ)(t)>0 (0il), (–1)^i–lx(t)x(ⁱ)(t)>0 (lin) for some integer l, 0ln, (–1)^n–l–1=1. In this paper, for a given l such that 0n–l–1=1, necessary conditions and sufficient conditions are found for (1) to have a solution x(t) which satisfies (2), and a necessary and sufficient condition is established in order that for every >0 the equation x(ⁿ)(t)+p(t)x(g(t))=0 (t>a) has a solution x(t) which satisfies (2). Related results are also contained.     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

Zur Kennzeichnung der Kollineationen zwischen Quadriken

In projective spaces Pⁿ(K), Pⁿ(K) over commutative fields K, K of characteristic 2 we consider the quadrics Q^n–1Pⁿ(K), Q,^n–1Pⁿ(K) which have the same rank and index and each of which is different from its singularity space. A one-to-one correspondence of Q^n–1 onto Q'^n–1 can be extended uniquely to a collineation of Pⁿ(K) onto Pⁿ(K) if and only if satisfies the following condition: maps the (n–2)-dimensional quadrics of Q,^n–1 of maximal rankonto (n–2)-dimensional quadrics of Q^n–1 of maximal rank (Theorem 1).

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Herrn Prof. Dr. Helmut Karzel zum 60. Geburtstag gewidmet     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

G8 estimator of solutions of systems of linear algebraic equations

For linear forms of regularized solutions (x, c)=Re c' · Re[I + i)+A'A_n ^–1]^–1 A'_nb of systems of equations Ax=b, where A is an n×m matrix, x, c, b are vectors, and _n is a sequence of constants, we propose the estimator , where is any measurable solution of the equation ()Re[1+₁a(())]²+ (₁–₂)(1+₁(gq()))=, a(y)=n^–1 Sp[Iy+^–1Z_s'Z^s+ iI]^–1, , _i=n_n ²_n ^–1s_n ^–1, ⁿ=mI_n ²_n ^–1s_n ^–1, X_i are independent observations on the matrix A. Under certain conditions, it is proved that G₈ is a consistent estimator for n and 0.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 111–119, 1988.     

Knaster's problem on continuous maps of a sphere into Euclidean space

A survey of known results and additional new ones on Knaster's problem: on the standard sphere S^n–1Rⁿ find configurations of points A₁, , A_k, such that for any continuous map fS^n–1R^m one can find a rotation a of the sphere S^n–1 such that f(a(A₁)==f(a(A_k)) and some problems closely connected with it. We study the connection of Knaster's problem with equivariant mappings, with Dvoretsky's theorem on the existence of an almost spherical section of a multidimensional convex body, and we also study the set {a S0(n)f(a(A₁))==f(a(A_k))} of solutions of Knaster's problem for a fixed configuration of points A₁, , A_kS^n–1 and a map fS^n–1R^m in general position. Unsolved problems are posed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 169–178, 1987.     

On a Parabolic Variational Inequality That Generalizes the Equation of Polytropic Filtration

We obtain conditions for the existence and uniqueness of a solution of a parabolic variational inequality that is a generalization of the equation of polytropic elastic filtration without initial conditions. The class of uniqueness of a solution of this problem consists of functions that increase not faster than e ^–t , > 0, as t –.     

On simplexes inscribed in the minor arc of a curve

On an arc L of length h, of class Cⁿ, and in Euclidean Eⁿ, the set of n+1 points (the partition of the arc) P={0, ₁ h, , _n–1h, h}, 0<₁<<_n–1<1 determines a simplex S_h(P) inscribed in the arc. For its volume V_h(P) we evaluate lim and prove that its greatest value is obtained for a unique choice of P=P_cr. The exact values for _i from P_cr are found for n=2, 3, 4, 5, 6.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 114–120, 1990.     

Phragmén-Lindelöf theorems for second-order quasilinear elliptic equations

Analogues are formulated of the well-known, in the theory of analytic functions, Phragmen-Lindelöf theorem for the gradients of solutions of a broad class of quasilinear equations of elliptic type. Examples are given illustrating the accuracy of the results obtained for the gradients of solutions of the equations of the form div(|U|^–2u)=f(x, u, u), where f(x, u, u) is a function locally bounded in ²ⁿ⁺¹. f(x, 0, u)=0, uf(x, u, u) c¦u¦1+q(1+ ¦u|), > 1, c > 0, q > 0, is an arbitrary real number, and n >- 2. The basic role in the technique employed in the paper is played by the apparatus of capacitary characteristics.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1376–1381, October, 1992.The author sincerely appreciates E. M. Landis's permanent attention and numerous useful discussions.     

The hyperbolic n-space as a graph in Euclidean (6n−6)-space

We construct for each n2 an isometric C-imbedding HⁿE^6n–6 of the n-dim. hyperbolic space into the Euclidean space E^6n–6 whose image is the graph of a C-map RⁿR^5n–6. This generalizes a well known result of D. Blanusa 1955 (the case n=2).     

Discrete Series Characters for GL(n, q)

The discrete series characters of the finite general linear group GL(n, q) are expressed as uniquely defined integral linear combinations of characters induced from linear characters on certain subgroups Hd, n of GL(n, q). The coefficients in these linear combinations are determined (for all n, q) by a family of polynomials r(T) Z[T] indexed by the set of all partitions .     

Probabilities of large deviations in a certain scheme of summation of dependent random variables

In the paper one investigates questions regarding the CLT for an Ising scheme. If , where S _n = ₌₁ ⁿ X _n, while (X_nk) is a triangular array of random variables, constituting an Ising scheme, then in the zone one proves the equalities lim(1 — F_n(x))(1 — (x))^–1=1, lim F_n(–x)(–1))^–1=1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 138–144, 1989.     

A relation between Chung's and Strassen's laws of the iterated logarithm

Summary Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)^–1/2–f(x)¦, 0x1 suitably normalized as T.This extends Chung's result valid for f(x)0, stating that lim inf.[ sup ¦(2T loglogT)^–1/2 W(xT)¦(loglog T)^–1]=/4 a.s. T 0x1     

Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+[1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line [0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .     

Acceleration of convergence for finite element solutions of the Poisson equation

Summary Letu _h be the finite element solution to–u=f with zero boundary conditions in a convex polyhedral domain . Fromu _h we calculate for eachz and ||1 an approximationu _h ^– (z) toD u(z) with |D u(z)–u _h ^– (z)|=O(h ^2k–2) wherek is the order of the finite elements. The same superconvergence order estimates are obtained also for the boundary flux. We need not work on a regular mesh but we have to compute averages ofu _h where the diameter of the domain of integration must not depend onh.     

Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds

Necessary and sufficient geometric conditions are proved for the equation u–Q(x)u=0, Q(x)0, to have a bounded nontrivial solution on a noncompact Riemannian manifold. The results imply as corollaries conditions for parabolicity and stochastic completeness of a manifold, previously established by other methods.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 66–77, 1989.     

The intuitionistic propositional calculus with quantifiers

Let L be the language of the intuitionistic propositional calculus J completed by the quantifiers and , and let calculus 2J in language L contain, besides the axioms of J, the axioms xB (x) B(y) and B(y) xB (x). A Kripke semantics is constructed for 2J and a completeness theorem is proven. A result of D. Gabbay is generalized concerning the undecidability of C2J⁺-extension of 2J by schemes x (x B) and x(A B(x))A xB (x) specificially: the undecidability is proven of each T theory in language L such that [2J]T [C2J+] ([2J] ([2J] denotes the set of all theorems of calculus 2J).Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 69–76, July, 1977.     

Spherical multipliers

It is proven in the paper that if functionf(x)L^p(Rⁿ), where 1/p> 1/2 + 1/(n + 1), then the restriction of the Fourier transform f() to the unit sphere S^n–1 lies in L²(S^n–1). As was shown by Fefferman [1], it follows from this that, when > (n –1)/(2(n + 1)), the Riesz-Bochner multiplier acts in LP(Rⁿ) if (n –1–2)/(2n) <1/p < (n + 1 + 2)/(2n).Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 105–112, January, 1978.The author wishes to thank B. S. Mityagin for his attention to this work.     

Maximal subgroups of symmetric groups defined on projective spaces over finite fields

Let PL(n, q) be a complete projective group of semilinear transformations of the projective space P(n–1, q) of projective degree n–l over a finite field of q elements; we consider the group in its natural 2-transitive representation as a subgroup of the symmetric group S(P*(n–1, q)) on the setp*(n–1),q=p(n–1,q)/{O}. In the present note we show that for arbitrary n satisfying the inequality n>4[(qⁿ–1)/(q^n–1–1)] [in particular, for n>4(q +l)] and for an arbitrary substitutiong s (p*(n–1,q))pL(n,q) the group PL(n,q), g contains the alternating group A(P* (n–1,q)). Forq=2, 3 this result is extended to all n3.Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 91–100, July, 1974.The author expresses his sincere thanks to M. M. Glukhov for his interest in his work.