Extensions of an inequality of Bonnesen toD-dimensional space and curvature conditions for convex bodies

For convex bodies inE ^d (d 3) with diameter 2 we consider inequalitiesW _i – W _d–1 +( - 1) W _d 0 (i = 0, , d – 2) whereW _j are the quermassintegrals. In addition, for a ball, equality is attained for a body of revolution for which the elementary symmetric functions _d–1–i of main curvature radii is constant. The inequality is actually proved fori = d – 2 by means of Weierstrass's fundamental theorem of the calculus of variations.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Solution of one problem of nonlinear adsorption kinetics by the method of lines

The method of lines is constructed and proved for numerical solution of a nonlinear initial-boundary-value problem of parabolic type describing the adsorption of a substance from an aqueous solution of bounded volume by a spherical adsorbent. The method is developed under natural assumptions on the smoothness of the solution of the original problem. The rate of convergence of the method depends on the smoothness of the initial function and is of order O(h) if v₀(x) 0, O(h^1/2) if v₀(x) C¹[0, 1], and 0(|v ₀(x)|W ₂ ¹ (O,h)).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 24–30, 1987.     

Regularity properties of isometric immersions

We show that an isometric immersion y from a two-dimensional domain S with C^1,α boundary to ℝ³ which belongs to the critical Sobolev space W^2,2 is C¹ up to the boundary. More generally C¹ regularity up to the boundary holds for all scalar functions V ∈ W^2,2(S) which satisfy det ∇²V=0. If S has only Lipschitz boundary we show such V can be approximated in W^2,2 by functions V_k ∈ W^1,∞∩W^2,2 with det ∇²V_k=0.     

On the accuracy of a difference scheme for solving the nonstationary coupled thermoelastic problem in stresses

For numerical solution of the coupled one-dimensional problem of dynamic thermoelasticity in stresses (strains) we construct a second-order approximating difference scheme. We study its stability and obtain an a priori estimate. We prove that the solution of the scheme converges to a generalized solution of the original problem in the Sobolev class W ₂ ² (Q_T).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 95–99.     

Difference schemes for the dirichlet problem for an elliptical equation with discontinuous coefficients in an arbitrary region on a regular grid

Exact difference scheme operators are used to construct a difference scheme for a second-order elliptical equation with discontinuous coefficients. The solution of the scheme converges to the solution of the original problem at a rate O(h^1/2) in the grid norm W₂ ¹().Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 3–7, 1985     

On a nonlocal problem for a parabolic equation

We study a nonlocal boundary-value problem for a parabolic equation in a two-dimensional domain, establish ana priori estimate in the energy norm, prove the existence and uniqueness of a generalized solution from the classW ₂ ^1,0 (Q _T ), and construct a difference scheme for the second-order approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 790–800, June, 1995.     

Vacuum of the electroweak gauge theory in a strong magnetic field in A heat bath

Expressions are obtained for the thermodynamic potential of a gas of electrically charged vector bosons in a magnetic field and the effective potential of the Weinberg-salam theory in a strong magnetic field in a heat bath. In the single-loop approximation, an estimate is obtained for the value of the critical temperatureT _cr corresponding to the phase transition in the Weinberg-Salam theory in a magnetic field. It is shown that it is possible to determine correctly the total probability for production of aW ⁺W^– boson pair from the vacuum by an ultrastrong magnetic field in a heat bath, and an expression is obtained for this probability. It is argued that in the Weinberg-Salam theory the restoration of the spontaneously broken symmetry atT=0,B>B _cr ⁽¹⁾ is simultaneously accompanied by dynamical symmetry breaking of the theory.State University, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 446–456, March, 1995.     

Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights

A classic 1970 paper of B. Muckenhoupt established necessary and sufficient conditions for weightedL _p convergence of Hermite series, that is, orthogonal expansions corresponding to the Hermite weight. We generalize these to orthogonal expansions for a class of Freud weightsW ²:=e ^–2Q , by first proving a bound for the difference of the orthonormal polynomials of degreen+1 andn–1 of the weightW ². Our identical necessary and sufficient conditions close a slight gap in Muckenhoupt's conditions for the Hermite weight at least forp>1. Moreover, our necessary conditions apply whenQ(x)=|x|, >1 while our sufficient conditions apply at least for =2,4,6,....Communicated by Vilmos Totik.     

Existence and Regularity of a Class of Weak Solutions to the Navier–Stokes Equations

We construct a class of weak solutions to the Navier–Stokes equations, which have second order spatial derivatives and one order time derivatives, ofppower summability for 1 < p ≤ 5/4. Meanwhile, we show thatu L^s(0, T; W^{2, r}(Ω)) with 1/s + 3/2r = 2 for 1 < r ≤ 5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if u L^s(0, T; L^r(Ω)) with 1/s + 3/2r = 1 for Ω satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain inR³and 1 < s ≤ 2. For Ω = Rⁿwithn ≥ 3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412).     

Aircraft control for flight in an uncertain environment: Takeoff in windshear

The design of the control of an aircraft encountering windshear after takeoff is treated as a problem of stabilizing the climb rate about a desired value of the climb rate. The resulting controller is a feedback one utilizing only climb rate information. Its robustness vis-a-vis windshear structure and intensity is illustrated via simulations employing four different windshear models.Notations ARL aircraft reference line - D drag force, lb - g gravitational force per unit mass=const, ft sec^–2 - h vertical coordinate of aircraft center of mass (altitude), ft - L lift force, lb - m aircraft mass=const, lb ft^–1 sec² - O mass center of aircraft - S reference surface, ft² - t time, sec - T thrust force, lb - V aircraft speed relative to wind-based reference frame, ft sec^–1 - V _e aircraft speed relative to ground, ft sec^–1 - W _x horizontal component of wind velocity, ft sec^–1 - W _h vertical component of wind velocity, ft sec^–1 - x horizontal coordinate of aircraft center of mass, ft - relative angle of attack, rad - relative path inclination, rad - _e path inclination, rad - thrust inclination, rad - air density=const, lb ft² sec² Dot denotes time derivative.     

On Spin Models, Modular Invariance, and Duality

A spin model is a triple (X, W ⁺, W ^–), where W ⁺ and W ^– are complex matrices with rows and columns indexed by X which satisfy certain equations (these equations allow the construction of a link invariant from(X, W ⁺, W ^–) ). We show that these equations imply the existence of a certain isomorphism between two algebras and associated with (X, W ⁺, W ^–) . When is the Bose-Mesner algebra of some association scheme, and is a duality of . These results had already been obtained in [15] when W ⁺, W ^– are symmetric, and in [5] in the general case, but the present proof is simpler and directly leads to a clear reformulation of the modular invariance property for self-dual association schemes. This reformulation establishes a correspondence between the modular invariance property and the existence of spin models at the algebraic level. Moreover, for Abelian group schemes, spin models at the algebraic level and actual spin models coincide. We solve explicitly the modular invariance equations in this case, obtaining generalizations of the spin models of Bannai and Bannai [3]. We show that these spin models can be identified with those constructed by Kac and Wakimoto [20] using even rational lattices. Finally we give some examples of spin models at the algebraic level which are not actual spin models.     

Kolmogorov and Linear Widths of Weighted Sobolev-Type Classes on a Finite Interval, II

Let I be a finite interval, r and ρ(t)=dist{t, ∂I}, tI. Denote by Δ^s₊W^r_p, α, 0α<∞, the class of functions x on I with the seminorm x^(r)ρ^α_Lp1 for which Δ^s_τx, τ>0, is nonnegative on I. We obtain two-sided estimates of the Kolmogorov widths d_n(Δ^s₊W^r_p, α)_Lq and of the linear widths d_n(Δ^s₊W^r_p, α)^lin_Lq, s=0, 1, …, r+1.     

Estimates of Freud-Christoffel functions for some weights with the whole real line as support

Upper and lower bounds for generalized Christoffel functions, called Freud-Christoffel functions, are obtained. These have the form λ_n,p(W,j,x) = infPW_Lp(R)/|P^(j)(X)| where the infimum is taken over all polynomials P(x) of degree at most n − 1. The upper and lower bounds for λ_n,p(W,j,x) are obtained for all 0 < p ∞ and J = 0, 1, 2, 3,… for weights W(x) = exp(−Q(x)), where, among other things, Q(x) is bounded in [− A, A], and Q″ is continuous in β(−A, A) for some A > 0. For p = ∞, the lower bounds give a simple proof of local and global Markov-Bernstein inequalities. For p = 2, the results remove some restrictions on Q in Freud's work. The weights considered include W(x) = exp(− ¦x¦^α/2), α > 0, and W(x) = exp(− exp(¦x¦)), > 0.     

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.     

Accurate estimates of deviations of spline approximations to classes of differentiable functions

We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions s_r(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–s_r(f; x) ¦ and f(x)–s_r(f; x)|_c on the class W^mH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971.     

A rate of convergence bound of regular-grid difference schemes for elliptical equations with discontinuous coefficients

Exact difference scheme operators are applied to construct a difference scheme for the second-order elliptical equation. The solution of this difference scheme convergence to the solution of the original problem at a rate O(h) in theW ₂ ¹()- grid norm.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 21–26, 1985.     

Solvability of the mixed initial boundary-value problem for a parabolic equation with a nonlocal condition of first kind

The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W ₂ ^1,0 and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002.     

Accuracy of the “cross” difference scheme for the system of acoustic equations

A cross difference scheme is constructed by the integro-interpolation method for the system of acoustic equations and an a priori bound is derived in some norm weaker than L². The bound is used to prove convergence of the solution of the difference scheme at a rate 0(²+h²) to the solution of the original differential problem in the class W₂ ²(Q_T) and at a rate 0(+h) to the solution in W₂ ¹(Q_T).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 29–36, 1985     

Difference schemes for fourth-order one-dimensional equations with nonlinear boundary condition

A difference scheme is constructed for a boundary-value problem for a one-dimensional biharmonic equation with nonlinear boundary condition. Under the hypothesis that the exact solution of the problem belongs to the Sobolev space W ₂ ^k(),k [2, 4], in the lattice norm L ₂ (), an estimate is obtained of the precision of the difference scheme to O(h^k–1,5).Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 43–50, 1989.