Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.     

Morera type theorems on the unit disc

, . . : , , , , .     

Generalized One-Sided Laws for the Larger Observations of a Triangular Array

Consider independent and identically distributed random variables {X,X _nj , 1jn,n1} with density f(x)=px ^–p–1 I(x1), where p>0. We show that there exist unusual generalized Laws of the Iterated Logarithm involving the larger order statistics from our array.     

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ₁ + ₂ < /2, or 3/2 < ₁ + ₂ 2 and a destabilizing influence if /2 < ₁ + ₂ < 3/2, where 0 _j , (j=1, 2) and , is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles ₁ and ₂ play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.     

On the integersn for which Ω(n)=k (II)

Let (n) be the number of prime divisors ofn, counted with multiplicity. We denote byS(x, k) the set of thenx for which (n)=k, and byV _p(n) the exponent of the primep in the factorization ofn. In a previous paper we proved a result which implies that, ify=x/2 ^k tends to infinity withk>2loglogx where >1, then the distribution of the numbers on the setS(x, k) converges to the normal distribution of Gauss. Here, besides a slight improvement of that result, we give, for the moment of orderq of the above mentioned distribution, a formula which holds uniformly for 2loglogxklog (x/3)/log2 where 1<<3/2.     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

Solution of integral equations in fractional Sobolev spaces

We consider linear integral equations and Urysohn equations with constant integration limits. Sufficient conditions are given for the solutions of these equations to be in Sobolev spacesW ₂ (0,1), 0 2. Finite-difference schemes are constructed for approximate solution of the original equation by special averaging of the right-hand side kernel. The rate of convergence of the approximate solution to the averaged exact solution is shown to beO(h|ln h|(1/2,)⁺(3/2,)).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 3–19, 1987.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | _p ,P(e )| _p }, where is an algebraic number satisfying || _p <p ^–/(p–1), and 0 is ap-adic number with | | _p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || _p <p ^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

On the positive solutions of some nonlinear diffusion problems

We consider the nonlinear diffusion equationu _t –a(x, u _{x x} )+b(x, u)=g(x, u) with initial boundary conditions andu(t, 0)=u(t, 1)=0. Here,a, b, andg denote some real functions which are monotonically increasing with respect to the second variable. Then, the corresponding stationary problem has a positive solution if and only if(0, ^*) or(0, ^*]. The endpoint ^* can be estimated by , where ₁ u denotes the first eigenvalue of the stationary problem linearized at the pointu. The minimal positive steady state solutions are stable with respect to the nonlinear parabolic equation.

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Zusammenfassung Wir betrachten die nichtlineare Diffusionsgleichungu _t –a(x, u _x ) _x +b(x, u)=g(x, u) mit Randbedingungen undu (t, 0)=u (t, 1)=0. Dabei sinda, b, undg monoton wachsende Funktionen bzgl. des zweiten Argumentes. Das zugehörige stationäre Problem hat genau dann eine positive Lösung, falls (0, ^*) oder(0, ^*]. Der Endpunkt ^* kann durch abgeschätzt werden, wobei ₁ u den ersten Eigenwert des an der Stelleu linearisierten stationären Problems bezeichnet. Die minimale positive stationäre Lösung ist stabil bzgl. der obigen nichtlinearen parabolischen Gleichung.

     

Aufeinanderfolgende Elemente in multiplikativen Zahlenmengen

LetM be a multiplicative set with 1M andmnM if and only ifmM,nM for (m,n)=1. It is shown by elementary means that there exists the asymptotic density of the setM(M–1) for every multiplicative setM. The density is positive if and only ifM possesses a positive density and 2M for some . This result is slightly generalized to sums over multiplicative functionsf with |f|1.     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Rate of approximation by rectangular partial sums of double orthogonal series

( ) . .

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Dedicated to Professor K. Tandori on his seventieth birthday

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This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.     

Combined free and forced laminar convection in internally finned square ducts

Analysis is presented for the heat transfer performance of square ducts with internal fins from each wall in the case of combined free and forced convection by fully developed laminar flow. Numerical results are obtained for the Nusselt number and the pressure drop parameter for various values of finlengths and heat source parameter. For various values of Rayleigh numbers, the Nusselt number increases with the increase in finlength and decreases with the increase in heat source parameter.

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Zusammenfassung Es wird eine Analyse für den Wärmeaustausch von quadratischen Rohren mit inneren Rippen an jeder Wand im Falle einer Kombination von freier und erzwungener Konvektion bei voll entwickelter laminarer Strömung gegeben. Numerische Resultate für die Nusselt-Zahl und den Druckabfall-Koeffizienten für verschiedene Rippenbreiten und Parameter der Wärmequelle werden erhalten. Für einige Werte der Rayleighzahl wächst die Nusselt-Zahl mit der Rippenbreite und fällt mit wachsendem Parameter der Wärmequelle.

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Nomenclature A cross sectional area of the duct - B _2k Bernoulli numbers - c _p specific heat at constant pressure - D _h hydraulic diameter of finless duct - E _n complex constants (20) - F heat source parameter,Q/c _p - F _n () defined by Equation (14) - G(, , , ) Green's function (15, 16) - g gravitational acceleration - H() Heaviside function - h() defined by Equation (22) - i imaginary unit,i ²=–1 - ImW imaginary part ofW - K(,t) kernel of the integral equation, defined by (25) - k thermal conductivity - L pressure drop parameter, –D _h ² (p/x+ _w )/ - l fin length of each fin, Figure (1) - N _u Nusselt number, Equation (32) - p pressure - Q heat generation rate - R() defined by Equation (26) - R _A Rayleigh number, _w gc _p D _h ⁴ /k - ReW real part ofW - T dimensionless temperature, (t–t _w )/(c _p D _h ² /k) - T _mx dimensionless mixed mean temperature, Equation (33) - t fluid temperature - t ₀ reference temperature atx=0 - u local axial velocity - mean axial velocity - V u/ - W complex function defined by Equation (6) - w suffix denoting wall conditions - W ₀ defined by Equation (9) - W ₁ W–W ₀, Equation (18) - x axial coordinate along the length of the duct - y, z cross-sectional coordinates - constant temperature gradient, t/x - coefficient of thermal expansion of the fluid - fluid density - _n - dynamic viscosity - () Dirac delta function - ² Laplacian operator, ²/y ²/²/z ² - , y/D _h ,z/D _h      

i-perfectm-cycle systems,m ≤ 19

Summary For allm 19 and each meaningful value ofi (2 i m/2), the spectrum problem fori-perfectm-cycle systems is examined.     

A note on the independent domination number of subset graph

The independent domination number i(G) (independent number (G)) is the minimum (maximum) cardinality among all maximal independent sets of G. Haviland (1995) conjectured that any connected regular graph G of order n and degree 1/2n satisfies i(G) 2n/3 1/2. For 1 k l m, the subset graph S _m (k, l) is the bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i(S _m (k, l)) and prove that if k + l = m then Havilands conjecture holds for the subset graph S _m (k, l). Furthermore, we give the exact value of (S _m (k, l)).This work was supported by National Natural Sciences Foundation of China (19871036).     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).