Showing papers in "Mathematika in 1990"
TL;DR: In this article, a detailed comparison between the parallel and non-parallel theories is given and it is shown that at small vortex wavelengths, the parallel flow theories have some validity; unless the vortex wavelength is small, nonparallel effects are dominant.
Abstract: Goertler vortices are thought to be the cause of transition in many fluid flows of practical importance. A review of the different stages of vortex growth is given. In the linear regime, nonparallel effects completely govern this growth, and parallel flow theories do not capture the essential features of the development of the vortices. A detailed comparison between the parallel and nonparallel theories is given and it is shown that at small vortex wavelengths, the parallel flow theories have some validity; otherwise nonparallel effects are dominant. New results for the receptivity problem for Goertler vortices are given; in particular vortices induced by free stream perturbations impinging on the leading edge of the walls are considered. It is found that the most dangerous mode of this type can be isolated and it's neutral curve is determined. This curve agrees very closely with the available experimental data. A discussion of the different regimes of growth of nonlinear vortices is also given. Again it is shown that, unless the vortex wavelength is small, nonparallel effects are dominant. Some new results for nonlinear vortices of 0(1) wavelengths are given and compared to experimental observations.
138 citations
TL;DR: It was shown in this article that for suitable a and b, n ≥ 7, one can have Vn(An) = VnBn(Bn) and for every (n − 1)-dimensional subspace H of ℝn, where Bn is the unit ball of the subspace.
Abstract: LetIt is proved that for suitable a and b, n≥7, one can have Vn(An) = Vn(Bn) and for every (n–1)-dimensional subspace H of ℝn, where Bn is the unit ball of ℝn. This strengthens previous negative results on a problem of H. Busemann and C. M. Petty.
93 citations
TL;DR: In this article, a lower bound for the dimension of lim-sup sets of the form (1) for a fairly general class of families of positive integers has been obtained, which includes a range of results in the theory of Diophantine approximation.
Abstract: Sets of the general form where U is a subset of ℛ k and is a family of subsets of U indexed by a set J , are common in the theory of Diophantine approximation [4, 7, 18, 19]. They are also closely connected with exceptional sets arising in analysis and with sets of “small divisors” in dynamical systems [1, 8, 15”. When J is the set of positive integers ℕ, the set Λ(ℱ) is of course the lim-sup of the sequence of sets F j , j = 1, 2,… [11, p. 1]. We will also call sets of the form (1), with the more general index set J, lim-sup sets. When such lim-sup sets have Lebesgue measure zero, it is of interest to determine their Hausdorff dimension. It is usually difficult to obtain a good lower bound for the Hausdorff dimension (and it can be much harder to determine than an upper bound). In this paper we will obtain a lower bound for the dimension of lim-sup sets of the form (1) for a fairly general class of families ℕ which includes a range of results in the theory of Diophantine approximation. This lower bound depends explicitly on the geometric structure and distribution in U of the sets F α in ℕ.
91 citations
TL;DR: In this article, the authors considered the case k = 1 of the assertion that when the statement;(Ak) “For almost all there are infinitely many natural numbers q for which there exist integers a1,…, ak such that (a1, ak, q) = 1 and where holds, if, and only if, diverges.
Abstract: In 1941 Duffin and Schaeffer [2] considered the case k = 1 of the assertion that when the statement;(Ak) “For almost all there are infinitely many natural numbers q for which there exist integers a1,…, ak such that (a1 … ak, q) = 1 andwhere holds, if, and only if, diverges.
62 citations
TL;DR: In this article, an algebraic number field of degree n and discriminant d is considered and the conjugates of the number μ in K ( i ) are denoted by μ ( i ).
Abstract: Let K be an algebraic number field of degree n and discriminant d . Let K (1) ,…, K ( n ) be the embeddings of the field. Then n = r 1 + 2 r 2 where K (1) , …, K ( r l ) are real and the remainder complex, satisfying . The conjugates of the number μ in K ( i ) are denoted by μ ( i .
52 citations
TL;DR: In this article, the authors determined the maximum possible (by volume) portion of the three-dimensional Euclidean space that can be occupied by a family of congruent circular cylinders of infinite length in both directions.
Abstract: We determine what is the maximum possible (by volume) portion of the three-dimensional Euclidean space that can be occupied by a family of non-overlapping congruent circular cylinders of infinite length in both directions. We show that the ratio of that portion to the whole of the space cannot exceed π/√12 and it attains π/√12 when all cylinders are parallel to each other and each of them touches six others. In the terminology of the theory of packings and coverings, we prove that the space packing density of the cylinder equals π/√12, the same as the plane packing density of the circular disk.
45 citations
TL;DR: In this paper, some new Opial-type integrodifferential inequalities in one variable are established, which generalize the existing ones which have a wide range of applications in the study of differential and integral equations.
Abstract: In this paper some new Opial-type integrodifferential inequalities in one variable are established. These generalize the existing ones which have a wide range of applications in the study of differential and integral equations.
41 citations
TL;DR: In this article, le module des sommes trigonometriques sur la variete de dimension n -s definie par s formes en n variables, avec une forme lineaire en exposant.
Abstract: Nous estimons le module des sommes trigonometriques sur la variete de dimension n – s definie par s formes en n variables, avec une forme lineaire en exposant. Cela s'applique a l'etude de la distribution des points rationnels d'une telle variete definie sur un corps fini ou sur le corps des nombres rationnels.
33 citations
TL;DR: For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p } as discussed by the authors, whose cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra.
Abstract: For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p }. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.
22 citations
TL;DR: In this article, the authors gave a formula for the potential distribution due to a circular disc condenser with arbitrary spacing parameter к (the ratio of separation of the discs to their radius), which was simpler to calculate than the formulation which I gave in [8: 1949], but unfortunately it fails to satisfy two requirements, as the present paper shows.
Abstract: Atkinson, Young and Brezovich [1: 1983] gave a formula for the potential distribution due to a circular disc condenser with arbitrary spacing parameter к (the ratio of separation of the discs to their radius). This was simpler to calculate than the formulation which I gave in [8: 1949]; but unfortunately it fails to satisfy two requirements, as the present paper shows. Together with [8], this paper shows that the potential formulated in [8] satisfies all requirements.
TL;DR: In this article, it was shown that the Banach space admits an equivalent locally uniformly convex norm for a scattered compact space K such that its ω 1-th derived set K(ω 1) is empty.
Abstract: If a scattered compact space K is such that its ω1-th derived set K(ω1) is empty then the Banach space ℒ(K) admits an equivalent locally uniformly convex norm
TL;DR: In this paper, the authors introduced three expansions of Birget and Rhodes' identity functor for groups, and proved that the cut-down to generators of a subsemigroup of a group G is the free inverse semigroup on A.
Abstract: In the terminology of Birget and Rhodes [3], an expansion is a functor F from the category of semigroups into some special category of semigroups such that there is a natural transformation η from F to the identity functor for which η s is surjective for every semigroup S . The three expansions introduced in [3] have proved to be of particular interest when applied to groups. In fact, as shown in [4], Ĝ (2) are isomorphic for any group G , is an E -unitary inverse monoid and the kernel of the homomorphism η G is the minimum group congruence on . Furthermore, if G is the free group on A , then the “cut-down to generators” which is a subsemigroup of is the free inverse semigroup on A . Essentially the same result was given by Margolis and Pin [12].
TL;DR: In this article, the authors derived estimates for the dimension of product measures, where dim and Dim denote Hausdorff and packing dimensions, respectively, and Dim and Dim are the dimensions of sets.
Abstract: In recent papers on fractals attention has shifted from sets to measures [1, 5, 10]. Thus it seems interesting to know whether results for the dimension of sets remain valid for the dimension of measures. In the present paper we derive estimates for the dimension of product measures. Falconer [3] summarizes known results for sets and Tricot [8] gives a complete description in terms of Hausdorff and packing dimension. Let dim and Dim denote Hausdorff and packing dimension. If then
TL;DR: In this paper, the surface area and volume of a convex compact subset of ℝ d are denoted s ( K ) and v ( K ), respectively, and the support function of the convex body K is denned by h ( K, x ) = max y∈K x t y and the polar dual of K 0 = { x: |x t y|1, y ∈K }.
Abstract: A convex compact subset of ℝ d is called a convex body. The (Euclidean) surface area and volume of a convex body K are denoted s ( K ) and v ( K ) respectively. The support function of a convex body K is denned by h ( K, x ) = max y∈K x t y and the polar dual of K is given by K 0 = { x: |x t y|1, y∈K }. Double vertical bars shall denote the Euclidean length of a vector , and S shall denote the unit sphere (the Euclidean unit ball): S = {x: ║x║≤1}. We use for the mixed volume
TL;DR: In this paper, all rank 2 real valuations of K(x, y) which extend V 0 have been studied for algebraically closed fields of characteristic zero, where x, y being algebraically independent over the field K.
Abstract: Let V 0 be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane [3] gave a method of constructing all real valuations of K(x) which are extensions of V 0 .In this paper, we determine explicitly all rank 2 valuations of K(x) which extend V 0 .One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski [5] in the case when K is an algebraically closed field of characteristic zero.
TL;DR: In this paper, the sum of the m -th powers of the zeros of the Bessel polynomial y n (x ) was shown to be the Hawkins polynomials for m = 0, 1, 2, 3, 4.
Abstract: Let be the sum of the m -th powers of the zeros of the Bessel polynomial y n ( x ). It is known that for m = 0, 1, 2, …, where c m ( v ) is the Hawkins polynomial. In this paper we find rational functions w m ( v ) such that for m = 0, 1, 2, …
TL;DR: In this article, the unique extension of the (additive) p-adic valuation on to K, normalized with ord p = 1, is defined and denoted by "ord" by the authors.
Abstract: Let where a0≠0, m≥2, n = e1 + … + em and the ξi(1≤i≤m) are the distinct zeros of f in some algebraic closure of the p-adic field . Then K = (ξ1, ξ2, …, ξm) is a finite separable extension of and we denote by “ord” the unique extension of the (additive) p-adic valuation on to K, normalized with ord p = 1.
TL;DR: Penganggaran hasil tambah eksponen berganda $S(f;q) =\sigma e^{2\pi ǫ)-f(\underset{\sim}{x})/q$ ying lengkap menjadi tajuk kajian ramai penyelidik as mentioned in this paper.
Abstract: Penganggaran hasil tambah eksponen berganda $S(f;q) =\sigma e^{2\pi f(\underset{\sim}{x})/q}$ yang dinilaikan di atas set reja modulo $q$ yang lengkap menjadi tajuk kajian ramai penyelidik. Di dalam makalah ini anggaran kepada hasil tambah ini untuk $q = p^\alpha, p$ nombor perdana dan $\alpha > 0$ didapatkan melalui kaedah polihedron Newton yang telah dihasilkan terlebih dahulu. Didapati hasil tambah ini bersandar kepada anggaran kekardinalan set penyelesaian sistem persamaan kongruen tertentu modulo $p^{[\alpha/2]}$, dan juga kepada hasil tambah Gaussan tertentu bergantung kepada pariti $\alpha.$ Anggaran yang diperolehi adalah terbaik mungkin dengan kaedah polihedron Newton terutama apabila I± genap.
Katakunci: Modulo q; hasil tambah eksponen; hasil tambah Gaussan; kaedah polihedron Newton; kekardinalan