Reflecting a triangle in the plane

We prove that if the three angles of a triangleT in the plane are different from (60°, 60°, 60°), (30°, 30°, 120°), (45°,45°,90°),(30°,60°,90°), then the set of vertices of those triangles which are obtained fromT by repeating ‘edge-reflection’ is everywhere dense in the plane.     

Pressure and flow in two dimensional numerical bifurcation models

The finite element method has been used to solve the Navier-Strokes equations for steady flow conditions in bifurcations. The results are presented as pressure, velocity and streamline plots at different Reynolds number. The three bifurcations considered have rigid walls and bifurcation angles of 0°, 20° and 180°. For the bifurcation with branch angles 0° and 20° there is flow separation along the inner wall of the outlet branches and large spatial pressure variations, these phenomena being more pronounced at the higher Reynolds numbers. For the bifurcation with a branch angle of 180° the high pressure gradients occured at the outer corner and for the high Reynolds number a vortex formation developed downstream of this corner.     

Oxidation of aldehydes by aqueous Ce (IV) in nitric acid

The kinetics of oxidation of acetaldehyde (at 20°, 31° and 40° C.), propionaldehyde (at 15°, 25° and 35° C.) and chloral hydrate (40°, 50° and 60° C.) by ceric nitrate in nitric acid medium [(H⁺) ≈ 0 · 5?1 · 5 M] were studied. The reactions were followed by determining rates of ceric disappearance for variations in [Ce⁴⁺], [RCHO], [H⁺], μ, temperature, etc. The stoichiometry, ΔCe⁴⁺ / Δ CH₃CHO ≈ 2 was established. The reactions were found to be 2nd Order—first order each with respect to [Ce⁴⁺] and [RCHO]. No complex formation between Ce⁴⁺ and aldehydes was observed. Neutral Ce (NO₃)₄ (H₂O)₂ and diol of the aldehyde were assumed the active species. The rate and thermodynamic data were calculated and discussed. The rates of oxidation were correlated with the structures of the aldehydes.     

Four positive solutions for the semilinear elliptic equation: -\Delta u+u=a(x)u^p+f(x) in {\mathbb R}^N

We consider the existence of positive solutions of the following semilinear elliptic problem in : where , , , and . Under the conditions: 1° for all , 2° as , 3° there exist and such that 4°, we show that (*) has at least four positive solutions for sufficiently small but . Received December 11, 1998 / Accepted July 16, 1999 / Published online April 6, 2000     

The K°-relation on the Congruence Lattice of a Regular Semigroup with an Inverse Transversal

Let S be a regular semigroup with an inverse transversal S° and C(S) the congruence lattice of S. A relation K° on C(S) is introduced as follows: if ρ, θ∈ C(S), then we say that ρ and θ are K°-related if Ker ρ° = Ker θ° , where ρ° = ρ|S°. Expressions for the least and the greatest congruences in the same K°-class as ρ are provided. A number of equivalent conditions for K° being a congruence are given.     

西北太平洋热带气旋源地、强度及其关联性的统计研究

该研究使用方差分析、非参数检验等推断统计方法,对1949-2008年西北太平洋2029个热带气旋(TC)资料的气候特征进行统计分析。结果表明:(1)82.65%的热带气旋产生在纬度5°～25°N和经度110°～160°E区域中;其中14°～19°N、115°～120°E的区域(5°×5°面积)是热带气旋生成最多区域,其位于南中国海东北部、吕宋岛西侧的狭小海面。(2)西北太平洋热带气旋源地的季节变化:1-4月较偏南偏东,产生范围相对集中;5-6月源地中心向北变化;7-9月产生的热带气旋不仅频数多而且分布范围广;10-12月份源地的中心则向南转移。(3)热带气旋强度随季节变化,冬季1月,2月气旋强度最弱、春、夏气旋强度强于冬季,春夏季节内强度无显著变化、秋季台风最强。(4)统计显示,近60年来,热带气旋强度随生成源地纬度偏南而增强,随源地经度偏东增强,经度比纬度的影响更为显著;但是源地的经、纬度与热带气旋强度呈一定的交互影响。该研究是首次将方差分析、非参数检验等推断统计方法应用于热带气旋气候特征研究,提高了对西北太平洋热带气旋源地、强度及其关联性的认识。     

Paper folding and trigonometric ratios

The purpose of this note is to discuss how paper folding can be used to find the exact trigonometric ratios of the following four angles: 22.5°, 67.5°, 27°, and 63°.     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.     

Thermal expansion of ferroelectrics

Using the interference method the thermal expansion of lithium hydrazinium sulphate [Li(N₂H₅)SO₄] has been investigated in the temperature range ?170° C. to +220° C. The principal expansion coefficients along the crystallographic axes have been determined and at room temperature (25° C.) the values ofa _a,a _b anda _c are respectively 17·7, 13·5, and 43·0, ×10^?6 (°C.)^?1. The expansion coefficients are found to vary nonlinearly, and, in particular, along thec-axis the thermal expansion shows an anomalous behaviour between ?160°C. and ?60°C. and also in the neighbourhood of +130°C. These anomalies are explained as due to homomorphous transitions in which there is a reorientation of the ?NH ₊ ³ group about the N-N axis or a re-alignment of the ?NH₂ group.     

Some studies on wave refraction in relation to beach erosion along the kerala coast

Using the British Admiralty bathymetric charts off the West Coast of India and employing the graphical method of constructing wave refraction diagrams, an attempt is made to study the behaviour of the shortperiod waves (4, 5 and 6 seconds) which are found to affect the coast generally in the neighbourhood of Cochin Port entrance. Nineteen stations, at intervals of roughly one mile, are chosen around the three-fathom line in this area. Considering a probable field of approach of deep-water waves, limited to a cone of 90°, five directions of approach are chosen at intervals of 22 1/2;° in the range of 202 1/2° to 292 1/2°. Refraction diagrams are prepared for these directions and periods, and from these, the refraction functions and directional parameters are evaluated for each station. The possible directions of flow of long-shore current and the areas vulnerable to erosion and sedimentation are investigated.     

完备矩阵代数Ⅰ——乘法连续性问题

<正> 完备空间与完备矩阵环是Kothe.G.于1934年引入的,近年来由于在分析中最有用的一类线性拓扑空间——核空间的种种研究的影响,又给Kothe理论带来了许多新的进展,出现了大量文章.然而,至今为止还未见到有把完备矩阵环当作拓扑代数来加以探讨的工作,为此,我们准备着手对这样一类显然有其重要性的拓扑代数进行详细的讨论.     

Delayed effects of sudden ionospheric disturbances on 164 Kc/s transmission between Tashkent and Ahmedabad

Continuous recording of the field strength of 164 Kc/s CW transmission from Tashkent (42° N, 69° E) has been going on at Ahmedabad (23° 01′ N, 72° 36′ E) since March 1960. The present communication deals with a delayed effect of S.I.D.’s which has been observed on the 164 Kc/s transmissions.     

Spectral radius of Hadamard product versus conventional product for non-negative matrices

We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all n×n non-negative matrices A and B, ρ(A°B)?ρ((A°A)(B°B))^1/2?ρ(AB), in which ° represents the Hadamard product.     

Ionospheric total electron content and slab-thickness at low latitudes in Indian zone

Observations of Faraday rotation of beacon signals from low orbiting satellite BE-B recorded at one station near the dip equator (Kodaikanal, dip 3·4° N) and at another station near the peak of the equatorial anomaly (Ahmedabad, dip 34° N) give a complete coverage of the equatorial anomaly belt in Indian zone. Contours of total elctron content (TEC) are obtained on a grid of latitudeversus local time for the different seasons of low (1964–66) and high (1967–69) solar activity epochs in the latitude belt 10° S to 26° N dip latitude. The development of the equatorial anomaly and its dependence on season and solar activity are discussed. Using similar contours of F₂ layer critical frequency, f₀F₂ contours of equivalent slab-thickness, τ are also constructed. The dependence of τ on season and solar activity and its implications on temperature are discussed.     

The relationship between the vortical structure and the wall shear stress in a blind trough cavity subject to a jet impingement

In a blind trough cavity subject to a jet– impingement, the jet switching phenomenon has been known for decades. The large vortices, formed by the Kelvin– Helmholtz instability play a crucial role in this phenomenon– they are the main cause of the fluid entrainment phenomenon, which consequently evoke the Coanda effect and finally results in the jet switch. As the jet swings, the vortical structure inside the cavity changes together with the impinigement point. The aim of this article is to describe the relationship between the wall shear stress, i.e. the motion of the impingement point, and the vortical structure.The analysed data were obtained using the TR PIV system, the caity aspect ratios 16:24 and 18:24, jet hydraulic diameter based Reynolds numbers 3000, 11200 and 19600, and the impingement angles 0°, 5°, 10° and 15°. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Multiplicative Quasi-Adequate Transversals of an Abundant Semigroup

In this article, we explore the multiplicative quasi-adequate transversals of an abundant semigroup. Let S be an abundant semigroup with multiplicative quasi-adequate transversals. Then the product of any two multiplicative quasi-adequate transversals is also a multiplicative quasi-adequate transversal. Moreover, all multiplicative quasi-adequate transversals of S form a rectangular band. Let S°and S ^? be multiplicative quasi-adequate transversals of S, and let δ°(δ^?) be the δ-relation on S°(S ^?). Then there exists a bijection ? from S°/δ°onto S ^?/δ^?. In particular, if δ°and δ^? are congruences, then the bijection ? is an isomorphism.     

Force constants and mean amplitudes for some anions of MX6 type (X=F,Cl, Br)

Force constants of the Urey Bradley type for twelve MX₆ ^n? ions ofOh symmetry have been reported. Generalized mean square amplitudes and mean amplitudes of vibration for bonded and non-bonded distances have also been calculated. Thermodynamic quantities for these molecules have been calculated at various temperatures from 50° K to 1600° K using a rigid rotator, harmonic oscillator approximation.     

Study of microwave absorption and molecular structure of some substituted naphthalene sulphonic acids and their dipolar mixture

The dielectric constant (∈′) and the loss factor (∈″) of naphthalene sulphonic acid, 0-naphthol sulphonic acid, l-naphthylamine 8-sulphonic acid and 2–4 dinitro naphthol 7-sulphoinc acid and their dipolar mixture in dilute solution of dioxane have been determined at three temperatures (20°, 30° and 40° C) in 3·13 cm (9585 Mc/sec.) microwave region by concentration variation method. The data obtained have been used for determining the relaxation times. The values of relaxation times at different temperatures have been used to determine free energy, heat and entropy of activation for the process of dipole orientation. It has been concluded that the dipole orientation in the molecules is contributed by both molecular as well as intra-molecular rotations. In case of dipolar mixture the relaxation time is average of the two components.     

On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices

We prove the spectral radius inequality ρ(A₁°A₂°?°A_k)?ρ(A₁A₂?A_k) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖A°B‖?ρ(A^TB) for nonnegative matrices, which improves Schur’s classical inequality ‖A°B‖?‖A‖‖B‖, where ‖·‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.     

Numerical Simulation of Spot-Weld Joints under Crash Loading

This paper compares FE simulations of spot-weld joints for dual-phase steel under different load cases by using damage models of Gurson-Tvergaard-Needleman (GTN) and its two extensions, GTN-Johnson-Cook and GTN-Hutchinson. Spot-weld specimens have three zones depending with different material properties: Base material, heat-affected zone and fusion zone. The characterization of the base material is straightforward. The other two zones are characterized with specifically heat-treated specimens. For each zone, flat smooth tensile, flat notched tensile and Iosipescu-shear specimen are used in order to obtain the damage behavior for different triaxiality values. GTN damage model parameters are calibrated with the help of smooth and notched flat tensile specimens. The parameters of the above mentioned extensions of GTN damage model are identified with the help of Iosipescu-shear specimen. Finally, the calibrated material models are used in the FE simulation of the spot-weld specimens under quasi static-load case (10 mm/min) for loading directions of 0°, 30°, 60°, 90°. The numerical force-displacement results are in good agreement with experimental results. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)