On the circles of Apollonius associated with a triangle

Given a ratio , >>0, and a triangle ABC, on the sides and , using ratios , and , three circles of Apollonius are denned. In this paper, we will show that the three centers are collinear, the circles are coaxal and develop a necessary and sufficient condition that these circles intersect. J. A. Hoskins, W. D. Hoskins and R. G. Stanton obtained these results in a recent paper using algebraic computation. Our aim is to establish all these results using only results from elementary Euclidean geometry and thereby uncovering more geometric insights and avoid lengthy calculations.     

On the type of triangle groups

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.      

Triangle in a triangle: On a problem of Steinhaus

A necessary and sufficient condition on the sidesp, q, r of a trianglePQR and the sidesa, b, c of a triangleABC in order thatABC contains a congruent copy ofPQR is the following: At least one of the 18 inequalities obtained by cyclic permutation of {a, b, c} and arbitrary permutation of {itp, q, r} in the formula $$\begin{array}{l} Max\{ F(q^2 + r^2 - p^2 ), F'(b^2 + c^2 - a^2 )\} \\ + Max\{ F(p^2 + r^2 - q^2 ), F'(a^2 + c^2 - b^2 )\} \le 2Fcr \\ \end{array}$$ is satisfied. In this formulaF andF′ denote the surface areas of the triangles, i.e. $$\begin{array}{l} F = {\textstyle{1 \over 4}}(2a^2 b^2 + 2b^2 c^2 + 2c^2 a^2 - a^4 - b^4 - c^4 )^{1/2} \\ F' = {\textstyle{1 \over 4}}(2p^2 q^2 + 2q^2 r^2 + 2r^2 p^2 - p^4 - q^4 - r^4 )^{1/2} . \\ \end{array}$$      

On convex triangle functions

We prove that the strongest (largest convex) solution of the functional inequality $$\tau \left( {\frac{{F + G}}{2},\frac{{H + K}}{2}} \right) \le \frac{{\tau (F,H) + \tau (G,K)}}{2},$$ whereF, G, H andK are arbitrary distribution functions, is the triangle function τ(F, G)(x) = Max(F(x) +G(x) ? 1, 0).     

On best quadratic triangle inequalities

Contrary to published results, it is shown that there do not exist ‘strongest’ (or ‘best possible’) homogeneous quadratic polynomial triangle inequalities of the form $$q(R,r) \leqslant s^2 \leqslant Q(R,r)$$ without further restrictions. Also, several best inequalities for symmetric functions of three positive variables are considered.     

On the approximation of derivatives of the interpolation Hermite polynomial on a triangle

Bounds on the deviation of the directional derivatives of the Hermite polynomial in the directions of a triangle sides are obtained; it is proved that these bounds are sharp. As a consequence, bounds on the deviations of the partial derivatives up to the third order inclusive are obtained.     

On the principal minors of a matrix with a multiple eigenvalue

The property of a Hermitian n × n matrix A that all its principal minors of order n − 1 vanish is shown to be a purely algebraic implication of the fact that the lowest two coefficients of its characteristic polynomial are zero. To prove this assertion, no information on the rank or eigenvalues of A is required. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 47–49.     

On the finite principal bundles

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V ₀. A holomorphic principal G-bundle E _G over a compact connected Kähler manifold X is called finite if for each subquotient W of the G-module V ₀, the holomorphic vector bundle E _G (W) over X associated to E _G for W is finite. Given a holomorphic principal G-bundle E _G over X, we prove that the following four statements are equivalent: (1) The principal G-bundle E _G admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite étale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*E _G is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle E _G (W) = E ×  ^G W over X, associated to the principal G-bundle E _G for the G-module W, is finite. (4) The principal G-bundle E _G is finite.     

On principal T-bands in a Banach lattice

It is shown that for every positive order continuous Riesz operatorT, defined on an order complete complex Banach latticeE which is separated by its Köthe dual, there exists a Frobenius decomposition ofE into a countable number of disjoint principalT-bands and a band on whichT is quasi-nilpotent. A basis for the generalized eigenspace ofT pertaining to its maximal eigenvalue is constructed and the positivity properties of its elements are studied. The distinguished eigenvalues ofT are characterized and it is also shown that the theory ofT-bands is symmetric with respect to the duality which exists betweenE and its Köthe dual. This generalizes aspects of work done by H.D. Victory and R.-J. Jang-Lewis.     

On principal submatrices

It is shown that if A[ω] is a principal submatrix of the positive definite Hermitian matrix A, then A ^?1[ω] ?(A[ω])^?1is a positive semidefinite hermitian matrix. This fact is used to give a brief proof of a result of Saburou Saitoh concerning Hadamard products.     

On principal submatrices

It is shown that if A[ω] is a principal submatrix of the positive definite Hermitian matrix A, then A ^-1[ω] -(A[ω])^-1is a positive semidefinite hermitian matrix. This fact is used to give a brief proof of a result of Saburou Saitoh concerning Hadamard products.     

On a generalization of principal weak flatness property

In this article a generalization of principal weak flatness of acts is defined. Using this new property the characterization of some new classes of monoids are investigated. Furthermore, many known results are generalized.     

On the Ramsey number of the triangle and the cube

The Ramsey number r(K ₃,Q _n ) is the smallest integer N such that every red-blue colouring of the edges of the complete graph K _N contains either a red n-dimensional hypercube, or a blue triangle. Almost thirty years ago, Burr and Erd?s conjectured that r(K ₃,Q _n )=2 ⁿ⁺¹?1 for every n∈?, but the first non-trivial upper bound was obtained only recently, by Conlon, Fox, Lee and Sudakov, who proved that r(K ₃,Q _n )?7000·2 ⁿ . Here we show that r(K ₃,Q _n )=(1+o(1))2 ⁿ⁺¹ as n→∞.