In 1957 N.C. Ankeny provided a new proof of the three squares theorem using geometry of numbers. This paper generalizes Ankeny’s technique, proving exactly which integers are represented by x2 + 2y2 + 2z2 and x2 + y2 + 2z2 as well as proving su cient conditions for an integer to be represented by x2 + y2 + 3z2 and x2 + y2 + 7z2.

f((1− t)x1 + tx2) ≤ (1− t)f(x1) + tf(x2), x1, x2 ∈ C, 0 ≤ t ≤ 1, then f : X → R is convex. Proof. Let (x1, α1), (x2, α2) ∈ epi f and 0 ≤ t ≤ 1. The fact that the pairs (xi, αi) belong to epi f means in particular that f(xi) < ∞, and hence that xi ∈ C, as otherwise we would have f(xi) =∞. But (1− t)(x1, α1) + t(x2, α2) = ((1− t)x1 + tx2, (1− t)α1 + tα2), and, as x1, x2 ∈ C, f((1− t)x1 + tx2) ≤ (...

Proposition 0.1. Let R be a commutative ring. TFAE: 1. Every ideal in R is finitely generated. 2. Every chain of ideals I1 ⊆ I2 ⊆ · · · is stable. That is, there exists n such that In = In+1 = · · · . 3. Every nonempty set of ideals has a maximal element wrt inclusion. Proof. 1. (1) =⇒ (2). I = ⋃ k Ik ideal in R. I is generated by x1, x2, . . . xm. There exists n such that x1, . . . , xm ∈ In. ...

Two new problems of bivalent propositional logic are proposed here: firstly, to distinguish the sense of propositions, besides the logical value and secondly, to analyze the ”ponderal” difference between two parts of a proposition: subject-predicate. 1 Relational projections and extensions Let r = (A1, A2, A3, R) be a ternary relation (see [3]). Starting with this, we may define three binary re...

The capacity of the class of relay channels with sender xl, a relay sender x2, a relay receiver y, = /(x1, x2), and ultimate receiver y is proved to be The relay channel (%, X ‘&, p(y,y, 1 x,, x,), 9 X 9,) is a model for the communication between a sender x, and a receiver y through two paths; a direct path from x, to y, and a path from Manuscript received September 5, 1979; revised June 16, 19...

Louis W. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 1 1− ax − x2 ∗ 1 1− bx − x2 = 1− x2 1− abx − (2 + a2 + b2)x2 − abx3 + x4 , where ∗ denotes the Hadamard product. In a similar way, by considering tilings of a 2× n rectangle with 1× 1 and 1× 2 bricks in the top row, and 1× 1 and 1× n bricks in the bottom row, we find...

Let aj, 1 <. j <: n + 1, be n + 1 positive real numbers. The random vector (X1,X2,...,Xn) has a Dirichlet D(n; au a2,...,an, att+1) distribution if its probability density function (p.d.f.) is given by (cf. Wilks [14]): (i) f(xt,x2,...,xn) = Kx'rM ' ••• < " _ 1 ( i x , x 2 . . . X T 1 for every (xx,x2,...,xn)e Sn, where Sn -= {(*,., x2,..., xn) | Xi > 0, 1 <: r g n; xt + x2 + ... + x„ < 1} K = ...

In this paper, we shall write ! using a notation, item notation, which enables one to make more redexes visible, and shall extend -reduction to all visible redexes. We will prove that ! written in item notation and accommodated with extended reduction, satis es all its original properties (such as Church Rosser, Subject Reduction and Strong Normalisation). The notation itself is very simple: if...

Here the simulation design is the same as that in scenario 1 described in the main text with two partially missing variables X1 and X2. Missing data were created by following the two non-ignorable mechanisms, 1) logit{pr(Rk = 1|X1, X2, Y, Z)} = 0.25Y + 0.25Z + X2 + X1 and 2) logit{pr(Rk = 1|X1, X2, Y, Z)} = 0.75 + Y + 0.25Z −X1 + X2, for k = 1, 2. Although in both mechanisms Rk strongly depends...