Let X be a Banach space and let G be a closed bounded subset of X. For x1, x2, . . . , xm ∈ X, we set ρ x1, x2, . . . , xm,G sup{max1≤i≤m‖xi−y‖ : y ∈ G}. The setG is called simultaneously remotal if, for any x1, x2, . . . , xm ∈ X, there exists g ∈ G such that ρ x1, x2, . . . , xm,G max1≤i≤m‖xi−g‖. In this paper, we show that if G is separable simultaneously remotal in X, then the set of ∞Bochn...

We study Minkowski’s inequality Da b(x1 + x2, y1 + y2) ≤ Da b(x1, y1) +Da b(x2, y2) (x1, x2, y1, y2 ∈ R+) and its reverse where Da b is the difference mean introduced by Stolarsky. We give necessary and sufficient conditions (concerning the parameters a, b) for the inequality above (and for its reverse) to hold.

We study the problem of finding all integer solutions of the Diophantine equations x2 − 5Fnxy − 5 (−1) y2 = ±Ln, x2 − Lnxy + (−1) y2 = ±5F 2 n , and x2 − Lnxy + (−1) y2 = ±F 2 n . Using these equations, we also explore all integer solutions of some other Diophantine equations.

Suppose we want to approximate some complicated distribution p(X) by a simpler distribution q(X). The approximating distribution can, for example, have a simpler parameteric form (e.g. Gaussian), or capture simpler dependencies (e.g. factored). One way of choosing the best approximation within some class, is to minimise the variational free-energy. It is then often the case that the approximati...

We find a new Tribonacci-like sequence of positive integers 〈x0, x1, x2, . . .〉 given by xn = xn−1 + xn−2 + xn−3 , n ≥ 3, and gcd(x0, x1, x2) = 1 that contains no prime numbers. We show that the sequence with initial values x0 = 151646890045, x1 = 836564809606, x2 = 942785024683 is the current record in terms of the number of digits.

Let x1 and x2 be integers divisible only by some fixed primes. Is it possible that x1 + x2 is a perfect power? Special cases of the equation x1 +x2 = yk were formerly considered over Z. In this paper we develop an algorithm to solve this equation over global algebraic function fields. Our method is illustrated by two explicit examples.

∃! Τ name(T) = name(X) + "_" + name(rel(x,y)) + "_" + name(Y) ∧ ∃! F1 ∈ Τ name(F1) = name(X) + "$" ∧ ∃! F2 ∈ Τ (is_entity(Y) → name(F2) = name(Y) + "$" ∨ is_value(Y) → name(F2) = name(Y)) ∧ ∃! F3 ∈ Τ name(F3) = name(validity_period(rel(X,Y))) + "$" ∧ is_entity(X) ∃! Τ name(T) = name(X) + "_" + name(rel(x,y)) + "_" + name(Y) ∧ ∃! F1 ∈ Τ name(F1) = name(X) + "$" ∧ ∃! F2 ∈ Τ (is_entity(Y) → name(F...

In this article λ-method of Mitrinović-Vasić [1] is applied to improve the upper bound for the arc sin function of L. Zhu [4]. 1. Inequalities of Shafer-Fink’s type D. S. Mitrinović in [1] considered the lower bound of the arc sin function, which belongs to R. E. Shafer. Namely, the following statement is true. Theorem 1.1 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ 6( √...

n In this paper we investigate the behaviour of the solutions of equations ~i=t aix, = b, where ~--~=~ ai = 0 and b # 0, with respect to colorings of the set N of positive integers. It tunas out that for any b # 0 there exists an 8-coloring of N, admitting no monochromatic solution of x3 -x2 = x2 x l + b. For this equation, for b odd and 2-colorings, only an odd-even coloring prevents a monochr...