A graph G is called a sum graph if there is a so-called sum labeling of G, i.e. an injective function l : V (G) → N such that for every u, v ∈ V (G) it holds that uv ∈ E(G) if and only if there exists a vertex w ∈ V (G) such that l(u) + l(v) = l(w). We say that sum labeling l is minimal if there is a vertex u ∈ V (G) such that l(u) = 1. In this paper, we show that if we relax the conditions (ei...

Hilbert posed the following problem as the 17th in the list of 23 problems in his famous 1900 lecture: Given a multivariate polynomial that takes only non-negative values over the reals, can it be represented as a sum of squares of rational functions? In 1927, E. Artin gave an affirmative answer to this question. His result guaranteed the existence of such a finite representation and raised the...

be a positive definite quadratic form with determinant D R and integer coefficients a ;Call it an even form if all a ; ; are even ., an odd form if at least one a ;; is odd . Then Í,, is called non-decomposable, if it cannot be expressed as a sum of two non-negative quadratic forms with integer coefficients . Mordell 1 ) proved that f, can always be decomposed into a sum of five squares of line...

Let X1,. .. , XN be independent non-negative integer valued random variables. In this paper we present a sufficient condition for the collection (X1,. .. , XN) to be stochastically increasing in the sum X1 + · · · + XN. Thus, if the condition is satisfied then (X1,. .. , XN) are negatively associated when conditioned on the sum X1 + · · · + XN. The result is a generalization of the main theorem...

Suppose u, v are different currencies, and the exchange rate is C(u, v) (i.e., 1 unit of v is worth C(u, v) units of u). We set the length of the edge as w(u, v) = logC(u, v). We take the log because, in currency exchange, the rates are multiplicative factors, but in shortest path, path length is defined as sum of edge lengths. By this conversion, the length of a path is log of the total exchan...

Under the assumption of the Riemann hypothesis, the Linear Independence hypothesis, and a bound on negative discrete moments of the Riemann zeta function, we prove the existence of a limiting logarithmic distribution of the normalisation of the weighted sum of the Liouville function, Lα(x) = ∑ n≤x λ(n)/n α, for 0 ≤ α < 1/2. Using this, we conditionally show that these weighted sums have a negat...

In this paper we investigate angle geometry in the universal Teichmüller space. We construct three examples of triangles bounded by geodesic segments such that the first example has the sum of the three inner angles less than π, the second example has the sum of the three angles equal to π, and the third example has the sum of the three angles greater than π. Our result gives a negative answer ...

We completely classify the Q-factorial terminal toric Fano three-folds such that sum of squared torus invariant prime divisors is non-negative.