58 research outputs found

    On the Sum of Dilations of a Set

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    We show that for any relatively prime integers 1p<q1\leq p<q and for any finite AZA \subset \mathbb{Z} one has |p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$

    A low-energy decomposition theorem

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    We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly non-multiplicative.Comment: 16 page

    Challenges of marketing management in the non-governmental sector of South-Eastern Europe in the context of global recession

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    During the process of social and economic transition in the SEE countries numerous non-governmental organisations found a fertile ground for their activities. In spite of unstable political and general conditions these organisations have managed to survive and contribute to the creation of social capital. Non-governmental organisations have gradually become efficient subjects of civil society (or the third sector) and their role in the processes of social and economic transition of post-communist societies was recognized and accepted by all the social participants, both domestic and from abroad. It is generally assumed that the turbulent processes of social and economic transition of these countries would have probably had a different course without the social engagement of non-governmental organisations that catalysed and amortized the social processes in which they took part. The main goal of this paper is to point out the importance of applying the marketing concept integrated into the crisis management of non-governmental organisations (NGOs) in transitional countries of South-Eastern Europe (SEE) during the economic recession starting in 2008. Although the main purpose of marketing is to secure the organisational survival on the public scene, this is also an opportunity for non-governmental organisations to additionally profile themselves and improve the image they have in society

    On the Number of Solutions of Exponential Congruences

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    For a prime pp and an integer aZa \in \Z we obtain nontrivial upper bounds on the number of solutions to the congruence xxa(modp)x^x \equiv a \pmod p, 1xp11 \le x \le p-1. We use these estimates to estimate the number of solutions to the congruence xxyy(modp)x^x \equiv y^y \pmod p, 1x,yp11 \le x,y \le p-1, which is of cryptographic relevance

    On the Convex Hull of the Integer Points

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    Let P r denote the convex hull of the integer points in the disc of radius r . We prove that the number of vertices of P r is essentially r 2 /3 as r approaches infinity

    Sum of dilates in vector spaces

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    Let d2d\geq 2, AZdA \subset \mathbb{Z}^d be finite and not contained in a translate of any hyperplane, and qZq \in \mathbb{Z} such that q>1|q| > 1. We show |A+ q \cdot A| \geq (|q|+d+1)|A| - O_{q,d}(1).$

    New Sum-Product Estimates for Real and Complex Numbers

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    A variation on the sum-product problem seeks to show that a set which is defined by additive and multiplicative operations will always be large. In this paper, we prove new results of this type. In particular, we show that for any finite set A of positive real numbers, it is true that |{a+bc+d:a,b,c,d∈A}|≥2|A|2-1.As a consequence of this result, it is also established that |4k-1A(k)|:=|A…A⏟ktimes+⋯+A…A⏟4k-1times|≥|A|k.Later on, it is shown that both of these bounds hold in the case when A is a finite set of complex numbers, although with smaller multiplicative constants. © 2015, Springer Science+Business Media New York

    On the Convex Hull of the Integer Points

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    Let P_{r} denote the convex hull of the integer points in the disc of radius r. We prove that the number of vertices of P_{r} is essentially r^{2/3} as r approaches infinity.

    A low-energy decomposition theorem

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