On the restriction problem for discrete paraboloid in lower dimension

We apply geometric incidence estimates in positive characteristic to prove the optimal $L^2 \to L^3$ Fourier extension estimate for the paraboloid in the four-dimensional vector space over a prime residue field. In three dimensions, when $-1$ is not a square, we prove an $L^2 \to L^{\frac{32}{9} }$ extension estimate, improving the previously known exponent $\frac{68}{19}.$Comment: Final versio

New sum-product type estimates over finite fields

Let $F$ be a field with positive odd characteristic $p$. We prove a variety of new sum-product type estimates over $F$. They are derived from the theorem that the number of incidences between $m$ points and $n$ planes in the projective three-space $PG(3,F)$, with $m\geq n=O(p^2)$, is $$O( m\sqrt{n} + km ),$$ where $k$ denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that \begin{equation}\label{mres} |A\pm A|+|A\cdot A| =\Omega \left(|A|^{1+\frac{1}{5}}\right), \end{equation} for any $A$ such that $|A|<p^{\frac{5}{8}}.$Comment: This is a revised version: Theorem 1 was incorrect as stated. We give its correct statement; this does not seriously affect the main arguments throughout the paper. Also added is a seres of remarks, placing the result in the context of the current state of the ar

Creation of Supply Chain Management Aimed At Improving the Quality and Accessibility of Municipal Services

Abstract— Supply chain management of regulation are mainly used in the field of municipal services management. Coordination between local governments and other organizations that control the provision of municipal services is poorly developed on the territory of municipal entities. The solution to this problem becomes particularly relevant in the context of creating the supply chain system for the management.  It is quite obvious that a review of approaches and methods in organizing the provision of municipal services is necessary under supply chain strategy. First of all, we implemented the effective supply chain for allocation of resources in the context of the transition to medium-term budget planning, and also to the integration of supply chain mechanism for organizing the provision of municipal services in the existing territorial development management system. This article provides an insight into supply chain management practices in a municipal context. It highlights the status, challenges and way forward for the implementation of supply chain management in a municipal environment

Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show that the Poincar\'e-Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.Comment: 17 pages, 2 figure

Eta photoproduction off the neutron at GRAAL: Evidence for a resonant structure at W=1.67 GeV

New (preliminary) data on eta photoproduction off the neutron are presented. These data reveal a resonant structure at W=1.67 GeV.Comment: 8 pages, 4 figures. Published in Proceedings of Workshop on the Physics of Excited Nucleons NSTAR2004, Grenoble, France, March 24 - 27, pg.19