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In this work we study the question: for a given r, is there any point (x; y) on the plane such that the deviation of the number of lattice points, contained inside the circle of radius r centered at this point, from the area of the disc has order O(r1/2+ε) for arbitrary positive number "? We show that the answer for this question is positive. [ABSTRACT FROM AUTHOR]

In this article, we consider some metric questions on Manifolds defined by the system of equations. We obtain estimates in terms of products of singular numbers of some matrices defined by taking all partial derivatives of entries of a given matrix of the same order. [ABSTRACT FROM AUTHOR]

In this article we give a new proof of the Fundamental Theorem of Algebra. Our proof is algebraic. We simplify the known proof of the Fundamental Theorem considering special case of polynomials of odd degree with real coeffcients. This case allows us to apply the method of mathematical induction to get the proof in general case without using infiniteness of the field. [ABSTRACT FROM AUTHOR]