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In this paper we introduce the concept of the Hamilton triangle of a given triangle in an isotropic plane and investigate a number of important properties of this concept. We prove that the Hamilton triangle is homological with the observed triangle and with its contact and complementary triangles. We also consider some interesting statements about the relationships between the Hamilton triangle and some other significant elements of the triangle, like e.g. the Euler and the Feuerbach line, the Steiner ellipse and the tangential triangle.

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We prove certain Menon-type identities associated with the subsets of the set {1, 2,..., n} and related to the functions f, fk , Ф and Ф k , defined and investigated by Nathanson.

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Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers.

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In the 1980’s the author proved lower bounds for the mean value of the modulus of the error term of the prime number theorem and other important number theoretic functions whose oscillation is in connection with the zeros of the Riemann zeta function. In the present work a general theorem is shown in a simple way which gives a lower bound for the mentioned mean value as a function of a hypothetical pole of the Mellin transform of the function. The conditions are amply satisfied for the Riemann zeta function. In such a way the results recover the earlier ones (even in a slightly sharper form). The obtained estimates are often optimal apart from a constant factor, at least under reasonable conditions as the Riemann Hypothesis. This is the case, in particular, for the error term of the prime number theorem.

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In this paper we establish some Ostrowski type inequalities for double integral mean of absolutely continuous functions. An application for special means is given as well.

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We prove the weak consistency of the trimmed least square estimator of the covariance parameter of an AR(1) process with stable errors.

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The ultrapower T* of an arbitrary ordered set T is introduced as an infinitesimal extension of T. It is obtained as the set of equivalence classes of the sequences in T, where the corresponding relation is generated by a free ultrafilter on the set of natural numbers. It is established that T* always satisfies Cantor’s property, while one can give the necessary and sufficient conditions for T so that T* would be complete or it would fulfill the open completeness property, respectively. Namely, the density of the original set determines the open completeness of the extension, while independently, the completeness of T* is determined by the cardinality of T.

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We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer–Mrowka–Ozsváth–Szabó. Moreover, we prove a gluing formula relating our invariant with the first author’s Bauer–Furuta type invariant, which refines Kronheimer–Mrowka’s invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.

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We extend the construction of Y-type invariants to null-homologous knots in rational homology three-spheres. By considering m-fold cyclic branched covers with m a prime power, this extension provides new knot concordance invariants Y m C ( K ) of knots in S3. We give computations of some of these invariants for alternating knots and reprove independence results in the smooth concordance group.

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We prove a theorem on the preservation of inequalities between functions of a special form after differentiation on an ellipse. In particular, we obtain generalizations of the Duffin–Schaeffer inequality and the Vidensky inequality for the first and second derivatives of algebraic polynomials to an ellipse.

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