arXiv subjects

Yagub N. Aliyev

Publications and source records attributed to Yagub N. Aliyev.

13 recordsLinked to original sources

arXiv2026

Equivalence of the minimality conditions for the root functions of Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

We study the minimality of the system of root functions associated with a Sturm--Liouville problem whose boundary condition depends linearly on the eigenparameter. Two different criteria for minimality were previously obtained using independent approaches. In this paper, we establish the equivalence of these criteria and provide a unified characterization of the exceptional cases in which the removal of certain associated functions fails to preserve minimality. The theoretical results are illustrated by several examples involving multiple eigenvalues, demonstrating the consistency of the two approaches and clarifying the structure of the corresponding root function systems.

math.CA↗

arXiv2026

Optimal bounds for the ratio of differences of quadratic, arithmetic, and harmonic means

We determine the optimal constants in inequalities comparing the differences of the quadratic, arithmetic, and harmonic means of $n$ nonnegative real numbers. Specifically, we prove that for $n\ge3$ the sharp double inequality \[ \frac{1}{\sqrt{n}}\le \frac{A_n-H_n}{Q_n-H_n}\le \sqrt{\frac{n-1}{n}} \] holds true. This extends earlier results by T. Mitev, which established the sharp bounds only for the cases $n=3,4,$ and $5$. Our approach is based on a variant of the classical optimization method of Cauchy and Maclaurin, in which a quadratic symmetric function is optimized under simultaneous constraints on the arithmetic and harmonic means.

math.CA↗

arXiv2026

Minimality of the root functions of Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

We consider a Sturm--Liouville problem in which the spectral parameter appears linearly in one of the boundary conditions. The study focuses on the root functions of the problem, including eigenfunctions and associated functions corresponding to multiple eigenvalues. By employing the characteristic function of the boundary value problem, explicit representations are obtained for the biorthogonal system and for several special associated functions that play a crucial role in the spectral analysis. These representations allow previously established criteria for the basis and minimality properties of the system of root functions to be reformulated directly in terms of the characteristic function and its derivatives at the eigenvalues. As a consequence, the investigation of particular boundary value problems becomes considerably simpler. Several illustrative examples are analyzed to demonstrate the effectiveness of the proposed approach and to show its agreement with known results in the literature.

math.CA↗

arXiv2026

Sturm-Liouville problems with a boundary condition depending linearly on an eigenparameter

This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends linearly on the spectral parameter. The differential equation is considered on the interval $(0,1)$ with a classical boundary condition at one endpoint and an eigenparameter--dependent boundary condition at the other. Explicit formulas for the inner products and norms of the root functions are obtained. These relations make it possible to analyze the structure of the system of root functions and the corresponding biorthogonal system. Using these results, the minimality of the system of root functions in $L_2(0,1)$ is established. Furthermore, the basis properties of the system of root functions in the spaces $L_p(0,1)$, $1<p<\infty$, are investigated. Necessary and sufficient conditions under which the system forms a basis are derived. Special attention is given to the cases of multiple eigenvalues and the case when the eigenvalue coincides with the critical value $-d/c$. The obtained results reveal a symmetry between different spectral cases and provide a simpler approach that avoids the use of the exit space $L_2(0,1) \oplus \mathbb{C}$. Several examples are presented to illustrate the theoretical results.

math.CA↗

arXiv2026

The Maximum of the Volume of a Cevian Simplex and its Parts

The cevian triangle corresponding to an interior point $M$ of a triangle is the triangle determined by the feet of the three cevians concurrent at $M$. It is known that the area of the cevian triangle for an interior point $M$ of a triangle is at most $\frac{1}{4}$ of the area of the triangle, with maximum attained when $M$ is the triangle's centroid. This can be generalized from triangles to $n$-dimensional simplices, with $\frac{1}{4}$ replaced by $\frac{1}{n^n}$, using barycentric coordinates. We also use this method to solve two optimization problems about the parts of this simplex.

math.MG↗

arXiv2025

The dual of Philo's shortest line segment problem

We study the dual of Philo's shortest line segment problem and find the optimal line segments passing through two given points, with a common endpoint, and with the other endpoints on a given line. This problem is dual, in a point-and-line-exchanging sense, to a famous problem of antiquity used to solve the problem of duplicating the cube. The provided solution uses multivariable calculus and elementary geometry methods. Interesting connections with the angle bisector of the triangle are explored. A generalization of the problem using $l_p$ ($p\ge 1$) norm is proposed. The particular case $p=\infty$ is also studied. It is shown that in the cases $p=0$ and $p=2$ the median and the symedian, respectively, of a triangle do not always give a solution for the corresponding optimization problems. The general case $p\ne 1$ and related problems are proposed as open problems.

cs.CG↗

arXiv2024

The number of inscribed and circumscribed graphs of a convex polyhedron

In the paper we prove that the number of graphs inscribed into graph of a convex polyhedron and circumscribed around another graph does not exceed 4. For this we first studied Poncelet type problem about the number of convex $n$-gons inscribed into one convex $n$-gon and circumscribed around another convex $n$-gon. It is proved that their number is also at most 4. This contrasts with Poncelet type porisms where usually infinitude of such polygons is proved, provided that one such polygon already exists. An inequality involving ratio of lengths of line segments is used. Alternative way of using Maclaurin-Braikenridge's conic generation method is also discussed. Properties related to constructibility with straightedge and compass are also studied. A new proof, based on mathematical induction, of generalized Maclaurin- Braikenridge's theorem is given. We also gave examples of regular polygons and a polyhedron for which number 4 is realized.

math.GM↗

arXiv2024

Caustics of a Paraboloid and Apollonius Problem

We study caustics of an elliptical paraboloid and the history of their various representations from 3D models in XIX century to the recent computer graphics. In the paper two ways of generating the surface, one with cartesian coordinates using formula for principal curvatures, and the other one with parabolic coordinates using Seidel's formula were demonstrated. By finding the intersection curves of these caustics with the paraboloid we extend the solution of F. Caspari for classical Apollonius problem about the number of concurrent normals to the points of the paraboloid itself. A complete classification of all possible cases of intersections of these caustics with their paraboloid is given.

math.DG↗

arXiv2024

Inequalities about the area bounded by three cevian lines of a triangle

In the paper we prove generalization of Schlömilch's and Zetel's theorems about concurrent lines in a triangle. This generalization is obtained as a corollary of sharp geometric inequality about the ratio of triangular areas which is proved using discrete variant of Hölder's inequality. Also a new sharp refinement of J.F. Rigby's inequality, which itself generalized Möbius theorem about the areas of triangles formed by cevians of a triangle, is proved.

math.MG↗

arXiv2023

The Best Constant For Inequality Involving Sum Of The Reciprocals And Product Of Positive Numbers With Unit Sum

In the paper we study a special parameter containing algebraic inequality involving sum of reciprocals and product of positive real numbers whose sum is 1. We determine the best values of the parameter using a new optimization argument. In the case of three numbers the algebraic inequality have some interesting geometric applications involving a generalization of Euler's inequality about the ratio of radii of circumscribed and inscribed circles of a triangle.

math.CA↗

arXiv2023

Apollonius Problem and Caustics of an Ellipsoid

In the paper we discuss Apollonius Problem on the number of normals of an ellipse passing through a given point. It is known that the number is dependent on the position of the given point with respect to a certain astroida. The intersection points of the astroida and the ellipse are used to study the case when the given point is on the ellipse. The problem is then generalized for 3-dimensional space, namely for Ellipsoids. The number of concurrent normals in this case is known to be dependent on the position of the given point with respect to caustics of the ellipsoid. If the given point is on the ellipsoid then the number of normals is dependent on position of the point with respect to the intersections of the ellipsoid with its caustics. The main motivation of this paper is to find parametrizations and classify all possible cases of these intersections.

math.HO↗

arXiv2023

On integer linear combinations of terms of rational cycles for the generalized 3x+1 problem

In the paper, some special linear combinations of the terms of rational cycles of generalized Collatz sequences are studied. It is proved that if the coefficients of the linear combinations satisfy some conditions then these linear combinations are integers. The discussed results are demonstrated on some examples. In some particular cases the obtained results can be used to explain some patterns of digits in $p$-adic representations of the terms of the rational cycles.

math.NT↗

arXiv2023

Geometric Properties of Planar and Spherical Interception Curves

In the paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants A,B and Gauss's constant G, which were used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of the properties of these curves as limiting cases of some plane and spherical geometry results.

math.DG↗