256 research outputs found

    CHEBYSHEV AND EQUILIBRIUM MEASURE VS BERNSTEIN AND LEBESGUE MEASURE

    No full text
    We show that Bernstein polynomials are related to the Lebesgue measure on [0, 1] in a manner similar as Chebyshev polynomials are related to the equilibrium measure of [−1, 1]. We also show that Pell's polynomial equation satisfied by Chebyshev polynomials, provides a partition of unity of [−1, 1], the analogue of the partition of unity of [0, 1] provided by Bernstein polynomials. Both partitions of unity are interpreted as a specific algebraic certificate that the constant polynomial "1" is positive-on [−1, 1] via Putinar's certificate of positivity (for Chebyshev), and-on [0, 1] via Handeman's certificate of positivity (for Bernstein). Then in a second step, one combines this partition of unity with an interpretation of a duality result of Nesterov in convex conic optimization to obtain an explicit connection with the equilibrium measure on [−1, 1] (for Chebyshev) and Lebesgue measure on [0, 1] (for Bernstein). Finally this connection is also partially established for the "d"-dimensional simplex

    2023-2024 Catalog

    Get PDF
    The 2023-2024 Governors State University Undergraduate and Graduate Catalog is a comprehensive listing of current information regarding:Degree RequirementsCourse OfferingsUndergraduate and Graduate Rules and Regulation

    Pell's equation, continued fractions and Diophantine approximations of irrational numbers

    No full text
    This bachelor's thesis deals with Pell's equation, while clearly presenting structured information from studied domestic and foreign books, articles, and other sources. The goal of this thesis is to create study material primarily for university students but also for inquisitive high school students, and thus explain as intuitively as possible what Pell's equation is, how to find its solutions, and how it is related, for example, to continued fractions, approximations of irrational numbers, and invertible elements in Z[√n ]. The main motivation for solving Pell's equation throughout the work is specifically that its solutions give best approximations of irrational square roots. Pell's equation is presented in a brief historical context. Further, it is proved that there is a non-trivial integer solution for every Pell equation, and the theory of continued fractions is used to find it. To make the creation of continued fractions easier, the so-called Tenner's algorithm is introduced. Specifically, the search for a solution to Pell's equation is derived using convergents and the periodicity of continued fractions of irrational roots. Subsequently, the structure of the solution is described: it is proved that there is a so-called minimal solution that generates all positive solutions, and a set of..

    Pell's Equation

    No full text
    U ovom diplomskom radu najprije smo naveli definicije te neke općenite rezultate vezane uz neke oblike diofantskih jednadžbi, a zatim se u ostatku rada usmjerili na specijalan oblik nelinearne diofantske jednadžbe: Pellovu jednadžbu. Osim toga, proučili smo pelovske jednadžbe x2dy2=±1,±4x^{2}-dy^{2}=\pm 1,\pm 4 . Naveli smo dokaze o nužnim i dovoljnim uvjetima postojanja njihovih rješenja, pokazali strukturu rješenja te naposlijetku pokazali kako te jednadžbe riješiti korištenjem verižnih razlomaka i drevne indijske Chakravala metode.In this final paper, we first stated the definitions and some general results related to some types of Diophantine equations. After that, we focused on a special type of nonlinear Diophantine equation: Pell’s equation. Additionally, we studied the equations x2dy2=±1,±4x^{2}-dy^{2}=\pm 1,\pm 4. We presented the necessary and sufficient conditions for the existence of their solutions, showed the structure of the solutions and finally showed how to solve these equations using the continued fractions method and the ancient Indian Chakravala method

    The Negative Pell's Equation in Positive Characteristic

    Get PDF
    The Pell’s equation and the negative Pell’s equation are two of the most well-studied topics in number theory. While the solvability of the Pell’s equation over the integers is well known for centuries, the solvability problem of the negative Pell’s equation over the integers, especially the density problem of how likely the negative Pell’s equation is solvable, was not fully answered until recently. In this thesis, we consider these equations over the polynomial ring F[t] where F is a finite field whose characteristic is greater than 2. In Chapter 3 of this thesis, two different well-known proofs of the solvability of the Pell’s equation over F[t] are presented. In Chapter 4, we present conditions and examples of when the negative Pell’s equation is solvable and the function field analogue of the density problem on the negative Pell’s equation

    Pell's Equation

    No full text
    U ovom diplomskom radu najprije smo naveli definicije te neke općenite rezultate vezane uz neke oblike diofantskih jednadžbi, a zatim se u ostatku rada usmjerili na specijalan oblik nelinearne diofantske jednadžbe: Pellovu jednadžbu. Osim toga, proučili smo pelovske jednadžbe x2dy2=±1,±4x^{2}-dy^{2}=\pm 1,\pm 4 . Naveli smo dokaze o nužnim i dovoljnim uvjetima postojanja njihovih rješenja, pokazali strukturu rješenja te naposlijetku pokazali kako te jednadžbe riješiti korištenjem verižnih razlomaka i drevne indijske Chakravala metode.In this final paper, we first stated the definitions and some general results related to some types of Diophantine equations. After that, we focused on a special type of nonlinear Diophantine equation: Pell’s equation. Additionally, we studied the equations x2dy2=±1,±4x^{2}-dy^{2}=\pm 1,\pm 4. We presented the necessary and sufficient conditions for the existence of their solutions, showed the structure of the solutions and finally showed how to solve these equations using the continued fractions method and the ancient Indian Chakravala method

    2023-2024 Boise State University Undergraduate Catalog

    Get PDF
    This catalog is primarily for and directed at students. However, it serves many audiences, such as high school counselors, academic advisors, and the public. In this catalog you will find an overview of Boise State University and information on admission, registration, grades, tuition and fees, financial aid, housing, student services, and other important policies and procedures. However, most of this catalog is devoted to describing the various programs and courses offered at Boise State

    Rational bisectors and solutions of Pell's equations

    Full text link
    On the coordinate plane, the slopes a,a, bb of two straight lines and the slope cc of one of the angle bisectors between these lines satisfy the equation (ac)2(b2+1)=(bc)2(a2+1).(a-c)^2(b^2+1) = (b-c)^2(a^2+1). Recently, a certain formula of non-trivial integral solutions of this equation by using solutions of negative Pell's equation was discovered by the author. In this article, we describe the integeral solutions of x2dy2=k|x^2-dy^2| = k (kk: integer >1> 1) with the fundamental unit of Q(d)\mathbb Q(\sqrt d) and the fundamental solutions of x2dy2=pn|x^2-dy^2| = p^n (pp: prime number, nn: positive integer), and clarify the structure of the rational solutions of the above equation as its application.Comment: 12 page

    CHEBYSHEV AND EQUILIBRIUM MEASURE VS BERNSTEIN AND LEBESGUE MEASURE

    No full text
    We show that Bernstein polynomials are related to the Lebesgue measure on [0, 1] in a manner similar as Chebyshev polynomials are related to the equilibrium measure of [−1, 1]. We also show that Pell's polynomial equation satisfied by Chebyshev polynomials, provides a partition of unity of [−1, 1], the analogue of the partition of unity of [0, 1] provided by Bernstein polynomials. Both partitions of unity are interpreted as a specific algebraic certificate that the constant polynomial "1" is positive-on [−1, 1] via Putinar's certificate of positivity (for Chebyshev), and-on [0, 1] via Handeman's certificate of positivity (for Bernstein). Then in a second step, one combines this partition of unity with an interpretation of a duality result of Nesterov in convex conic optimization to obtain an explicit connection with the equilibrium measure on [−1, 1] (for Chebyshev) and Lebesgue measure on [0, 1] (for Bernstein). Finally this connection is also partially established for the "d"-dimensional simplex

    Pell's equation, continued fractions and Diophantine approximations of irrational numbers

    Get PDF
    This bachelor's thesis deals with Pell's equation, while clearly presenting structured information from studied domestic and foreign books, articles, and other sources. The goal of this thesis is to create study material primarily for university students but also for inquisitive high school students, and thus explain as intuitively as possible what Pell's equation is, how to find its solutions, and how it is related, for example, to continued fractions, approximations of irrational numbers, and invertible elements in Z[√n ]. The main motivation for solving Pell's equation throughout the work is specifically that its solutions give best approximations of irrational square roots. Pell's equation is presented in a brief historical context. Further, it is proved that there is a non-trivial integer solution for every Pell equation, and the theory of continued fractions is used to find it. To make the creation of continued fractions easier, the so-called Tenner's algorithm is introduced. Specifically, the search for a solution to Pell's equation is derived using convergents and the periodicity of continued fractions of irrational roots. Subsequently, the structure of the solution is described: it is proved that there is a so-called minimal solution that generates all positive solutions, and a set of...Katedra matematiky a didaktiky matematikyFaculty of EducationPedagogická fakult
    corecore