Joseph Bak - Academia.edu (original) (raw)
Papers by Joseph Bak
Albanian journal of mathematics, Dec 14, 2021
Israel Journal of Mathematics, 1977
ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ... more ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ,n = 1,2, …, ¦a k ≦ Akk}. We consider the rate of approximation by elements ofP A , of continuous functions in [0, 1] which vanish at x = 0. Also a classP A is called “efficient” if globally it guarantees the Jackson rate of approximation. Some necessary conditions for efficiency and some sufficient ones are derived.
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Jp Journal of Algebra Number Theory and Applications, 2005
The American Mathematical Monthly
Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Inte... more Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Integrales de línea y funciones enteras. Propiedades de las F. enteras. De las F. Analíticas. Dominios conectado simplemente. Singularidades aisladas de una f. analítica. El teorema de residuo. Aplicación del teorema a la evaluación de sumas de integrales. Contorno de las técnicas integrales. Intruducción al mapeo conformal. El teorema de mapeo de Riemann. Teoremas de módulos máximos de dominios no limitados. Funciones armónicas. Distintas formas de f. analíticas. Continuación analítica, f. gamma y zeta. Aplicaciones.
ABSTRACT Incluye bibliografía e índice
Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune ... more Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune either reaches a predetermined goal or is entirely lost. The probability of success is the probability that he will reach his goal. The duration of play is the expected number of bets until the contest is over. This article deals with the effect on both the probability of success and the duration of play if the size of the individual bets is reduced by an integral factor. In particular, we determine when the duration of play is increased most dramatically, and we obtain an upper bound for the ratio of increased duration.
Undergraduate Texts in Mathematics, 2010
ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} &a... more ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} & {z \ne a} \\{f'(a)} & {z = a} \\ } \right. $ g(z) = \left\{ {\begin{array}{*{20}{c}} {f(z) - f(a)} & {z \ne a} \\{f'(a)} & {z = a} \\ \end{array} } \right. then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to g as well as to f. (Note that since f is entire, g is continuous; however, it is not obvious that g is entire.)We begin by showing that the Rectangle Theorem applies to g.
Mathematics Magazine, 2001
Journal of Approximation Theory, 1977
Journal of Approximation Theory, 1982
Undergraduate Texts in Mathematics, 2010
ABSTRACT In the next section, we will see how various types of (real) definite integrals can be a... more ABSTRACT In the next section, we will see how various types of (real) definite integrals can be associated with integrals around closed curves in the complex plane, so that the Residue Theorem will become a handy tool for definite integration.
Undergraduate Texts in Mathematics, 2010
ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f ca... more ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f can be continued analytically to a region D 1 that intersects D if there exists a function g, analytic in D 1 and such that g = f throughout D1 ÇD2 D_1 \cap D_2 . By the Uniqueness Theorem (6.9) any such continuation of f is uniquely determined. (It is possible, however, to have two analytic continuations g 1 and g 2 of a function f to regions D 1 and D 2 respectively with g1 ¹ g2 g_1 \ne g_2 throughout D1 ÇD2 D_1 \cap D_2 . See Exercise 1.)
The American Mathematical Monthly
This note examines the relation among critical points, saddle points, and extremal points of a co... more This note examines the relation among critical points, saddle points, and extremal points of a complex analytic function and thereby presents an aspect of function theory that is in marked contrast to its well-known real-analytic counterpart. Also shown is an interesting application of this relation to the estimation of certain real sums.
Albanian journal of mathematics, Dec 14, 2021
Israel Journal of Mathematics, 1977
ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ... more ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ,n = 1,2, …, ¦a k ≦ Akk}. We consider the rate of approximation by elements ofP A , of continuous functions in [0, 1] which vanish at x = 0. Also a classP A is called “efficient” if globally it guarantees the Jackson rate of approximation. Some necessary conditions for efficiency and some sufficient ones are derived.
[
Jp Journal of Algebra Number Theory and Applications, 2005
The American Mathematical Monthly
Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Inte... more Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Integrales de línea y funciones enteras. Propiedades de las F. enteras. De las F. Analíticas. Dominios conectado simplemente. Singularidades aisladas de una f. analítica. El teorema de residuo. Aplicación del teorema a la evaluación de sumas de integrales. Contorno de las técnicas integrales. Intruducción al mapeo conformal. El teorema de mapeo de Riemann. Teoremas de módulos máximos de dominios no limitados. Funciones armónicas. Distintas formas de f. analíticas. Continuación analítica, f. gamma y zeta. Aplicaciones.
ABSTRACT Incluye bibliografía e índice
Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune ... more Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune either reaches a predetermined goal or is entirely lost. The probability of success is the probability that he will reach his goal. The duration of play is the expected number of bets until the contest is over. This article deals with the effect on both the probability of success and the duration of play if the size of the individual bets is reduced by an integral factor. In particular, we determine when the duration of play is increased most dramatically, and we obtain an upper bound for the ratio of increased duration.
Undergraduate Texts in Mathematics, 2010
ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} &a... more ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} & {z \ne a} \\{f'(a)} & {z = a} \\ } \right. $ g(z) = \left\{ {\begin{array}{*{20}{c}} {f(z) - f(a)} & {z \ne a} \\{f'(a)} & {z = a} \\ \end{array} } \right. then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to g as well as to f. (Note that since f is entire, g is continuous; however, it is not obvious that g is entire.)We begin by showing that the Rectangle Theorem applies to g.
Mathematics Magazine, 2001
Journal of Approximation Theory, 1977
Journal of Approximation Theory, 1982
Undergraduate Texts in Mathematics, 2010
ABSTRACT In the next section, we will see how various types of (real) definite integrals can be a... more ABSTRACT In the next section, we will see how various types of (real) definite integrals can be associated with integrals around closed curves in the complex plane, so that the Residue Theorem will become a handy tool for definite integration.
Undergraduate Texts in Mathematics, 2010
ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f ca... more ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f can be continued analytically to a region D 1 that intersects D if there exists a function g, analytic in D 1 and such that g = f throughout D1 ÇD2 D_1 \cap D_2 . By the Uniqueness Theorem (6.9) any such continuation of f is uniquely determined. (It is possible, however, to have two analytic continuations g 1 and g 2 of a function f to regions D 1 and D 2 respectively with g1 ¹ g2 g_1 \ne g_2 throughout D1 ÇD2 D_1 \cap D_2 . See Exercise 1.)
The American Mathematical Monthly
This note examines the relation among critical points, saddle points, and extremal points of a co... more This note examines the relation among critical points, saddle points, and extremal points of a complex analytic function and thereby presents an aspect of function theory that is in marked contrast to its well-known real-analytic counterpart. Also shown is an interesting application of this relation to the estimation of certain real sums.