In this paper, we develop the notion of c-almost periodicity for functions defined on vertical strips in the complex plane. As a generalization of Bohr's concept of almost periodicity, we study the main properties...
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In this paper, we develop the notion of c-almost periodicity for functions defined on vertical strips in the complex plane. As a generalization of Bohr's concept of almost periodicity, we study the main properties of this class of functions which was recently introduced for the case of one real variable. In fact, we extend some important results of this theory which were already demonstrated for some particular cases. In particular, given a non-null complex number c, we prove that the family of vertical translates of a prefixed c-almost periodic function defined in a vertical strip U is relatively compact on any vertical substrip of U, which leads to proving that every c-almost periodic function is also almost periodic and, in fact, c(m)-almost periodic for each integer number m.
Based on Bohr's equivalence relation for general Dirichlet series, in this paper we connect the families of equivalent exponential polynomials with a geometrical point of view related to lines in crystal-like stru...
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Based on Bohr's equivalence relation for general Dirichlet series, in this paper we connect the families of equivalent exponential polynomials with a geometrical point of view related to lines in crystal-like structures. In particular we characterize this equivalence relation, and give an alternative proof of Bochner's property referring to these functions, through this new geometrical perspective.
As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of c-almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results...
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As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of c-almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results of this theory which were already demonstrated for the standard cases. In particular, we prove that every c-almost periodic function in the sense of Stepanov approach (in the sense of equiWeyl or Weyl approaches, respectively) is also c(m) -almost periodic in the sense of Stepanov approach (in the sense of equiWeyl or Weyl approaches, respectively) for each non-zero integer number m. This study is performed for both representative cases of functions defined on the real axis and with values in a Banach space and the complexfunctions defined on vertical strips in the complex plane.
Recent results by R. Szász, individually and in collaboration with E. Deniz and M. Çaglar, and foundational work by U. Jayatilake collectively combine to provide a proof of Brannan's longstanding coeffic...
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Recent results by R. Szász, individually and in collaboration with E. Deniz and M. Çaglar, and foundational work by U. Jayatilake collectively combine to provide a proof of Brannan's longstanding coefficient conjecture for the case that β = 1 . The resulting cumulative proof of the conjecture relies on a computer-assisted argument. In this paper, we present new results and use them to provide a complete direct analytical proof of the conjecture.
The inverse Mills ratio is R := phi/Psi, where phi and are Psi respectively, the probability density function and the tail function of the standard normal distribution. Exact bounds on R(z) for complex z with Rz >=...
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The inverse Mills ratio is R := phi/Psi, where phi and are Psi respectively, the probability density function and the tail function of the standard normal distribution. Exact bounds on R(z) for complex z with Rz >= 0 are obtained, which then yield logarithmically exact upper bounds on high-order derivatives of R. These results complement the many known bounds on the (inverse) Mills ratio of the real argument. The main idea of the proof is a non-asymptotic version of the so-called stationary-phase method. This study was prompted by a recently discovered alternative to the Euler-Maclaurin formula.
This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling t...
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ISBN:
(数字)9783319940632
ISBN:
(纸本)9783319940625
This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling this text to serve well in courses for students in engineering or applied sciences. The overall aim in designing this text is to accommodate students of different mathematical backgrounds and to achieve a balance between presentations of rigorous mathematical proofs and applications.;The text is adapted to enable maximum flexibility to instructors and to students who may also choose to progress through the material outside of coursework. Detailed examples may be covered in one course, giving the instructor the option to choose those that are best suited for discussion. Examples showcase a variety of problems with completely worked out solutions, assisting students in working through the exercises. The numerous exercises vary in difficulty from simple applications of formulas to more advanced project-type problems. Detailed hints accompany the more challenging problems. Multi-part exercises may be assigned to individual students, to groups as projects, or serve as further illustrations for the instructor. Widely used graphics clarify both concrete and abstract concepts, helping students visualize the proofs of many results.;Freely accessible solutions to every-other-odd exercise are posted to the book’s Springer website. Additional solutions for instructors’ use may be obtained by contacting the authors directly.
This monograph examines rotation sets under the multiplication by;(mod 1) map and their relation to degree d polynomial maps of the complex plane. These sets are higher-degree analogs of the corresponding sets under t...
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ISBN:
(数字)9783319788104
ISBN:
(纸本)9783319788098
This monograph examines rotation sets under the multiplication by;(mod 1) map and their relation to degree d polynomial maps of the complex plane. These sets are higher-degree analogs of the corresponding sets under the angle-doubling map of the circle, which played a key role in Douady and Hubbard's work on the quadratic family and the Mandelbrot set. Presenting the first systematic study of rotation sets, treating both rational and irrational cases in a unified fashion, the text includes several new results on their structure, their gap dynamics, maximal and minimal sets, rigidity, and continuous dependence on parameters. This abstract material is supplemented by concrete examples which explain how rotation sets arise in the dynamical plane of complex polynomial maps and how suitable parameter spaces of such polynomials provide a complete catalog of all such sets of a given degree. As a main illustration, the link between rotation sets of degree 3 and one-dimensional families of cubic polynomials with a persistent indifferent fixed point is outlined.
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