New Inequalities of Cusa–Huygens Type

Using the power series expansions of the functions cotx,1/sinx and 1/sin2x, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on 0,π/2. Our results are much better than those in the existing literature.

Power series expansions of modular forms and p-adic interpolation of the square roots of Rankin–Selberg special values

Let [Formula: see text] be a newform of even weight [Formula: see text] for [Formula: see text], where [Formula: see text] is a possibly split indefinite quaternion algebra over [Formula: see text]. Let [Formula: see text] be a quadratic imaginary field splitting [Formula: see text] and [Formula: see text] an odd prime split in [Formula: see text]. We extend our theory of [Formula: see text]-adic measures attached to the power series expansions of [Formula: see text] around the Galois orbit of the CM point corresponding to an embedding [Formula: see text] to forms with any nebentypus and to [Formula: see text] dividing the level of [Formula: see text]. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic [Formula: see text]-level structure. Also, we restrict these [Formula: see text]-adic measures to [Formula: see text] and compute the corresponding Euler factor in the formula for the [Formula: see text]-adic interpolation of the “square roots”of the Rankin–Selberg special values [Formula: see text], where [Formula: see text] is the base change to [Formula: see text] of the automorphic representation of [Formula: see text] associated, up to Jacquet-Langland correspondence, to [Formula: see text] and [Formula: see text] is a compatible family of grössencharacters of [Formula: see text] with infinite type [Formula: see text].

Parallel Software to Offset the Cost of Higher Precision

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

Analytic continuation and incomplete data tomography

A unique feature of medical imaging is that the object to be imaged has a compact support. In mathematics, the Fourier transform of a function that has a compact support is an entire function. In theory, an entire function can be uniquely determined by its values in a small region, using, for example, power series expansions. Power series expansions require evaluation of all orders of derivatives of a function, which is an imposable task if the function is discretely sampled. In this paper, we propose an alternative method to perform analytic continuation of an entire function, by using the Nyquist–Shannon sampling theorem. The proposed method involves solving a system of linear equations and does not require evaluation of derivatives of the function. Noiseless data computer simulations are presented. Analytic continuation turns out to be extremely ill-conditioned.

On the automaticity of sequences defined by the Thue–Morse and period-doubling Stieltjes continued fractions

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in [Formula: see text] modulo [Formula: see text]. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo [Formula: see text] are [Formula: see text]-automatic.

Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region

Using the operator L c a defined by Carlson and Shaffer, we defined a new subclass of analytic functions ML c a ( λ ; ψ ) defined by a subordination relation to the shell shaped function ψ ( z ) = z + 1 + z 2 . We determined estimate bounds of the four coefficients of the power series expansions, we gave upper bound for the Fekete–SzegőSzegő functional and for the Hankel determinant of order two for f ∈ ML c a ( λ ; ψ ) .

On the Role of Discretization Errors in the Quantification of Parameter Uncertainties

The independence of numerical and parameter uncertainties is investigated for the flow around the KVLCC2 tanker at Re = 4.6 × 106 using the time-averaged RANS equations supplemented by the k–ω two-equation SST model. The uncertain input parameter is the inlet velocity that varies ±0.25% and ±0.50% for the determination of sensitivity coefficients using finite-difference approximations. The quantities of interest are the friction and pressure coefficients of the ship and the Cartesian velocity components and turbulence kinetic energy at the propeller plane. A grid refinement study is performed for the nominal conditions to allow the estimation of the discretization error with power series expansions. However, for grids between 6 × 106 and 47.6 × 106 cells, not all the selected quantities of interest exhibit monotonic convergence. Therefore, the estimates of the sensitivity coefficients of the selected quantities of interest using the local sensitivity method and finite-differences performed for refinement levels that correspond to 0.764 × 106, 6 × 106 and 47.6 × 106 cells lead to significantly different values. Nonetheless, for a given grid, negligible differences are obtained for the sensitivity coefficients obtained with two different intervals in the finite-differences approximation. Discrepancies between sensitivity coefficients are compared with the estimated numerical uncertainties. Results obtained in the study suggest that uncertainty quantification performed in coarse grids may be significantly affected by discretization errors.

Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis

We study the three-term recurrence coefficients [Formula: see text] of polynomial sequences orthogonal with respect to a perturbed linear functional depending on a variable [Formula: see text] We obtain power series expansions in [Formula: see text] and asymptotic expansions as [Formula: see text] We use our results to settle some conjectures proposed by Walter Van Assche and collaborators.

КЭЭ БИР ЭЛЕМЕНТАРДЫК ФУНКЦИЯЛАРДЫ ДАРАЖАЛУУ КАТАРГА АЖЫРАТУУДА СТУДЕНТТЕРДИН ЧЫГАРМАЧЫЛЫК ОЙ ЖҮГҮРТҮҮСҮН ЖОГОРУЛАТУУНУН ПЕДАГОГИКАЛЫК ЫКМАЛАРЫ

Аннотация: Биздин заманда билим алууга болгон көз караш өзгөрдү: мурун маалымат алуу абдан маанилүү болсо, азыр маалыматтарды колдонууну билиш керек. Себеби, азыркы турмушта Google сыяктуу маалымат булактары бар. Биз биргелешкен математика курсу синергияны пайда кылып, алгебра менен геометриянын элементтерин өздөштүрүүгө жардам берет деп ишенебиз. Алгебралык, дифференциалдык жана интегралдык теӊдемелердин жакындаштырылган чыгарылыштарын тургузууда жана ошондой эле ар кандай интегралдарды баалоодо параметрдин же көз карандысыз өзгөрүлмөнүн даражасы бар катарлар менен иштөөгө туура келет. Негизинен даражалуу катарга ажыратуу Ньютондун биномунун формуласынын жардамы менен же Тейлордун катарын колдонуу жолу аркылуу тургузулат. Бул илимий макалада ошол тууралуу сөз болот. Түйүндүү сөздөр: Тейлордун катары, Маклорендин катары, катарга ажыратуу, көрсөткүчтүү функция, тригонометриялык функциялар, сумма, интервал, бардык чыныгы сандардын огу, жыйналуучу катар, Коши-Адамардын формуласы, Лагранж формуласындагы калдык мүчө, көрсөткүчү бар биномдук катар, логарифмикалык функция, барабардык, касиеттер, аргументтин мааниси, даража, тактык, тартип, баалоо. Аннотация: В области математики знание точных формулировок определений, теорем и т.п. теперь не столь важно, как умение их использовать для решения задач, связанных с окружающей действительностью. Мы убеждены в том, что курс математики, объединяющий элементы алгебры и геометрии поможет повысить уровень усвоения материала за счет эффекта синергии, возникающего при этом. При построении приближенных решений алгебраических, дифференциальных и интегральных уравнений, а также при оценке различных интегралом нам приходится иметь дело с рядами по степеням параметра или независимой переменной. Такие разложения в степенные ряды строятся обычно либо с помощью формулы бинома Ньютона, либо путем использования рядов Тейлора. О них и пойдет речь ниже. Ключевые слова: Ряд Тейлора, ряд Маклорена, разложения в ряд, Показательная функция, тригонометрические функции, сумма, интервал, на всей действительной оси, сходящийся ряд, формула Коши-Адамара, остаточный член в формуле Лагранжа, биноминальный ряд с показателем , логарифмическая функция, равенства, свойства, значение аргумента, степень, точность, порядок, оценка. Аnnotation: Nowadays, getting general information is easy an ditisim portant to beable to correctly interpretand use existing data. In the field of mathematics, knowledge of exact formulations of definitions, theorems, etc. now it is not so important as the ability to use them for solving problems related to the surround dingreality. We are convinced that the course of mathematics, combining the elements of Algebra and Geometry, will help to in crease the level of mastering matterdueto the synergy effect thatarises. In constructing approximate solutions of algebraic differential, and integral equations, as well as in estimating various integrals, we have to deal with series in powers of a parameter or an independent variable. Such power series expansions are usually constructed either using the Newton binomial formula, or by using the Taylor series. About them find it below. Keywords: Taylor series, Maclaurin series, series expansions, Exponential function, trigonometric functions, sum, interval, on the whole real axis, convergent series, Cauchy-Hadamard formula, residual term in Lagrange formula, binomial series with exponent μ, logarithm function, equalities, properties, argument value, degree, accuracy, order, evaluation.