On the growth of series in system of functions and Laplace-Stieltjes integrals

For a regularly convergent in ${\Bbb C}$ series $A(z)=\sum\nolimits_{n=1}^{\infty}a_nf(\lambda_nz)$ in the system ${f(\lambda_nz)}$, where$f(z)=\sum\nolimits_{k=0}^{\infty}f_kz^k$ is an entire transcendental function and $(\lambda_n)$is a sequence of positive numbers increasing to $+\infty$, it isinvestigated the relationship between the growth of functions $A$ and $f$ in terms of a generalized order. It is proved that if$a_n\ge 0$ for all $n\ge n_0$, $\ln \lambda_n=o\big(\beta^{-1}\big(c\alpha(\frac{1}{\ln \lambda_n}\ln \frac{1}{a_n})\big)\big)$ for each $c\in (0, +\infty)$ and $\ln n=O(\Gamma_f(\lambda_n))$ as $n\to\infty$ then $\displaystyle\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_A(r))}{\beta(\ln r)}=\varlimsup\limits_{r\to+\infty}\frac{\alpha(\ln M_f(r))}{\beta(\ln r)},$ where $M_f(r)=\max\{|f(z)|\colon |z|=r\}$, $\Gamma_f(r):=\frac{d\ln M_f(r)}{d\ln r}$ and positive continuous on $(x_0, +\infty)$ functions $\alpha$and $\beta$ are such that $\beta((1+o(1))x)=(1+o(1))\beta(x)$, $\alpha(c x)=(1+o(1))\alpha(x)$ and$\frac{d\beta^{-1}(c\alpha(x))}{d\ln x}=O(1)$ as $x\to+\infty$ for each $c\in(0, +\infty)$.\A similar result is obtained for the Laplace-Stieltjes type integral $I(r) = \int\limits_{0}^{\infty}a(x)f(rx) dF(x)$.

Relative Growth of Series in Systems of Functions and Laplace—Stieltjes-Type Integrals

For a regularly converging-in-C series A(z)=∑n=1∞anf(λnz), where f is an entire transcendental function, the asymptotic behavior of the function Mf−1(MA(r)), where Mf(r)=max{|f(z)|:|z|=r}, is investigated. It is proven that, under certain conditions on the functions f, α, and the coefficients an, the equality limr→+∞α(Mf−1(MA(r)))α(r)=1 is correct. A similar result is obtained for the Laplace–Stiltjes-type integral I(r)=∫0∞a(x)f(rx)dF(x). Unresolved problems are formulated.

Multi-Regge limit of the two-loop five-point amplitudes in $$ \mathcal{N} $$ = 4 super Yang-Mills and $$ \mathcal{N} $$ = 8 supergravity

In previous work, the two-loop five-point amplitudes in $$ \mathcal{N} $$ N = 4 super Yang-Mills theory and $$ \mathcal{N} $$ N = 8 supergravity were computed at symbol level. In this paper, we compute the full functional form. The amplitudes are assembled and simplified using the analytic expressions of the two-loop pentagon integrals in the physical scattering region. We provide the explicit functional expressions, and a numerical reference point in the scattering region. We then calculate the multi-Regge limit of both amplitudes. The result is written in terms of an explicit transcendental function basis. For certain non-planar colour structures of the $$ \mathcal{N} $$ N = 4 super Yang-Mills amplitude, we perform an independent calculation based on the BFKL effective theory. We find perfect agreement. We comment on the analytic properties of the amplitudes.

The Earth and Pregivenness in Transcendental Phenomenology

The doctrine of the pregivenness of the world features prominently in Husserl’s numerous phenomenological analyses and descriptions of the role the world plays in our experience. Properly evaluating its function within the overall system of transcendental phenomenology is, however, by no means a straightforward task, as evidenced by many manuscripts from the 1930s. These detail various epistemological and metaphysical difficulties and potential paradoxes encumbering the notion of the pre-given world. This paper contends that some of these difficulties can be alleviated by revisiting Husserl’s late concept of the earth and, more specifically, disclosing its transcendental function in the constitution of pregivenness. To test this claim, I turn to Husserl’s 1931 manuscript describing the paradox of “the originary acquisition of the world.” I argue that the paradox is dissolved by introducing the transcendental-phenomenological concept of the earth.

The First Solution for the Helical Flow of a Generalized Maxwell Fluid within Annulus of Cylinders by New Definition of Transcendental Function BNrrn

Most articles choose the transcendental function B1rrn to define the finite Hankel transform, and very few articles choose B0rrn. The derivations of B0rrn and B1rrn are also considered the same. In this paper, we find that the derivative formulas for the transcend function BNrrn are different and prove the derivative formulas for B0rrn and B1rrn. Based on the exact formulas of B0rrn and B1rrn, we keep on studying the helical flow of a generalized Maxwell fluid between two boundless coaxial cylinders. In this case, the inner and outer cylinders start to rotate around their axis of symmetry at different angular frequencies and slide at different linear velocities at time t=0+. We deduced the velocity field and shear stress via Laplace transform and finite Hankel transform and their inverse transforms. According to generalized G and R functions, the solutions we obtained are given in the form of integrals and series. The solution of ordinary Maxwell fluid has been also obtained by solving the limit of the general solution of fractional Maxwell fluid.

Board Role Games as a Factor in the Development of the Transcendent Function of the Psyche

The article covers issues related to the psychology of KG Jung, namely with the transcendent function of the psyche. The basic concepts and provisions of the analytical approach of KG Jung are considered such as archetype, collective unconsciousness, player position, transcendence, transcendental function and others. The authors investigates such things as transcendent function of psyche as a necessary mental factor for the successful beginning and transference of the individuation process and relations between it and board role games. The article also pays attention to such an issue as the importance of creative activity manifestations for the normalization of the work of the transcendental function and the psyche as a whole. The phenomenon of board role games as manifestations of creative activity is considered. The study of the phenomenon of board role games and their possible connection with the state of the transcendent function of the psyche is considered due to the fact that board role games have gained significant popularity among young people. Peculiarities of game as a type of human activity and board role games in the process of transcendental function development of psyche are determined. The main types of role-playing games in general are named. Definitions of the basic concepts inherent in the subculture of board role-playing games today are given. The authors consider that during the board role-playing game, there is a violent activity of the collective unconscious archetypes, which in turn entails the need to use the transcendental function of the psyche of the players in order to become aware of these undefined mental contents. Based on the above, we can say that the article may be useful for professionals interested in the relationship of the Jungian approach with creative processes, namely its practical side, the importance of modern youth subcultures. Considering the results of the research, the conclusion was made that board role games are the factor in the development of the transcendental function of the psyche.