Proof of the functional equation for the Riemann zeta-function

In this article, we shall prove a result which enables us to transfer from finite to infinite Euler products. As an example, we give two new proofs of the infinite product for the sine function depending on certain decompositions. We shall then prove some equivalent expressions for the functional equation, i.e. the partial fraction expansion and the integral expression involving the generating function for Bernoulli numbers. The equivalence of the infinite product for the sine functions and the partial fraction expansion for the hyperbolic cotangent function leads to a new proof of the functional equation for the Riemann zeta function.

An Initial-Controlled Double-Scroll Hyperchaotic Map

Initial condition-dominated offset boosting provides a special channel to arrange coexisting orbits. Due to the nonlinearity and inherent periodicity, sinusoidal function is often introduced into a dynamical system for multistability design. In this paper, an initial-controlled double-scroll hyperchaotic map is constructed based on two sine functions. Four patterns of the double-scroll hyperchaotic orbits are found as 0-degree, 90-degree, 45-degree and 135-degree. Consequently, different modes for attractor growing are demonstrated. Finally, hardware experiments based on STM32 are carried out to verify the theoretical analysis and numerical simulation.

On the order of the non-primes of type 6n+5 and 6n+1

The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed using three parametric sine functions. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. Furthermore, I will show an exact primality test using these three parametric sine functions.

Dissecting the Roles of the Autonomic Nervous System and Physical Activity on Circadian Heart Rate Fluctuations in Mice

Heart rate (HR) and blood pressure as well as adverse cardiovascular events show clear circadian patterns, which are linked to interdependent daily variations in physical activity and cardiac autonomic nerve system (ANS) activity. We set out to assess the relative contributions of the ANS (alone) and physical activity to circadian HR fluctuations. To do so, we measured HR (beats per minute, bpm) in mice that were either immobilized using isoflurane anesthesia or free-moving. Nonlinear fits of HR data to sine functions revealed that anesthetized mice display brisk circadian HR fluctuations with amplitudes of 47.1±7.4bpm with the highest HRs in middle of the dark (active) period (ZT 18: 589±46bpm) and lowest HRs in the middle of the light (rest) period (ZT 6: 497±54bpm). The circadian HR fluctuations were reduced by ~70% following blockade of cardiac parasympathetic nervous activity (PNA) with atropine while declining by <15% following cardiac sympathetic nerve activity (SNA) blockade with propranolol. Small HR fluctuation amplitudes (11.6±5.9bpm) remained after complete cardiac ANS blockade. Remarkably, circadian HR fluctuation amplitudes in freely moving, telemetrized mice were only ~32% larger than in anesthetized mice. However, after gaining access to running wheels for 1week, circadian HR fluctuations increase to 102.9±12.1bpm and this is linked directly to increased O2 consumption during running. We conclude that, independent of physical activity, the ANS is a major determinant of circadian HR variations with PNA playing a dominant role compared to SNA. The effects of physical activity to the daily HR variations are remarkably small unless mice get access to running wheels.

Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications

In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions for real powers of the inverse cosine (sine) function and the inverse hyperbolic cosine (sine) function. By applying different series expansions for the square of the inverse cosine function, the author not only finds infinite series representations of the circular constant Pi and its square, but also derives two combinatorial identities involving central binomial coefficients.

On partition functions of refined Chern-Simons theories on S3

We present a new expression for the partition function of the refined Chern-Simons theory on S3 with an arbitrary gauge group, which is explicitly equal to 1 when the coupling constant is zero. Using this form of the partition function we show that the previously known Krefl-Schwarz representation of the partition function of the refined Chern-Simons theory on S3 can be generalized to all simply laced algebras.For all non-simply laced gauge algebras, we derive similar representations of that partition function, which makes it possible to transform it into a product of multiple sine functions aiming at the further establishment of duality with the refined topological strings.

Moment functions and exponential monomials on commutative hypergroups

The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions.

Free Vibration Investigations of Rotating FG Beams Resting on Elastic Foundation with Initial Geometrical Imperfection in Thermal Environments

In practical operations of mechanical structures, it is not difficult to meet some large components such as helicopter rotors, gas turbine blades of marine engines, and rotating railway bridges, where these elements can be seen as beam models rotating around one fixed axis. Therefore, mechanical explorations of these structures with and without the effect of temperature will guide the design, manufacture, and use of them in practice. This is the first paper that uses the shear deformation theory-type hyperbolic sine functions and the finite element method to analyze the free vibration response of rotating FGM beams with initial geometrical imperfections resting on elastic foundations considering the effect of temperature. The material properties are assumed to be varied in the thickness direction of the beam based on the power law function and temperature changes The proposed theory and mathematical model are verified by comparing the results with other exact solutions. The numerical investigations have taken into account some geometrical and material parameters to evaluate the effects on the vibration behavior of the structure such as the rotational speed, temperature, as well as initial geometrical imperfections. The drawn comments have numerous scientific and practical implications for rotating beam structures.

On the order of the non-primes of type 6n+5 and 6n+1

The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. By using parametric sine functions, I will show here that all assumptions known today regarding the distribution of prime numbers larger than 3 are wrong. In particular, this refutes Riemann´s hypothesis of random distribution of prime numbers. Furthermore, I will show an exact primality test based on these parametric sine functions, which only uses one parameter.

On the order of the non-primes of type 6n+5 and 6n+1

The fundamental importance of prime numbers for mathematics is that all other natural numbers can be represented as a unique product of prime numbers[1]. For centuries, mathematicians have been trying to find an order in the occurrence of prime numbers. Since Riemann´s paper on the number of primes under a given size, the distribution of the primes is assumed to be random[2]. Here, I show that prime numbers are not randomly distributed. Their positions are determined by the order of occurrence of the nonprimes of types 6n+5 and 6n+1. By using parametric sine functions, I will show here that all assumptions known today regarding the distribution of prime numbers larger than 3 are wrong. In particular, this refutes Riemann´s hypothesis of random distribution of prime numbers. Furthermore, I will show an exact primality test based on these parametric sine functions, which only needs the calculation modes +, -, · , : and only uses one parameter. There is no such deterministic primality test existing until today[3] [4].