A New Look at Generalized Means

Since antiquity, the relationships between 2-tuples and their Pythagorean means have been represented in geometric forms. In this paper, we extend the practice to generalized power means through new representations, and also to 3-tuples. These geometric forms give rise to new algebraic expressions for summary statistics of 2- and 3-tuples.

Sharp power mean bounds for Seiffert mean

In this paper, we find the greatest value p = log 2/（log Tr - log 2） = 1.53.- and the least value q -- 5/3 - 1.66.. such that the double inequality Mp（a,b） 〈 T（a,b） 〈 Mq（a,b） holds for all a, b 〉 0 with a # b. Here, Mp（a, b） and T（a, b） are the p-th power and Seiffertmeans of two positive numbers a and b, respectively.

Generalizations of mixed weighted power mean inequality

Tarnavas established mixed weighted power mean inequality in 1999. A separation of weighted power mean inequslity was derived in this paper. As its applications, some separations of other inequalities were given.

Sharps Bounds for Power Mean in Terms of Contraharmonic Mean

In this research work, we consider the below inequalities: (1.1). The researchers attempt to find an answer as to what are the best possible parameters α, β that (1.1) can be held? The main tool is the optimization of some suitable functions that we seek to find out. Without loss of generality, we have assumed that a > b and let for 1) and a < b, (t small) for 2) to determine the condition for α and β to become f(t) ≤ 0 and g(t) ≥ 0.

Research on Input Power Testing Technology of Smart Toilet

Input power is an important indicator for the safety testing of electrical and electronic products.Smart toilets have been frequently detected that the actual input power is inconsistent with the nominal power.Research and analysis show that the accuracy of the input power measurement results of the smart toilet mainly depends on the temperature control principle and power test method of the product.This article mainly discusses the characteristics of product temperature control,the testing method of mean and peak power and its characteristics,which will provide reference for the input power test of the smart toilet,and will improve the accuracy of power measurement.

RIGHT MEAN FOR THEα-z BURES-WASSERSTEIN QUANTUM DIVERGENCE

The optimization problem to minimize the weighted sum ofα-z Bures-Wasserstein quantum divergences to given positive definite Hermitian matrices has been solved.We call the unique minimizer theα-z weighted right mean,which provides a new non-commutative version of generalized mean(H?lder mean).We investigate its fundamental properties,and give many interesting operator inequalities with the matrix power mean including the Cartan mean.Moreover,we verify the trace inequality with the Wasserstein mean and provide bounds for the Hadamard product of two right means.

The Hybrid Power Mean Involving the Character Sum of Polynomials and a Sum Analogous to Kloosterman Sum

The main purpose of this paper is using the properties of the classical Gauss sum and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sum of polynomials and a sum analogous to Kloosterman sum mod p,an odd prime,and give two sharp asymptotic formulae for them.

THE OPTIMAL GENERALIZED LOGARITHMIC MEAN BOUNDS FOR SEIFFERT'S MEAN

For p ∈ R, the generalized logarithmic mean Lp（a, b） and Seiffert＇s mean T（a, b） of two positive real numbers a and b are defined in （1.1） and （1.2） below respectively. In this paper, we find the greatest p and least q such that the double-inequality Lp（a, b） 〈 T（a,b） 〈 Lq（a,b） holds for all a,b 〉 0 and a ≠ b.

A Symbol Timing Recovery Algorithm of M-PSK Signals for Burst Modem Applications with Small Packet Size

Feedforward symbol timing recovery techniques are particularly important for initial acquisition in burst modems. However, these techniques either have large calculation burden or sensitive to frequency offsets. In this paper, we proposed an efficient symbol timing recovery algorithm of MPSK signals named OMQ(Ordered Maximum power using Quadratic approximation partially) algorithm which is based on the Quadratic Approximation(QA) algorithm. We used ordered statistic sorting method to reduce the computational complexity further, meanwhile maximum mean power principle was used to decrease frequency offset sensitivity. The proposed algorithm adopts estimation-down sampling structure which is suitable for small packet size transmission. The results show that, while comparing with the QA algorithm, the computational complexity is reduced by 75% at most when 8 samples per symbol are used. The proposed algorithm shows better performance in terms of the jitter variance and sensitivity to frequency offsets.

Magic Numbers of the Great Pyramid: A Surprising Result

Recently attention has been drawn to the frequently observed fifth power of the golden mean in many disciplines of science and technology. Whereas in a forthcoming contribution the focus will be directed towards Fibonacci number-based helical structures of living as well as inorganic matter, in this short letter the geometry of the Great Pyramid of the ancient Egyptians was investigated once more. The surprising main result is that the ratio of the in-sphere volume of the pyramid and the pyramid volume itself is given by π⋅φ5, where φ = 0.618033987⋅⋅⋅ is nature’s most important number, the golden mean. In this way not only phase transitions from microscopic to cosmic scale are connected with φ5, also ingenious ancient builders have intuitively guessed its magic before.

Golden Quartic Polynomial and Moebius-Ball Electron

A symmetrical quartic polynomial, named golden one, can be connected to coefficients of the icosahedron equation as well as to the gyromagnetic correction of the electron and to number 137. This number is not a mystic one, but is connected with the inverse of Sommerfeld’s fine-structure constant and this way again connected with the electron. From number-theoretical realities, including the reciprocity relation of the golden ratio as effective pre-calculator of nature’s creativeness, a proposed closeness to the icosahedron may point towards the structure of the electron, thought off as a single-strand compacted helically self-confined charged elemantary particle of less spherical but assumed blunted icosahedral shape generated from a high energy double-helix photon. We constructed a chiral Moebius “ball” from a 13 times 180˚twisted double helix strand, where the turning points of 12 generated slings were arranged towards the vertices of a regular icosahedron, belonging to the non-centrosymmetric rotation group I532. Mathematically put, we convert the helical motion of an energy quantum into a stationary motion on a Moebius stripe structure. The radius of the ball is about the Compton radius. This chiral closed circuit Moebius ball motion profile can be tentatively thought off as the dominant quantum vortex structure of the electron, and the model may be named CEWMB (Charged Electromagnetic Wave Moebius Ball). Also the gyromagnetic factor of the electron (ge = 2.002319) can be traced back to this special structure. However, nature’s energy infinity principle would suggest a superposition with additional less dominant (secondary) structures, governed also by the golden mean. A suggestion about the possible structure of delocalized hole carriers in the superconducting state is given.

Beyond a Quartic Polynomial Modeling of the DNA Double-Helix Genetic Code

By combination of finite number theory and quantum information, the complete quantum information in the DNA genetic code has been made likely by Planat et al. (2020). In the present contribution a varied quartic polynomial contrasting the polynomial used by Planat et al. is proposed that considered apart from the golden mean also the fifth power of this dominant number of nature to adapt the code information. The suggested polynomial is denoted as g(x) = x4 - x3 - (4 - ϕ2 )x2 + (4 – ϕ2)x + 1, where is the golden mean. Its roots are changed to more golden mean based ones in comparison to the Planat polynomial. The new coefficients 4 – ϕ2 instead of 4 would implement the fifth power of the golden mean indirectly applying . As an outlook, it should be emphesized that the connection between genetic code and resonance code of the DNA may lead us to a full understanding of how nature stores and processes compacted information and what indeed is consciousness linking everything with each other suggestedly mediated by all-pervasive dark constituents of matter respectively energy. The number-theoretical approach to DNA coding leads to the question about the helical structure of the electron.

An Extension of the Mitrinovi c-Djokovi e Inequality

A certain generalization of the Mitrinovi c-Djokovi c inequality is proved by means of elementary calculus.

Another Extension of the Mitrinovic-Djokovic Inequality

Another generalization of the Mitrinovic-Djokovic inequality is proved by elementary means.

Expanded Relative Operator Entropies and Operator Valued α-Divergence

For strictly positive operators A and B, and for x ∈ [0,1] and r ∈[-1,1], we investigate the operator power mean A#x,rB=A1/2{（1-x）/＋x（a-1＊2BA-1/2）r}1/rA1/2 If r = O, this is reduced to the geometric operator mean A#x,rB=A1/2{（1-x）/＋x（a-1＊2BA-1/2）r}1/rA1/2 Since A #0,r B = A and A #l,r B = B, weregard A#t,rB as apath combining A and B.Our aim is to show the essential properties of St,r （AIB）. The Tsallis relative operator entropy by Yanagi, Kuriyama and Furuichi can also be expanded, and by using this, we can give an expanded operator valued a-divergence and obtain its properties.

L^2 NORM INEQUALITY WITH POWER WEIGHTS FOR THE MAXIMAL RIESZ SPHERICAL MEANS

We have pointed out in [1] that so far the L^2 norm inequalities with power weights for the Riesz means σ_R~δ(g)(x) of multiple Fourier integrals have been obtained only by Hirschman and J. L. Rubio de Francia respectively:

Comparison of Some Means

The known results on comparison of extended mean values are used to compare power means, Stolarsky means and Heron mean. Some proofs of well known results are simplified with several new results obtained.

On the Fourth Power Mean of the Character Sums Over Short Intervals

The main purpose of this paper is to study the mean value properties of the character sums over the interval [1,p/8） by using the mean value theorems of the Dirichlet L-functions, and give an interesting mean value formula for this study.

On the Sixth Power Mean of the Two-term Exponential Sums

The main purpose of this article is to study the calculating problem of the sixth power mean of the two-term exponential sums,and give an interesting calculating formula for it.At the same time,the paper also provides a new and effective method for the study of the high order power mean of the exponential sums.

ON THE GENERAL k-TH KLOOSTERMAN SUMS AND ITS FOURTH POWER MEAN

The main purpose of this paper is to study the asymptotic property of the fourth power mean of the general k-th Kloosterman sums, and give an interesting asymptotic formula.