The First Five Years of a Phase Theory for Complex Systems and Networks

In this paper,we review the development of a phase theory for systems and networks in its first five years,represented by a trilogy:Matrix phases and their properties;The MIMO LTI system phase response,its physical interpretations,the small phase theorem,and the sectored real lemma;The synchronization of a multi-agent network using phase alignment.Towards the end,we also summarize a list of ongoing research on the phase theory and speculate what will happen in the next five years.

How an Effective “Cosmological Constant” May Affect a Minimum Scale Factor, to Avoid a Cosmological Singularity (Breakdown of the First Singularity Theorem)

We once again reference Theorem6.1.2of the book by Ellis, Maartens, and MacCallum in order to argue that if there is a non zero initial scale factor, that there is a partial breakdown of the Fundamental Singularity theorem which is due to the Raychaudhuri equation. Afterwards, we review a construction of what could happen if we put in what Ellis, Maartens, and MacCallum call the measured effective cosmological constant and substitute Λ→Λeffective in the Friedman equation. i.e. there are two ways to look at the problem, i.e. after Λ→Λeffective, set ΛVac as equal to zero, and have the left over as scaled to background cosmological temperature, as was postulated by Park (2002) or else have ΛVac as proportional to ΛVac^1038GeV2 which then would imply using what we call a 5-dimensional contribution to Λ as proportional to Λ≈Λ5D^-const/Tβ. We find that both these models do not work for generating an initial singularity. Λ removal as a non zero cosmological constant is most easily dealt with by a Bianchi I universe version of the generalized Friedman equation. The Bianchi I universe case almost allows for use of Theorem 6.1.2. But this Bianchi 1 Universe model almost in fidelity with Theorem 6.1.2 requires a constant non zero shear for initial fluid flow at the start of inflation which we think is highly unlikely.

Some Applications of the First Sylow Theorem

In this research,numerical examples of the first Sylow theorem are discussed.Groups,subgroups,cyclic groups,p-group,Sylow p-subgroup and,Cauchy’s theorem were used to illustrate the results.

The discrete variational principle and the first integrals of Birkhoff systems＊

This paper shows that first integrals of discrete equation of motion for Birkhoff systems can be determined explicitly by investigating the invariance properties of the discrete Pfaffian. The result obtained is a discrete analogue of theorem of Noether in the calculus of variations. An example is given to illustrate the application of the results.

The first genetically gene-edited babies: It's “irresponsible and too early”

A scientist, Jiankui He of Southern University of Science and Technology of China,recently claimed at the Second International Summit on Human Genome Editing in Hong Kong on 29 November that he has created the world's first genetically altered babies using CRISPR. This announcement sparked controversy and criticism. The newly developed CRISPR/Cas9 technique has been applied to genetic modification of many kinds of animals. However, the technique is still in its infancy and many questions remain to be answered before it can be used for clinical purposes, especially for reproductive purposes.

Nonlinear Riemann-Hilbert Problem for the First Order Elliptic System with the General Form

Nonlinear boundary problem for a class of the first order elliptic partial differential system with the general form are studied in this paper.And the equivalent nonlinear singular integral equations are established.Then the existence theorem of solution of the problem are also obtained.

New Generating Sets of the First Order Lane-Emden Differential Equations in <i>N</i>-Dimensional Radially Symmetric Polytropes

In the present paper, two new generating sets, of homology invariant functions will be established. Moreover, by the aid of two independent homology invariant functions of each set we established the transformed first order Lane-Emden equation. The first equation for polytropic index n ≠–1, ±∞ depends on five free parameters, while the other equation is for, n = ±∞ and depends on three free parameters.

Positive solutions to (n-1,1) m-point boundary value problems with dependence on the first order derivative

Using the extension of Krasnoselskii＇s fixed point theorem in a cone, we prove the existence of at least one positive solution to the nonlinear nth order m-point boundary value problem with dependence on the first order derivative. The associated Green＇s function for the nth order m-point boundary value problem is given, and growth conditions are imposed on the nonlinear term f which ensures the existence of at least one positive solution. A simple example is presented to illustrate applications of the obtained results.

Transformation Formulas for the First Kind of Lauricella’s Function of Several Variables

Very recently Atash and Al-Gonah [1] derived two extension formulas for Lauricella’s function of the second kind of several variables and . Now in this research paper we derive two families of transformation formulas for the first kind of Lauricella’s function of several variables and with the help of generalized Dixon’s theorem on the sum of the series obtained earlier by Lavoie et al. [2]. Some new and known results are also deduced as applications of our main formulas.

The First Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes a connection between the two branches of calculus:differential calculus and integral calculus.Integral calculus arose from the area problem.Differential calculus arose from the tangent problem.Two parts seem unrelated.But the great Mathematician Newton realized that differentiation and integration are inverse processes.

Dilation,discrimination and Uhlmann's theorem of link products of quantum channels

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.

Does the provision of live black soldier fly and yellow mealworm larvae improve Muscovy duck welfare?

Background The provision of environmental enrichments to Muscovy ducks could reduce the expression of the aggressive behaviors.The aim of the present study was to evaluate the effects of black soldier fly(BSF)and yellow mealworm(YM)live larva provision on Muscovy duck performance,excreta corticosterone metabolites(ECM),behavior,and blood parameters.Methods A total of 1263-day-old female Muscovy ducklings were allotted to 18 pens(6 replicates/treatment,7 birds/pen)and assigned to 3 experimental treatments:a control group fed commercial feed,and two experimental treatments fed commercial feed plus the 5%(based on the expected daily feed intake,as fed basis)of BSF and YM live larvae(BSF and YM groups,respectively).A two-phase feeding program was applied:starter(from 3 to 31 days of age)and grower-finisher(from 32 to 55 days of age).The live weight,average daily gain,average daily feed intake,and feed conversion ratio were calculated.Larva consumption times were collected,and video recordings were performed during 3 periods(P)each day:the hour before(P1),during(P2),and after(P3)the larva administration.ECM were evaluated at 3,31,and 55-day-old.Finally,the total red and white blood cell counts,serum proteins,lipids,and liver and renal function serum enzymes were evaluated on 12 birds/treatment.Results The experimental treatment did not affect the growth performance of the birds(P>0.05).Larva consumption times were always similar between the two insect species,except at 14–18 days of age,were BSF larvae were consumed faster than YM larvae(P<0.001).The birds showed less walking activity during P2,and preening behavior increased in YM birds during P3.The C birds increased the attack behavior over the weeks(P<0.05).During weeks 1–3 the YM group reduced the attack frequency(P1>P3;P<0.05).Finally,the provision of live BSF and YM larvae significantly reduced the ECM at 55 days of age and the heterophil to lymphocyte ratio(P<0.05).Conclusions Live BSF and YM larva supplementation in Muscovy duck improves duck welfare,without impairing birds'growth performance.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

SOME LIOUVILLE-TYPE THEOREMS FOR THE STATIONARY 3D MAGNETO-MICROPOLAR FLUIDS

In this paper,we prove some Liouville-type theorems for the stationary magnetomicropolar fluids under suitable conditions in three space dimensions.We first prove that the solutions are trivial under the assumption of certain growth conditions for the mean oscillations of the potentials.And then we show similar results assuming that the solutions are contained in L^(p)(R^(3))with p∈[2,9/2).Finally,we show the same result for lower values of p∈[1,9/4)with the further assumption that the solutions vanish at infinity.

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

AAK Theorem and a Design of Multidimensional BIBO Stable Filters

Rational approximation theory occupies a significant place in signal processing and systems theory. This research paper proposes an optimal design of BIBO stable multidimensional Infinite Impulse Response filters with a realizable (rational) transfer function thanks to the Adamjan, Arov and Krein (AAK) theorem. It is well known that the one dimensional AAK results give the best approximation of a polynomial as a rational function in the Hankel semi norm. We suppose that the Hankel matrix associated to the transfer function has a finite rank.

Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra

In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

Vanishing Theorems for p-Harmonic Forms on Submanifolds in Spheres

In this paper,we present some vanishing theorems for p-harmonic forms on-super stable complete submanifold M immersed in sphere Sn+m.When 2≤1≤n-2,M has a flat normal bundle.Assuming that M is a minimal submanifold andδ>1(n-1)p2/4n[p-1+(p-1)2kp],we prove a vanishing theorem for p-harmonicℓ-forms.