NEW PERIODIC SOLUTIONS OF ITO'S 5th-ORDER mKdV EQUATION AND ITO'S 7th-ORDER mKdV EQUATION

Based on the modified Jocobi elliptic function expansion method and the modified extended tanh function method,a new algebraic method is presented to obtain mu ltiple travelling wave solutions for nonlinear wave equations.By using the metho d,Ito's 5th order and 7th order mKdV equations are studied in detail and more new exact Jocobi elliptic function periodic solutions are found.With modulus m→1 or m→0,these solutions degenerate into corresponding solitary wave s olutions,shock wave solutions and trigonometric function solutions.

The Proof of the Generalized Piemann’s Hypothesis

The article presents the proof of the validity of the generalized Riemann hypothesis on the basis of adjustment and correction of the proof of the Riemanns hypothesis in the work?[1], obtained by a finite exponential functional series and finite exponential functional progression.

SYNTHESIS OF FUNCTIONALIZED CALIX[4]ARENE BEARING [M]DAZOLE MOIETY TO SERVE AS CATALYTIC FUNCTION

Synthesis of catix[4]arene with one imidazolyl group on the upper rim is presented.

Calculation of Response Function for Bonner Sphere Spectrometer Based on Geant4

In order to make further studies on fusion neutron diagnosis on HL-2A /HL-2M,we have developed and succeeded in the calculation of the Response Function for a Bonner sphere spectrometer,which consists of eight polyethylene spheres with ^3He proportional counters inside.The response function of the Bonner spectrometer to neutrons is of fundamental importance for its neutron spectrum unfolding procedure and is directly related to the quality of the unfolded spectrum.In this paper,we calculated the response function to neutrons from 10^-9 MeV to100 MeV by Geant4.In order to test the accuracy of the Geant4 simulation,we apply it to measure an ^241Am-Be neutron source,and the measured neutron counts of the spectrometer and simulated counts are found to be highly consistent,with a relative error up to 9.3%.This has proven the calculation of the neutron response of the Bonner sphere spectrometer by Geant4 to be quite accurate.

Theoretical Investigation of the Mechanism of Rh(III)-catalyzed Annulation of 2-Biphenylboronic Acid with Activated Alkene

The mechanism is investigated for Cp^(tBu)Rh(OH)_(2)-catalyzed annulation of 2-biphenylboronic acid with three activated alkenes using M06-2X functional.The reaction comprises transmetalation via two steps and following C-H activation producing reactive Rh-biphenyl complex with two Rh—Cσbonds.After the coordination/insertion of alkenes,respective fused or bridged cyclic products are yielded depending on different alkenes accompanied by the release of CptBuRh.The promotion of Cp^(tBu)Rh(OH)_(2) lies in the barrier decrease of transmetalation and C-H activation ready for coordination/insertion ensuring the smooth progress of common rate-limiting reductive elimination.The stereoselective transfer and ring rotation are specific for benzoquinone and cyclopropenone.The role of Rh(III)catalyst and release of Rh(I)is supported by Multiwfn analysis on frontier molecular orbital(FMO)of specific transiton states(TSs)and Mayer bond order(MBO)value of vital bonding,breaking.

About Spherical Functions on SL(2,R)

In this paper,we will discuss some properties of the(n,m)-spherical fuctions on the Lie group G=SL(2, R),and obtain the decomposition of f in C_c^4(G)into these functions.Also we give the Fourier inversion formula for the(n,m)-spherical functions in C_c^3(G).

Positive Periodic Solution of a Discrete Predator-prey Patch-system

A periodic difference predator-prey model with Holling-（m ＋ 1）（m 〉 2） type functional response and impulses is established. Sufficient conditions are derived for the existence of periodic solutions by using a continuation theorem in coincidence degree.

Non-commutative Arithmetics on Thompson's Monoid

The main purpose of this paper is to define prime and introduce non-commutative arithmetics based on Thompson's group F.Defining primes in a non-abelian monoid is highly non-trivial,which relies on a concept called"castling".Three types of castlings are essential to grasp the arithmetics.The divisor functionτon Thompson's monoid S satisfiesτ(uv)≤τ(u)τ(v)for any u,v∈S.Then the limitτ_(0)(u)=lim_(n→∞)(τ(u~n))^(1/n)exists.The quantityC(S)=sup_(1≠u∈S)τ_(0)(u)/τ(u)describes the complexity for castlings in S.We show thatC(S)=1.Moreover,the MCbius function on S is calculated.And the Liouville functionCon S is studied.