A New Proof for Congruent Number’s Problem via Pythagorician Divisors

Considering Pythagorician divisors theory which leads to a new parameterization, for Pythagorician triplets ( a,b,c )∈ ℕ 3∗ , we give a new proof of the well-known problem of these particular squareless numbers n∈ ℕ ∗ , called congruent numbers, characterized by the fact that there exists a right-angled triangle with rational sides: ( A α ) 2 + ( B β ) 2 = ( C γ ) 2 , such that its area Δ= 1 2 A α B β =n;or in an equivalent way, to that of the existence of numbers U 2 , V 2 , W 2 ∈ ℚ 2∗ that are in an arithmetic progression of reason n;Problem equivalent to the existence of: ( a,b,c )∈ ℕ 3∗ prime in pairs, and f∈ ℕ ∗ , such that: ( a−b 2f ) 2 , ( c 2f ) 2 , ( a+b 2f ) 2 are in an arithmetic progression of reason n;And this problem is also equivalent to that of the existence of a non-trivial primitive integer right-angled triangle: a 2 + b 2 = c 2 , such that its area Δ= 1 2 ab=n f 2 , where f∈ ℕ ∗ , and this last equation can be written as follows, when using Pythagorician divisors: (1) Δ= 1 2 ab= 2 S−1 d e ¯ ( d+ 2 S−1 e ¯ )( d+ 2 S e ¯ )=n f 2;Where ( d, e ¯ )∈ ( 2ℕ+1 ) 2 such that gcd( d, e ¯ )=1 and S∈ ℕ ∗ , where 2 S−1 , d, e ¯ , d+ 2 S−1 e ¯ , d+ 2 S e ¯ , are pairwise prime quantities (these parameters are coming from Pythagorician divisors). When n=1 , it is the case of the famous impossible problem of the integer right-angled triangle area to be a square, solved by Fermat at his time, by his famous method of infinite descent. We propose in this article a new direct proof for the numbers n=1 (resp. n=2 ) to be non-congruent numbers, based on an particular induction method of resolution of Equation (1) (note that this method is efficient too for general case of prime numbers n=p≡a ( ( mod8 ) , gcd( a,8 )=1 ). To prove it, we use a classical proof by induction on k , that shows the non-solvability property of any of the following systems ( t=0 , corresponding to case n=1 (resp. t=1 , corresponding to case n=2 )): ( Ξ t,k ){ X 2 + 2 t ( 2 k Y ) 2 = Z 2 X 2 + 2 t+1 ( 2 k Y ) 2 = T 2 , where k∈ℕ;and solutions ( X,Y,Z,T )=( D k , E k , f k , f ′ k )∈ ( 2ℕ+1 ) 4 , are given in pairwise prime numbers.2020-Mathematics Subject Classification 11A05-11A07-11A41-11A51-11D09-11D25-11D41-11D72-11D79-11E25 .

The determination of 52 elements in marine geological samples by an inductively coupled plasma optical emission spectrometry and an inductively coupled plasma mass spectrometry with a high-pressure closed digestion method

An improved analytical method to determine the content of 52 major, minor and trace elements in marine geological samples, using a HF-HCl-HNO_3 acid system with a high-pressure closed digestion method（HPCD）, is studied by an inductively coupled plasma optical emission spectrometry（ICP-OES） and an inductively coupled plasma mass spectrometry（ICP-MS）. The operating parameters of the instruments are optimized, and the optimal analytical parameters are determined. The influences of optical spectrum and mass spectrum interferences, digestion methods and acid systems on the analytical results are investigated. The optimal spectral lines and isotopes are chosen, and internal standard element of rhodium is selected to compensate for matrix effects and analytical signals drifting. Compared with the methods of an electric heating plate digestion and a microwave digestion, a high-pressure closed digestion method is optimized with less acid, complete digestion,less damage for digestion process. The marine geological samples are dissolved completely by a HF-HCl-HNO_3 system, the relative error（RE） for the analytical results are all less than 6.0%. The method detection limits are 2–40μg/g by the ICP-OES, and 6–80 ng/g by ICP-MS. The methods are used to determine the marine sediment reference materials（GBW07309, GBW07311, GBW07313）, rock reference materials（GBW07103, GBW07104,GBW07105）, and cobalt-rich crust reference materials（GBW07337, GBW07338, GBW07339）, the obtained analytical results are in agreement with the certified values, and both of the relative standard deviation（RSD） and the relative error（RE） are less than 6.0%. The analytical method meets the requirements for determining 52 elements contents of bulk marine geological samples.

Magnetic transition of ferromagnetic material at high pressure using a novel system

A system for the investigation of the magnetic properties of materials under high pressure is fabricated based on diamond anvil cell （DAC） technology. The system is designed with an improved coil arranged around the diamond of a non-magnetic DAC. Using this system, the magnetic transition of ferromagnetic （Fe） sample under increasing pressure can be observed. We successfully obtain the evolution of magnetic properties as a function of applied pressure reaching 26.9 GPa in the Fe sample. A magnetic transition is observed at approximately 13 GPa, which is consistent with the theoretical prediction.

A Comparative Study on Rhetoric of University Commencement Speech⁃es Between China and America

The commencement speech becomes pivotal because it inspires,educates,and enlightens the graduates.Therefore,it is of great value to make a study on the speeches.This article analyzes two commencement speeches between China and America from the perspective of rhetoric,namely Aristotle’s appeals as the well as deductive and inductive method of reasoning and finds out that the similarities and differences of those two speeches concerning Aristotle’s appeals as the well as deductive and inductive method of reasoning.

Novel Wind Power Grid-connection System Using Inductive Filtering Technology

Herein,a novel wind power grid-connection system based on inductive filtering is proposed to improve grid-connection compatibility,and is implemented in a 50-MW real system.First,the topology and wiring configuration of the proposed system are discussed.Thereafter,an equivalent circuit and mathematical model are established to reveal the filtering characteristics and resonance damping mechanism of the proposed system.Finally,a 50-MW wind farm-based experimental study,which is conducted to validate the effectiveness and availability of the system is discussed.The experimental results show that the main harmonics,power factor,voltage fluctuation,and flicker satisfy national standards.