Quick Reads: Searching for Pythagorean Triples

2012 ◽  
Vol 18 (3) ◽  
pp. 140-143
Author(s):  
Natalya Vinogradova
Keyword(s):  
Pythagorean Triples ◽  
Algebraic Representations

This method emphasizes the strong connections between geometric and algebraic representations.

On the Origin of Four Pi in Physics

2016 ◽  
pp. 3994-4013
Author(s):  
Aaron Hanken
Keyword(s):  
Black Hole ◽  
Surface Area ◽  
Single Particles ◽  
Pythagorean Triples ◽  
Free Particles ◽  
Close Relationship ◽  
Dimensional Parameters ◽  
Gaussian Surface ◽  
Planck Energy ◽  
The Universe

We find the highest symmetry between the fields intrinsic to free particles (free particles having only mass, charge and spin), and show these fields symmetries and their close relationship to force and entropy. The Boltzmann Constant is equal to the natural entropy, in that it is The Planck Energy over The Planck Temperature. This completes a needed symmetry in The Bekenstein-Hawking Entropy. Upon substitution of Planck Units into The Schwarzschild Radius, we find that the mass and radius of any black hole define both the gravitational constant and the natural force. We find that the Gaussian Surface area about a particle is equal to the surface area of an equally massed black hole if we define the gravitational field of that particle to be the quotient of The Planck Force and the particles mass. By these simple substitutions we find that gravity is quantized in units of surface entropy. We also find Pythagorean Triples are resting within the dimensional parameters of Special Relativity, and show this to be the dimensional aspects of single particles observing one another, coupled with the intrinsic Hubble nature of the universe.

scholarly journals TOPOLOGICAL NOETHERIANITY FOR ALGEBRAIC REPRESENTATIONS OF INFINITE RANK CLASSICAL GROUPS

2021 ◽  
Author(s):  
R. H. EGGERMONT ◽  
A. SNOWDEN
Keyword(s):  
Classical Groups ◽  
Infinite Rank ◽  
Polynomial Representations ◽  
Algebraic Representations

AbstractDraisma recently proved that polynomial representations of GL∞ are topologically noetherian. We generalize this result to algebraic representations of infinite rank classical groups.

scholarly journals On Polymorphic Sessions and Functions

2021 ◽  
Vol 43 (2) ◽  
pp. 1-55
Author(s):  
Bernardo Toninho ◽  
Nobuko Yoshida
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Linear Formulation ◽  
Session Types ◽  
Logical Foundation ◽  
Π Calculus ◽  
Soundness And Completeness ◽  
System F ◽  
Algebraic Representations

This work exploits the logical foundation of session types to determine what kind of type discipline for the Λ-calculus can exactly capture, and is captured by, Λ-calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session π-calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the theory of the Λ-calculus: (1) we obtain a characterisation of inductive and coinductive session types via their algebraic representations in System F; and (2) we extend our results to account for value and process passing, entailing strong normalisation.

Proof Without Words: Parametric Representation of Primitive Pythagorean Triples

1996 ◽  
Vol 69 (3) ◽  
pp. 189-189
Author(s):  
Raymond A. Beauregard ◽  
E. R. Suryanarayan
Keyword(s):  
Parametric Representation ◽  
Pythagorean Triples

85.26 Another Method for Pythagorean Triples

2001 ◽  
Vol 85 (503) ◽  
pp. 273
Author(s):  
Hassan A. Shah Ali
Keyword(s):  
Pythagorean Triples

Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function

ZDM ◽  
2010 ◽  
Vol 42 (6) ◽  
pp. 607-619 ◽  
Author(s):  
Michael O. J. Thomas ◽  
Anna J. Wilson ◽  
Michael C. Corballis ◽  
Vanessa K. Lim ◽  
Caroline Yoon
Keyword(s):  
Cognitive Neuroscience ◽  
Algebraic Representations ◽  
Understanding Function

scholarly journals Pythagorean Triples in Cryptography and Associated Networks

2019 ◽  
Vol 8 (12) ◽  
pp. 832-837
Keyword(s):  
Plain Text ◽  
Data Preservation ◽  
Pythagorean Triples ◽  
Increasing Demand ◽  
Secured Data

Any organization is obliged to ensure secrecy of data from hacking criminals complying with the increasing demand for secured data. So data preservation is indispensablethrough cryptographic methods. It is adoptedseveralreal life applications such as ecommerce, In this paper we indicated a procedure for generating different Pythagorean triples and with the help of C++ coding developed a mechanism for both encoding and decoding of a plain text in English alphabets. We also demonstrated them with illustrative examples

THE SHUFFLE VARIANT OF JEŚMANOWICZ’ CONJECTURE CONCERNING PYTHAGOREAN TRIPLES

2011 ◽  
Vol 90 (3) ◽  
pp. 355-370
Author(s):  
TAKAFUMI MIYAZAKI
Keyword(s):  
Unique Solution ◽  
Pythagorean Triples ◽  
Pythagorean Triple ◽  
Positive Integers

AbstractLet (a,b,c) be a primitive Pythagorean triple such that b is even. In 1956, Jeśmanowicz conjectured that the equation ax+by=cz has the unique solution (x,y,z)=(2,2,2) in the positive integers. This is one of the most famous unsolved problems on Pythagorean triples. In this paper we propose a similar problem (which we call the shuffle variant of Jeśmanowicz’ problem). Our problem states that the equation cx+by=az with x,y and z positive integers has the unique solution (x,y,z)=(1,1,2) if c=b+1 and has no solutions if c>b+1 . We prove that the shuffle variant of the Jeśmanowicz problem is true if c≡1 mod b.

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