Some finiteness results for algebraic groups and unramified cohomology over higher-dimensional fields

Abstract Methods are available to measure conductivity, charge, surface potential, carrier density, piezo-electric and other electrical properties with nanometer scale resolution. One of these methods, scanning microwave impedance microscopy (sMIM), has gained interest due to its capability to measure the full impedance (capacitance and resistive part) with high sensitivity and high spatial resolution. This paper introduces a novel data-cube approach that combines sMIM imaging and sMIM point spectroscopy, producing an integrated and complete 3D data set. This approach replaces the subjective approach of guessing locations of interest (for single point spectroscopy) with a big data approach resulting in higher dimensional data that can be sliced along any axis or plane and is conducive to principal component analysis or other machine learning approaches to data reduction. The data-cube approach is also applicable to other AFM-based electrical characterization modes.

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PERSPECTIVES OF ELLIPTICAL CRYPTOGRAPHY TO ENSURE THE INTEGRITY AND CONFIDENTIALITY OF INFORMATION

Visnyk Universytetu “Ukraina” ◽

10.36994/2707-4110-2019-2-23-26 ◽

2019 ◽

Author(s):

Anna ILYENKO ◽

Sergii ILYENKO ◽

Yana MASUR

Keyword(s):

Information Systems ◽

Elliptic Curve ◽

Elliptic Curves ◽

Finite Fields ◽

Computational Efficiency ◽

Digital Signature ◽

Discrete Logarithm ◽

Algebraic Groups ◽

Schnorr Signature ◽

Stability Of Algorithms

In this article, the main problems underlying the current asymmetric crypto algorithms for the formation and verification of electronic-digital signature are considered: problems of factorization of large integers and problems of discrete logarithm. It is noted that for the second problem, it is possible to use algebraic groups of points other than finite fields. The group of points of the elliptical curve, which satisfies all set requirements, looked attractive on this side. Aspects of the application of elliptic curves in cryptography and the possibilities offered by these algebraic groups in terms of computational efficiency and crypto-stability of algorithms were also considered. Information systems using elliptic curves, the keys have a shorter length than the algorithms above the finite fields. Theoretical directions of improvement of procedure of formation and verification of electronic-digital signature with the possibility of ensuring the integrity and confidentiality of information were considered. The proposed method is based on the Schnorr signature algorithm, which allows data to be recovered directly from the signature itself, similarly to RSA-like signature systems, and the amount of recoverable information is variable depending on the information message. As a result, the length of the signature itself, which is equal to the sum of the length of the end field over which the elliptic curve is determined, and the artificial excess redundancy provided to the hidden message was achieved.

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HIGHER DIMENSIONAL ROBERTSON WALKER UNIVERSES INTERACTING WITH BRANS-DICKE FIELD

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.10.80 ◽

2020 ◽

Vol 9 (10) ◽

pp. 8545-8557

Author(s):

K. P. Singh ◽

T. A. Singh ◽

M. Daimary

Keyword(s):

Higher Dimensional

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Celestial Tapestry

10.1093/oso/9780198851950.001.0001 ◽

2020 ◽

Author(s):

Nicholas Mee

Keyword(s):

Mathematics Curriculum ◽

Dimensional Space ◽

Golden Section ◽

Single Point ◽

Historical Context ◽

Computer Art ◽

Lightning Strike ◽

Secret Message ◽

Axiomatic Method ◽

Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

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