This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

This article is intended to give a review of the history of the classical aspects of unified field theories in the 20th century. It includes brief technical descriptions of the theories suggested, short biographical notes concerning the scientists involved, and an extensive bibliography. The present first installment covers the time span between 1914 and 1933, i.e., when Einstein was living and working in Berlin - with occasional digressions into other periods. Thus, the main theme is the unification of the electromagnetic and gravitational fields augmented by short-lived attempts to include the matter field described by Schrodinger's or Dirac's equations. While my focus lies on the conceptual development of the field, by also paying attention to the interaction of various schools of mathematicians with the research done by physicists, some prosopocraphical remarks are included.

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On the History of Unified Field Theories.

A hidden agenda

Adetailed survey ispresentedoftheliteratureonattituderepresentationdatingfromtheearlyworkofEulerand Hamilton to recent publications in e elds such as navigation and control. The scope is limited to the development of the aircraft kinematic transformation equations in terms of four different attitude representations, including the well-known Euler angles, the Euler-axis rotation parameters, the direction cosines, and the Euler ‐Rodrigues quaternion.Theemphasisisdirectedattheapplicationofthequaternionformulationtoaircraftkinematics.Results are presented that reinforce observations that the quaternion formulation, typically implemented to eliminate singularities associated with the Euler angle formulation, is far superior to the other commonly used formulations based on computational efe ciency alone. A development of quaternion constraints necessary to independently constrain roll, pitch, yaw, bank angle, elevation angle, and/or azimuth angle is presented. For verie cation of simulation codes, a general closed-form solution to the quaternion formulation, for the case of constant rotation, is also presented. Additionally, a discussion is provided of numerical integration methods and numerical errors for the quaternion formulation. This discussion is especially important for simulations that may still utilize a common error reduction scheme originally developed for analog computers.

Review of Attitude Representations Used for Aircraft Kinematics

Preface 1. Introduction: a role for history Part I. Human and Artificial Mathematicians: 2. Communicating with automated theorem provers 3. Automated conjecture formation 4. The role of analogy in mathematics Part II. Plausibility, Uncertainty and Probability: 5. Bayesianism in mathematics 6. Uncertainty in mathematics and science Part III. The Growth of Mathematics: 7. Lakatos's philosophy of mathematics 8. Beyond the methodology of mathematical research programmes 9. The importance of mathematical conceptualisation Part IV. The Interpretation of Mathematics: 10. Higher dimensional algebra Appendix Bibliography Index.

/pdf/towards-a-philosophy-of-real-mathematics-55g4j0zop6.pdf

Towards a Philosophy of Real Mathematics

A History of Algebra.

/pdf/on-the-toral-structure-of-lie-p-algebras-3zv3dksg0l.pdf

On the toral structure of Lie p-algebras

Symmetric Lie Algebras

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/23330/0000270.pdf;sequence=1

Cartan Decompositions and Engel Subalgebra Triangulability

Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. L...

David J. Winter

Papers

A History of Algebra.

On the toral structure of Lie p-algebras

Symmetric Lie Algebras

Cartan Decompositions and Engel Subalgebra Triangulability

LIE algops and simple lie algebras