Abraham A Ungar - Academia.edu (original) (raw)
Books by Abraham A Ungar
Progress in Physics has been created for publications on advanced studies in theoretical and expe... more Progress in Physics has been created for publications on advanced studies in theoretical and experimental physics, including related themes from mathematics and astronomy.
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Papers by Abraham A Ungar
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Springer eBooks, 2014
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Springer eBooks, 2010
Einstein addition admits scalar multiplication between any real number and any relativistically a... more Einstein addition admits scalar multiplication between any real number and any relativistically admissible velocity vector, giving rise to the Einstein gyrovector spaces. As an example, Einstein scalar multiplication enables hyperbolic lines to be calculated with respect to Cartesian coordinates just as Euclidean lines are calculated with respect to Cartesian coordinates. Along with remarkable analogies that Einstein scalar multiplication shares with
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Springer eBooks, 2010
ABSTRACT In Chap. 3, we have seen two important theorems in mechanics. These are Theorem 3.3, p. ... more ABSTRACT In Chap. 3, we have seen two important theorems in mechanics. These are Theorem 3.3, p. 69, about the mass and the center of momentum velocity of a particle system in classical mechanics, and Theorem 3.2, p. 64, about the mass and the center of momentum velocity of a particle system in relativistic mechanics. Theorem 3.3 naturally suggests the introduction of the concept of barycentric coordinates into Euclidean geometry. Guided by analogies, we will see in this chapter how Theorem 3.2 naturally suggests the introduction of the concept of barycentric coordinates into hyperbolic geometry, where they are called gyrobarycentric coordinates.
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Elsevier eBooks, 2018
This chapter results from a change of parameter, changing the parameter P∈ℝn×m P ∈ ℝ n × m , stud... more This chapter results from a change of parameter, changing the parameter P∈ℝn×m P ∈ ℝ n × m , studied in Chapter 4, into a new parameter, V ∈ ℝ c n × m , the space of which is the c -ball ℝ c n × m of the ambient space ℝ n × m , m , n ∈ N . The c -ball, endowed with Einstein V -parameter addition of signature ( m , n ), forms a bi-gyrogroup of signature ( m , n ). The latter admits a scalar multiplication, turning itself into an Einstein bi-gyrovector space of signature ( m , n ). In the special case when m = 1, Einstein bi-gyrogroups and Einstein bi-gyrovector spaces of signature (1, n ) descend to Einstein gyrogroups and Einstein gyrovector spaces studied in Chapters 2 and 3.
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American Mathematical Monthly, May 1, 1991
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Bulletin of the Seismological Society of America, Dec 1, 1973
A method for obtaining a type of progressing waves is introduced. The method is applied to show t... more A method for obtaining a type of progressing waves is introduced. The method is applied to show that [ ( c t − z cosh α ) 2 + r 2 sinh 2 α ] − 1 / 2 F [ sinh − 1 ( c t − z coshα r sinh α ) + i θ ] ( α being a constant) is a progressing wave satisfying the wave equation c 2∇2 φ = ∂2φ/∂ t 2 in cylindrical coordinates r , θ and z , for an arbitrary analytic function F of a complex variable. In terms of this and other similar progressing waves, we consider the problem of wave propagation from a moving point source in two semi-infinite fluid spaces. Both the subsonic and supersonic cases are included. The solutions for a fixed line source and for a stationary point source are obtained as limiting cases.
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This book presents a powerful way to study Einstein's special theory of relativity and its un... more This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. It introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors . Newtonian velocity addition is the common vector addition, which is both commutative and associative. The resulting vector spaces, in turn, form the algebraic setting for the standard model of Euclidean geometry. In full analogy, Einsteinian velocity addition is a gyrovector addition, which is both gyrocom
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Mathematics Interdisciplinary Research, 2016
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Recently discovered tools to study analytic hyperbolic geometry in terms of analogies with analyt... more Recently discovered tools to study analytic hyperbolic geometry in terms of analogies with analytic Euclidean geometry are presented and employed. Special attention is paid to the study of two novel hyperbolic triangle centers that we call hyperbolic Cabrera points of a hyperbolic triangle and to the way they descend to their novel Euclidean counterparts. The two novel hyperbolic Cabrera points are the (1) Cabrera gyrotriangle ingyrocircle gyropoint and the (2) Cabrera gyrotriangle exgyrocircle gyropoint. Accordingly, their Euclidean counterparts to which they descend are the two novel Euclidean Cabrera points, which are the (1) Cabrera triangle incircle point and the (2) Cabrera triangle excircle point.
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![Research paper thumbnail of Erratum: The abstract complex Lorentz transformation group with real metric. I. Special relativity formalism to deal with the holomorphic automorphism group of the unit ball in any complex Hilbert space [J. Math. Phys. <b>35</b>, 1408–1426 (1994)]](https://attachments.academia-assets.com/118141033/thumbnails/1.jpg)
](Journal of Mathematical Physics, Jul 1, 1994
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Computers & mathematics with applications, Feb 1, 2003
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Advances in Applied Clifford Algebras, Oct 12, 2012
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WORLD SCIENTIFIC eBooks, Jun 1, 1991
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Euclidean Barycentric Coordinates The Classical Triangle Centers Triangle Incircle and Excircles ... more Euclidean Barycentric Coordinates The Classical Triangle Centers Triangle Incircle and Excircles Cartesian Models of Hyperbolic Geometry The Interplay of Einstein and Vector Addition Hyperbolic Barycentric Coordinates Hyperbolic Triangle Centers Hyperbolic Triangle Incircle and Excircles
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Physical Review A, May 8, 2002
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Journal of Group Theory, Jan 5, 2000
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arXiv (Cornell University), Feb 22, 2013
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Progress in Physics has been created for publications on advanced studies in theoretical and expe... more Progress in Physics has been created for publications on advanced studies in theoretical and experimental physics, including related themes from mathematics and astronomy.
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
Springer eBooks, 2014
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Springer eBooks, 2010
Einstein addition admits scalar multiplication between any real number and any relativistically a... more Einstein addition admits scalar multiplication between any real number and any relativistically admissible velocity vector, giving rise to the Einstein gyrovector spaces. As an example, Einstein scalar multiplication enables hyperbolic lines to be calculated with respect to Cartesian coordinates just as Euclidean lines are calculated with respect to Cartesian coordinates. Along with remarkable analogies that Einstein scalar multiplication shares with
Bookmarks Related papers MentionsView impact
Springer eBooks, 2010
ABSTRACT In Chap. 3, we have seen two important theorems in mechanics. These are Theorem 3.3, p. ... more ABSTRACT In Chap. 3, we have seen two important theorems in mechanics. These are Theorem 3.3, p. 69, about the mass and the center of momentum velocity of a particle system in classical mechanics, and Theorem 3.2, p. 64, about the mass and the center of momentum velocity of a particle system in relativistic mechanics. Theorem 3.3 naturally suggests the introduction of the concept of barycentric coordinates into Euclidean geometry. Guided by analogies, we will see in this chapter how Theorem 3.2 naturally suggests the introduction of the concept of barycentric coordinates into hyperbolic geometry, where they are called gyrobarycentric coordinates.
Bookmarks Related papers MentionsView impact
Elsevier eBooks, 2018
This chapter results from a change of parameter, changing the parameter P∈ℝn×m P ∈ ℝ n × m , stud... more This chapter results from a change of parameter, changing the parameter P∈ℝn×m P ∈ ℝ n × m , studied in Chapter 4, into a new parameter, V ∈ ℝ c n × m , the space of which is the c -ball ℝ c n × m of the ambient space ℝ n × m , m , n ∈ N . The c -ball, endowed with Einstein V -parameter addition of signature ( m , n ), forms a bi-gyrogroup of signature ( m , n ). The latter admits a scalar multiplication, turning itself into an Einstein bi-gyrovector space of signature ( m , n ). In the special case when m = 1, Einstein bi-gyrogroups and Einstein bi-gyrovector spaces of signature (1, n ) descend to Einstein gyrogroups and Einstein gyrovector spaces studied in Chapters 2 and 3.
Bookmarks Related papers MentionsView impact
American Mathematical Monthly, May 1, 1991
Bookmarks Related papers MentionsView impact
Bulletin of the Seismological Society of America, Dec 1, 1973
A method for obtaining a type of progressing waves is introduced. The method is applied to show t... more A method for obtaining a type of progressing waves is introduced. The method is applied to show that [ ( c t − z cosh α ) 2 + r 2 sinh 2 α ] − 1 / 2 F [ sinh − 1 ( c t − z coshα r sinh α ) + i θ ] ( α being a constant) is a progressing wave satisfying the wave equation c 2∇2 φ = ∂2φ/∂ t 2 in cylindrical coordinates r , θ and z , for an arbitrary analytic function F of a complex variable. In terms of this and other similar progressing waves, we consider the problem of wave propagation from a moving point source in two semi-infinite fluid spaces. Both the subsonic and supersonic cases are included. The solutions for a fixed line source and for a stationary point source are obtained as limiting cases.
Bookmarks Related papers MentionsView impact
This book presents a powerful way to study Einstein's special theory of relativity and its un... more This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. It introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors . Newtonian velocity addition is the common vector addition, which is both commutative and associative. The resulting vector spaces, in turn, form the algebraic setting for the standard model of Euclidean geometry. In full analogy, Einsteinian velocity addition is a gyrovector addition, which is both gyrocom
Bookmarks Related papers MentionsView impact
Mathematics Interdisciplinary Research, 2016
Bookmarks Related papers MentionsView impact
Recently discovered tools to study analytic hyperbolic geometry in terms of analogies with analyt... more Recently discovered tools to study analytic hyperbolic geometry in terms of analogies with analytic Euclidean geometry are presented and employed. Special attention is paid to the study of two novel hyperbolic triangle centers that we call hyperbolic Cabrera points of a hyperbolic triangle and to the way they descend to their novel Euclidean counterparts. The two novel hyperbolic Cabrera points are the (1) Cabrera gyrotriangle ingyrocircle gyropoint and the (2) Cabrera gyrotriangle exgyrocircle gyropoint. Accordingly, their Euclidean counterparts to which they descend are the two novel Euclidean Cabrera points, which are the (1) Cabrera triangle incircle point and the (2) Cabrera triangle excircle point.
Bookmarks Related papers MentionsView impact
Journal of Mathematical Physics, Jul 1, 1994
Bookmarks Related papers MentionsView impact
Computers & mathematics with applications, Feb 1, 2003
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact
Advances in Applied Clifford Algebras, Oct 12, 2012
Bookmarks Related papers MentionsView impact
WORLD SCIENTIFIC eBooks, Jun 1, 1991
Bookmarks Related papers MentionsView impact
Euclidean Barycentric Coordinates The Classical Triangle Centers Triangle Incircle and Excircles ... more Euclidean Barycentric Coordinates The Classical Triangle Centers Triangle Incircle and Excircles Cartesian Models of Hyperbolic Geometry The Interplay of Einstein and Vector Addition Hyperbolic Barycentric Coordinates Hyperbolic Triangle Centers Hyperbolic Triangle Incircle and Excircles
Bookmarks Related papers MentionsView impact
Physical Review A, May 8, 2002
Bookmarks Related papers MentionsView impact
Journal of Group Theory, Jan 5, 2000
Bookmarks Related papers MentionsView impact
arXiv (Cornell University), Feb 22, 2013
Bookmarks Related papers MentionsView impact
Bookmarks Related papers MentionsView impact