Elemer Elad Rosinger

University of Pretoria | UP · Department of Mathematics and Applied Mathematics

Physics depends on “physical intuition”, much of which is formulated in terms of Mathematics. Mathematics itself depends on Logic. The paper presents three latest novelties in Logic which have major consequences in Mathematics. Further, it presents two possible significant departures in Mathematics itself. These five departures can have major impli...

It was shown in De Raedt, Hess and Michielsen [J.Comp.Theor.Nanosci. 8, 1011 (2011) ], cited in the sequel as DRHM, that upon a correct use of the respective statistical data, the celebrated Bell inequalities cannot be violated by quantum systems. This paper presents in more detail the surprisingly elementary, even if rather subtle related basic ar...

It has for long been been overlooked that, quite easily, infinitely many {\it ultrapower} field extensions $\mathbb{F}_{\cal U}$ can be constructed for the usual field $\mathbb{R}$ of real numbers, by using only elementary algebra. This allows a simple and direct access to the benefit of both infinitely small and infinitely large scalars, {\it with...

Infinitely many {\it ultrapower} field extensions $\mathbb{F}_{\cal U}$ are constructed for the usual field $\mathbb{R}$ of real numbers by using only elementary algebra, thus allowing for the benefit of both infinitely small and infinitely large scalars, and doing so {\it without} the considerable usual technical difficulties involved in setting u...

Recently, [3], it was shown that in certain composite quantum systems with time independent potentials, the extent of the entanglement in an initial state is conserved during the time evolution under the Schr\"{o}dinger equation, and thus in the absence of any measurement. Here the {\it extent} of entanglement is meant in the sense of the {\it grad...

So far, the order completion method for solving PDEs, introduced in 1990, can solve by far the most general linear and nonlinear systems of PDEs, with possible initial and/or boundary data. Examples of solving various PDEs with the order completion method are presented. Some of such PDEs do not have global solutions by any other known methods, or a...

A more careful consideration of the recently introduced "Grossone Theory" of Yaroslav Sergeev, [1], leads to a considerable enlargement of what can constitute possible legitimate mathematical theories by the introduction here of what we may call the {\it Syntactic - Semantic Axiomatic Theories in Mathematics}. The usual theories of mathematics, eve...

It is shown that the description as a "frog" of John von Neumann in a recent item by the Princeton celebrity physicist Freeman Dyson does among others miss completely on the immesnely important revolution of the so called "von Neumann architecture" of our modern electronic digital computers.

For the first time in known literature, one studies {\it entanglement dynamics} which is the way the complexity of entanglement may change in time, for instance, in the solution of a Schr/"{o}dinger equation giving the state of a composite quantum system. The paper is a preliminary study which gives the rigorous definition of the respective general...

In this series of essays, I explore and discuss spiritualization of materialistic atheism in support of Andre Comte-Sponville. These essays are further dedicated to Marie-Louise Nykamp. Regardless of being adepts of Darwinism, or not, the presence of animals on our Planet Earth can constitute a most fortunate opportunity to learn about ourselves as...

It appears not to be known that subjecting the axioms to certain conditions, such as for instance to be physically meaningful, may interfere with the logical essence of axiomatic systems, and do so in unforeseen ways, ways that should be carefully considered and accounted for. Consequently, the use of ''physical intuition" in building up axiomatic...

This paper presents the phenomenon of {\it disconnect} in the axiomatic approach to theories of Physics, a phenomenon which appears due to the insistence on axioms which have a physical meaning. This insistence introduces a restriction which is {\it foreign} to the abstract nature of axiomatic systems as such. Consequently, it turns out to introduc...

The often cited book [11] of Asher Peres presents Quantum Mechanics without the use of the Heisenberg Uncertainty Principle, a principle which it calls an "ill-defined notion". There is, however, no argument in this regard in the mentioned book, or comment related to the fact that its use in the realms of quanta is not necessary, let alone, unavoid...

Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized en...

We consider the grading structure on the tensor product corresponding to the tensor rank; the relation to the notion of entanglement is discussed. We also study a complex problem of finding the minimal (with respect to the aforementioned grading structure) representation of elements of the tensor product. The general construction is presented over...

One is reminded in this paper of the often overlooked fact that the geometric straight line, or GSL, of Euclidean geometry is not necessarily identical with its usual Cartesian coordinatisation given by the real numbers in R. Indeed, the GSL is an abstract idea, while the Cartesian, or for that matter, any other specific coordinatisation of it is...

We consider a non-negative integer valued grading function on tensor products which aims to measure the extent of entanglement. This grading, unlike most of the other measures of entanglement, is defined exclusively in terms of the tensor product. It gives a possibility to approach the notion of entanglement in a more refined manner, as the non-ent...

Information being a relatively new concept in science, the likelihood is pointed out that we do not yet have a good enough grasp of its nature and relevance. This likelihood is further enhanced by the ubiquitous use of information which creates the perception of a manifest, yet in fact, rather superficial familiarity. The paper suggests several asp...

It is shown that, unknown to nearly everyone, modern theoretical Physics is significantly {\it constrained} by the tacit acceptance of the ancient Archimedean Axiom imposed upon Geometry by Euclid more than two millennia ago, an axiom which does not seem to have any modern physical motivation. By freeing oneself of this axiom a large variety of sca...

It is mentioned that in physics, much like in everyday life, we are vitally interested in certain abstract concepts, such as, geometry, number, time, or for that matter, monetary value. And contrary to usual views, we can never ever really know what such abstract concepts are. Instead, all that we may know are specific models of such concepts. This...

Two highly consequential limitations in ETI studies are briefly mentioned.

As argued earlier elsewhere, what is the Geometric Straight Line, or in short, the GSL, we shall never know, and instead, we can only deal with various mathematical models of it. The so called standard model, given by the usual linearly ordered field $\mathbb{R}$ of real numbers is essentially based on the ancient Egyptian assumption of the Archime...

It took two millennia after Euclid and until in the early 1880s, when we went beyond the ancient axiom of parallels, and inaugurated geometries of curved spaces. In less than one more century, General Relativity followed. At present, physical thinking is still beheld by the yet deeper and equally ancient Archimedean assumption. In view of that, it...

A family of {\it quantumness spaces} is identified and precisely defined. They are spaces which characterize the difference between states given by classical compositions of systems, and on the other hand, states corresponding to their quantum compositions. Consequently, the quantum composition of systems is {\it reduced} to two classical compositi...

It is shown that under suitable compositions of systems, arbitrary large amounts of entangled type states can easily be obtained.

Recently, P Krugman has suggested that psychology should be included in the theory of economics in view of its critical role played in the behaviour of the large masses of people whose day to day participation impacts so heavily upon economic affairs. Here it is argued that such an inclusion of psychology as an important component of economic theor...

This is a two part paper. The first part, written somewhat earlier, pre-sented standard processes which cannot so easily be accommodated within what are presently considered as physical type realms. The second part further elaborates on that fact. In particular, it is ar-gued that quantum superposition and entanglement may better be understood in e...

Four comments are presented on the book of Roger Penrose entitled "The Road to Reality, A Complete Guide to the Laws of the Uni-verse". The first comment answers a concern raised in the book. The last three point to important omissions in the book. 1. Preliminaries As argued in Appendix 2 of [4], and in a version truncated by the editors in [3], th...

A class of {\it non-Cartesian} physical systems, [7], are those whose composite state spaces are given by significantly extended tensor products. A more detailed presentation of the way such extended tensor products are constructed is offered, based on a step by step comparison with the construction of usual tensor products. This presentation clari...

The following open problem is presented and motivated : Are there physical systems whose state spaces do not compose according to ei-ther the Cartesian product, as classical systems do, or the usual tensor product, as quantum systems do ? "History is written with the feet ..." Ex-Chairman Mao 1. Non-Cartesian Systems The state spaces of systems wit...

The recently proposed and partly developed "Grossone Theory" of Y D Sergeyev is analyzed and partly clarified.

Four comments are presented on the book of Roger Penrose entitled "The Road to Reality, A Complete Guide to the Laws of the Universe". The first comment answers a concern raised in the book. The last three point to important omissions in the book.

Two somewhat long overdue arguments presented here may help in further clarifying the so called "Black Whole War", and beyond that, may be useful in Physics at large. The concept of information while fundamental in Physics, tends to be misunderstood by reductionist slogans such as "information is physical".

The infinitely many differential algebras of generalized functions are naturally subjected to a basic dichotomic sheaf theoretic singularity test regarding their significantly different abilities to deal with large classes of singularities. The property of a vector space or algebra of generalized functions of being a {\it flabby sheaf} proves to be...

Recently, [3], it was shown that Special Relativity is in fact based on one single physical axiom which is that of Reciprocity. Originally, Einstein, [1], established Special Relativity on two physical axioms, namely, the Galilean Relativity and the Constancy of the Speed of Light in inertial reference frames. Soon after, [4,5], it was shown that t...

Mathematical theories are classified in two distinct classes : {\it rigid}, and on the other hand, {\it non-rigid} ones. Rigid theories, like group theory, topology, category theory, etc., have a basic concept - given for instance by a set of axioms - from which all the other concepts are defined in a unique way. Non-rigid theories, like ring theor...

It is a rather universal tacit and unquestioned belief—and even more so among physicists—that there is one and only one real line, namely, given by the coodinatisation of Descartes through the usual field R of real numbers. Such a dramatically limiting and thus harmful belief comes, unknown to equally many, from the similarly tacit acceptance of th...

What makes sets, or more precisely, the category {\bf Set} important in Mathematics are the well known {\it two} specific ways in which arbitrary mappings $f : X \longrightarrow Y$ between any two sets $X, Y$ can {\it fail} to be bijections. Namely, they can fail to be injective, and/or to be surjective. As for bijective mappings they are rather tr...

Much of Mathematics, and therefore Physics as well, have been limited by four rather consequential restrictions. Two of them are ancient taboos, one is an ancient and no longer felt as such bondage, and the fourth is a surprising omission in Algebra. The paper brings to the attention of those interested these four restrictions, as well as the fact...

Singularities appear in numerous important mathematical models used in Physics. And in most of such cases singularities are involved in essentially nonlinear contexts. For more than four decades, general enough nonlinear theories of singularities have been developed. A critically important related feature is that, above certain levels in singularit...

Arguments on the need, and usefulness, of going beyond the usual Hausdorff-Kuratowski-Bourbaki, or in short, HKB concept of topology are presented. The motivation comes, among others, from well known {\it topological type processes}, or in short TTP-s, in the theories of Measure, Integration and Ordered Spaces. These TTP-s, as shown by the classica...

A further significant extension is presented of the infinitely large class of differential algebras of generalized functions which are the basic structures in the nonlinear algebraic theory listed under 46F30 in the AMS Mathematical Subject Classification. These algebras are constructed as {\it reduced powers}, when seen in terms of Model Theory. T...

Two somewhat overlooked aspects of information, namely, total involvement and simultaneous presence, are presented, aspects that may be useful in theories of Physics.

It is a rather universal tacit and unquestioned belief - and even more so among physicists - that there is one and only one set of real scalars, namely, the one given by the usual field $\mathbb{R}$ of real numbers, with its usual linear order structure on the geometric line. Such a dramatically limiting and thus harmful belief comes, unknown to eq...

The presence of infinitesimals is traced back to some of the most general algebraic structures, namely, semigroups, and in fact, magmas, [1], in which none of the structures of linear order, field, or the Archimedean property need to be present. Such a clarification of the basic structures from where infinitesimals can in fact emerge may prove to h...

Deficiencies in Kauffman's proposal regarding a new way for building scientific theories are pointed out. A suggestion to overcome them, and in fact, independently construct mathematical theories which are beyond the reach of Goedel's incompleteness theorem is presented. This suggestion is based on bringing together recent developments in literatur...

Recently delivered lectures on Self-Referential Mathematics, [2], at the Department of Mathematics and Applied Mathematics, University of Pretoria, are briefly presented. Comments follow on the subject, as well as on Inconsistent Mathematics.

There has for longer been an interest in finding equivalent conditions which define inner product spaces, and the respective literature is considerable, see for instance Amir, which lists 350 such results. Here, in this tradition, an alternative definition of orthogonality is presented which does not make use of any inner product. This definition,...

The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of scalars, namely, within one or the other of the infinitely many reduced power algebras whi...

The No-Cloning property in Quantum Computation is known not to depend on the unitarity of the operators involved, but only on their linearity. Based on that fact, here it is shown that the No-Cloning property remains valid when Quantum Mechanics is re-formulated within far wider frameworks of {\it scalars}, namely, one or the other of the infinitel...

Recently, [10,11], the Heisenberg Uncertainty relation and the No-Cloning property in Quantum Mechanics and Quantum Computation, respectively, have been extended to versions of Quantum Mechanics and Quantum Computation which are re-formulated using scalars in {\it reduced power algebras}, [2-9], instead of the usual real or complex scalars. Here, t...

Quantum computation has suggested, among others, the consideration of "non-quantum" systems which in certain respects may behave "quantum-like". Here, what algebraically appears to be the most general possible known setup, namely, of {\it magmas} is used in order to construct "quantum-like" systems. The resulting magmatic composition of systems has...

A method is presented for using the consistent part of inconsistent axiomatic systems.

A characterization of congruences in free semigroups is presented.

Part 1 : For more than two millennia, ever since Euclid's geometry, the so called Archimedean Axiom has been accepted without sufficiently explicit awareness of that fact. The effect has been a severe restriction of our views of space-time, a restriction which above all affects Physics. Here it is argued that, ever since the invention of Calculus b...

The group invariance of entanglement is obtained within a very general and simple setup of the latter, given by a recently introduced considerably extended concept of tensor products. This general approach to entanglement - unlike the usual one given in the particular setup of tensor products of vector spaces - turns out not to need any specific al...

Two successive generalizations of the usual tensor products are given. One can be constructed for arbitrary binary operations, and not only for semigroups, groups or vector spaces. The second one, still more general, is constructed for arbitrary generators on sets.

For a large class of nonlinear evolution PDEs, and more generally, of nonlinear semigroups, as well as their approximating numerical methods, two rather natural stability type convergence conditions are given, one being necessary, while the other is sufficient. The gap between these two stability conditions is analyzed, thus leading to a general no...

Semigroups generated by topological operations such as closure, interior or boundary are considered. It is noted that some of these semigroups are in general finite and noncommutative. The problem is formulated whether they are always finite.

The role of mathematical models in physics has for longer been well established. The issue of their proper building and use appears to be less clear. Examples in this regard from relativity and quantum mechanics are mentioned. Comments concerning a more appropriate way in setting up and using mathematical models in physics are presented.

Two concepts of being Archimedean are defined for arbitrary categories.

The Principle of Relativity has so far been understood as the {\it covariance} of laws of Physics with respect to a general class of reference frame transformations. That relativity, however, has only been expressed with the help of {\it one single} type of mathematical entities, namely, the scalars given by the usual continuum of the field $\mathb...

The problem is posed to find out for arbitrary nonvoid sets $X$ which are all the mappings $T : X \longrightarrow X$ that can be defined and each separately identified through means of categories alone. As argued, this problem may have a certain foundational relevance.

Inverse limits, unlike direct limits, can in general be void, [1]. The existence of fixed points for arbitrary mappings $T : X \longrightarrow X$ is conjectured to be equivalent with the fact that related direct limits of all finite partitions of X are not void.

A rather general ergodic type scheme is presented on arbitrary sets X, as they are generated by arbitrary mappings T : X \longrightarrow X. The structures considered on X are given by suitable subsets of the set of all of its finite partitions. Ergodicity is studied not with respect to subsets of X, but with the {\it inverse limits} of families of...

The parts contributed by the author in recent discussions with several physicists and mathematicians are reviewed, as they have been occasioned by the 2006 book "The Trouble with Physics", of Lee Smolin. Some of the issues addressed are the possible and not yet sufficiently explored relationship between modern Mathematics and theoretical Physics, a...

The comments relate to the often overlooked, if not in fact intentionally disregarded depths of what the so called internal aspects of mathematical knowledge may involve, depths concerning among others issues such as its unreasonable effectiveness in natural sciences, to use the terms of Eugene Wigner, suggested more than four decades ago.

One of the two basic theorems in [5] on the existence of solutions of PDEs is improved with the use of a group analysis type argument.

It took two millennia after Euclid and until in the early 1880s, when we went beyond the ancient axiom of parallels, and inaugurated geometries of curved spaces. In less than one more century, General Relativity followed. At present, physical thinking is still beheld by the yet deeper and equally ancient Archimedean assumption which entraps us into...

This is a two part paper which discusses various issues of cosmic contact related to what so far appears to be a self-imposed censorship implied by the customary acceptance of the Archimedean assumption on space-time.

Polynomial functions $f : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ are studied for which sums of arbitrary length $f (1) + f (2) + f (3) + >... + f (n)$, with $n \in \mathbb{N}_+$, can be expressed by polynomial functions $g : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ which involve a bounded number of operations, thus not depending on $n$. Open...

It is argued that the Copenhagen Interpretation of Quantum Mechanics, founded ontologically on the concept of probability, may be questionable in view of the fact that within Probability Theory itself the ontological status of the concept of probability has always been, and is still under discussion.

A recent general model of entanglement, [5], that goes much beyond the usual one based on tensor products of vector spaces is further developed here. It is shown that the usual Cartesian product can be seen as two extreme particular instances of non-entanglement. Also the recent approach to entanglement in [8] is incorporated in the general model i...

Entanglement is a well known fundamental resource in quantum information. Here the following question is addressed : which are the deeper roots of entanglement that may help in its better understanding and use ? The answer is that one can reproduce the phenomenon of entanglement in a far more general and simple way, a way that goes much beyond the...

Galilean Relativity and Einstein's Special and General Relativity showed that the Laws of Physics go deeper than their representations in any given reference frame. Thus covariance, or independence of Laws of Physics with respect to changes of reference frames became a fundamental principle. So far, all of that has only been expressed within one si...

A particularly easy, even if for long overlooked way is presented for defining globally arbitrary Lie group actions on smooth functions on Euclidean domains. This way is based on the appropriate use of the usual parametric representation of functions. As a further benefit of this way, one can define large classes of genuine Lie semigroup actions. H...

For large classes of systems of polynomial nonlinear PDEs necessary and sufficient conditions are given for the existence of solutions which are discontinuous across hyper-surfaces. These PDEs contain the Navier-Stokes equations, as well as those of General Relativity and Magneto-Hydrodynamics. 1. Preliminary Remarks There has for longer been an in...

The essentials of a new method in solving very large classes of nonlinear systems of PDEs, possibly associated with initial and/or boundary value problems, are presented. The PDEs can be defined by continuous, not necessarily smooth expressions, and the solutions obtained cab be assimilated with usual measurable functions, or even with Hausdorff co...

Several thoughts are presented on the long ongoing difficulties both students and academics face related to Calculus 101. Some of these thoughts may have a more general interest.

A variety of possible extensions of mappings between posets to their Dedekind order completion is presented. One of such extensions has recently been used for solving large classes of nonlinear systems of partial differential equations with possibly associated initial and/or boundary value problems.

A method based on order completion for solving general equations is presented. In particular, this method can be used for solving large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems.

It is shown that the standard Kolmogorov model for probability spaces cannot in general allow the elimination but of only a small amount of probabilistic redundancy. This issue, a purely theoretical weakness, not necessarily related to empirical reality, appears not to have received enough attention in foundational studies of Probability Theory.

The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from...

Ideals of continuous functions which satisfy an off diagonality condition proved to be important connected with the solution of large classes of nonlinear PDEs, and more recently, in General Relativity and Quantum Gravity. Maximal ideals within those which satisfy that off diagonality condition are important since they lead to differential algebras...

Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called space-time foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras has been presented, among them, a global Cauchy-Kovalevskaia t...

Any Lie group G acting on a Euclidean nonvoid open subset M can be seen as a subgroup of the smooth diffeomorphisms Diff^\infty(M,M) of M into itself. Thus actions by such Lie groups G correspond to smooth coordinate transforms on M which, in particular, have smooth inverses. In Rosinger [1, chap. 13], the study of Lie semigroups G in the vastly la...

It is argued that Goedel's incompleteness theorem should be seen as self-evident, rather than unexpected or surprising.

Nonrigid mathematical structures may no longer form usual Eilenberg - Mac Lane categories, but more general ones, as illustrated by pseudo-topologies. A rather general concept of pseudo-topology was used in constructing differential algebras of generalized functions containing the Schwartz distributions, [4-6,8-11]. These algebras proved to be conv...

It is shown for a simple ODE that it has many symmetry groups beyond its usual Lie group symmetries, when its generalized solutions are considered within the nowhere dense differential algebra of generalized functions. 0. Idea and Motivation The standard Lie group theory applied to symmetries of solutions of PDEs, Olver [1-3], deals with classical,...

Two additional reasons are suggested for the seeming lack of progress in producing quantum algorithms.

The role of impossibilities in theories of Physics is mentioned and a recent result is recalled in which Quantum Mechanics is characterized by three information-theoretic impossibilities. The inconvenience of the asymmetries established by such impossibilities is pointed out.

Two questions are suggested as having priority when trying to bring together Quantum Mechanics and General Relativity. Both questions have a scope which goes well beyond Physics, and in particular Quantum Mechanics and General Relativity.

A transition of focus from state space to frames of reference and their transformations is argued as being the appropriate setup for ensuring the covariance of physical laws. Such an approach can not only simplify and clarify aspects of General Relativity, but can possibly help in the development of a Grand Unified Theory as well.

The lack of the Max Born interpretation of the wave function as a probability density describing the localization of a quantum system in configuration space is pointed out related to the recent category based model of quantum mechanics suggested in Abramski & Coecke [1,2] and Coecke.

Following our previous work, we suggest here a large class of algebras of scalars in which simultaneous and correlated computations can be performed owing to the existence of surjective algebra homomorphisms. This may replace the currently used traditional computations in which only real or complex scalars are used, or occasionally, nonstandard one...

There have lately been a variety of attempts to connect, or even explain, if not in fact, reduce human consciousness to quantum mechanical processes. Such attempts tend to draw a sharp and fundamental distinction between the role of consciousness in classical mechanics, and on the other hand, in quantum mechanics, with an insistence on the assumed...