136 research outputs found
Noncommutative lattices
The extended study of non-commutative lattices was begun in 1949 by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began 40 years later with Jonathan Leech\u27s 1989 paper "Skew lattices in rings." Recently, noncommutative generalizations of lattices and related structures have seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this activity is derived in some way from the initiation, over thirty years ago, of Jonathan Leech\u27s program of research that studied noncommutative variations of lattices. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to the renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as handy reference to researchers of noncommutative algebras
Higher-Order Tarski Grothendieck as a Foundation for Formal Proof
We formally introduce a foundation for computer verified proofs based on higher-order Tarski-Grothendieck set theory. We show that this theory has a model if a 2-inaccessible cardinal exists. This assumption is the same as the one needed for a model of plain Tarski-Grothendieck set theory. The foundation allows the co-existence of proofs based on two major competing foundations for formal proofs: higher-order logic and TG set theory. We align two co-existing Isabelle libraries, Isabelle/HOL and Isabelle/Mizar, in a single foundation in the Isabelle logical framework. We do this by defining isomorphisms between the basic concepts, including integers, functions, lists, and algebraic structures that preserve the important operations. With this we can transfer theorems proved in higher-order logic to TG set theory and vice versa. We practically show this by formally transferring Lagrange\u27s four-square theorem, Fermat 3-4, and other theorems between the foundations in the Isabelle framework
Kaehler Differential Algebras for 0-Dimensional Schemes and Applications
The aim of this dissertation is to investigate Kaehler differential algebras and their Hilbert functions for 0-dimensional schemes in P^n. First we give relations between Kaehler differential 1-forms of fat point schemes and another fat point schemes. Then we determine the Hilbert polynomial and give a sharp bound for the regularity index of the module of Kaehler differential m-forms, for 0<m<n+2. Next, we examine the Kaehler differential algebras for fat point schemes whose supports lie on non-singular conics in P^2. Finally, we prove the Segre bounds for equimultiple fat point schemes in P^4, this result allows us to determine the regularity index of the module of Kaehler differential 1-forms, and a sharp bound for the regularity index of the module of Kaehler differential m-forms, for 1<m<6
Exponential Ordering for Nonautonomous Neutral Functional Differential Equations
We study monotone skew-product semiflows generated by families of
nonautonomous neutral functional differential equations with infinite delay and
stable D-operator, when the exponential ordering is considered. Under adequate
hypotheses of stability for the order on bounded sets, we show that the
omega-limit sets are copies of the base to explain the long-term behavior of
the trajectories. The application to the study of the amount of material within
the compartments of a neutral compartmental system with infinite delay, shows
the improvement with respect to the standard ordering.Comment: 29 pages. arXiv admin note: text overlap with arXiv:2401.1770
Closed subgroups of the infinite symmetric group
Let S=Sym(\Omega) be the group of all permutations of a countably infinite
set \Omega, and for subgroups G_1, G_2\leq S let us write G_1\approx G_2 if
there exists a finite set U\subseteq S such that = .
It is shown that the subgroups closed in the function topology on S lie in
precisely four equivalence classes under this relation. Which of these classes
a closed subgroup G belongs to depends on which of the following statements
about pointwise stabilizer subgroups G_{(\Gamma)} of finite subsets
\Gamma\subseteq\Omega holds:
(i) For every finite set \Gamma, the subgroup G_{(\Gamma)} has at least one
infinite orbit in \Omega.
(ii) There exist finite sets \Gamma such that all orbits of G_{(\Gamma)} are
finite, but none such that the cardinalities of these orbits have a common
finite bound.
(iii) There exist finite sets \Gamma such that the cardinalities of the
orbits of G_{(\Gamma)} have a common finite bound, but none such that
G_{(\Gamma)}=\{1\}.
(iv) There exist finite sets \Gamma such that G_{(\Gamma)}=\{1\}.
Some questions for further investigation are discussed.Comment: 33 pages. See also http://math.berkeley.edu/~gbergman/papers and
http://shelah.logic.at (pub. 823). To appear, Alg.Univ., issue honoring
W.Taylor. Main results as before (greater length due to AU formatting), but
some new results in \S\S11-12. Errors in subscripts between displays (12) and
(13) fixed. Error in title of orig. posting fixed. 1 ref. adde
Rules and Meaning in Quantum Mechanics
This book concerns the metasemantics of quantum mechanics (QM). Roughly, it
pursues an investigation at an intersection of the philosophy of physics and
the philosophy of semantics, and it offers a critical analysis of rival
explanations of the semantic facts of standard QM. Two problems for such
explanations are discussed: categoricity and permanence of rules. New results
include 1) a reconstruction of Einstein's incompleteness argument, which
concludes that a local, separable, and categorical QM cannot exist, 2) a
reinterpretation of Bohr's principle of correspondence, grounded in the
principle of permanence, 3) a meaning-variance argument for quantum logic,
which follows a line of critical reflections initiated by Weyl, and 4) an
argument for semantic indeterminacy leveled against inferentialism about QM,
inspired by Carnap's work in the philosophy of classical logic.Comment: 150 page
Structure and Applicability : an Analysis of the Problem of the Applicability of Mathematics
From the introduction: [...] In the first part (Historical Considerations), I will deal with
the historical problem of understanding why the applicability problem has been
dismissed after Frege\u2019s and logicists\u2019 analysis (Chapter 1: A Neglected Problem).
I will show that their answer is no longer satisfying and that such a dismissal
was not due to a real overcoming of the problem.
The second part (Philosophical Problems) will be devoted to the analysis
of the specific philosophical problems lying behind the applicability of mathematics.
First I will discuss Wigner\u2019s (1960) famous analysis (Chapter 2: Do
Miracles Occur? ), and then I will deal with Steiner\u2019s (1998) fundamental work
on the topic (Chapter 3: Applicabilities of Mathematics). As we will see, there
are many philosophical problems concerning the applicability of mathematics.
A further chapter (Chapter 4: Applicability and Ontological Issues) will be
devoted to an analysis of the relations between ontological questions in mathematics
and its applicability and effectiveness in science, in order to remove
any misunderstanding about the possibility that the problems of mathematical
applicability are nothing but a consequence of a certain ontological choice.
Finally, in the third part (An Account for Mathematical Representativeness)
I will offer an original account for one of the main roles played by
mathematics in science: the representative role, which is at the very base of so
many scientific discoveries and improvements in contemporary physics. First,
(Chapter 5: Structures and Applicability) I will present my account in a purely
theoretical way, and then I will offer some concrete examples in support of such
an account (Chapter 6: Some Concrete Examples)
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