404 research outputs found
On ideals of a skew lattice
Ideals are one of the main topics of interest to the study of the order
structure of an algebra. Due to their nice properties, ideals have an important
role both in lattice theory and semigroup theory. Two natural concepts of ideal
can be derived, respectively, from the two concepts of order that arise in the
context of skew lattices. The correspondence between the ideals of a skew
lattice, derived from the preorder, and the ideals of its respective lattice
image is clear. Though, skew ideals, derived from the partial order, seem to be
closer to the specific nature of skew lattices. In this paper we review ideals
in skew lattices and discuss the intersection of this with the study of the
coset structure of a skew lattice.Comment: 16 page
Quantum Exotic PDE's
Following the previous works on the A. Pr\'astaro's formulation of algebraic
topology of quantum (super) PDE's, it is proved that a canonical Heyting
algebra ({\em integral Heyting algebra}) can be associated to any quantum PDE.
This is directly related to the structure of its global solutions. This allows
us to recognize a new inside in the concept of quantum logic for microworlds.
Furthermore, the Prastaro's geometric theory of quantum PDE's is applied to the
new category of {\em quantum hypercomplex manifolds}, related to the well-known
Cayley-Dickson construction for algebras. Theorems of existence for local and
global solutions are obtained for (singular) PDE's in this new category of
noncommutative manifolds. Finally the extension of the concept of exotic PDE's,
recently introduced by A.Pr\'astaro, has been extended to quantum PDE's. Then a
smooth quantum version of the quantum (generalized) Poincar\'e conjecture is
given too. These results extend ones for quantum (generalized) Poincar\'e
conjecture, previously given by A. Pr\'astaro.Comment: 52 page
The type III manufactory
This paper of 9-10 pages is just a first draft, it contains very few proofs. It is possible that some propositions are false, or that some proofs are incomplete or trivially false.Using unusual objects in the theory of von Neumann algebra, as the chinese game Go or the Conway game of life (generalized on finitely presented groups), we are able to build, by hands, many type III factors
Uniqueness and non-uniqueness in percolation theory
This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on and,
more generally, on transitive graphs. For iid percolation on ,
uniqueness of the infinite cluster is a classical result, while on certain
other transitive graphs uniqueness may fail. Key properties of the graphs in
this context turn out to be amenability and nonamenability. The same problem is
considered for certain dependent percolation models -- most prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for others the
proofs are merely sketched, and for yet others they are omitted. Several open
problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The Octonions
The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction
Tensor Network Methods for Invariant Theory
Invariant theory is concerned with functions that do not change under the
action of a given group. Here we communicate an approach based on tensor
networks to represent polynomial local unitary invariants of quantum states.
This graphical approach provides an alternative to the polynomial equations
that describe invariants, which often contain a large number of terms with
coefficients raised to high powers. This approach also enables one to use known
methods from tensor network theory (such as the matrix product state
factorization) when studying polynomial invariants. As our main example, we
consider invariants of matrix product states. We generate a family of tensor
contractions resulting in a complete set of local unitary invariants that can
be used to express the R\'enyi entropies. We find that the graphical approach
to representing invariants can provide structural insight into the invariants
being contracted, as well as an alternative, and sometimes much simpler, means
to study polynomial invariants of quantum states. In addition, many tensor
network methods, such as matrix product states, contain excellent tools that
can be applied in the study of invariants.Comment: 21 page
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