173,700 research outputs found
On relativistic elements of reality
Several arguments have been proposed some years ago, attempting to prove the
impossibility of defining Lorentz-invariant elements of reality. I find that a
sufficient condition for the existence of elements of reality, introduced in
these proofs, seems to be used also as a necessary condition. I argue that
Lorentz-invariant elements of reality can be defined but, as Vaidman pointed
out, they won't satisfy the so-called product rule. In so doing I obtain
algebraic constraints on elements of reality associated with a maximal set of
commuting Hermitian operators.Comment: Clarifications, reference added; published versio
A total order in [0,1] defined through a 'next' operator
A `next' operator, s, is built on the set R1=(0,1]-{ 1-1/e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R1. Besides, the orbits {sn(a)}n are all dense in R1 and are constituted by elements of the same arithmetical character: if a is an algebraic irrational of degree k all the elements in a's orbit are algebraic of degree k; if a is transcendental, all are transcendental. Moreover, the asymptotic distribution function of the sequence formed by the elements in any of the half-orbits is a continuous, strictly increasing, singular function very similar to the well-known Minkowski's ?(Ă—) function.Total orders, pierce series, singular functions
An algebraic property of Hecke operators and two indefinite theta series
We prove an algebraic property of the elements defining Hecke operators on
period polynomials associated with modular forms, which implies that the
pairing on period polynomials corresponding to the Petersson scalar product of
modular forms is Hecke equivariant. As a consequence of this proof, we derive
two indefinite theta series identities which can be seen as analogues of
Jacobi's formula for the theta series associated with the sum of four squares.Comment: 11 pages. Published version. Forum Math., published online February
201
Combinatorial cohomology of the space of long knots
The motivation of this work is to define cohomology classes in the space of
knots that are both easy to find and to evaluate, by reducing the problem to
simple linear algebra. We achieve this goal by defining a combinatorial graded
cochain complex, such that the elements of an explicit submodule in the
cohomology define algebraic intersections with some "geometrically simple"
strata in the space of knots. Such strata are endowed with explicit
co-orientations, that are canonical in some sense. The combinatorial tools
involved are natural generalisations (degeneracies) of usual methods using
arrow diagrams.Comment: 20p. 9 fig
On semisimple classes and semisimple characters in finite reductive groups
In this article, we study the elements with disconnected centralizer in the
Brauer complex associated to a simple algebraic group G defined over a finite
field with corresponding Frobenius map F and derive the number of F-stable
semisimple classes of G with disconnected centralizer when the order of the
fundamental group has prime order. We also discuss extendibility of semisimple
characters to their inertia group in the full automorphism group. As a
consequence, we prove that "twisted" and "untwisted" simple groups of type E_6
are "good" in defining characteristic, which is a contribution to the general
program initialized by Isaacs, Malle and Navarro to prove the McKay Conjecture
in representation theory of finite groups
Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras
We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
- …