146 research outputs found
A note on the diagonalizable algebras of PA and ZF
AbstractWe prove that the diagonalizable algebras of PA and ZF are not isomorphic
Matrix functions that preserve the strong Perron-Frobenius property
In this note, we characterize matrix functions that preserve the strong
Perron-Frobenius property using the real Jordan canonical form of a real
matrix.Comment: To appear in The Electronic Journal of Linear Algebr
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
Strongly solvable spherical subgroups and their combinatorial invariants
A subgroup H of an algebraic group G is said to be strongly solvable if H is
contained in a Borel subgroup of G. This paper is devoted to establishing
relationships between the following three combinatorial classifications of
strongly solvable spherical subgroups in reductive complex algebraic groups:
Luna's general classification of arbitrary spherical subgroups restricted to
the strongly solvable case, Luna's 1993 classification of strongly solvable
wonderful subgroups, and the author's 2011 classification of strongly solvable
spherical subgroups. We give a detailed presentation of all the three
classifications and exhibit interrelations between the corresponding
combinatorial invariants, which enables one to pass from one of these
classifications to any other.Comment: v3: 58 pages, revised according to the referee's suggestions; v4:
numbering of sections changed to agree with the published versio
Explicit methods for Hilbert modular forms
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations
- …