74,103 research outputs found
Infinite Descent
Once one accepts that certain things metaphysically depend upon, or are metaphysically explained by, other things, it is natural to begin to wonder whether these chains of dependence or explanation must come to an end. This essay surveys the work that has been done on this issue—the issue of grounding and infinite descent. I frame the discussion around two questions: (1) What is infinite descent of ground? and (2) Is infinite descent of ground possible? In addressing the second question, I will consider a number of arguments that have been made for and against the possibility of infinite descent of ground. When relevant, I connect the discussion to two important views about the way reality can be structured by grounding: metaphysical foundationalism and metaphysical infinitism
Between Atomism and Superatomism
There are at least three vaguely atomistic principles that have come up in the literature, two explicitly and one implicitly. First, standard atomism is the claim that everything is composed of atoms, and is very often how atomism is characterized in the literature. Second, superatomism is the claim that parthood is well-founded, which implies that every proper parthood chain terminates, and has been discussed as a stronger alternative to standard atomism. Third, there is a principle that lies between these two theses in terms of its relative strength: strong atomism, the claim that every maximal proper parthood chain terminates. Although strong atomism is equivalent to superatomism in classical extensional mereology, it is strictly weaker than it in strictly weaker systems in which parthood is a partial order. And it is strictly stronger than standard atomism in classical extensional mereology and, given the axiom of choice, in such strictly weaker systems as well. Though strong atomism has not, to my knowledge, been explicitly identified, Shiver appears to have it in mind, though it is unclear whether he recognizes that it is not equivalent to standard atomism in each of the mereologies he considers. I prove these logical relationships which hold amongst these three atomistic principles, and argue that, whether one adopts classical extensional mereology or a system strictly weaker than it in which parthood is a partial order, standard atomism is a more defensible addition to one’s mereology than either of the other two principles, and it should be regarded as the best formulation of the atomistic thesis
Semigroups, rings, and Markov chains
We analyze random walks on a class of semigroups called ``left-regular
bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon,
and Rockmore. Using methods of ring theory, we show that the transition
matrices are diagonalizable and we calculate the eigenvalues and
multiplicities. The methods lead to explicit formulas for the projections onto
the eigenspaces. As examples of these semigroup walks, we construct a random
walk on the maximal chains of any distributive lattice, as well as two random
walks associated with any matroid. The examples include a q-analogue of the
Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are
``generalized derangement numbers'', which may be of independent interest.Comment: To appear in J. Theoret. Proba
Analogs of Schur functions for rank two Weyl groups obtained from grid-like posets
In prior work, the authors, along with M. McClard, R. A. Proctor, and N. J.
Wildberger, studied certain distributive lattice models for the "Weyl
bialternants" (aka "Weyl characters") associated with the rank two root
systems/Weyl groups. These distributive lattices were uniformly described as
lattices of order ideals taken from certain grid-like posets, although the
arguments connecting the lattices to Weyl bialternants were case-by-case
depending on the type of the rank two root system. Using this connection with
Weyl bialternants, these lattices were shown to be rank symmetric and rank
unimodal, and their rank generating functions were shown to have beautiful
quotient-of-products expressions. Here, these results are re-derived from
scratch using completely uniform and elementary combinatorial reasoning in
conjunction with some new combinatorial methodology developed elsewhere by the
second listed author.Comment: 15 page
Integrable Open Spin Chains and the Doubling Trick in N = 2 SYM with Fundamental Matter
We demonstrate that the one-loop anomalous dimension matrix in N = 2 SYM with
a single chiral hypermultiplet of fundamental matter, which is dual to AdS_5 X
S^5 with a D7-brane filling AdS_5 and wrapped around an $^3 in the S^5, is an
integrable open spin chain Hamiltonian. We also use the doubling trick to
relate these open spin chains to closed spin chains in pure N = 4 SYM. By using
the AdS/CFT correspondence, we find a relation between the corresponding open
and closed strings that differs from a simple doubling trick by terms that
vanish in the semiclassical limit. We also demonstrate that in some cases the
closed string is simpler and easier to study than the corresponding open
string, and we speculate on the nature of corrections due to the presence of
D-branes that this implies.Comment: 30 pages, 14 figure
Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces
We define a Toledo number for actions of surface groups and complex
hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which
allows us to define maximal representations. When the target is not of tube
type we show that there cannot be Zariski-dense maximal representations, and
whenever the existence of a boundary map can be guaranteed, the representation
preserves a finite dimensional totally geodesic subspace on which the action is
maximal. In the opposite direction we construct examples of geometrically dense
maximal representation in the infinite dimensional Hermitian symmetric space of
tube type and finite rank. Our approach is based on the study of boundary maps,
that we are able to construct in low ranks or under some suitable
Zariski-density assumption, circumventing the lack of local compactness in the
infinite dimensional setting.Comment: Comments are welcome! The maximality assumption was unfortunately
missing in Theorem 1.1 and 1.4 of the first versio
- …