42,906 research outputs found
Almost Settling the Hardness of Noncommutative Determinant
In this paper, we study the complexity of computing the determinant of a
matrix over a non-commutative algebra. In particular, we ask the question,
"over which algebras, is the determinant easier to compute than the permanent?"
Towards resolving this question, we show the following hardness and easiness of
noncommutative determinant computation.
* [Hardness] Computing the determinant of an n \times n matrix whose entries
are themselves 2 \times 2 matrices over a field is as hard as computing the
permanent over the field. This extends the recent result of Arvind and
Srinivasan, who proved a similar result which however required the entries to
be of linear dimension.
* [Easiness] Determinant of an n \times n matrix whose entries are themselves
d \times d upper triangular matrices can be computed in poly(n^d) time.
Combining the above with the decomposition theorem of finite dimensional
algebras (in particular exploiting the simple structure of 2 \times 2 matrix
algebras), we can extend the above hardness and easiness statements to more
general algebras as follows. Let A be a finite dimensional algebra over a
finite field with radical R(A).
* [Hardness] If the quotient A/R(A) is non-commutative, then computing the
determinant over the algebra A is as hard as computing the permanent.
* [Easiness] If the quotient A/R(A) is commutative and furthermore, R(A) has
nilpotency index d (i.e., the smallest d such that R(A)d = 0), then there
exists a poly(n^d)-time algorithm that computes determinants over the algebra
A.
In particular, for any constant dimensional algebra A over a finite field,
since the nilpotency index of R(A) is at most a constant, we have the following
dichotomy theorem: if A/R(A) is commutative, then efficient determinant
computation is feasible and otherwise determinant is as hard as permanent.Comment: 20 pages, 3 figure
Persistence Modules on Commutative Ladders of Finite Type
We study persistence modules defined on commutative ladders. This class of
persistence modules frequently appears in topological data analysis, and the
theory and algorithm proposed in this paper can be applied to these practical
problems. A new algebraic framework deals with persistence modules as
representations on associative algebras and the Auslander-Reiten theory is
applied to develop the theoretical and algorithmic foundations. In particular,
we prove that the commutative ladders of length less than 5 are
representation-finite and explicitly show their Auslander-Reiten quivers.
Furthermore, a generalization of persistence diagrams is introduced by using
Auslander-Reiten quivers. We provide an algorithm for computing persistence
diagrams for the commutative ladders of length 3 by using the structure of
Auslander-Reiten quivers.Comment: 48 page
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
The T-algebra spectral sequence: Comparisons and applications
In previous work with Niles Johnson the author constructed a spectral
sequence for computing homotopy groups of spaces of maps between structured
objects such as G-spaces and E_n-ring spectra. In this paper we study special
cases of this spectral sequence in detail. Under certain assumptions, we show
that the Goerss-Hopkins spectral sequence and the T-algebra spectral sequence
agree. Under further assumptions, we can apply a variation of an argument due
to Jennifer French and show that these spectral sequences agree with the
unstable Adams spectral sequence.
From these equivalences we obtain information about filtration and
differentials. Using these equivalences we construct the homological and
cohomological Bockstein spectral sequences topologically. We apply these
spectral sequences to show that Hirzebruch genera can be lifted to
E_\infty-ring maps and that the forgetful functor from E_\infty-algebras in
H\overline{F}_p-modules to H_\infty-algebras is neither full nor faithful.Comment: Minor revisions and more than a few typo corrections. To appear in
Algebraic and Geometric Topolog
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