2,232 research outputs found
The injective Leavitt complex
For a finite quiver without sinks, we consider the corresponding finite
dimensional algebra with radical square zero. We construct an explicit
compact generator for the homotopy category of acyclic complexes of injective
-modules. We call such a generator the injective Leavitt complex of .
This terminology is justified by the following result: the differential graded
endomorphism algebra of the injective Leavitt complex of is
quasi-isomorphic to the Leavitt path algebra of . Here, the Leavitt path
algebra is naturally Z-graded and viewed as a differential graded algebra with
trivial differential.Comment: 23 page
The injective and projective Leavitt complexes
For a certain finite graph E, we consider the corresponding finite
dimensional algebra A with radical square zero. An explicit compact generator
for the homotopy category of acyclic complexes of injective (resp. projective)
modules over A, called the injective (resp. projective) Leavitt complex of E,
was constructed in [18] (resp. [19]). We overview the connection between the
injective (resp. projective) Leavitt complex and the Leavitt path algebra of E.
A differential graded bimodule structure, which is right quasi-balanced, is
endowed to the injective (resp. projective) Leavitt complex in [18] (resp.
[19]). We prove that the injective (resp. projective) Leavitt complex is not
left quasi-balanced.Comment: arXiv admin note: text overlap with arXiv:1512.04178,
arXiv:1610.00144; text overlap with arXiv:1301.0195 by other author
Graded Steinberg algebras and partial actions
Given a graded ample Hausdorff groupoid, we realise its graded Steinberg
algebra as a partial skew inverse semigroup ring. We use this to show that for
a partial action of a discrete group on a locally compact Hausdorff topological
space, the Steinberg algebra of the associated groupoid is graded isomorphic to
the corresponding partial skew group ring. We show that there is a one-to-one
correspondence between the open invariant subsets of the topological space and
the graded ideals of the partial skew group ring. We also consider the
algebraic version of the partial -algebra of an abelian group and
realise it as a partial skew group ring via a partial action of the group on a
topological space. Applications to the theory of Leavitt path algebras are
given.Comment: 17 page
Relative Singularity Categories
We study the properties of the relative derived category
() of an abelian category
relative to a full and additive subcategory . In particular, when
\mathscr{A}=A{\text -}\mod for a finite-dimensional algebra over a field
and is a contravariantly finite subcategory of -\mod which
is admissible and closed under direct summands, the -singularity
category ()=()/
is studied. We give a sufficient condition when this category is triangulated
equivalent to the stable category of the Gorenstein category
of .Comment: 17 pages, to appear in Journal of Pure and Applied Algebr
Periodic solutions for neutral evolution equations with delays
The aim is to study the periodic solution problem for neutral evolution
equation
in Banach
space , where is a closed linear operator,
and generates a compact analytic operator semigroup .
With the aid of the analytic operator semigroup theories and some fixed point
theorems, we obtain the existence and uniqueness of periodic mild solution for
neutral evolution equations. The regularity of periodic mild solution for
evolution equation with delay is studied, and some the existence results of the
classical and strong solutions are obtained. In the end, we give an example to
illustrate the applicability of abstract results. Our works greatly improve and
generalize the relevant results of existing literatures
Graded -Theory, Filtered -theory and the classification of graph algebras
We prove that an isomorphism of graded Grothendieck groups of two
Leavitt path algebras induces an isomorphism of a certain quotient of algebraic
filtered -theory and consequently an isomorphism of filtered -theory of
their associated graph -algebras. As an application, we show that, since
for a finite graph with no sinks, of the Leavitt
path algebra coincides with Krieger's dimension group of its adjacency
matrix , our result relates the shift equivalence of graphs to the
filtered -theory and consequently gives that two arbitrary shift equivalent
matrices give stably isomorphic graph -algebras. This result was only
known for irreducible graphs.Comment: 41 pages, comments are very welcome
Applications of Balanced Pairs
Let , be a balanced pair in an abelian category.
We first introduce the notion of cotorsion pairs relative to ,
, and then give some equivalent characterizations when a relative
cotorsion pair is hereditary or perfect. We prove that if the
-resolution dimension of (resp.
-coresolution dimension of ) is finite, then the
bounded homotopy category of (resp. ) is contained
in that of (resp. ). As a consequence, we get that
the right -singularity category coincides with the left
-singularity category if the -resolution dimension of
and the -coresolution dimension of are
finite.Comment: 17 pages, accepted for publication in Science China Mathematic
An explicit projective bimodule resolution of a Leavitt path algebra
We construct an explicit projective bimodule resolution for the Leavitt path
algebra of a row-finite quiver. We prove that the Leavitt path algebra of a
row-countable quiver has Hochschild cohomolgical dimension at most one, that
is, it is quasi-free in the sense of Cuntz-Quillen. The construction of the
resolution relies on an explicit derivation of the Leavitt path algebra
Simple flat Leavitt path algebras are von Neumann regular
For a unital ring, it is an open question whether flatness of simple modules
implies all modules are flat and thus the ring is von Neumann regular. The
question was raised by Ramamurthi over 40 years ago who called such rings
SF-rings (i.e., simple modules are flat). In this note we show that a SF
Steinberg algebra of an ample Hausdorff groupoid, graded by an ordered group,
has an aperiodic unit space. For graph groupoids this implies that the graphs
are acyclic. Combining with the Abrams-Rangaswamy Theorem, it follows that SF
Leavitt path algebras are regular, answering Ramamurthi's question in positive
for the class of Leavitt path algebras
Extreme Learning Machine with Local Connections
This paper is concerned with the sparsification of the input-hidden weights
of ELM (Extreme Learning Machine). For ordinary feedforward neural networks,
the sparsification is usually done by introducing certain regularization
technique into the learning process of the network. But this strategy can not
be applied for ELM, since the input-hidden weights of ELM are supposed to be
randomly chosen rather than to be learned. To this end, we propose a modified
ELM, called ELM-LC (ELM with local connections), which is designed for the
sparsification of the input-hidden weights as follows: The hidden nodes and the
input nodes are divided respectively into several corresponding groups, and an
input node group is fully connected with its corresponding hidden node group,
but is not connected with any other hidden node group. As in the usual ELM, the
hidden-input weights are randomly given, and the hidden-output weights are
obtained through a least square learning. In the numerical simulations on some
benchmark problems, the new ELM-CL behaves better than the traditional ELM
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