arXiv subjects
Publications and source records attributed to Liping Li.
Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI$_G$ and VI$_q$, and answer positively an open question of Nagpal on representation stability theory.
In this paper we study representation theory of the category FI$^m$ introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI$^m$ in a natural way. In particular, we prove the representation stability property of finitely generated FI$^m$-modules over fields of characteristic 0.
We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the generic shift functor, then all finitely generated representations of the category have the property (P). In this way, we obtain simple criteria for properties such as Noetherianity, finiteness of Castelnuovo-Mumford regularity, and polynomial growth of dimension to hold. This gives a systemetic and uniform proof of such properties for representations of the categories $\FI_G$ and $\OI_G$ which appear in representation stability theory.
The construction of polar codes for channels other than BECs requires sorting of all bit channels and then selecting the best $K$ of them for a block length $N=2^n$. In this paper, two types of partial orders (PO) of polar codes are incorporated in the construction process to decrease the required computations. Three sets, corresponding to the good bit channels ($\mathcal{I}$), the frozen bit channels ($\mathcal{F}$), and the undetermined bit channels ($\mathcal{U}$), are selected by applying PO relations. The POs are channel independent and are therefore universal for all binary-input discrete memoryless channels. For a given specific channel, a new process, called Dimension Reduction (DR), is proposed in this paper to further reduce the size of $\mathcal{U}$. Our studies show that for $N=10$ and the code rate $R=0.5$ (being the worst code rate), incorporating PO relations alone can determine 50% of the bit channels ($|\mathcal{I}| + |\mathcal{F}| \approx N/2$). With our proposed DR, this number of the determined bit channels goes up to 82%, which brings a significant reduction of computations in the construction of polar codes.
The notion of central stability was first formulated for sequences of representations of the symmetric groups by Putman. A categorical reformulation was subsequently given by Church, Ellenberg, Farb, and Nagpal using the notion of FI-modules, where FI is the category of finite sets and injective maps. We extend the notion of central stability from FI to a wide class of categories, and prove that a module is presented in finite degrees if and only if it is centrally stable. We also introduce the notion of $d$-step central stability, and prove that if the ideal of relations of a category is generated in degrees at most $d$, then every module presented in finite degrees is $d$-step centrally stable.
Consider the regular Dirichlet extension $(\mathcal{E},\mathcal{F})$ for one-dimensional Brownian motion, that $H^1(\mathbb{R})$ is a subspace of $\mathcal{F}$ and $\mathcal{E}(f,g)=\frac12\mathbf{D}(f,g)$ for $f,g\in H^1(\mathbb{R})$. Both $H^1(\mathbb{R})$ and $\mathcal{F}$ are Hilbert spaces under $\mathcal{E}_α$ and hence there is $α$-orthogonal compliment $\mathcal{G}_α$. We give the explicit expression for functions in $\mathcal{G}_α$ which then can be described by another two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the darning method, their regular representations are given.
Let $k$ be a commutative Noetherian ring. In this paper we consider filtered modules of the category FI firstly introduced by Nagpal. We show that a finitely generated FI-module $V$ is filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated FI-modules $V$ whose projective dimension is finite, and describe an upper bound for it. Furthermore, we give a new proof for the fact that $V$ induces a finite complex of filtered modules, and use it as well as a result of Church and Ellenberg to obtain another upper bound for homological degrees of $V$.
In this paper, we shall treat the Schrödinger forms with non-local perturbations. We first extend the definitions of subcriticality, criticality and supercriticality for the Schrödinger forms by Takeda in [IJM, 2015] to the non-local cases in the context of quasi-regular Dirichlet forms. Then we prove an analytic characterization of these definitions via the bottom of the spectrum set.
In this paper we describe a machinery for homological calculations of representations of FI_G, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI_G-modules in terms of torsion degrees of their local cohomology groups.
We give two homological proofs of the Noetherianity of the category $FI$, a fundamental result discovered by Church, Ellenberg, Farb, and Nagpal.
The regular Dirichlet extension is the dual concept of regular Dirichlet subspace. The main purpose of this paper is to characterize all the regular Dirichlet extensions of one-dimensional Brownian motion and to explore their structures. It is shown that every regular Dirichlet extension of one-dimensional Brownian motion may essentially decomposed into at most countable disjoint invariant intervals and an $\mathcal{E}$-polar set relative to this regular Dirichlet extension. On each invariant interval the regular Dirichlet extension is characterized uniquely by a scale function in a given class. To explore the structure of regular Dirichlet extension we apply the idea introduced in [17], we formulate the trace Dirichlet forms and attain the darning process associated with the restriction to each invariant interval of the orthogonal complement of $H^1_\mathrm{e}(\mathbb{R})$ in the extended Dirichlet space of the regular Dirichlet extension. As a result, we find an answer to a long-standing problem whether a pure jump Dirichlet form has proper regular Dirichlet subspaces.
We show that the FI-homology of an FI-module can be computed via a Koszul complex. As an application, we prove that the Castelnuovo-Mumford regularity of a finitely generated torsion FI-module is equal to its degree.
The main purpose of this paper is to explore the structure of regular subspaces of 1-dim Brownian motion. As outlined in \cite{FMG} every such regular subspace can be characterized by a measure-dense set $G$. When $G$ is open, $F=G^c$ is the boundary of $G$ and, before leaving $G$, the diffusion associated with the regular subspace is nothing but Brownian motion. Their traces on $F$ still inherit the inclusion relation, in other words, the trace Dirichlet form of regular subspace on $F$ is still a regular subspace of trace Dirichlet form of one-dimensional Brownian motion on $F$. Moreover we have proved that the trace of Brownian motion on $F$ may be decomposed into two part, one is the trace of the regular subspace on $F$, which has only the non-local part and the other comes from the orthogonal complement of the regular subspace, which has only the local part. Actually the orthogonal complement of regular subspace corresponds to a time-changed Brownian motion after a darning transform.
Let $A$ be a finite dimensional algebra and $D^b(A)$ be the bounded derived category of finitely generated left $A$-modules. In this paper we consider lengths of compact exceptional objects in $D^b(A)$, proving a sufficient condition such that these lengths are bounded by the number of isomorphism classes of simple $A$-modules. Moreover, we show that algebras satisfying this condition is bounded derived simple.
In this paper we use a homological approach to obtain upper bounds for a few homological invariants of $FI_G$-modules $V$. These upper bounds are expressed in terms of the generating degree and torsion degree, which measure the top and socle of $V$ under actions of non-invertible morphisms in the category respectively.
We study the coinduction functor on the category of FI-modules and its variants. Using the coinduction functor, we give new and simpler proofs of (generalizations of) various results on homological properties of FI-modules. We also prove that any finitely generated projective VI-module over a field of characteristic 0 is injective.
It's known that the bit errors of polar codes with successive cancellation (SC) decoding are coupled. We call the coupled information bits the correlated bits. In this paper, concatenation schemes are studied for polar codes (as inner codes) and LDPC codes (as outer codes). In a conventional concatenation scheme, to achieve a better BER performance, one can divide all $N_l$ bits in a LDPC block into $N_l$ polar blocks to completely de-correlate the possible coupled errors. In this paper, we propose a novel interleaving scheme between a LDPC code and a polar code which breaks the correlation of the errors among the correlated bits. This interleaving scheme still keeps the simple SC decoding of polar codes while achieves a comparable BER performance at a much smaller delay compared with a $N_l$-block delay scheme.
Besides the determined construction of polar codes in BEC channels, different construction techniques have been proposed for AWGN channels. The current state-of-the-art algorithm starts with a design-SNR (or an operating SNR) and then processing is carried out to approximate each individual bit channel. However, as found in this paper, for fading channels, an operating SNR can not be directly used in approximating the bit channels. To achieve a better BER performance, the input SNR for the polar code construction in fadding channels is derived. A selection of the design-SNR for both the AWGN and the fading channels from an information theoretical point of view is studied. Also presented in this paper is the study of sacrificing a small data rate to gain orders of magnitude increase in the BER performance.