arXiv subjects
Publications and source records attributed to Liping Li.
We answer positively a question of Church, Miller, Nagpal and Reinhold on existence of a linear bound on the presentation degree of the homology of a complex of FI-modules. This implies a linear stable range for the homology of congruence subgroups of general linear groups.
Polar codes with memory (PCM) are proposed in this paper: a pair of consecutive code blocks containing a controlled number of mutual information bits. The shared mutual information bits of the succeeded block can help the failed block to recover. The underlying polar codes can employ any decoding scheme such as the successive cancellation (SC) decoding (PCM-SC), the belief propagation (BP) decoding (PCM-BP), and the successive cancellation list (SCL) decoding (PCM-SCL). The analysis shows that the packet error rate (PER) of PCM decreases to the order of PER squared while maintaining the same complexity as the underlying polar codes. Simulation results indicate that for PCM-SC, the PER is comparable to (less than 0.3 dB) the stand-alone SCL decoding with two lists for the block length $N=256$. The PER of PCM-SCL with $L$ lists can match that of the stand-alone SCL decoding with $2L$ lists. Two hardware decoders for PCM are also implemented: the in-serial (IS) decoder and the low-latency interleaved (LLI) decoder. For $N=256$, synthesis results show that in the worst case, the latency of the PCM LLI decoder is only $16.1\%$ of the adaptive SCL decoder with $L=2$, while the throughput is improved by 13 times compared to it.
In this paper, we are concerned with polymer models based on $α$-stable processes, where $α\in (\frac{d}{2},d\wedge 2)$ and $d$ stands for dimension. They are attached with a delta potential at the origin and the associated Gibbs measures are parametrized by a constant $γ$ playing the role of inverse temperature. Phase transition exhibits with critical value $γ_{cr}=0$. Our first object is to formulate the associated Dirichlet form of the canonical Markov process $X^{(γ)}$ induced by the Gibbs measure for a globular state $γ>0$ or the critical state $γ=0$. Approach of Dirichlet forms also leads to deeper descriptions of probabilistic counterparts of globular and critical states. Furthermore, we will characterize the behaviour of polymer near the critical point from probabilistic viewpoint by showing that $X^{(γ)}$ is convergent to $X^{(0)}$ as $γ\downarrow 0$ in a certain meaning.
We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.
In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring $k$, via introducing inductive functors which generalize important properties of shift functors of $\mathrm{FI}$-modules. In particular, a sufficient criterion for finiteness of Castelnuovo-Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory are equipped with inductive functors, and hence the finiteness of Castelnuovo-Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when $k$ is a field of characteristic 0.
For a given polar code construction, the existing literature on puncturing for polar codes focuses in finding the optimal puncturing pattern, and then re-selecting the information set. This paper devotes itself to find the optimal puncturing pattern when the information set is fixed. Puncturing the coded bits corresponding to the worst quality bit channels, called the worst quality puncturing (WQP), is proposed, which is analyzed to minimize the bit channel quality loss at the punctured positions. Simulation results show that WQP outperforms the best existing puncturing schemes when the information set is fixed.
The aim of this paper is two-fold. On one hand, we will study the distorted Brownian motion on $\mathbb{R}$, i.e. the diffusion process $X$ associated with a regular and strongly local Dirichlet form obtained by the closure of $\mathscr{E}(f,g)=\frac{1}{2}\int_\mathbb{R} f'(x)g'(x)ρ(x)dx$ for $f,g\in C_c^\infty(\mathbb{R})$ on $L^2(\mathbb{R}, \mathfrak{m})$, where $\mathfrak{m}(dx)=ρ(x)dx$ and $ρ$ is a certain positive function. After figuring out the irreducible decomposition of $X$, we will present a characterization of that $X$ becomes a semi-martingale by virtue of so-called Fukushima's decomposition. Meanwhile, it is also called a general skew Brownian motion, which turns out to be a weak solution to the stochastic differential equation with certain $μ$: \[ dY_t=dW_t+\int_\mathbb{R}μ(dz) dL^z_t(Y),\quad (*) \] where $(W_t)_{t\geq 0}$ is a standard Brownian motion and $(L^z_t(Y))_{t\geq 0}$ is the symmetric semi-martingale local time of the unknown semi-martingale $Y$ at $z$. On the other hand, the stochastic differential equation (*) will be considered further. The main purpose is to find the conditions on $μ$ equivalent to that there exist general skew Brownian motions being weak solution to (*). Moreover, the irreducibility and the equivalence in distribution of expected general skew Brownian motions will be characterized. Finally, several special cases will be paid particular attention to and we will prove or disprove the pathwise uniqueness for (*).
It is known that the bit errors of polar codes with successive cancellation (SC) decoding are coupled. However, existing concatenation schemes of polar codes with other error correction codes rarely take this coupling effect into consideration. To achieve a better error performance of concatenated systems with polar codes as inner codes, one can divide all bits in an outer block into different polar blocks to completely de-correlate the possible coupled errors in the transmitter side. We call this interleaving a blind interleaving (BI) which serves as a benchmark. Two BI schemes, termed BI-DP and BI-CDP, are proposed in the paper. To better balance performance, memory size, and the decoding delay from the de-interleaving, a novel interleaving scheme, named the correlation-breaking interleaving (CBI), is proposed. The CBI breaks the correlated information bits based on the error correlation pattern proposed and proven in this paper. The proposed CBI scheme is general in the sense that any error correction code can serve as the outer code. In this paper, Low-Density Parity-Check (LDPC) codes and BCH codes are used as two examples of the outer codes of the interleaving scheme. The CBI scheme 1) can keep the simple SC polar decoding while achieving a better error performance than the state-of-the-art (SOA) direct concatenation of polar codes with LDPC codes and BCH codes; 2) achieves a comparable error performance as the BI-DP scheme with a smaller memory size and a shorter decoding delay. Numerical results are provided to verify the performance of the BI schemes and the CBI scheme.
Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to $C_c^\infty(\mathbf{R}^3\setminus \{0\})$. We will prove that this energy form is a regular Dirichlet form with core $C_c^\infty(\mathbf{R}^3)$. The associated diffusion $X$ behaves like a $3$-dimensional Brownian motion with a mild radial drift when far from $0$, subject to an ever-stronger push toward $0$ near that point. In particular $\{0\}$ is not a polar set with respect to $X$. The diffusion $X$ is rotation invariant, and admits a skew-product representation before hitting $\{0\}$: its radial part is a diffusion on $(0,\infty)$ and its angular part is a time-changed Brownian motion on the sphere $S^2$. The radial part of $X$ is a "reflected" extension of the radial part of $X^0$ (the part process of $X$ before hitting $\{0\}$). Moreover, $X$ is the unique reflecting extension of $X^0$, but $X$ is not a semi-martingale.
It is shown in [10] that a regular and local Dirichlet form on an interval can be represented by so-called effective intervals with scale functions. This paper focuses on how to operate on effective intervals to obtain regular Dirichlet subspaces. The first result is a complete characterization for a Dirichlet form to be a regular subspace of such a Dirichlet form in terms of effective intervals. Then we give an explicit road map how to obtain all regular Dirichlet subspaces from a local and regular Dirichlet form on an interval, by a series of intuitive operations on the effective intervals in the representation above. Finally applying previous results, we shall prove that every regular and local Dirichlet form has a special standard core generated by a continuous and strictly increasing function.
The stiff problem is concerned with a thermal conduction model with a singular barrier of zero volume. In this paper, we shall build the phase transitions for the stiff problems in one-dimensional space. It turns out that every phase transition definitely depends on the total thermal resistance of the barrier, and the three phases correspond to the so-called impermeable pattern, semi-permeable pattern and permeable pattern of thermal conduction respectively. For each pattern, the related boundary condition of the flux at the barrier is also derived. Mathematically, we shall introduce and explore the so-called snapping out Markov process, which is the probabilistic counterpart of semi-permeable pattern in the stiff problem.
Firstly, we shall introduce the so-called snapping out Walsh's Brownian motion and present its relation with Walsh's Brownian motion. Then the stiff problem related to Walsh's Brownian motion will be described and we shall build a phase transition for it. The snapping out Walsh's Brownian motion corresponds to the so-called semi-permeable pattern of this stiff problem.
The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric linear diffusions. Let $(\mathcal{E},\mathcal{F})$ be a regular and local Dirichlet form on $L^2(I,m)$, where $I$ is an interval and $m$ is a fully supported Radon measure on $I$. We shall first present a complete representation for $(\mathcal{E},\mathcal{F})$, which shows that $(\mathcal{E},\mathcal{F})$ lives on at most countable disjoint `effective' intervals with corresponding scale function on each interval, and any point outside these intervals is a trap of the linear diffusion. Furthermore, we shall give a necessary and sufficient condition for $C_c^\infty(I)$ being a special standard core of $(\mathcal{E},\mathcal{F})$ and identify the closure of $C_c^\infty(I)$ in $(\mathcal{E},\mathcal{F})$ when $C_c^\infty(I)$ is contained but not necessarily dense in $\mathcal{F}$ relative to the $\mathcal{E}_1$-norm. This paper is partly motivated by a result of [Hamza, 1975], stated in [FOT, Theorem 3.1.6] and provides a different point of view to this theorem. To illustrate our results, many examples are provided.
As the first error correction codes provably achieving the symmetric capacity of binary-input discrete memory-less channels (B-DMCs), polar codes have been recently chosen by 3GPP for eMBB control channel. Among existing algorithms, CRC-aided successive cancellation list (CA-SCL) decoding is favorable due to its good performance, where CRC is placed at the end of the decoding and helps to eliminate the invalid candidates before final selection. However, the good performance is obtained with a complexity increase that is linear in list size $L$. In this paper, the tailored CRC-aided SCL (TCA-SCL) decoding is proposed to balance performance and complexity. Analysis on how to choose the proper CRC for a given segment is proposed with the help of \emph{virtual transform} and \emph{virtual length}. For further performance improvement, hybrid automatic repeat request (HARQ) scheme is incorporated. Numerical results have shown that, with the similar complexity as the state-of-the-art, the proposed TCA-SCL and HARQ-TCA-SCL schemes achieve $0.1$ dB and $0.25$ dB performance gain at frame error rate $\textrm{FER}=10^{-2}$, respectively. Finally, an efficient TCA-SCL decoder is implemented with FPGA demonstrating its advantages over CA-SCL decoder.
In the first part of this paper, we study Koszul property of directed graded categories. In the second part of this paper, we prove a general criterion for an infinite directed category to be Koszul. We show that infinite directed categories in the theory of representation stability are Koszul over a field of characteristic zero.
Study of polar codes in fading channels is of great importance when applying polar codes in wireless communications. Channel estimation is a fundamental step for communication to be possible in fading channels. For both systematic and non-systematic polar codes, construction of them is based on an information set and the known frozen bits. Efficient implementation of systematic and non-systematic polar codes exists. When it comes to channel estimation or channel tracking, additional pilot symbols are inserted in the codeword traditionally. In this paper, to improve the performance of polar codes in the finite domain, pilot symbols are selected from the coded symbols themselves. In order to keep the existing efficient structure of polar code encoding, pilot selection is critical since not all selections can reuse the existing structure. In this paper, two pilot selections denoted as Uneven Pilot Selection (UEPS) and Even Pilot Selection (EPS) are proposed, which do not change the efficient polar encoding structure. The proposed UEPS and EPS is proven to satisfy the efficient construction condition. The performance of EPS is shown in this paper to outperform both the UEPS and the traditional pilot insertion scheme. Simulation results are provided which verify the performance of the proposed pilot selection schemes.
We develop a local cohomology theory for FI$^m$-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI$^m$-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.
Given a regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on a fixed domain $E$ of $\mathbb{R}^d$, we first indicate that the basic assumption $C_c^\infty(E)\subset \mathcal{F}$ is equivalent to the fact that each coordinate function $f^i(x)=x_i$ locally belongs to $\mathcal{F}$ for $1\leq i\leq d$. Our research starts from these two different viewpoints. On one hand, we shall explore when $C_c^\infty(E)$ is a special standard core of $\mathcal{F}$ and give some useful characterizations. On the other hand, we shall describe the Fukushima's decompositions of $(\mathcal{E},\mathcal{F})$ with respect to the coordinates functions, especially discuss when their martingale part is a standard Brownian motion and what we can say about their zero energy part. Finally, when we put these two kinds of discussions together, an interesting class of stochastic differential equations are raised. They have uncountable solutions that do not depend on the initial condition.