A Fast and Practical Method to Estimate Volumes of Convex Polytopes

The volume is an important attribute of a convex body. In general, it is quite difficult to calculate the exact volume. But in many cases, it suffices to have an approximate value. Volume estimation methods for convex bodies have been extensively studied in theory, however, there is still a lack of practical implementations of such methods. In this paper, we present an efficient method which is based on the Multiphase Monte-Carlo algorithm to estimate volumes of convex polytopes. It uses the coordinate directions hit-and-run method, and employs a technique of reutilizing sample points. The experiments show that our method can efficiently handle instances with dozens of dimensions with high accuracy

Homology groups of simplicial complements: A new proof of Hochster theorem

In this paper, we consider homology groups induced by the exterior algebra generated by a simplicial compliment of a simplicial complex $K$. These homology groups are isomorphic to the Tor-groups $\mathrm{Tor}_{i, J}^{\mathbf{k}[m]}(\mathbf{k}(K),\mathbf{k})$ of the face ring $\mathbf{k}(K)$, which is very useful and much studied in toric topology. By using $\check{C}ech$ homology theory and Alexander duality theorem, we prove that these homology groups have dualities with the simplicial cohomology groups of the full subcomplexes of $K$. Then we give a new proof of Hochster's theorem.Comment: 5 page

BB-Rigidity of flag 22-spheres without 44-belt

Associated to every finite simplicial complex $K$, there is a moment-angle complex $\mathcal {Z}_{K}$. In this paper, we use some algebraic invariants to solve the $B$-rigidity problem for some special simplicial compelexes.Comment: 11 pages, 3 figure

A Tool for Computing and Estimating the Volume of the Solution Space of SMT(LA)

There are already quite a few tools for solving the Satisfiability Modulo Theories (SMT) problems. In this paper, we present \texttt{VolCE}, a tool for counting the solutions of SMT constraints, or in other words, for computing the volume of the solution space. Its input is essentially a set of Boolean combinations of linear constraints, where the numeric variables are either all integers or all reals, and each variable is bounded. The tool extends SMT solving with integer solution counting and volume computation/estimation for convex polytopes. Effective heuristics are adopted, which enable the tool to deal with high-dimensional problem instances efficiently and accurately

Some rigidity problems in toric topology: I

We study the cohomological rigidity problem of two families of manifolds with torus actions: the so-called moment-angle manifolds, whose study is linked with combinatorial geometry and combinatorial commutative algebra; and topological toric manifolds, which can be seen as topological generalizations of toric varieties. These two families are related by the fact that a topological toric manifold is the quotient of a moment-angle manifold by a subtorus action. In this paper, we prove that when a simplicial sphere satisfies some combinatorial condition, the corresponding moment-angle manifold and topological toric manifolds are cohomological rigid, i.e. their homeomorphism classes in their own families are determined by their cohomology rings. Our main strategy is to show that the combinatorial types of these simplicial spheres (or more generally, the Gorenstein$^*$ complexes in this class) are determined by the $\mathrm{Tor}$-algebras of their face rings. This is a solution to a classical problem (sometimes know as the $B$-rigidity problem) in combinatorial commutative algebra for a class of Gorenstein$^*$ complexes in all dimensions $\geqslant 2$.Comment: 56 pages, Some figures taken from arXiv:1002.0828 by other authors; In this version an omitted case added in the proof of Proposition E.2 and minor inaccuracies fixed in the proof of Proposition F.

Moment-angle manifolds and connected sums of sphere products

This paper investigates the moment-angle manifolds whose cohomology ring is isomorphic to that of a connected sum of sphere products. We first give a example of moment-angle manifolds corresponding to a 4 dimentional simplicial polytope. It has the property that its cohomology ring is isomorphic to that of a connected sum of sphere products with one produt of thress spheres. Finally, we give some general properties of this kind of moment-angle manifolds.Comment: 13 pages, 9 figure

Beamforming Network Optimization for Reducing Channel Time Variation in High-Mobility Massive MIMO

Communications in high-mobility environments have caught a lot of attentions recently. In this paper, fast time-varying channels for massive multiple-input multiple-output (MIMO) systems are addressed. We derive the exact channel power spectrum density (PSD) for the uplink from a high-speed railway (HSR) to a base station (BS) and propose to further reduce the channel time variation via beamforming network optimization. A large-scale uniform linear array (ULA) is equipped at the HSR to separate multiple Doppler shifts in angle domain through high-resolution transmit beamforming. Each branch comprises a dominant Doppler shift, which can be compensated to suppress the channel time variation, and we derive the channel PSD and the Doppler spread to assess the residual channel time variation. Interestingly, the channel PSD can be exactly expressed as the product of a pattern function and a beam-distortion function. The former reflects the impact of array aperture and is the converted radiation pattern of ULA, while the latter depends on the configuration of beamforming directions. Inspired by the PSD analysis, we introduce a common configurable amplitudes and phases (CCAP) parameter to optimize the beamforming network, by partly removing the constant modulus quantized phase constraints of matched filter (MF) beamformers. In this way, the residual Doppler shifts can be ulteriorly suppressed, further reducing the residual channel time variation. The optimal CCAP parameter minimizing the Doppler spread is derived in a closed form. Numerical results are provided to corroborate both the channel PSD analysis and the superiority of beamforming network optimization technique.Comment: Double columns, 13 pages, 10 figures, transactions pape

On the Fixed-Parameter Tractability of Some Matching Problems Under the Color-Spanning Model

Given a set of $n$ points $P$ in the plane, each colored with one of the $t$ given colors, a color-spanning set $S\subset P$ is a subset of $t$ points with distinct colors. The minimum diameter color-spanning set (MDCS) is a color-spanning set whose diameter is minimum (among all color-spanning sets of $P$). Somehow symmetrically, the largest closest pair color-spanning set (LCPCS) is a color-spanning set whose closest pair is the largest (among all color-spanning sets of $P$). Both MDCS and LCPCS have been shown to be NP-complete, but whether they are fixed-parameter tractable (FPT) when $t$ is a parameter is still open. Motivated by this question, we consider the FPT tractability of some matching problems under this color-spanning model, where $t=2k$ is the parameter. The problems are summarized as follows: (1) MinSum Matching Color-Spanning Set, namely, computing a matching of $2k$ points with distinct colors such that their total edge length is minimized; (2) MaxMin Matching Color-Spanning Set, namely, computing a matching of $2k$ points with distinct colors such that the minimum edge length is maximized; (3) MinMax Matching Color-Spanning Set, namely, computing a matching of $2k$ points with distinct colors such that the maximum edge length is minimized; and (4) $k$-Multicolored Independent Matching, namely, computing a matching of $2k$ vertices in a graph such that the vertices of the edges in the matching do not share common edges in the graph. We show that the first three problems are polynomially solvable (hence in FPT), while problem (4) is W[1]-hard.Comment: 12 pages, 2 figures, earlier version appeared in FAW'1

Time-Varying Downlink Channel Tracking for Quantized Massive MIMO Networks

This paper proposes a Bayesian downlink channel estimation algorithm for time-varying massive MIMO networks. In particular, the quantization effects at the receiver are considered. In order to fully exploit the sparsity and time correlations of channels, we formulate the time-varying massive MIMO channel as the simultaneously sparse signal model. Then, we propose a sparse Bayesian learning (SBL) framework to learn the model parameters of the sparse virtual channel. To reduce complexity, we employ the expectation maximization (EM) algorithm to achieve the approximated solution. Specifically, the factor graph and the general approximate message passing (GAMP) algorithms are used to compute the desired posterior statistics in the expectation step, so that high-dimensional integrals over the marginal distributions can be avoided. The non-zero supporting vector of a virtual channel is then obtained from channel statistics by a k-means clustering algorithm. After that, the reduced dimensional GAMP based scheme is applied to make the full use of the channel temporal correlation so as to enhance the virtual channel tracking accuracy. Finally, we demonstrate the efficacy of the proposed schemes through simulations.Comment: 30 Pages, 11 figure

Diversified Top-k Partial MaxSAT Solving

We introduce a diversified top-k partial MaxSAT problem, a combination of partial MaxSAT problem and enumeration problem. Given a partial MaxSAT formula F and a positive integer k, the diversified top-k partial MaxSAT is to find k maximal solutions for F such that the k maximal solutions satisfy the maximum number of soft clauses of F. This problem can be widely used in many applications including community detection, sensor place, motif discovery, and combinatorial testing. We prove the problem is NP-hard and propose an approach for solving the problem. The concrete idea of the approach is to design an encoding EE which reduces diversified top-k partial MaxSAT problem into partial MaxSAT problem, and then solve the resulting problem with state-of-art solvers. In addition, we present an algorithm MEMKC exactly solving the diversified top-k partial MaxSAT. Through several experiments we show that our approach can be successfully applied to the interesting problem