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Da Zhao

Publications and source records attributed to Da Zhao.

At least 19 recordsLinked to original sources
arXiv2026

CuBot-STT: Simultaneously Transport and Tighten for Lattice-type Modular Reconfiguration Robot

Lattice-type modular reconfigurable robots rearrange standardized modules across tasks. Dedicated transport robots reduce module weight and cost by eliminating module level reconfiguration actuators. But existing systems use low capacity connections or internal fastening robots that occupy space for functional components. This paper presents CuBot-STT, which integrates module transport, interface locking, and release in one external robot. Cubic modules provide structural support and $24\,\mathrm{V}$ power distribution. On each face, a gear transmission converts one external input into motion of four helical locking points, preserving internal space. The robot uses alternating foot support, a gripper for module handling, and a tightening head for interface actuation. The controller coordinates two-foot support and module locking through continuous constraint transfers. The planner decomposes relocation costs into an analytical lower bound and a residual learned through expert-guided, goal conditioned Q-learning. Contact face $A^*$ search expands relocation sequences into robot primitives satisfying connectivity, asymmetric manipulation, and foot stance constraints. Experiments demonstrate normal and tangential interface capacities of $300\,\mathrm{N}$ and $490\,\mathrm{N}$, electrical continuity across six modules, and autonomous reconfiguration of an 11-module lattice. Planning benchmarks show superior search efficiency over analytical guidance and bidirectional heuristic search, with generalization beyond training lattice sizes without fine-tuning.

cs.RO↗
arXiv2026

Factor Model Estimation for High-Dimensional Time Series with Heteroskedastic Noise

Factor modeling provides a framework for extracting common components from high-dimensional time series. We develop an estimator of the factor loading space that combines contemporaneous covariance with lagged autocovariance information. Under cross-sectionally heteroskedastic white noise, the contemporaneous covariance contains heterogeneous diagonal noise contributions. We apply the existing HeteroPCA algorithm to the combined estimation matrix to correct this contamination while retaining information from serial dependence. The framework also accommodates noise contamination on a known set of entries subject to sparsity and incoherence conditions. Under regularity conditions for dependent observations, we establish consistency and derive convergence rates, identifying regimes in which incorporating contemporaneous covariance improves the rate relative to the lagged-autocovariance estimator. Simulations evaluate loading-space estimation error and the effects of contemporaneous covariance inclusion and diagonal correction. An application to S\&P 500 stock returns illustrates the method.

stat.ME↗
arXiv2026

Doubly robust estimation of while-alive estimands in individually-randomized and cluster-randomized trials

Randomized trials in chronic disease settings often measure treatment benefit through recurrent non-fatal events that are truncated by death, where conventional summaries either discard recurrences, conflate the treatment effect with survival, or treat death as censoring and forfeit a causal interpretation. While-alive estimands measure event burden per unit time alive, but doubly robust estimation for the exposure-weighted while-alive rate remains undeveloped, particularly in cluster-randomized trials (CRTs). We develop a doubly robust estimator based on a local Nelson-Aalen representation, with augmented estimating equations targeting the marginal hazard of the terminal event and the weighted recurrent event rate among those alive; the estimator accommodates multiple event types through prespecified clinical weights and remains consistent if either the censoring model or the outcome working models are correctly specified. For CRTs, we define a new pair of individual-average and cluster-average estimands under informative cluster size, with inference based on cluster-level influence functions. We establish component-wise double robustness and asymptotic normality, corroborate the theory in simulations, and illustrate the methods with reanalyses of data from two completed randomized trials.

stat.ME↗
arXiv2026

Bipartite graphs, random graphs, and Lin--Lu--Yau curvature

Let $G = (X, Y; E)$ be a bipartite graph with parts $X$ and $Y$ where $|X|=m$ and $|Y|=n$. We show that every bipartite graph with more than $mn - D(m,n)$ edges has positive Lin--Lu--Yau curvature, where $D(m,n)=m-2+\lceil{\frac {n}{2}\rceil} \text{ if $n\geq 2m$}, \mbox{and} \ n-1 \text{ if $m\leq n< 2m$}.$ We also show that every bipartite graph of order $m+n$ with $m \geq n$ and minimum degree at least $\min\{n, \lfloor{\frac{m+n}{3}\rfloor}+1\}$ has positive Lin--Lu--Yau curvature. Both bounds are sharp. Meanwhile probabilistically we can relax the edge density conditions in above results. It is shown that relatively dense random bipartite graph is positively curved. All of our proofs are based on a new formula for Lin--Lu--Yau curvature of bipartite graphs.

math.CO↗
arXiv2026

On Neumaier Cayley graphs

In the present paper, we study Neumaier Cayley graphs. First, we give a criterion for a Cayley graph to be a Neumaier graph with a spread given by the cosets of a subgroup. Further, we construct a new infinite family of Neumaier Cayley graphs of unbounded nexus. Finally, we provide an algorithm for enumerating Neumaier Cayley graphs and computational results obtained by this algorithm.

math.CO↗
arXiv2026

Morphogenetic Assembly and Adaptive Control for Heterogeneous Modular Robots

This paper presents a closed-loop automation framework for heterogeneous modular robots, covering the full pipeline from morphological construction to adaptive control. In this framework, a mobile manipulator handles heterogeneous functional modules including structural, joint, and wheeled modules to dynamically assemble diverse robot configurations and provide them with immediate locomotion capability. To address the state-space explosion in large-scale heterogeneous reconfiguration, we propose a hierarchical planner: the high-level planner uses a bidirectional heuristic search with type-penalty terms to generate module-handling sequences, while the low level planner employs A* search to compute optimal execution trajectories. This design effectively decouples discrete configuration planning from continuous motion execution. For adaptive motion generation of unknown assembled configurations, we introduce a GPU accelerated Annealing-Variance Model Predictive Path Integral (MPPI) controller. By incorporating a multi stage variance annealing strategy to balance global exploration and local convergence, the controller enables configuration-agnostic, real-time motion control. Large scale simulations show that the type-penalty term is critical for planning robustness in heterogeneous scenarios. Moreover, the greedy heuristic produces plans with lower physical execution costs than the Hungarian heuristic. The proposed annealing-variance MPPI significantly outperforms standard MPPI in both velocity tracking accuracy and control frequency, achieving real time control at 50 Hz. The framework validates the full-cycle process, including module assembly, robot merging and splitting, and dynamic motion generation.

cs.RO↗
arXiv2025

Generalized block diagonal Laplacian spectrum of graphs

We reduce the $p^2$ block all-one matrices in the generalized block Laplacian spectrum of graphs to $p$ block all-one matrices in the generalized block diagonal Lapalcian spectrum of graphs introduced by Wang and the second author (\textit{Adv. Appl. Math.} 173B (2026)). In this case the matrices are all real symmetric, and hence the spectrum is real, which does not hold for the generalized block Laplacian spectrum. We also investigate the analogue by Hermitian adjacency matrix of digraphs.

math.CO↗
arXiv2025

Graph isomorphism and multivariate graph spectrum

We provide a criterion to distinguish two graphs which are indistinguishable by $2$-dimensional Weisfeiler-Lehman algorithm for almost all graphs. Haemers conjectured that almost all graphs are identified by their spectrum. Our approach suggests that almost all graphs are identified by their generalized block Laplacian spectrum.

math.CO↗
arXiv2025

Estimates of the first Dirichlet eigenvalue of graphs

Inspired by the Li--Yau eigenvalue-diameter estimates, we investigate lower bounds for the first Dirichlet eigenvalue in terms of the diameter (or inscribed radius) of a graph. Let $G = (V, E)$ be a graph with boundary $B$. Assume that the interior $Ω= V \setminus B$ is connected. Let $r$ be the inscribed radius of $(G, B)$ and $d$ be the maximum degree of $G$. We prove that $$λ_1(G, B) \geq \frac{d - 1}{r d^r},$$ which can be viewed as an analogue of the Lin--Yau bound and the Meng--Lin bound for normalized Dirichlet/Laplacian eigenvalues. We also derive the inequality $$λ_1(G, B) \geq \frac{1}{r |Ω|}.$$ In particular, for a tree $T$ with at least $3$ vertices, we show that $$λ_1(T) \geq 4 \sin^2 \fracπ{4r + 6} \geq \frac{1}{(r + 1)^2}.$$ Notably, both of the two preceding bounds are sharp up to a constant factor. We additionally examine upper bounds on the first Dirichlet eigenvalue under constraints on the numbers of interior and boundary vertices.

math.CO↗
arXiv2025

Almost all graphs have no cospectral mate with fixed level

Haemers conjectures that almost all graphs are determined by their spectra. Suppose $G \sim \mathcal{G}(n, p)$ is a random graph with each edge chosen independently with probability $p$ with $0 < p < 1$. Then $$\Pr(G \text{ is not controllable}) + \sum_{\ell = 2}^{n^{n^2}} \Pr(G \text{ has a generalized copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ implies that almost all graphs are determined by their generalized spectra. It is known that almost all graphs are controllable. We show that almost all graphs have no cospectral mate with fixed level $\ell$, namely $$\Pr(G \text{ has a copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ for every $\ell \geq 2$. Consequently, $$\Pr(G \text{ has a generalized copsectral mate with level } \ell) \to 0$$ as $n \to \infty$ for every $\ell \geq 2$. The result can also be interpreted in the framework of random integral matrices.

math.CO↗
arXiv2025

Comparison between the first Steklov eigenvalue and algebraic connectivity on trees

Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply-connected Riemannian manifolds of non-positive sectional curvature. In this paper, we compare the first (non-trivial) Steklov eigenvalue and algebraic connectivity of trees with prescribed number of boundary vertices and matching number. It is particularly noteworthy that while the extremal trees coincide for both operators, their corresponding eigenvalues differ significantly.

math.CO↗
arXiv2025

The first Steklov eigenvalue of planar graphs and beyond

The Steklov eigenvalue problem was introduced over a century ago, and its discrete form attracted interest recently. Let $D$ and $δΩ$ be the maximum vertex degree and the set of vertices of degree one in a graph $\mathcal{G}$ respectively. Let $λ_2$ be the first (non-trivial) Steklov eigenvalue of $(\mathcal{G}, δΩ)$. In this paper, using the circle packing theorem and conformal mapping, we first show that $λ_2 \leq 8D / |δΩ|$ for planar graphs. This can be seen as a discrete analogue of Kokarev's bound, that is, $λ_2 < 8π/ |\partial Ω|$ for compact surfaces with boundary of genus $0$. Let $B$ and $L$ be the maximum block size and the diameter of a block graph $\mathcal{G}$ respectively. Secondly, we prove that $λ_2 \leq 4 (B-1) (D-1)/ |δΩ|$ and $λ_2 \leq B/L$ for block graphs, which extend the results on trees by He and Hua. In the end, for trees with fixed leaf number and maximum degree, candidates that achieve the maximum first Steklov eigenvalue are given.

math.CO↗
arXiv2025

Maximize the Steklov eigenvalue of trees

We study the maximal Steklov eigenvalues of trees with given number of boundary vertices and total number of vertices. Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply-connected Riemannian manifolds of non-positive sectional curvature. Let $σ_{k,\text{max}}(b, n)$ be the maximal of $k$-th Steklov eigenvalue of trees with $b$ leaves as boundary and $n$ vertices. We determine that $$ σ_{2, \text{max}} (b, n) = \begin{cases} \frac{2}{n-1}, & b=2, n\geq 3, \frac{1}{r}, & b \geq 3, n = br + m, 3 - b \leq m \leq 1, r \in \mathbb{Z}_+, \frac{1}{r+1-\frac{1}{b}}, & b \geq 3, n = br + 2, r \in \mathbb{Z}_+, \end{cases} $$ and we characterize the trees attaining this bound. For $k \geq 3$, we show that $σ_{k, \text{max}} (b, n) = 1$. We also give a lower bound on the maximal Steklov eigenvalues of trees with given diameter and total number of vertices. Our work can be regarded as a completion of the work by He--Hua [Upper bounds for the Steklov eigenvalues on trees, Calc. Var. Partial Differential Equations (2022)] and Yu--Yu [Monotonicity of Steklov eigenvalues on graphs and applications, Calc. Var. Partial Differential Equations (2024)].

math.CO↗
arXiv2025

Spectral Clustering on Multilayer Networks with Covariates

The community detection problem on multilayer networks have drawn much interest. When the nodal covariates ar also present, few work has been done to integrate information from both sources. To leverage the multilayer networks and the covariates, we propose two new algorithms: the spectral clustering on aggregated networks with covariates (SCANC), and the spectral clustering on aggregated Laplacian with covariates (SCALC). These two algorithms are easy to implement, computationally fast, and feature a data-driven approach for tuning parameter selection. We establish theoretical guarantees for both methods under the Multilayer Stochastic Blockmodel with Covariates (MSBM-C), demonstrating their consistency in recovering community structure. Our analysis reveals that increasing the number of layers, incorporating covariate information, and enhancing network density all contribute to improved clustering accuracy. Notably, SCANC is most effective when all layers exhibit similar assortativity, whereas SCALC performs better when both assortative and disassortative layers are present. On the simulation studies and a primary school contact data analysis, our method outperforms other methods. Our results highlight the advantages of spectral-based aggregation techniques in leveraging both network structure and nodal attributes for robust community detection.

stat.ME↗
arXiv2024

Upper bounds of Steklov eigenvalues on graphs

Let $Δ$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $σ_2$ of $G$ with boundary $B$. Using metrical deformation via flows, we first show that $σ_2 = \mathcal{O}\left(\frac{Δ(g+1)^3}{|B|}\right)$ for graphs of orientable genus $g$ if $|B| \geq \max\{3 \sqrt{g},|V|^{\frac{1}{4} + ε}, 9\}$ for some $ε> 0$. This can be seen as a discrete analogue of Karpukhin's bound. Secondly, we prove that $σ_2 \leq \frac{8Δ+4X}{|B|}$ based on planar crossing number $X$. Thirdly, we show that $σ_2 \leq \frac{|B|}{|B|-1} \cdot δ_B$, where $δ_B$ denotes the minimum degree for boundary vertices in $B$. At last, we compare several upper bounds on Laplacian eigenvalues and Steklov eigenvalues.

math.CO↗
arXiv2024

Bivariate $Q$-polynomial structures for the nonbinary Johnson scheme and the association scheme obtained from attenuated spaces

The study of $P$-polynomial association schemes (distance-regular graphs) and $Q$-polynomial association schemes, and in particular $P$- and $Q$-polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of $P$- and $Q$-polynomial association schemes are described by Askey-Wilson orthogonal polynomials or their relatives. These polynomials are one-variable orthogonal polynomials. It seems that the new attempt to define and study higher rank $P$- and $Q$-polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Crampé-d'Andecy-Vinet-Zaimi [arXiv:2212.10824], and then followed by Bannai-Kurihara-Zhao-Zhu [arXiv:2305.00707]. The general theory and some explicit examples of families of higher rank (multivariate) $P$- and/or $Q$-polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate $Q$-polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the attenuated space association schemes are bivariate $Q$-polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study.

math.CO↗
arXiv2023

Multivariate P- and/or Q-polynomial association schemes

The classification problem of $P$- and $Q$-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of $P$- and $Q$-polynomial association schemes to multivariate cases, namely to consider higher rank $P$- and $Q$-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definition nor results. Very recently, Bernard, Crampé, d'Andecy, Vinet, and Zaimi [arXiv:2212.10824], defined bivariate $P$-polynomial association schemes, as well as bivariate $Q$-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate $P$- and/or $Q$-polynomial association schemes.

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