arXiv subjects

Qingying Deng

Publications and source records attributed to Qingying Deng.

At least 19 recordsLinked to original sources

arXiv2026

A Characterization of Trees with Trinomial Partial Petrial Polynomials

The partial Petrial polynomial of a bouquet can be computed from the coranks over GF(2) of matrices obtained by varying the diagonal entries of the adjacency matrix of its intersection graph. Motivated by this matrix formulation, we study the corresponding polynomial for simple graphs and determine when it has exactly two or three nonzero terms. A key tool is the interpolating property: the exponents of the nonzero terms are consecutive. Using this property together with local complementation minors of grafts, we extend the known characterization of the binomial case from connected circle graphs to all connected simple graphs, showing that such a graph has a binomial partial Petrial polynomial if and only if it is a path. Our main result characterizes the trinomial case for trees: a tree has a trinomial partial Petrial polynomial if and only if it is a T-shape tree or an H-shape tree. Here, a T-shape tree has maximum degree 3 and exactly one vertex of degree 3, whereas an H-shape tree has maximum degree 3 and exactly two vertices of degree 3, which are adjacent. We also derive explicit formulas for both families in terms of Jacobsthal numbers.

math.CO↗

arXiv2026

Partial-twuality polynomials of matrices

The study of partial-twuality polynomials originates from the classical operations of geometric duality and Petrie duality on cellularly embedded graphs. These involutions generate the symmetric group $S_3$, and applying them to subsets of edges yields the notions of partial-(geometric) duality, partial-Petriality, and more generally, partial-twuality. In this paper, we generalize this theory of partial-twuality polynomials within the framework of matrix algebra. The key observation that the Euler genus of a bouquet under a partial-twuality can be expressed as a rank function of its adjacency matrix motivates and leads to the definition of a partial-twuality polynomial for an arbitrary square matrix over any field, thereby providing a universal algebraic counterpart to the topological polynomials. We then investigate basic properties of these polynomials, including product formulas, recursion relations, degrees, interpolation behaviors, and invariance and duality theorems under the matrix operations of pivoting and inversion. We conclude by posing some problems for further research.

math.CO↗

arXiv2026

Test-Time Adaptation for Non-stationary Time Series: From Synthetic Regime Shifts to Financial Markets

Time series encountered in practice are rarely stationary. When the data distribution changes, a forecasting model trained on past observations can lose accuracy. We study a small-footprint test-time adaptation (TTA) framework for causal timeseries forecasting and direction classification. The backbone is frozen, and only normalization affine parameters are updated using recent unlabeled windows. For classification we minimize entropy and enforce temporal consistency; for regression we minimize prediction variance across weak time-preserving augmentations and optionally distill from an EMA teacher. A quadratic drift penalty and an uncertainty triggered fallback keep updates stable. We evaluate this framework in two stages: synthetic regime shifts on ETT benchmarks, and daily equity and FX series (SPY, QQQ, EUR/USD) across pandemic, high-inflation, and recovery regimes. On synthetic gradual drift, normalization-based TTA improves forecasting error, while in financial markets a simple batch-normalization statistics update is a robust default and more aggressive norm-only adaptation can even hurt. Our results provide practical guidance for deploying TTA on non-stationary time series.

q-fin.ST↗

arXiv2025

Matrix Quasi-tree Theorem

Building on prior work that established Matrix Quasi-tree Theorems for special embedded graphs, in this paper, we develop a comprehensive theory applicable to all embedded graphs. We introduce symbolic skew-adjacency matrices and reduction maps as key innovations, and prove that a specific polynomial derived from these matrices encodes all spanning quasi-trees of a bouquet. This result provides a complete analogue of the Matrix Tree Theorem for topological graph theory, with applications to quasi-tree enumeration in both orientable and non-orientable embedded graphs.

math.CO↗

arXiv2025

Proof of a conjecture of Fomichev and Karev

We prove a conjecture of Fomichev and Karev [{European J. Combin.} 127 (2025) 104160] by showing the equality of two graph invariants: $\varphi$, defined via graph colorings, and $\psi$, derived from the $\mathfrak{sl}(2)$-weight system of its 2-dimensional irreducible representation.

math.CO↗

arXiv2025

Introducing a vertex polynomial invariant for embedded graphs

The ribbon group action extends geometric duality and Petrie duality by defining two embedded graphs as twisted duals precisely when they lie within the same orbit under this group action. Twisted duality yields numerous novel properties of fundamental graph polynomials. In this paper, we resolve a problem raised by Ellis-Monaghan and Moffatt [Trans. Amer. Math. Soc. 364 (2012), 1529--1569] for vertex counts by introducing the vertex polynomial: a generating function quantifying vertex distribution across orbits under the ribbon group action. We establish its equivalence via transformations of boundary component enumeration and derive recursive relations through edge deletion, contraction, and twisted contraction. For bouquets, we prove the polynomial depends only on signed intersection graphs. Finally, we provide topological interpretations for the vertex polynomial by connecting this polynomial to the interlace polynomial and the topological transition polynomial.

math.CO↗

arXiv2024

Hamiltonian Properties of Hybrid-Faulty Burnt Pancake Graphs

We investigate the combined occurrence of edge faults and vertex faults in the burnt pancake graph ($ BP_n $). In this paper, we prove that $ BP_n - F $, where $ F $ includes pairs of end-vertices of matching edges and fault-tolerant edges, contains a Hamiltonian cycle when $ |F| \leq n-2 $ and a Hamiltonian path when $ |F| \leq n-3 $. This establishes that $ BP_n $ is $(n-2)$-hybrid fault Hamiltonian and $(n-3)$-hybrid fault Hamiltonian connected for $ n \geq 3 $. These results are demonstrated to be optimal under the given conditions, with all bounds shown to be tight.

math.CO↗

arXiv2024

Thistlethwaite Theorems for Knotoids and Linkoids

The classical Thistlethwaite theorem for links can be phrased as asserting that the Kauffman bracket of a link can be obtained from an evaluation of the Bollob\'as-Riordan polynomial of a ribbon graph associated to one of the link's Kauffman states. In this paper, we extend this result to knotoids, which are a generalization of knots that naturally arises in the study of protein topology. Specifically we extend the Thistlethwaite theorem to the twisted arrow polynomial of knotoids, which is an invariant of knotoids on compact, not necessarily orientable, surfaces. To this end, we define twisted knotoids, marked ribbon graphs, and their arrow- and Bollob\'as-Riordan polynomials. We also extend the Thistlethwaite theorem to the loop arrow polynomial of knotoids in the plane, and to spherical linkoids.

math.GT↗

arXiv2024

The number of quasi-trees of bouquets with exactly one non-orientable loop

Recently, Merino extended the classical relation between the $2n$-th Fibonacci number and the number of spanning trees of the $n$-fan graph to ribbon graphs, and established a relation between the $n$-associated Mersenne number and the number of quasi-trees of the $n$-wheel ribbon graph. Moreover, Merino posed a problem of finding the Lucas numbers as the number of spanning quasi-trees of a family of ribbon graphs. In this paper, we solve the problem and give the Matrix-Quasi-tree Theorem for a bouquet with exactly one non-orientable loop. Furthermore, this theorem is used to verify that the number of quasi-trees of some classes of bouquets is closely related to the Fibonacci and Lucas numbers. We also give alternative proofs of the number of quasi-trees of these bouquets by using the deletion-contraction relations of ribbon graphs.

math.CO↗

arXiv2024

Twisted braids

Twisted knot theory, introduced by M.O. Bourgoin, is a generalization of virtual knot theory. It naturally yields the notion of a twisted braid, which is closely related to the notion of a virtual braid due to Kauffman. In this paper, we first prove that any twisted link can be described as the closure of a twisted braid, which is unique up to certain basic moves. This is the analogue of the Alexander Theorem and the Markov Theorem for classical braids and links. Then we also give reduced presentations for the twisted braid group and the flat twisted braid group. These reduced presentations are based on the fact that these twisted braid groups on $n$ strands are generated by a single braiding element and a single bar element plus the generators of the symmetric group on $n$ letters.

math.GT↗

arXiv2024

Twist polynomial as a weight system for set systems

Recently, Chmutov proved that the partial-dual polynomial considered as a function on chord diagrams satisfies the four-term relation. Deng et al. then proved that this function on framed chord diagrams also satisfies the four-term relation, i.e., is a framed weight system. In this paper, we extend their results to the twist polynomial of a set system by proving that the twist polynomial on set systems satisfies the four-term relation.

math.CO↗

arXiv2024

Partial-dual polynomial as a framed weight system

Recently, Chmutov proved that the partial-dual polynomial considered as a function on chord diagrams satisfies the four-term relations. In this paper, we show that this function on framed chord diagrams also satisfies the four-term relations, i.e., is a framed weight system.

math.CO↗

arXiv2022

Unknotting twisted knots with Gauss diagram forbidden moves

Twisted knot theory, introduced by M.O.Bourgoin, is a generalization of virtual knot theory. It is easily shown that any virtual knot can be deformed into a trivial knot by a finite sequence of generalized Reidemeister moves and two "forbidden moves" $F_{1}$ and $F_{2}$. Similarly, we show that any twisted knot also can be deformed into a trivial knot or a trivial knot with a bar by a finite sequence of extended Reidemeister moves and three "forbidden moves" $T_{4}$, $F_{1}$ (or $F_{2}$) and $F_{3}$ (or $F_{4}$) .

math.GT↗

arXiv2021

One conjecture on cut points of virtual links and the arrow polynomial of twisted links

Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut points can be used as a tool to extend other classical invariants to virtual links. We prove that one of the conjectures in Dye's paper is correct. Moreover, we analyze the connection and difference between checkerboard framing obtained from virtual link diagram by adapting cut points and twisted link diagram obtained from virtual link diagram by introducing bars. By adjusting the normalized arrow polynomial of virtual links, we generalize it to twisted links. And we show that it is an invariant for twisted link. Finally, we figure out three characteristics of the normalized arrow polynomial of a checkerboard colorable twisted link, which is a tool of detecting checkerboard colorability of a twisted link. The latter two characteristics are the same as in the case of checkerboard colorable virtual link diagram.

math.GT↗

arXiv2021

Projectors in the Virtual Temperley-Lieb Algebra

We present a method of defining projectors in the virtual Temperley-Lieb algebra, that generalizes the Jones-Wenzl projectors in Temperley-Lieb algebra. We show that the projectors have similar properties with the Jones-Wenzl projectors, and contain an extra property which is associated with the virtual generator elements, that is, the product of a projector with a virtual generator is unchanged. We also show the uniqueness of the projector $f_n$ in terms of its axiomatic properties in the virtual Temperley-Lieba algebra $VTL_n(d)$. Finally, we find the coefficients of $f_n$ and give an explicit formula for the projector $f_n$.

math.GT↗

arXiv2020

On arrow polynomials of checkerboard colorable virtual links

In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

math.GT↗

arXiv2019

Distributed Deep Learning Model for Intelligent Video Surveillance Systems with Edge Computing

In this paper, we propose a Distributed Intelligent Video Surveillance (DIVS) system using Deep Learning (DL) algorithms and deploy it in an edge computing environment. We establish a multi-layer edge computing architecture and a distributed DL training model for the DIVS system. The DIVS system can migrate computing workloads from the network center to network edges to reduce huge network communication overhead and provide low-latency and accurate video analysis solutions. We implement the proposed DIVS system and address the problems of parallel training, model synchronization, and workload balancing. Task-level parallel and model-level parallel training methods are proposed to further accelerate the video analysis process. In addition, we propose a model parameter updating method to achieve model synchronization of the global DL model in a distributed EC environment. Moreover, a dynamic data migration approach is proposed to address the imbalance of workload and computational power of edge nodes. Experimental results showed that the EC architecture can provide elastic and scalable computing power, and the proposed DIVS system can efficiently handle video surveillance and analysis tasks.

cs.CV↗

arXiv2018

The generalized Yamada polynomials of virtual spatial graphs

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is similar to the topological definition of a virtual link. Our main goal is to generalize the classical Yamada polynomial that is defined for a spatial graph. We define a generalized Yamada polynomial for a virtual spatial graph and prove that it can be normalized to a rigid vertex isotopic invariant and to a pliable vertex isotopic invariant for graphs with maximum degree at most 3. We consider the connection and difference between the generalized Yamada polynomial and the Dubrovnik polynomial of a classical link. The generalized Yamada polynomial specializes to a version of the Dubrovnik polynomial for virtual links such that it can be used to detect the non-classicality of some virtual links. We obtain a specialization for the generalized Yamada polynomial (via the Jones-Wenzl projector $P_2$ acting on a virtual spatial graph diagram), that can be used to write a program for calculating it based on Mathematica Code.

math.GT↗