Search Results (14)

This paper constructs a universal algebraic realization model for Hermitian parameter matrices whose entries have modulus at most one and whose diagonal entries are normalized. The entries of the parameter matrix are used as weights of oriented crossings in averaged Wick-type moment formulas. The construction separates each crossing parameter into its unit complex factor and its modulus. We first construct a universal free algebraic unimodular factor model whose moments are defined by a balanced pair-oriented crossing formula. The modulus factors are then encoded by an auxiliary commutative algebra and recombined with the unimodular factors by tensorization. The resulting normalized sums converge in joint algebraic moments to the averaged weighted Wick moment functional, whose moments are given by a fully averaged pair-partition formula. In the general complex Hermitian case, the construction is purely algebraic, and no positivity, traciality, operator boundedness, or operator algebraic realization is claimed. In the real symmetric specialization, the averaged oriented formula reduces to the standard mixed Gaussian pair-partition formula with color-dependent crossing parameters, so the construction contains the known Fock-representable mixed Gaussian cases as positive Fock-space examples. Moreover, the averaged functional satisfies a uniform Gaussian-type moment growth bound. Full article

The efficient minus domination problem (EMDP) and the efficient signed domination problem (ESDP) are domination-type problems in graphs. These problems are known to be NP-complete on chordal graphs and polynomially solvable on chain interval graphs, while the complexity on proper interval graphs remained open. By exploiting the totally unimodular structure of the closed-neighborhood matrix induced by a proper interval ordering, we obtain linear programming formulations under which both the EMDP and ESDP become polynomially solvable. The same perspective naturally extends to vertex-weighted settings and to other domination variants defined by similar neighborhood constraints. Full article

Maritime transportation companies operate in highly volatile environments, where data-driven decision-making is critical to navigating fluctuating freight revenue, fuel and transit costs, and dynamic trade-related policies. This study addresses the liner service network design and container flow management problem, with the objective of maximizing weekly profit, calculated as total freight revenue minus comprehensive operational costs associated with fuel, berthing, transit, and policy-driven extra fees. We formulate a mixed-integer programming (MIP) model for the problem and demonstrate that the constraint matrix associated with vessel leasing is totally unimodular. This property permits the reformulation of the original MIP model into a semi-relaxed MIP model, which maintains optimality while improving computational efficiency. Using shipping data in a realistic liner service network, the proposed model demonstrates its practical applicability in balancing complex trade-offs to optimize profitability. Sensitivity analyses provide actionable insights for data-driven decision-making, including when to expand service networks, discontinue unprofitable routes, and strategically deploy vessel leasing to mitigate rising operational costs and regulatory penalties. This study provides a practical, computationally efficient, and data-driven framework to support liner shipping companies in making robust tactical decisions amid economic and regulatory dynamics. Full article

Container liner shipping companies operate within a complex environment where they must balance profitability and service reliability. Meanwhile, evolving regulatory policies, such as surcharges imposed on ships of a particular origin or type on specific trade lanes, introduce new operational challenges. This study addresses the heterogeneous ship routing and demand acceptance problem, aiming to maximize two conflicting objectives: weekly profit and total transport volume. We formulate the problem as a bi-objective mixed-integer programming model and prove that the ship chartering constraint matrix is totally unimodular, enabling the reformulation of the model into a partially relaxed MIP that preserves optimality while improving computational efficiency. We further analyze key mathematical properties showing that the Pareto frontier consists of a finite union of continuous, piecewise linear segments but is generally non-convex with discontinuities. A case study based on a realistic liner shipping network confirms the model’s effectiveness in capturing the trade-off between profit and transport volume. Sensitivity analyses show that increasing freight rates enables higher profits without large losses in volume. Notably, this paper provides a practical risk management framework for shipping companies to enhance their adaptability under shifting regulatory landscapes. Full article

Inland waterway transportation is critical for the movement of hazardous liquid cargoes. To prevent contamination when transporting different types of liquids, certain shipments necessitate tank cleaning at designated stations between tasks. This process often requires detours, which can decrease operational efficiency. This study addresses the Tank Cleaning Station Location and Cleaning Task Assignment (TCSL-CTA) problem, with the objective of minimizing total system costs, including the construction and operational costs of tank cleaning stations, as well as the detour costs incurred by ships visiting these stations. We formulate the problem as a mixed-integer programming (MIP) model and prove that it can be reformulated into a partially relaxed MIP model, preserving optimality while enhancing computational efficiency. We further analyze key mathematical properties, showing that the assignment constraint matrix is totally unimodular, enabling efficient relaxation, and that the objective function exhibits submodularity, reflecting diminishing returns in facility investment. A case study on the Yangtze River confirms the model’s effectiveness, where the optimized plan resulted in detour costs accounting for only 5.2% of the total CNY 4.23 billion system cost and achieved an 89.1% average station utilization. Managerial insights reveal that early construction and balanced capacity allocation significantly reduce detour costs. This study provides a practical framework for long-term tank cleaning infrastructure planning, contributing to cost-effective and sustainable inland waterway logistics. Full article

Domination problems are fundamental problems in graph theory with diverse applications in optimization, network design, and computational complexity. This paper investigates $\left\{k\right\}$-domination, k-tuple domination, and their total domination variants in weighted strongly chordal graphs and chordal bipartite graphs. Specifically, the $\left\{k\right\}$-domination problem in weighted strongly chordal graphs and the total $\left\{k\right\}$-domination problem in weighted chordal bipartite graphs are shown to be solvable in $\mathcal{O}(n+m)$ time. For weighted proper interval graphs and convex bipartite graphs, we solve the k-tuple domination and total k-tuple domination problems in $\mathcal{O}(n^{2.371552}{log}^2\left(n\right)log(n/\delta ))$, where $\delta$ is the desired accuracy. Furthermore, for weighted unit interval graphs, the k-tuple domination problem achieves a significant complexity improvement, reduced from $\mathcal{O}\left(n^{k+2}\right)$ to $\mathcal{O}(n^{2.371552}{log}^2\left(n\right)log(n/\delta ))$. These results are achieved through a combination of linear and integer programming techniques, complemented by totally balanced matrices, totally unimodular matrices, and graph-specific matrix representations such as neighborhood and closed neighborhood matrices. Full article

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from ${\mathbb{Z}}^E$ orthogonal to rows of the matrix form a regular chain group N. Assume that $\psi$ is a homomorphism from N into a finite additive Abelian group A and let $A_{\psi }\left[N\right]$ be the set of vectors g from ${(A-0)}^E$, such that ${\sum }_{e\in E}g\left(e\right)·f\left(e\right)=\psi \left(f\right)$ for each $f\in N$ (where · is a scalar multiplication). We show that $|A_{\psi }\left[N\right]|$ can be evaluated by a polynomial function of $\left|A\right|$. In particular, if $\psi \left(f\right)=0$ for each $f\in N$, then the corresponding assigning polynomial is the classical characteristic polynomial of M. Full article

In this paper, we mainly study the embedding problem of unimodular row vectors, focusing on avoiding the identification of polynomial zeros. We investigate the existence of the minimal syzygy module of the ZLP polynomial matrix and demonstrate that the minimal syzygy module has structural properties that are similar to the fundamental solution system of homogeneous linear equations found in linear algebra. Finally, we provide several embedding methods for unimodular vectors in certain cases. Full article

Simultaneously polarimetric radar (SPR) realizes the rapid measurement of a target’s polarimetric scattering matrix by transmitting orthogonal radar waveforms of good ambiguity function (AF) properties and receiving their echoes via two orthogonal polarimetric channels at the same time, e.g., horizontal (H) and vertical (V) channels (antennas) sharing the same phase center. The orthogonality of the transmitted waveforms can be realized using low-correlated phase-coded sequences in the H and V channels. However, the Doppler tolerances of the waveforms composed by such coded sequences are usually quite low, and it is hard to meet the requirement of accurate measurement regarding moving targets. In this paper, a joint design approach for unimodular orthogonal complementary sequences along with the optimal receiving filter is proposed based on the majorization–minimization (MM) method via alternate iteration for obtaining simultaneously polarimetric waveforms (SPWs) of good orthogonality and of the desired AF. During design, the objective function used for minimizing the sum of the complementary integration sidelobe level (CISL) and the complementary integration isolation level (CIIL) is constructed under the mismatch constraint of signal-to-noise ratio (SNR) loss. Different SPW examples are given to show the superior performance of our design in comparison with other designs. Finally, practical experiments implemented with different SPWs are conducted to show our advantages more realistically. Full article

The equivalence of systems is a crucial concept in multidimensional systems. The Smith normal forms of multivariate polynomial matrices play important roles in the theory of polynomial matrices. In this paper, we mainly study the unimodular equivalence of some special kinds of multivariate polynomial matrices and obtain some tractable criteria under which such matrices are unimodular equivalent to their Smith normal forms. We propose an algorithm for reducing such $nD$ polynomial matrices to their Smith normal forms and present an example to illustrate the availability of the algorithm. Furthermore, we extend the results to the non-square case. Full article

Here, we considerably develop the methods of power geometry for a system of partial differential equations and apply them to two different fluid dynamics problems: computing the boundary layer on a needle in the first approximation and computing the asymptotic forms of solutions to the problem of evolution of the turbulent flow. For each equation of the system, its Newton polyhedron and its hyperfaces with their normals and truncated equations are calculated. To simplify the truncated systems, power-logarithmic transformations are used and the truncated systems are further extracted. Here, we propose algorithms for computing unimodular matrices of power transformations for differential equations. Results: (1) the boundary layer on the needle is absent in liquid, while in gas it is described in the first approximation; (2) the solutions to the problem of evolution of turbulent flow have eight asymptotic forms, presented explicitly. Full article

We propose a phenomenological model of two-zeros Majorana neutrino mass matrix based on the $A_4$ symmetry, where the structure of mixing matrix is a unimodular second scheme of trimaximal $T{M}_2$, and the charged lepton mass matrix is diagonal. We show that, among seven possible two-zero textures with $A_4$ symmetry, only two textures, namely the texture with $M_{ee}=0$ and $M_{e\mu }=0$ and its permutation, are acceptable in the non-perturbation method, since the results associated with these two textures are consistent with the experimental data. We obtain a unique relation between our phases, namely $\rho +\sigma =\varphi \pm \pi$, and an effective equation ${sin}^2{\theta }_{13}=\frac{2}{3}{R}_{\nu }$ where $R_{\nu }=\frac{\delta m^2}{\Delta m^2}$. Then, only by using the experimental ranges of $R_{\nu }$, we obtain the allowable range of the unknown parameter $\varphi$ as the phase of $T{M}_2$ mixing matrix, which leads to obtaining not only the ranges of all neutrino oscillation parameters of the model (which agree well with experimental data) but also with the masses of neutrinos, the Dirac and Majorana phases and the Jarlskog parameter, and to predict the normal neutrino mass hierarchy. Finally, we show that all the predictions regarding our two specific textures agree with the corresponding data reported from neutrino oscillation, cosmic microwave background and neutrinoless double beta decay. Full article

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix $A\in {\mathbb{Z}}^{n\times n}$, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix $U\in {\mathbb{Z}}^{n\times n}$. With the property that $det(U)=\pm 1$, then $U^{-1}\in {\mathbb{Z}}^{n\times n}$ is guaranteed such that $U{U}^{-1}=I$, where $I\in {\mathbb{Z}}^{n\times n}$ is an identity matrix. In this paper, we propose a new integer matrix $\tilde{G}\in {\mathbb{Z}}^{n\times n}$, which is referred to as an almost-unimodular matrix. With $det(\tilde{G})\ne \pm 1$, the inverse of this matrix, ${\tilde{G}}^{-1}\in {\mathbb{R}}^{n\times n}$, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal $\pm 1$. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix. Full article

Clustering is to group data so that the observations in the same group are more similar to each other than to those in other groups. k-means is a popular clustering algorithm in data mining. Its objective is to optimize the mean squared error (MSE). The traditional k-means algorithm is not suitable for applications where the sizes of clusters need to be balanced. Given n observations, our objective is to optimize the MSE under the constraint that the observations need to be evenly divided into k clusters. In this paper, we propose an iterative method for the task of clustering with balanced size constraints. Each iteration can be split into two steps, namely an assignment step and an update step. In the assignment step, the data are evenly assigned to each cluster. The balanced assignment task here is formulated as an integer linear program (ILP), and we prove that the constraint matrix of this ILP is totally unimodular. Thus the ILP is relaxed as a linear program (LP) which can be efficiently solved with the simplex algorithm. In the update step, the new centers are updated as the centroids of the observations in the clusters. Assuming that there are n observations and the algorithm needs m iterations to converge, we show that the average time complexity of the proposed algorithm is $O(m{n}^{1.65})$ – $O(m{n}^{1.70})$ . Experimental results indicate that, comparing with state-of-the-art methods, the proposed algorithm is efficient in deriving more accurate clustering. Full article