A general nonlinear projection equation is investigated in this paper. Under mild conditions, the uniqueness of the solution, a bound of the residual and an error bound for this equation are given. Based on the properties of the general nonlinear projection equation, a neurodynamic model for solving this equation is proposed and the fixed-time stability is proved. As an application, the model is applied to solve the semi-smooth projection equation. Numerical results demonstrate the effectiveness of the proposed method.
In this paper, we determine the exact values of spc-number of circulant digraphs $Cn([k]),$ which completes the known bounds on them. Next, we prove that the spc-number of strong tournaments is no more than three. Finally, we prove that the pc-number of the digraphs $D$ is equal to the pvc-number of the line digraphs $L(D).$
In this paper, we study the graded discrepancy of graphs. We show that for any fixed $p$ in $(0,1),$ there exists a sequence of $n$-vertex graphs $\{G_n\}$ with edge densities tending to $p$ such that $G_n$ admits a vertex ordering whose graded discrepancy is bounded by a constant independent of $n.$ The construction combines a structured graph layout with a round-robin vertex ordering and blockwise interleaving.
An r-uniform hypergraph is linear if any pair of edges of the hypergraph has at most one common vertex. In this paper, we focus on Turan type problems for linear hypergraphs. For a graph F, the r-expansion F+ is the r-graph obtained from F by enlarging each edge of F with r-2 new vertices disjoint from V(F) such that distinct edges of F are enlarged by distinct vertices. First, we prove that a K+_{s,t}-free r-partite linear r-graph of order n has at most ((t-1)/(r-1))^{1/s} n^{2-1/s} + O(n^{2-2/s}) edges, which strengthens the results of Lazebnik, Verstraete [Electron. J. Comb., 10(2003), #R25] and Timmons [Electron. J. Combin. 29(3)(2022) #P3.46]. Second, we give a sharp upper bound for the number of edges of linear hypergraphs containing no expansion of double star, where the double star is a tree with two vertices of degree greater than one.
Let t be a non-negative real number. If a graph G has toughness t, and deleting any edge of G decreases its toughness, then G is a minimally t-tough graph. Katona et al. conjectured that the minimum degree of every minimally t-tough graph is d2te. Although the conjecture is disproved in general, authors attempt to confirm it for some classes of graphs. In this paper, for each positive integer k ≥ 2, we prove that every minimally 1/k-tough graph whose matching number is at most 3 has a vertex of degree one.
This paper concerns the Cauchy problem of three-dimensional incompressible magneto-micropolar equations with partially mixed velocity dissipation and magnetic diffusion. Under smallness assumption on initial data, we first establish the global existence of smooth solution, and then derive the long-time decay estimates for the solution. The proof is based on energy methods, and some new weighted energy functionals and bootstrap argument are introduced. We remark that, despite the lack of dissipation in certain directions, the solution exhibits power decay rates as time tends to infinity.
The facility location problem of urban vertiports is one of the crucial issues in urban low-altitude traffic optimization decisions. In order to effectively alleviate ground traffic congestion and improve passenger travel quality, this study focuses on the urban vertiport location optimization problem considering multi-modal transport. An integer programming model with capacity constraints and vertiport classification constraints is established. The model aims to minimize both vertiport construction cost and passenger travel cost. The RLT method is applied to linearize the non-convex quadratic terms in the objective function. For the reformulated optimal model, a depth-first branch-and-bound algorithm is developed. A branching strategy using priority functions is defined to enhance solving efficiency. Simulation experiments are conducted in the main urban area of Nanjing in China to verify the effectiveness of the model and algorithm. Three location schemes are proposed from the perspective of passenger, investor and bilateral equilibrium. Numerical results indicate that the bilateral equilibrium scheme is satisfied with both passenger travel demand and vertiport construction requirement. A comparative analysis of multimodal transport and ground traffic demonstrates that multimodal transport significantly reduces travel time while keeping cost increases within acceptable limits. These findings fully validate the feasibility and effectiveness of the proposed vertiport location scheme. The proposed schemes provide decision-making references for the strategic planning of urban air traffic infrastructure.
Denote *diam(G) (*rad(G)) as the minimum directed diameter (radius) among all orientations of the bridgeless graph G. Denote G as a triangulation with n vertices. Mondal, Parthiban and Rajasingh proved that *diam(G) ≤ n/2 + O(√n). Ge, Liu and Wang improved it to n/2. In this paper, we first prove that *rad(G) ≤ 2r, where r is the radius of G. We also prove that for s-connected triangulation G, there exists an integer c such that *rad(G) ≤ n/s + c. As a corollary, we improve the upper bound of the oriented diameter of 5-connected triangulations to 2n/5 + c. Furthermore, we prove that under some connecting condition of the triangulation G, *rad(G) ≤ r+1, where r is the radius of G. Then for s-connected triangulation G under some connecting condition, there exists an integer c such that *rad(G) ≤ n/(2s) + c and *diam(G) ≤ n/s + c, which are tight apart from a constant.
In this paper, we investigate the fast rotation limit of the density-dependent incompressible Euler equations in two-dimensional bounded domain. The case considered here is the so-called quasi-homogeneous regime in which the initial density is a small perturbation of a constant state. We show that solution of the original system will convergence to the solution of the quasi-homogeneous incompressible Euler system. The proof is based on a combination of uniform estimates in $H^s$ norms with a compensated compactness argument. No well-prepared restrictions are imposed on the initial data.
A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differs by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by a well-known result of Edwards about Max-Cut, we study the maximum bisections on two types of the network graphs. For each $i\ge 1$, let $PN(i)$ be an $n$-vertex pyramid network graph with $m$ edges and let $CQ_i$ be an $n$-vertex crossed hypercube graph with $m$ edges. We show that $PN(i)$ admits a bisection of size at least $m/2+n-\sqrt{3n+1}+1$ and this bound is tight. We also prove that $CQ_i$ admits a bisection of size at least $m/2+(i-1)n/4$ and this bound is tight. Both of the lower bounds are larger than the Edwards' bound.
Erdős raised the following problem according to Steinberg’s conjecture: Is there an integer such that every planar graph without cycles of length from 4 to k is 3-colorable. By far, the result about the problem was improved to $k$≤ 7 by Borodin et al. However, by permitting the existence of adjacent triangles except $K_4$, for an arbitrary integer $k$≥5, there exists a planar graph without cycles of length from 5 to $k$ such that $G$ is not 3-colorable. Let d denote the minimum distance between two diamonds in $G$, where a diamond is the union of two adjacent triangles. In this paper, we prove that a planar graph $G$ with $d$≥2 and without cycles of length from 5 to 18 is 3-colorable. The reader is invited to find the smallest integer $k$ such that a planar graph $G$ with $d$≥2 and without cycles of length from 5 to $k$ is 3-colorable.
This paper explores the properties and mathematical formulations of multidimensional simple waves, extending the well-established theory of one-dimensional simple waves to higher dimensions. The study focuses on the connection between simple waves and the Monge-Amp`ere equation, particularly in the context of gas dynamics and potential flows. Key aspects include the characterization of simple waves in unsteady and steady flows, the role of characteristic lines, and the application of Hodograph and Legendre transformations to derive solutions. The paper also addresses the challenges and open questions in extending simple wave theory to more complex systems, such as non-reducible systems, radiative heat transfer, and chemical reactions. The research highlights both theoretical advancements and practical applications, providing a foundation for future studies in this area.
A classical theorem of Ahlswede and Katona determines the maximum density of the 2-edge star in a graph with a given edge density. Motivated by its application in hypergraph Turán problems, we establish a refinement of their result under the additional assumption that the graph contains a large independent set in which every vertex has high degree.
Chao and Yu introduced an entropy method for hypergraph Turán problems, and used it to show that the family of ⌊ k/2⌋ k-uniform tents have Turán density k!/k^k. Il'kovič and Yan improved this by reducing to a subfamily of ⌈ k/e⌉ tents. In this note, enhancing Il'kovič-Yan's result, we give a significantly shorter entropy proof, with optimal bounds within this framework.
The classical Andrásfai–Erdős–Sós Theorem states that for ℓ≥ 2, every n-vertex K_ℓ+1-free graph with minimum degree greater than 3ℓ-4/3ℓ-1n must be ℓ-partite. We establish a simple criterion for r-graphs, r ≥ 2, to exhibit an Andrásfai–Erdős–Sós type property, also known as degree-stability. This leads to a classification of most previously studied hypergraph families with this property. An immediate application of this result, combined with a general theorem by Keevash–Lenz–Mubayi, solves the spectral Turán problems for a large class of hypergraphs. For every r-graph F with degree-stability, there is a simple algorithm to decide the F-freeness of an n-vertex r-graph with minimum degree greater than (π(F) - ε_F)nr-1 in time O(n^r), where ε_F >0 is a constant. In particular, for the complete graph K_ℓ+1, we can take ε_K_ℓ+1 = (3ℓ^2-ℓ)^-1, and this bound is tight up to some multiplicative constant factor unless 𝐖[1] = 𝐅𝐏𝐓. Based on a result by Chen–Huang–Kanj–Xia, we further show that for every fixed C > 0, this problem cannot be solved in time n^o(ℓ) if we replace ε_K_ℓ+1 with (Cℓ)^-1 unless 𝐄𝐓𝐇 fails. Furthermore, we apply the degree-stability of K_ℓ+1 to decide the K_ℓ+1-freeness of graphs whose size is close to the Turán bound in time (ℓ+1)n^2, partially improving a recent result by Fomin–Golovach–Sagunov–Simonov. As an intermediate step, we show that for a specific class of r-graphs F, the (surjective) F-coloring problem can be solved in time O(n^r), provided the input r-graph has n vertices and a large minimum degree, refining several previous results.
As a popular and easy-to-implement machine learning method for solving differential equations, the physics-informed neural network (PINN) sometimes may fail and find poor solutions which bias against the exact ones. In this paper, we establish a framework of modified equation to explain the failure phenomenon and characterize the implicit bias of a general residual minimization (RM) method. We provide a simple way to derive the modified equation which models the numerical solution obtained by RM methods. Next, we show the modified solution deviates from the original exact solution. The proof uses a by-product of this paper, that is, a necessary and sufficient condition on characterizing the singularity of the coefficients. This equivalent condition can be extended to other types of equations in the future. Finally, we prove, as a complete characterization of the implicit bias, that RM method implicitly biases the numerical solution against the exact solution and towards a modified solution. In this work, we focus on elliptic equations with discontinuous coefficients, but our approach can be extended to other types of equations and our understanding of the implicit bias may shed light on further development of deep learning based methods for solving equations.
The efficiency of three Krylov subspace methods with their ILU0-preconditioned version in solving the systems with the nonadiagonal sparse matrix is examined. The systems have arisen from the discretization of Poisson's equation using the 4th and 6th-order compact schemes. Four matrix-vector multiplication techniques based on four sparse matrix storage schemes are considered in the algorithm of the Krylov subspace methods and their effects are explored. The convergence history, error reduction, iteration-resolution relation and CPU-time are addressed. The efficacy of various methods is evaluated against a benchmark scenario in which the conventional second-order central difference scheme is employed to discretize Poisson's equation. The Krylov subspace methods, paired with four distinct matrix-vector multiplication strategies across three discretization approaches, are tested and implemented within an incompressible fluid flow solver to solve the elliptic segment of the equations. The resulting solution process CPU-time surface gives a new vision regarding speeding up a CFD code with proper selection of discretization stencil and matrixvector multiplication technique.
In this paper, the method of fundamental solutions (MFS) is first developed for solving direct problems in bi-layer materials in the biomedical field of optical fluorescence. The governing system of second-order linear partial differential equations (PDEs) for the emission and excitation fluences is transformed into a single fourth-order PDE with appropriate boundary and interface matching conditions. The MFS is subsequently further developed, in conjunction with a constrained minimization regularization procedure, to solve nonlinear inverse optical fluorescence tomography problems. Numerical results confirm the accuracy, stability and versaof the meshless technique
We consider a Cahn-Hilliard gradient flow model with a free energy functional, which contains a non-local term in addition to linear and non-linear local terms. The non-local terms can be based on smooth and weakly singular kernel operators. We establish the well-posedness of this problem, construct an unconditional energy stable scheme, and carry out a stability and convergence analysis. Several numerical results are presented to illustrate the efficiency and robustness of the proposed scheme.
In this work, we modify a conjugate gradient（CG） method recently proposed in the literature, where a PRP conjugate gradient method is modified using trust region. Particularly, we propose a hybrid CG method that incorporates the parameters β PRP , β FR and β CD , and this new search direction satisfies both the trust region feature and the sufficient descent conditions.Furthermore, under suitable conditions the developed method is proved to be globally convergent. The method is tested on some benchmark problems from the literature and numerical results show that it is quite efficient in solving large scale problems.
