arXiv Math @arxiv_math@qoto.org

Let $H$ be a graph with vertex set $V(H)=\{v_1, v_2, \cdots, v_k\}$. The generalized blow-up graph $H_{p_1,\ldots,p_k}^{q_1,\ldots,q_k}$ is constructed by replacing each vertex $v_i \in V(H)$ with the graph $G_i = p_iK_t \cup q_iK_1$$(i=1,2,\cdots,k)$, then connecting all vertices between $G_i$ and $G_j$ whenever $v_iv_j \in E(H)$. In this paper, we enumerate the spanning trees in generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Ge [Discrete Appl. Math. 305 (2021) 145-153], Cheng, Chen and Yan [Discrete Appl. Math. 320 (2022) 259-269]. Furthermore, we determine the resistance distances and Kirchhoff indices of generalized blow-up graphs $H_{p_1, p_2, \cdots, p_k}^{q_1, q_2, \cdots, q_k}$, which extends the results of Sun, Yang and Xu [Discrete Math. 348 (2025) 114327], Xu and Xu [Discrete Appl. Math. 362 (2025) 18-33], Ni, Pan and Zhou [Discrete Appl. Math. 362 (2025) 100-108].

The two-layer quasi-geostrophic equations (2QGE) serve as a simplified model for simulating wind-driven, stratified ocean flows. However, their numerical simulation remains computationally expensive due to the need for high-resolution meshes to capture a wide range of turbulent scales. This becomes especially problematic when several simulations need to be run because of, e.g., uncertainty in the parameter settings. To address this challenge, we propose a data-driven reduced order model (ROM) for the 2QGE that leverages randomized proper orthogonal decomposition (rPOD) and long short-term memory (LSTM) networks. To efficiently generate the snapshot data required for model construction, we apply a nonlinear filtering stabilization technique that allows for the use of larger mesh sizes compared to a direct numerical simulations (DNS). Thanks to the use of rPOD to extract the dominant modes from the snapshot matrices, we achieve up to 700 times speedup over the use of deterministic POD. LSTM networks are trained with the modal coefficients associated with the snapshots to enable the prediction of the time- and parameter-dependent modal coefficients during the online phase, which is hundreds of thousands of time faster than a DNS. We assess the accuracy and efficiency of our rPOD-LSTM ROM through an extension of a well-known benchmark called double-gyre wind forcing test. The dimension of the parameter space in this test is increased from two to four.

Recent literature reports two sectional techniques, the finite volume method [Das et al., 2020, SIAM J. Sci. Comput., 42(6): B1570-B1598] and the fixed pivot technique [Kushwah et al., 2023, Commun. Nonlinear Sci. Numer. Simul., 121(37): 107244] to solve one-dimensional collision-induced nonlinear particle breakage equation. It is observed that both the methods become inconsistent over random grids. Therefore, we propose a new birth modification strategy, where the newly born particles are proportionately allocated in three adjacent cells, depending upon the average volume in each cell. This modification technique improves the numerical model by making it consistent over random grids. A detailed convergence and error analysis for this new scheme is studied over different possible choices of grids such as uniform, nonuniform, locally-uniform, random and oscillatory grids. In addition, we have also identified the conditions upon kernels for which the convergence rate increases significantly and the scheme achieves second order of convergence over uniform, nonuniform and locally-uniform grids. The enhanced order of accuracy will enable the new model to be easily coupled with CFD-modules. Another significant advancement in the literature is done by extending the discrete model for two-dimensional equation over rectangular grids.

Let $Λ$ be an order in a division algebra over a number field. We prove, under some conditions, that $SL_3(Λ)$ is boundedly generated by elementary matrices.

The past decade has seen notable advances in our understanding of structured error-correcting codes, particularly binary Reed--Muller (RM) codes. While initial breakthroughs were for erasure channels based on symmetry, extending these results to the binary symmetric channel (BSC) and other binary memoryless symmetric (BMS) channels required new tools and conditions. Recent work uses nesting to obtain multiple weakly correlated "looks" that imply capacity-achieving performance under bit-MAP and block-MAP decoding. This paper revisits and extends past approaches, aiming to simplify proofs, unify insights, and remove unnecessary conditions. By leveraging powerful results from the analysis of boolean functions, we derive recursive bounds using two or three looks at each stage. This gives bounds on the bit error probability that decay exponentially in the number of stages. For the BSC, we incorporate level-k inequalities and hypercontractive techniques to achieve the faster decay rate required for vanishing block error probability. The results are presented in a semitutorial style, providing both theoretical insights and practical implications for future research on structured codes.

We continue the development of the theory of capturing schemes over $ω_1$ by analyzing the relation between the capturing construction schemes (whose existence is implied by Jensen's $\Diamond$-principle) and both the Continuum Hypothesis and Ostaszewski's $\clubsuit$-principle. Formally, we show that the property of being capturing can be viewed as the conjunction of two properties, one of which is implied by $\clubsuit$ and the other one by CH. We apply these principles to construct multiple gaps, entangled sets and metric spaces without uncountable monotone subspaces.

We consider the computation of internal solutions for a time domain plasma wave equation with unknown coefficients from the data obtained by sampling its transfer function at the boundary. The computation is performed by transforming known background snapshots using the Cholesky decomposition of the data-driven Gramian. We show that this approximation is asymptotically close to the projection of the true internal solution onto the subspace of background snapshots. This allows us to derive a generally applicable bound for the error in the approximation of internal fields from boundary data for a time domain plasma wave equation with an unknown potential $q$. For general $q\in L^\infty$, we prove convergence of these data generated internal fields in one dimension for two examples. The first is for piecewise constant initial data and sampling $τ$ equal to the pulse width. The second is piecewise linear initial data and sampling at half the pulse width. We show that in both cases the data generated solutions converge in $L^2$ at order $\sqrtτ$. We present numerical experiments validating the result and the sharpness of this convergence rate.

We introduce two notions of barrier certificates that use multiple functions to provide a lower bound on the probabilistic satisfaction of safety for stochastic dynamical systems. A barrier certificate for a stochastic dynamical system acts as a nonnegative supermartingale, and provides a lower bound on the probability that the system is safe. The promise of such certificates is that their search can be effectively automated. Typically, one may use optimization or SMT solvers to find such barrier certificates of a given fixed template. When such approaches fail, a typical approach is to instead change the template. We propose an alternative approach that we dub interpolation-inspired barrier certificates. An interpolation-inspired barrier certificate consists of a set of functions that jointly provide a lower bound on the probability of satisfying safety. We show how one may find such certificates of a fixed template, even when we fail to find standard barrier certificates of the same template. However, we note that such certificates still need to ensure a supermartingale guarantee for one function in the set. To address this challenge, we consider the use of $k$-induction with these interpolation-inspired certificates. The recent use of $k$-induction in barrier certificates allows one to relax the supermartingale requirement at every time step to a combination of a supermartingale requirement every $k$ steps and a $c$-martingale requirement for the intermediate steps. We provide a generic formulation of a barrier certificate that we dub $k$-inductive interpolation-inspired barrier certificate. The formulation allows for several combinations of interpolation and $k$-induction for barrier certificate. We present two examples among the possible combinations. We finally present sum-of-squares programming to synthesize this set of functions and demonstrate their utility in case studies.

We study highest weight vectors for symmetric and alternating spaces of tensors, whose dimensions are given by generalized Kronecker coefficients. We present a unified explicit construction for corresponding spanning sets of highest weight vectors and completely describe the linear relations among them. We prove that these highest weight vectors satisfy a natural yet nontrivial duality. As applications, we also give conceptual interpretations to power expansions of Cayley's first hyperdeterminant and its dual exterior Cayley form.

We bring to light a new connection between dynamics and Heegaard Floer homology. On a closed 3-manifold $Y$ we consider a pseudo-Anosov flow $ϕ$ with no perfect fits with respect to its singularity locus $L \subset Y$, or perhaps a larger collection of closed orbits. Using work of Agol and Guéritaud on veering branched surfaces we produce a chain complex computing the link Floer homology of $L$ in the framing specified by the degeneracy curves of the flow. Using work of Landry, Minsky, and Taylor we show that the generators of the chain complex correspond to certain closed multi-orbits of $ϕ$. We prove that two canonical generators $\mathbf{x}^\mathrm{top}$ and $\mathbf{x}^\mathrm{bot}$ determine non-trivial homology classes located in the $\text{spin}^\text{c}$-grading of the flow, and its opposite. Finally, we observe that our specific model of the chain complex for link Floer homology naturally supports a grading with dynamical significance. This grading, a modification of the regular Maslov grading, is shown to categorify a suitable normalization of the zeta function associated to $ϕ$.

The author proves that there are infinitely many primes $p$ such that $\| αp - β\| < p^{-\frac{28}{87}}$, where $α$ is an irrational number and $β$ is a real number. This sharpens a result of Jia (2000) and provides a new triple $(γ, θ, ν)=(\frac{59}{87}, \frac{28}{87}, \frac{1}{29})$ that can produce primes in Ford and Maynard's work on prime-producing sieves. Our minimum amount of Type-II information required ($ν= \frac{1}{29}$) is less than any previous work on this topic using only traditional Type-I and Type-II information.

We characterize intermediate $\mathbb{R}$-algebras $A$ between the ring of semialgebraic functions ${\mathcal S}(X)$ and the ring ${\mathcal S}^*(X)$ of bounded semialgebraic functions on a semialgebraic set $X$ as rings of fractions of ${\mathcal S}(X)$. This allows us to compute the Krull dimension of $A$, the transcendence degree over $\mathbb{R}$ of the residue fields of $A$ and to obtain a Łojasiewicz inequality and a Nullstellensatz for archimedean $\mathbb{R}$-algebras $A$. In addition we study intermediate $\mathbb{R}$-algebras generated by proper ideals and we prove an extension theorem for functions in such $\mathbb{R}$-algebras.

It is proved that for any non-empty finite subset $Q$ of the square numbers, $ |Q+Q|\geq C'|Q|(\log |Q|)^{1/3+o(1)} $.

In this note, we give a characterization of immersed submanifolds of simply-connected space forms for which the quotient of the extrinsic diameter by the focal radius achieves the minimum possible value of $2$. They are essentially round spheres, or the ``Veronese'' embeddings of projective spaces. The proof combines the classification of submanifolds with planar geodesics due to K. Sakamoto with a version of A. Schur's Bow Lemma for space curves. Open problems and the relation to recent work by M. Gromov and A. Petrunin are discussed.

In this study, we address the challenges associated with accurately determining gaze location on a screen, which is often compromised by noise from factors such as eye tracker limitations, calibration drift, ambient lighting changes, and eye blinks. We propose the use of an extended Kalman filter (EKF) to smooth the gaze data collected during eye-tracking experiments, and systematically explore the interaction of different system parameters. Our results demonstrate that the EKF significantly reduces noise, leading to a marked improvement in tracking accuracy. Furthermore, we show that our proposed stochastic nonlinear dynamical model aligns well with real experimental data and holds promise for applications in related fields.

We present a comprehensive study on $SIM(2)$ and $ISIM(2)$ groups, their representations and algebraic aspects. After obtaining $SIM(2)$ through the Inönü-Wigner contraction procedure, a complete four-dimensional algebraic representation is shown for $\mathfrak{sim(2)}$ and $\mathfrak{isim(2)}$. Besides that, we apply Bargmann's formalism to investigate the (projective) representations for both cases, keeping track of the source of phase factors. We complete the study by presenting a particularly simple analysis to probe the existence of local phase factors, which is useful when dealing with non-abelian groups.

Let $G$ be an infinite countable amenable group and let $(X,G)$ be a $G$-subshift with specification, containing a free element. We prove that $(X,G)$ is universal, i.e., has positive topological entropy and for any free ergodic $G$-action on a standard probability space, $(Y,ν,G)$, with $h(ν)<h_{top}(X)$, there exists a shift-invariant measure $μ$ on $X$ such that the systems $(Y,ν,G)$ and $(X,μ,G)$ are isomorphic. In particular, any $K$-shift (consisting of the indicator functions of all maximal $K$-separated sets) containing a free element is universal.

For each of the following conditions, we characterize the pseudovarieties of semigroups V that satisfy it: (i) every epimorphism to a member of V is onto; (ii) every epimorphism to a finite semigroup with domain a member of V is onto; (iii) for every epimorphism from S to T with S in V and T finite, T is also a member of V.

We present second-order algorithms to approximate the solution of a nonlocal Gray-Scott model that is known to generate interesting spatio-temporal structures such as pulse and stripes solutions. Our algorithms rely on a quadrature method for the spatial discretization and the method of lines using a second-order Adams-Bashforth for the time marching. We focus on studying the impact of the type of boundary constraints, e.g. nonlocal Dirichlet/Neumann or local periodic, and the type of nonlocal diffusion, i.e. integral operator with thin- or fat-tailed kernels, on the generation of pulse solutions. Our numerical investigations show that when the spread of the kernel is large, i.e. when the model is nonlocal, both the type of kernels and type of boundary constraints have a strong impact on the solutions profiles.

We consider a polling system with two queues, where a single server is attending the queues in a cyclic order and requires non-zero switching times to switch between the queues. Our aim is to identify a fairly general and comprehensive class of Markovian switching policies that renders the system stable. Potentially a class of policies that can cover the Pareto frontier related to individual-queue-centric performance measures like the stationary expected number of waiting customers in each queue; for instance, such a class of policies is identified recently for a polling system near the fluid regime (with large arrival and departure rates), and we aim to include that class. We also aim to include a second class that facilitates switching between the queues at the instance the occupancy in the opposite queue crosses a threshold and when that in the visiting queue is below a threshold (this inclusion facilitates design of `robust' polling systems). Towards this, we consider a class of two-phase switching policies, which includes the above mentioned classes. In the maximum generality, our policies can be represented by eight parameters, while two parameters are sufficient to represent the aforementioned classes. We provide simple conditions to identify the sub-class of switching policies that ensure system stability. By numerically tuning the parameters of the proposed class, we illustrate that the proposed class can cover the Pareto frontier for the stationary expected number of customers in the two queues.