Papers

This note generalizes the port-Hamiltonian formulation of the continuous time primal-dual gradient algorithm for static constrained convex optimization to the convex optimal control problem.The resulting dynamics is shown to be a port-Hamiltonian system of partial differential equations, involving ordinary physical time as well 'algorithmic' time. Convergence to the optimal control solution is indicated, and it is argued that sub-optimal control strategies could be derived starting from the partial differential equation formulation.

It is known that the study of vibration characteristics at the tip of the micro-cantilever and its relationship to the sample plays a very important role in improving the resolution of an Atomic Force Microscopy (AFM). In this paper, within the Euler Bernoulli beam model, a mathematical model, defined as a model with two unknown inputs and two measured outputs, is considered for the simultaneous determination of the unknown bending moment and the shear force at the tip of the micro-cantilever from two feasible measured outputs at the same tip: the deflection and the slope. This model leads to the following inverse problem: find $M(t)$ and $g(t)$ in $\rho_A(x)u_{tt}+\mu(x) u_{t}+(r(x)u_{xx})_{xx}=0$, $(x,t)\in \Omega_T:=(0,\ell)\times (0,T)$ subject to the boundary conditions $u(0,t)=u_{x}(0,t)=0$, $\left(r(x)u_{xx}\right)_{x=\ell}=M(t)$, $\left((r(x)u_{xx})_x\right)_{x=\ell}=g(t)$, and the homogenous initial conditions, from the measured outputs $w_{\ell}(t):=u(\ell,t)$ and $\theta_{\ell}(t):=u_x(\ell,t)$. It is proved that the vector-form input-output map $\mathcal{P}:=\left (\Phi, \Psi \right )$, with $\left (\Phi q \right )(t):=u(0,t;q)$ and $\left (\Psi q\right )(t):=u_x(0,t;q)$, where $q(t):=\left (M(t),g(t)\right )$, corresponding to the inverse problem, is compact and Lipschitz continuous. This result allows us to prove the existence of a solution of the minimization problem for the Tikhonov functional. As a consequence, the existence of a quasi-solution to the inverse problem is established. Furthermore, a vector form expression for the Frechet gradient of the Tikhonov functional is derived, and the Lipschitz continuity of the Frechet gradient is rigorously proven. This crucial property ensures the monotonic behavior of iterative gradient-based numerical methods.

This paper investigates the exponential ergodicity of the exact solution and the tamed Euler solution for McKean-Vlasov stochastic differential equations driven by Lévy noise. First, we establish exponential ergodicity for both the original equation and the tamed Euler method. Then we prove the convergence of the numerical invariant measure to the exact invariant measure, which is obtained by combining the propagation of chaos (PoC) result with the strong convergence of the tamed Euler scheme. Furthermore, we derive a convergence rate for the numerical invariant measure by establishing uniform-in-time PoC and uniform-in-time convergence of the tamed Euler method. Finally, numerical experiments are presented to illustrate the theoretical results.

Color graphs and their subgraphs, referred to as bubble graphs, correspond bijectively to the simplicial complexes of pseudomanifolds and their subsimplices, respectively. In this paper, we introduce matrix representations for colored graphs and their associated bubble graphs. By using this correspondence, we define simplicial-complex matrices and subsimplex matrices that encode the simplicial complexes of pseudomanifolds and their subsimplices. Moreover, we formulate mutation and crossover operations on colored graphs. Through the established correspondence among simplicial complexes, colored graphs, and simplicial-complex matrices, we extend these operations to simplicial complexes and simplicial-complex matrices. We further implement an algorithm generating simplicial-complex matrices and a genetic algorithm performing mutation and crossover of them to produce pseudomanifolds exhibiting diverse topologies. In addition, we implement procedures for decomposing the generated simplicial-complex matrices into simplex matrices, reconstructing the simplicial complexes of the associated pseudomanifolds from this information, and computing geometric quantities such as the volume, circumcenter, and dual-simplex volume of each simplex.

Cooling-limited exothermic semi-batch reactors require coordinated feed and cooling control to shorten batch time while maintaining the prescribed temperature. We develop a theoretical basis for advanced regulatory control (ARC) design by combining minimum-time and local safety analyses. Minimum-time analysis leads to an economic valve position control structure that adjusts feed using the temperature control system's cooling request, while cooling regulates temperature. Local safety analysis specifies the controller form and tuning conditions for reducing feed during cooling overload and restoring it as capacity becomes available. We also provide guidelines for industrial implementation and tuning. A reduced benchmark verifies the analytical tuning conditions, and an industrial-scale polymerization model evaluates the design. In simulations with parameter mismatch and unmodeled reaction dynamics, ARC achieves batch times comparable to those of parameter adaptive nonlinear model predictive control, using regulatory feedback without online nonlinear optimization.

For a $T_0$-space $X$, let $Q (X)$ be the poset of nonempty compact saturated sets of $X$ with the reverse inclusion order. The space $X$ is said to have property Q if it satisfies the following two conditions: (1) $\wedge K$ exists for any $K\in Q(X)$, and (2) for any filtered family $\{K_d : d\in D\}\subseteq Q(X)$ and $x\in X$, if $\bigvee^{\uparrow}_{d\in D}\bigwedge K_d$ exists and $x\not\leq \bigvee^{\uparrow}_{d\in D}\bigwedge K_d$, then there is $\varphi\in \prod\limits_{d\in D}\!\!K_d$ and an upper bound $u$ of $\varphi(D)$ such that $x\not\leq u$. In this paper, we prove that every $d$-space with property Q is well-filtered and the Smyth power space of a $T_0$-space always has property Q. Hence the Smyth power construction preserves the well-filteredness. For a complete lattice $L$ and an order-compatible $d$-topology $\tau$ on it, we show that when $L$ possesses a certain distributivity, $(L, \tau)$ is well-filtered.

Inspired by research on the Cartesian product of two lattice polytopes, this paper investigates the Ehrhart theory of the join of two lattice polytopes. This is also a well-known open problem listed on the website of the American Institute of Mathematics. This paper resolves this open problem. We first construct counterexamples showing that the join of two Ehrhart positive polytopes is not necessarily Ehrhart positive. Then we prove that if two lattice polytopes have the integer decomposition property and the spanning property, then their join also has these two properties. However, the very ample property is not inherited under joins. Finally, we show that unimodular triangulations, regular triangulations, and quadratic triangulations are preserved under the join operation. As a byproduct, we state the necessary and sufficient condition for the Cartesian product of two Gorenstein lattice polytopes to remain Gorenstein.

We give a lower bound of the minimum crossing number of knots and links projected on a 2-sided surface in a 3-manifold using the rank of fundamental groups which means that the crossing number can take arbitrary large values. On the contrary, we show that for a branched surface, the minimum crossing number is always zero. As an application, we give an upper bound of the three-page index by the crossing number.

We present a numerical method for computing resonances of one-dimensional Schrödinger equations perturbed by a compactly supported potential, via finding zeros of the Wronskian associated with Jost solutions of the reference equation, computed through the resolution of Cauchy problems. All resonances located in a given domain are found efficiently using a defeated Newton algorithm. A key ingredient of the method is the choice of reference potential for which Jost solutions are known, which removes spurious resonances often encountered numerically. We test this method on three types of reference potentials and perturbations thereof: Pöschl-Teller potentials, exponentially decaying potentials, and potentials associated with Reissner-Nordströmde Sitter black holes. In particular we study the impact of perturbations on the resonances, and the stability of small resonances under perturbation. As an illustration, we use the method to numerically study the strong cosmic censorship hypothesis.

Let $f:X\to S$ be a smooth proper family of smooth projective varieties. An irreducible complex local system on a fiber admits an isomonodromic deformation, hence determines a holomorphic section of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence produces a real-analytic section $\sigma_{Dol}:S\to M_{Dol}(X/S)$ of the relative Dolbeault moduli space. In this paper, we investigate when this real-analytic section is holomorphic. The first approach uses the first-order infinitesimal deformation: we prove a Cauchy–Riemann type criterion showing that holomorphicity in a tangent direction of $S$ is measured by the composition of the Kodaira–Spencer map with the non-abelian Higgs field. The second approach involves higher-order derivatives: after restricting $\sigma_{Dol}$ to infinitesimal thickenings of the reference point in $S$, we introduce obstruction classes measuring the failure of holomorphicity and relate them to the Taylor expansion of the harmonic metric. We apply these criteria to three problems. First, we study the interaction between the $\mathbb C^*$-action on Higgs bundles and isomonodromic deformations.Second, for an initial polarized complex variation of Hodge structures, we consider the associated non-abelian Noether–Lefschetz locus. We prove that this locus is precisely the maximal complex analytic subvariety of $S$ on which the real-analytic isomonodromic deformation $\sigma_{Dol}$ becomes holomorphic. Both the first-order and higher-order methods yield proofs of this characterization. Lastly, we prove that if the initial Higgs bundle is generically regular nilpotent and the isomonodromic deformation is holomorphic, then every member of the family is represented by a nilpotent Higgs bundle.

The concept of geodesic-like curves was introduced by Chen in 2010 as a method for estimating shortest paths (geodesics) on parametric surfaces, with its convergence established theoretically. However, an efficient numerical computational framework has not yet been developed. In this paper, we propose an elegant and efficient approach for computing geodesic-like curves by leveraging deep learning and Physics-Informed Neural Networks (PINNs). Under the proposed framework, not only can single parametric surfaces be handled efficiently, but a broad class of complex parametric surfaces including multi-surface systems with $C^0$ or higher continuity and surfaces of revolution can also be robustly addressed.

In this paper, we study a new inverse spectral problem that consists in the recovery of the third-order differential equation from two spectra corresponding to the boundary conditions $y(0) = y(1) = y(2) = 0$ and $y(0) = y'(0) = y(1) = 0$. The uniqueness and existence theorems for the solution are obtained. To prove the results, we treat the inverse problem using a general approach that reconstructs higher-order differential operators from the Weyl-Yurko matrix.

A Dyck path of semilength $n$ is a lattice path from $(0,0)$ to $(n,n)$ consisting of $n$ right-steps $(1,0)$ and $n$ up-steps $(0,1)$ that never rises above the line $y=x$. These paths are enumerated by the Catalan numbers and play a central role in enumerative combinatorics. We color the cells of the integer grid in black and white according to two natural patterns, namely chessboard and column-alternating, and enumerate the Dyck paths having equal numbers of black and white cells beneath them.

The self-averaging of the replica overlap is proven in the Edwards-Anderson (EA) model under random field almost everywhere in the coupling constant space in any dimension. The EA order parameter is represented in terms of the derivative of the free energy density with respect to the random field strength, regardless of boundary conditions. Tasaki's correlation inequality for finite-dimensional spin glass models shows that the expectation of the squared replica overlap is bounded by the squared EA order parameter. These simple evaluations enable us to prove that the variance of the replica overlap vanishes in the infinite-volume limit. The self-averaging of the replica bond overlap is proven also in the EA model with Gaussian exchange interaction without random field. Short-range spin glass models have been shown to behave differently from mean-field spin glass models with RSB phase.

We present a survey on the bottom of the spectrum of the Hodge Laplacian on complete noncompact Kähler manifolds, with particular emphasis on Kähler hyperbolic manifolds and bounded symmetric domains. We also discuss theorems regarding the upper bounds for the bottom of the spectrum under Ricci and bisectional curvature assumptions, along with rigidity results for manifolds attaining the maximal bottom of the spectrum. Throughout the article, we propose several open problems.

We consider the operator–theoretic model for the group of biholomorphic automorphisms $Aut(B)$ of the unit ball $B$ of a complex Hilbert space $\H$ by representing each automorphism as a bounded linear operator on the augmented Hilbert space $\H\oplus \mathbb{C}$. Any member of $Aut(B)$ admits a natural block operator matrix representation acting on $\mathcal{H}\oplus\mathbb{C}$. We study the geometry of the subset $M(\mathcal{H})$ of $\mathcal{B}(\mathcal{H}\oplus\mathbb{C})$ consisting of these block operator matrices. It is shown that every element corresponding to a non-rotation automorphism is a smooth point of $\mathcal{B}(\mathcal{H}\oplus\mathbb{C})$. Orthogonality between two such matrices is characterized geometrically by the antipodality of the corresponding Möbius images of a boundary point of the ball. This orthogonality characterization is applied to show that an inner automorphism of $\Aut(B)$ that preserves Birkhoff–James orthogonality in both directions if and only if it is conjugation by a pure rotation, yielding a rigidity result. The normalized block matrices are $J$-unitary under a suitable normalization, where $J = \operatorname{diag}(I_{\mathcal{H}}, -1)$. We show that norm of such block matrices satisfy a submultiplicativity under a certain composition rule other than usual operator multiplication, and induce a metric on certain subsets of $Aut(B)$ which recover the hyperbolic metric on the Hilbert ball. The symmetric structure of Birkhoff–James orthogonality within $M(\mathcal{H})$ is also studied: there are no left-symmetric points, while the only right-symmetric points are pure rotations.

The Moving-Target Traveling Salesman Problem (MT-TSP) seeks a minimum cost trajectory for an agent that departs from a static depot, visits a set of moving targets, each within one of their assigned time windows, and returns to the depot. In this article, we study the Moving-Target Traveling Salesman Problem with Moving Obstacles (MT-TSP-MO), a generalization of the MT-TSP where the agent trajectory must avoid moving obstacles. We present a Mixed-Integer Conic Programming (MICP) formulation that can be solved using off-the-shelf solvers, as well as a fast and scalable Two-Phase Bilevel Search (TPBS) algorithm that computes high-quality feasible solutions for the problem. We evaluate our approaches against an existing baseline algorithm on a broad range of problem instances with up to 40 targets and 40 obstacles. The results demonstrate that both the proposed methods significantly outperform the baseline with respect to success rates, solution costs, and computation time.

We introduce and study the notion of norming approximate orthogonality, a two-parameter generalization of Birkhoff–James orthogonality in normed linear spaces. For $\delta, \varepsilon \in [0,1)$ with $\varepsilon < (1-\delta)^2$, we say $x \nperp y$ in $X$ if there exists $f \in X^*$ with $|f(x)| \geq (1-\delta)\|f\|\|x\|$ and $|f(y)| \leq \frac{\varepsilon}{1-\delta}\|f\|\|y\|$, simultaneously relaxing both the norming condition on $x$ and the vanishing condition on $y$. It is proved that \[ x\nperp y \iff \|x+\lambda y\|\geq (1-\delta)\|x\|-\frac{\varepsilon}{1-\delta}\|\lambda y\|~\qquad \forall ~\text{scalars}~\lambda. \] This framework interpolates between two notions of approximate orthogonality in normed linear spaces due to Chmieliński and Dragomir, and recovers three existing notions of orthogonality in extreme cases: exact Birkhoff–James orthogonality at $\delta = \varepsilon = 0$, the approximate orthogonality of Chmieliński at $\delta = 0$, and the approximate orthogonality of Dragomir at $\varepsilon = 0$. A two-parameter proximity result generalizing Chmlieński's characterization of $\perp_B^\varepsilon$ is established. The forward and converse implications are governed by the distinct thresholds $\frac{\varepsilon}{(1-\delta)^2}$ and $\frac{\varepsilon}{1-\delta}$, which collapse to $\varepsilon$ of Chmieliński precisely when $\delta=0$, and the strictness of this gap is confirmed by counterexamples in $\ell_\infty^2$. A dual formulation of norming approximate orthogonality is established with a complete equivalence in the reflexive case. We apply our results to (vector-valued) continuous function spaces, which extends some earlier results and recovers few operator theoretical results with alternative proofs using measure theoretic techniques.

In this paper, we establish generalized B.-Y. Chen inequalities for Riemannian submersions and Riemannian maps between Riemannian manifolds by employing the generalized $\delta$-invariants introduced by Chen. We derive optimal inequalities involving the generalized $\delta$-invariants associated with the vertical and horizontal distributions together with extrinsic invariants determined by the second fundamental tensors of the submersion and the Riemannian map. Furthermore, we characterize the equality cases through precise algebraic conditions on the corresponding shape operators, providing their geometric interpretation. As applications, we obtain explicit generalized Chen inequalities for Riemannian submersions and Riemannian maps whose total or target manifolds are real space forms and complex space forms. These results extend the classical Chen inequalities as well as several recent Chen-type inequalities for Riemannian submersions and Riemannian maps available in the literature.

This paper studies Nesterov accelerated methods for continuously differentiable convex-concave bilinear saddle point problems. For the continuous-time model, we analyze a second-order primal-dual dynamical system with vanishing damping $\alpha/t$, where $\alpha\geq 3$. Under the merely convex-concave setting, we prove convergence of the primal-dual trajectory to a saddle point. In the noncritical regime $\alpha>3$, we further obtain the improved rate $o(1/t^{2})$ for the primal-dual gap and $o(1/t)$ for the velocity, and, under an additional Lipschitz gradient assumption, $o(1/t)$ for the stationarity residual. We then derive a structure-preserving finite-difference discretization, which leads to a fast primal-dual algorithm with Nesterov extrapolation. For a general accelerated parameter sequence ${t_k}$ satisfying $t_{k+1}^2-t_k^2\le \rho t_{k+1}$ with $\rho\in(0,1]$, we prove the $O(1/t_k^{2})$ convergence rate for the primal-dual gap and convergence of the generated sequence. In the noncritical case $\rho<1$, we further establish the improved rate $o(1/t_k^{2})$ for the gap and $o(1/t_k)$ for the stationarity residual. These results provide continuous-discrete acceleration methods for bilinear saddle point problems in the merely convex-concave setting.

In this short article we consider a preferential attachment random graph model with edge steps, studied by Alves, Ribeiro and Sanchis. Starting with an initial graph $\mathbb{G}_1$ formed by a vertex with a self-loop attached to it, the model evolves as follows. At every subsequent (discrete) time step, either with probability $p$ we add a vertex to the graph and connect it to exactly one of the older vertices selected with probability proportional to its degree, or with probability $1-p$ we add one edge between two existing vertices, both selected (independently) with probability proportional to their degrees. Let $\omega(\mathbb{G})$ be the clique number of a graph $\mathbb{G}$, i.e.\ the number of vertices in a largest complete subgraph of $\mathbb{G}_{}$. Alves, Ribeiro and Sanchis showed that, for any given $\varepsilon>0$, we have $\omega(\mathbb{G}_{2t})\geq t^{\frac{1-p}{2-p}(1-\varepsilon)}$ with high probability (i.e.\ with probability tending to $1$ as $t\rightarrow \infty$). Here we strengthen this bound by showing that, for any function $f:\mathbb{N}\mapsto \mathbb{N}$ that satisfies $f(t)\rightarrow \infty$ as $t\rightarrow \infty$, with high probability \[\omega(\mathbb{G}_{2t}) = \Omega\left(t^{\frac{1-p}{2-p}}\Big(\log^{\frac{1}{2-p}}(t)f(t)\Big)^{-1}\right).\]

Perfect sphere packing in the Boolean space is a fundamental and complex problem with significant implications for coding theory, cryptography, and discrete mathematics. The classical solution to the perfect sphere packing problem was provided by Hamming via his well-known perfect codes. However, a major limitation of the traditional Hamming metric is its strict applicability, as it allows perfect partitioning only for spaces with specific, highly constrained dimensions. To address this structural limitation, this article introduces a novel distance metric specifically designed for Boolean hypercubes. The proposed metric modifies the topological properties of the space, making it mathematically viable to partition a Boolean space of any arbitrary dimension into disjoint, perfect spheres. We rigorously define the algebraic properties of this new distance function and demonstrate its consistency across various dimensions. Furthermore, we explore the structural characteristics of the resulting packings. This approach bypasses the classical dimensional constraints of Hamming codes, potentially opening new avenues for designing error-correcting codes and cryptographic primitives in non-traditional dimensions.

Fault-tolerant broadcasting in dense Gaussian networks is recovered by re-rooting the broadcast at a new source at maximum graph distance from the faulty nodes. This paper extends the re-rooting framework by replacing its boundary-search source-selection step with a quotient-lattice-aware algebraic construction. The first contribution is a constant-time counting method for valid new sources, formulated as an intersection of two diameter-$k$ boundary sets in the Gaussian quotient. The exact count is obtained by a fixed union of side-pair intervals over nine quotient-lattice copies, giving a closed-form procedure without scanning the network or boundary. The second contribution is a shifted direct selector for two arbitrary faulty nodes. Given faulty nodes $A$ and $B$, the problem is translated to $C=\operatorname{mod}_{G_k}(B-A)$, and the selector finds $P$ satisfying $d(P,0)=d(P,C)=k$. For each of nine quotient-lattice shifts, sixteen signed linear systems are checked. Nonparallel systems are solved via Cramer's rule; parallel systems are handled by interval-endpoint selection. At most $9\times16=144$ shifted sign cases are evaluated, giving $O(1)$ selection under the word-RAM model. Validation reports zero count mismatches over $26{,}623$ tested nodes, $500{,}000$ valid outputs over $500{,}000$ sampled fault pairs, and $40{,}000$ successful re-rooted broadcast trials. The shifted selector achieves a $5.92\times$ speedup over boundary search at $k=200$, remaining stable as $k$ increases. These results make new-source selection algebraic, bounded, and independent of network size.

Fault-tolerant broadcasting in dense Eisenstein–Jacobi networks requires efficient recovery when faulty nodes disrupt the original broadcast structure. A re-rooting-based method guarantees that, for any two faulty nodes, a valid new source exists at maximum graph distance from both faults. However, identifying such a source without scanning the network or testing all boundary candidates remains an open practical problem. This paper presents a closed-form, constant-time algorithm for counting and selecting a valid new source in dense Eisenstein–Jacobi networks under two node faults. The two-fault problem is reduced to a boundary-intersection problem involving the origin and a difference node. The distance-$t$ boundary, where $t$ is the network diameter, is partitioned into six directed sides of the Eisenstein–Jacobi hexagon. Since the network is a quotient structure, intersection equations are solved modulo the defining lattice, requiring evaluation of seven quotient-lattice shifts across all $6\times 6$ side pairs, yielding at most $252$ algebraic systems. The first algorithm counts all valid new sources for faults at $0$ and $A$. The second algorithm selects one valid new source for arbitrary fault pairs by solving translated side-pair systems, verifying each candidate, and shifting back. Each system is either a non-parallel $2\times 2$ linear system with at most one candidate, or a parallel system whose feasible candidates form an integer interval. Both algorithms run in $O(1)$ time under the fixed-word arithmetic model. Computational validation over $500{,}000$ sampled fault pairs and $40{,}000$ re-rooting trials confirms correctness: the selector always returns a valid new source, and the recovered broadcast reaches all non-faulty nodes.

Dense Eisenstein–Jacobi networks are degree-six algebraic interconnection topologies with regular structure, vertex symmetry, small diameter, and efficient communication algorithms. These properties make them suitable for parallel and on-chip communication systems in which collective operations such as one-to-all broadcasting are frequent. Existing optimal broadcasting algorithms for dense hexagonal/Eisenstein–Jacobi networks assume fault-free operation. However, a faulty internal forwarding node may interrupt message propagation and prevent complete delivery. This paper proposes a lightweight re-rooting-based fault-tolerant broadcasting method for dense Eisenstein–Jacobi networks. The main idea is to relocate the effective broadcast source to a new source node such that each faulty node is located at graph distance equal to the network diameter from the new source. Consequently, faulty nodes become leaf-level nodes in the broadcast process and are not required to forward the message. We present source-selection algorithms for one- and two-node failures and prove that for any pair of faulty nodes in a dense Eisenstein–Jacobi network there exists a common distance-diameter node that can serve as a valid re-rooted source. The source-selection procedure requires linear time in the network diameter. Equivalently, since $N=3t^2+3t+1$, the selection cost is $O(\sqrt{N})$ in the number of nodes. Since the standard one-to-all broadcast completes in one diameter time and the relocation phase is also bounded by one diameter, the proposed method completes in at most twice the network diameter. We also show that the two-fault guarantee does not generally extend to arbitrary three-fault configurations by giving an explicit counterexample. The proposed approach improves broadcast reliability without constructing redundant spanning trees, backup paths, or additional broadcast structures.

We study estimation and prediction in the Gamma model $\mathrm{Ga}(\alpha,\beta)$, where the shape parameter $\alpha$ is known and the scale parameter $\beta$ is unknown, under the Kullback–Leibler loss. For $\alpha\le1$, all scale-invariant estimators of $\beta$ have infinite risk, indicating a qualitative change in the estimation problem at the boundary $\alpha=1$. Our main result is that the Bayesian predictive density based on the Jeffreys prior is admissible for all $\alpha>0$. This resolves the admissibility problem for Bayesian predictive densities in Gamma models. As a related result, we also establish the admissibility of the corresponding Bayesian estimator for $\alpha>1$. To prove the predictive admissibility result, we develop an infinitesimal prediction framework based on Gamma processes. This framework naturally leads to a Kullback–Leibler loss for Lévy densities and establishes a connection between predictive distributions and Lévy measures. Under the resulting loss, the Bayesian predictive Lévy density is shown to be the posterior mean Lévy density. Unlike the normal and Poisson models, infinitesimal prediction in the Gamma model does not reduce to parameter estimation. Instead, it reduces to the estimation of a Lévy density. We relate this phenomenon to mean mixture curvature and discuss it from an information-geometric viewpoint.

In this work, we investigate a chemotaxis-driven fish–mussel ecosystem model described by a coupled system of partial differential equations subject to homogeneous Neumann boundary conditions. Under suitable assumptions on the system parameters and initial data, we establish the global existence of classical solutions by employing semigroup methods together with a priori estimates. We also examine the possible blow-up behavior of solutions in a three-dimensional domain. To support the theoretical analysis, a finite element method is developed for the numerical approximation of the system, and convergence studies based on mesh refinement are carried out to verify the accuracy and stability of the proposed numerical scheme. Furthermore, numerical simulations illustrating the blow-up behavior of solutions in the computational domain are presented.

This work is about two conjectures stated by Burity–Simis–Tohăneanu regarding the Jacobian ideal of the defining polynomial of a central arrangement of $m$ hyperplanes. One settles one of these conjectures referring to the Jacobian ideal being a minimal reduction of the ideal of $(m-1)$-fold products. The second conjecture claiming the linear type property of the Jacobian ideal is disproved in rank at least four, by means of an explicit counter-example. In the latter the corresponding Rees algebra admits a torsion defining equation which is a Pfaffian syzygetic obstruction in degree two. One also relates this Pfaffian obstruction to circuits and codimension-two flats of the arrangement.

Let $M$ be a relatively compact open subset of a complex manifold $M'$ with smooth boundary $X$ and let $L$ be a holomorphic line bundle over $M'$. Assuming that condition $Z(q)$ holds, we establish the semi-classical asymptotic behavior of $e^{-\frac{t}{k}\Box^{q}_k}$ near the boundary $X$ as $k\to\infty$, where $\Box^{q}_k$ is the $\bar{\partial}$-Neumann Laplacian acting on $(0,q)$-forms on $M$ with values in $L^k$. Our results extend the seminal work of Bismut to complex manifolds with boundary. As applications of our results, we provide a heat kernel-based proof of the holomorphic Morse inequalities for complex manifolds with boundary and derive a semi-classical Weyl law for the $\bar{\partial}$-Neumann Laplacian.

This article establishes the completeness of an axiomatization for the robust safety of dynamical systems with polynomial differential equations on bounded time horizons. Safety properties of robust systems are uniformly reduced to a sound axiomatization of polynomial invariants, resulting in reliable logical proofs of correctness. Approximate decidability results are also established: there is a computable algorithm such that, given any perturbation parameter $\delta$, it either produces a symbolic proof of robust safety (hence correctly decides the dynamical system to be robustly safe), or correctly decides that the system is not robustly safe under a perturbation of level $\delta$. In contrast to earlier works, this article crucially leverages results from subanalytic geometry to retain a level of exactness, thereby establishing positive results of provability/decidability allowing for arbitrary bounded (semialgebraic) initial/post conditions even without positive separation at their (topological) boundaries. This enables the generation of proofs of inductive safety beyond finite time horizons for general hybrid dynamical systems.