We introduce a rank-one Riemannian cometric update inducing a modification of the Riemannian metric that makes specific directions of motion cheaper to travel along. We establish basic completeness properties of this reward metric, and give an explicit characterization of its Levi--Civita connection. We propose a preconditioned trajectory-tracking strategy by adding the connection-difference term to a standard intrinsic PD control, and illustrate the construction on a connection control-affine system on the Special Euclidean group with a maze navigation experiment. When the nominal trajectory is an integral curve of the vector field used to define the reward metric, our methodology improves the overall tracking, which is demonstrated through simulation results.

This paper constructs a connection map on the second-order tangent bundle induced by a linear connection on the base manifold and uses it to define a generalized Sasaki metric. The associated geodesic equations are derived, and jet-constrained variational problems are shown to yield Riemannian quintics in tension. The construction is then specialized to rigid body attitude dynamics with first-order actuator dynamics, producing an intrinsic higher-order trajectory model on the rotation group. Numerical simulations compare quintics in tension with Riemannian cubics as nominal trajectories and show modest reductions in actuator-relevant cost with comparable tracking performance.

Duality relations for hypergeometric functions have reappeared as an active research topic several times, with the first instances tracing back to Euler and Gauss and the latest burst of activity occurring between 2015 and 2023. In this paper we present a common generalization of all relations of this type found in the existing literature both for hypergeometric and for $q$-hypergeometric functions. We cover both Gauss type and confluent generalized hypergeometric functions and their $q$-analogues. Our results entail a number of corollaries including multi-term relations for hypergeometric and $q$-hypergeometric series at a fixed argument.

Environment division multiple access (EDMA) exploits line-of-sight (LoS) availability, blockage diversity, and spatial isolation in the propagation environment to regulate inter-region interference, while non-orthogonal multiple access (NOMA) improves intra-region access efficiency through power-domain multiplexing and successive interference cancellation (SIC). This paper investigates heterogeneous environment-map-aware user matching and power allocation for pinching-antenna-assisted EDMA-NOMA systems. Based on large-scale path loss and an exponential LoS blockage model, an average LoS/NLoS effective large-scale channel gain is constructed, allowing the environment map to affect both desired service links and inter-region interference links. This avoids the overly optimistic assumption that environmental blockage only suppresses interference. A utility-based joint user-pairing and power-allocation algorithm (UBA-JPPA) is then proposed to jointly account for system throughput and the scheduled-user Jain fairness index under NOMA power constraints and an SIC-consistent rate definition. Simulation results show that, compared with the ablation baseline without a heterogeneous environment map, the proposed scheme provides consistent gains in sum rate, fairness, and the throughput--fairness operating region. Meanwhile, a strong EDMA-OMA benchmark remains competitive in some high-SNR regimes, indicating that the focus of this work is to verify the benefit of heterogeneous environment information for EDMA-NOMA resource allocation rather than to claim that NOMA universally outperforms OMA.

An ideal in a polynomial ring is symmetric if it is invariant under any permutation of variables. In this paper, we define and study the symmetric and isotypic Hilbert series for symmetric ideals in a polynomial ring with countably many variables. The symmetric Hilbert series is the limit of the Hilbert series of the invariant parts of the finite truncated quotients, while the isotypic Hilbert series records stable multiplicities of irreducible symmetric-group representations for each degree. Our main result proves that, under a mild support condition on the ideal, the symmetric Hilbert series is a rational function. We further show that this rationality extends to the isotypic Hilbert series for every irreducible representation. The proofs of these results rely on the monomial structure of the polynomials within the symmetric ideal, combined with Kostka inversion for the isotypic case.

The study of word maps and Waring-like problems has been widely pursued for finite simple groups, algebraic groups, and Lie groups. In this article, we study Engel word maps $e_{m}(x, y) = \left[\cdots\left[[x, y], y \right], \cdots, y \right]$ on certain linear groups over local rings, namely, $\mathrm{SL}_2(\mathcal R)$ and $\mathrm{PSL}_2(\mathcal R)$. We consider the commutative ring $\mathcal {R} $ to be either a complete, local principal ideal ring $\mathcal O$, or a local principal ideal ring of finite length $\mathcal O_\ell$. Suppose the characteristic of the residue field $k\cong \mathbb F_q$ is $\neq 2$. Under some mild conditions on $q$, we show that there exists a constant $q_0(m)$, such that for all $q \geq q_0(m)$, all lifts in $\mathrm{SL}_2(\mathcal{O})$ of non-scalar elements of $\mathrm{SL}_2(k)$, are in the image of the $m$-th Engel word over $\mathrm{SL}_2(\mathcal{O})$. We further show that all Engel word maps are surjective on $\mathrm{PSL}_2(\mathcal{O}_2)$ where $\mathcal{O}_2$ is a local principal ideal ring of length two. This work generalizes similar results about the Engel word map over fields.

The present paper considers McKean-Vlasov SDEs with density-dependent spatially unbounded drift, which may be viewed as a non-linear density-dependent perturbation of the Ornstein-Uhlenbeck process. We develop a comprehensive theoretical framework for this class of equations. First, we establish strong well-posedness and derive optimal Gaussian pointwise bounds for both the solution density and its gradient. Then we derive an explicit expression for the stationary density and show that it satisfies logarithmic Sobolev and Poincaré inequalities. Finally, we prove exponential convergence to equilibrium in the \(\chi^2\)-metric.

Let $x$ be uniformly distributed on $(0,1)$, and let $(q_n)_{n\geq1}$ be the digits of its Engel series expansion. We establish a Cramér-type moderate deviation expansion for $(\log q_n-n)/\sqrt n$. The proof is based on a martingale decomposition and asymptotic results for martingales. As consequences, we obtain a moderate deviation principle over the full range of scales between the central limit theorem and the law of large numbers, without the additional lower rate restriction required in several earlier works. We also derive a uniform Berry--Esseen bound of order $(\log n)/\sqrt n$.

We study the parametric subfamily p = 3m(m+1) + 1 with m = 2^a 3^b - 1, a,b in N*, a 3-smooth slice of the centred hexagonal numbers 3m^2 + 3m + 1 = (m+1)^3 - m^3, from the point of view of unconditional primality certification via the Pocklington-Lehmer criterion. The 3-smoothness of m+1 = 2^a 3^b yields, for every (a,b), a fully factored divisor F = 2^a 3^(b+1) of p-1 satisfying F > sqrt(p) unconditionally, reducing the certificate to two witnesses, for q = 2 and q = 3. Our main new contribution is a complete, deterministic characterisation of the two canonical witnesses. We prove that w_2 = 5 is a valid witness if and only if a - b = 1, 2 (mod 4), by quadratic reciprocity; and that w_3 = 7 is a valid witness if and only if m is not congruent to 2 (mod 7), by cubic reciprocity in Z[omega] using the explicit Eisenstein factorisation p = ((1+m) - m omega)((1+m) - m omega^2). These two results turn the heuristic "5 and 7 always work" (which is in fact false) into exact congruence conditions, and yield a deterministic witness-selection rule. Alongside, three elementary arithmetic filters (mod 6, a (-3) quadratic-residue sieve, and a mod-7 forbidden-class test) remove about 87% of candidates at negligible cost. As a demonstration, a multi-core implementation produced four unconditional certificates on consumer hardware, the largest a prime of 29998 decimal digits.

Linear problems often include many parameters that may be uncertain. Sensitivity analysis studies how these parameters impact optimal values. Instead of analyzing the objective function, we shift the focus to the optimal values of the variables. Three types of linear modifications are considered: on the cost vector, the right-hand side, and on the constraint matrix. Several theorems establish properties of these modifications, including conditions for continuity of optimal variable values, as well as local convexity and concavity properties.

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 this http URL 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.

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 crossing number of knots and links projected on a 2-sided surface in a 3-manifold using the rank of fundamental groups.

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, and let $\sigma_{\mathrm{Dol}}:\,S \to M_{\mathrm{Dol}}(X/S)$ be the real analytic family of Higgs bundles obtained from an isomonodromic deformation via the relative non-abelian Hodge correspondence. We study the interaction between isomonodromic deformation and the natural $\mathbb C^*$-action on Dolbeault moduli spaces. For $\lambda\in S^1$, we prove that, on any complex analytic subvariety $U\subset S$, the rescaled family $\lambda\cdot\sigma_{\mathrm{Dol}}|_U$ is again isomonodromic if $\sigma_{\mathrm{Dol}}|_U$ is holomorphic. Conversely, we prove that $\sigma_{\mathrm{Dol}}|_U$ must be holomorphic if there exists $\lambda\in S^1\backslash\{\pm 1\}$ such that $\lambda\cdot\sigma_{\mathrm{Dol}}|_U$ is isomonodromic. The proof is based on the study of real analytic deformations of Higgs bundles and the variation of harmonic metrics. As an application, we give a simplified proof of a local characterization of Simpson's non-abelian Noether--Lefschetz locus firstly proved in \cite[Theorem 1.2]{HSJZ}. Namely, if the initial local system underlies a polarized complex variation of Hodge structures, then the non-abelian Noether--Lefschetz locus is precisely the maximal complex analytic subvariety of $S$ on which the real analytic section $\sigma_{\mathrm{Dol}}$ becomes holomorphic. This gives an affirmative answer to a question of Esnault and Kerz.

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.

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.

We introduce and study the notion of \emph{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.

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.