Papers

We survey recent advancements on the realisation problem for mapping degree sets. In particular, we explain that any finite set containing zero is the mapping degree set between some $n$-manifolds, for each $n\geq3$. Our building factors for the construction of the realising manifolds will be aspherical manifolds. Thus, along the way, we review a couple of open questions related to maps of non-zero degree for aspherical manifolds.

We study smooth compact strictly convex hypersurfaces in the unit ball that meet the support sphere orthogonally and evolve by the $\alpha$-Gauss curvature flow $\partial_tX=-K^\alpha\nu$, $\alpha>0$. We prove that the solution remains strictly convex, becomes extinct in finite time and contracts to a single point on the support sphere. If $\alpha >\frac{1}{n+2}$, we apply a Cayley-type conformal map that sends the extinction point to the origin of a Euclidean half-space and then normalize the enclosed half-space volume. The resulting normalized hypersurfaces converge smoothly to the unit hemisphere. The proof combines boundary identities for the spherical free boundary, a boundary-adapted Tso estimate, an almost-monotonicity formula for a half-space entropy, and uniform curvature estimates for the normalized flow.

Universal scaling in phase separation is typically assumed to be isotropic in systems with conserved dynamics. Here we show that boundary forcing alone can break this dynamical scaling symmetry, leading to anisotropic coarsening, with different effective global growth laws observed parallel and perpendicular to the boundary. We consider a ternary mixture with two conserved components and a passive species undergoing surface evaporation, which provides a simple setting to investigate this effect. In this case, evaporation leads to a progressive mass loss and to the formation of macroscopic concentration gradients, which, in turn, drive anisotropic coarsening, with different effective growth laws observed parallel and perpendicular to the boundary. At the same time, bulk regions appear to retain the standard Model B scaling, suggesting that the observed anisotropy is mainly induced by boundary fluxes rather than by changes in the intrinsic dynamics. Our results indicate that boundary conditions can play an important role in breaking scaling symmetry and may offer a way to influence coarsening behavior in nonequilibrium phase separation.

We prove that every Severi-Brauer variety of dimension at least two over a number field contains a twisted elliptic normal curve. Consequently, every Brauer class over a number field is split by a genus one curve. We prove this using the fibration method, by showing that the rational points on a smooth compactification of the Hilbert scheme of twisted elliptic normal curves are dense in its Brauer-Manin set.

We investigate $L^p$ estimates for the $\bar{\partial}$-problem on rational Hartogs triangles $\mathbb{H}_{m/n} = \{ (z_1, z_2) \in \mathbb{C}^2 : |z_1|^m < |z_2|^n < 1 \}$. For $p \in (1, \infty)$, we establish the existence of a solution operator that is bounded on $L^p(\mathbb{H}_{m/n})$. Our approach avoid the need for any a priori condition on the data. We also show that the canonical solution $K_{\mathbb{H}_{m/n}}$ is bounded on $L^p(\mathbb{H}_{m/n})$ for $p \in (p_0, p_2)$, where $p_0=\frac{2m+2n}{m+n+1+\min\{m, n\}}$ and $p_2=\frac{2m+2n}{m+n-1}$. For classical Hartogs triangle, $\mathbb{H}_1$, this establishes boundedness for $p \in (1, 4)$.

We study CC-geodesic Kakeya sets in the first Heisenberg group, namely Borel sets $E$ such that, for every unit-speed CC-geodesic segment of length $1$ issuing from the identity, some left translate of the segment is contained in $E$. The natural analogue of the Kakeya conjecture would predict full Heisenberg Hausdorff dimension 4 for such sets. We show that this prediction fails: the sharp lower bound for their Heisenberg Hausdorff dimension is 3, and it remains sharp even among compact CC-geodesic Kakeya sets. By adjoining a Lebesgue-null set of full Heisenberg Hausdorff dimension, we obtain a CC-geodesic Kakeya set of full Heisenberg Hausdorff dimension 4 and zero Lebesgue measure. Finally, when one prescribes only geodesic segments with one fixed nonzero curvature parameter $\kappa$, rather than segments of all curvatures, the condition is weaker. For every $\kappa\in(0,2\pi]$, we construct a compact curvature-$\kappa$ Kakeya set of zero Lebesgue measure whose Euclidean and Heisenberg Hausdorff dimensions are both equal to $1$.

This paper is concerned with Remez-type inequalities and their applications in observability inequality. Our aim is twofold. First, we establish the following fractal Remez's inequality on the unit sphere $\mathbb{S}^{n-1}$ \begin{align*} \sup_{\mathbb{S}^{n-1}} |p|\le C(M,N,n,\delta)\sup_{M} |p|, \end{align*} where $M \subset \mathbb{S}^{n-1}$ ($n \ge 2$) is a fractal set of positive $(n-2+\delta)$-Hausdorff content for arbitrary $\delta \in (0,1)$, and $p$ is a spherical polynomial of degree at most $N\in \mathbb{Z}^+$. Second, building upon this fractal framework, we establish sharp observability inequalities for the heat equation on the sphere, again valid for all $\delta\in (0, 1)$, which improve the result of Burq and Moyano [J. Eur. Math. Soc. (JEMS), 25 (4) (2023)] in the spherical setting. Furthermore, as an additional application, we prove a lower-dimensional observability inequality for the heat equation with super-quadratic potentials $V(x) = |x|^{2m}$ ($m \in \mathbb{Z}^+, m\ge 2$) on the whole space $\mathbb{R}^n$.

Let $S_n(32\text{-}1)$ and $S_n(3\text{-}21)$ denote the sets of $n$-permutations avoiding the vincular patterns $32\text{-}1$ and $3\text{-}21$, respectively. Using Lehmer codes, we realize these families as weighted posets $L_n(32\text{-}1)$ and $L_n(3\text{-}21)$, where the weight of a code is the inversion number of its permutation. We show that the maximal elements of each of these posets, $\operatorname{Max} L_n(32\text{-}1)$ and $\operatorname{Max} L_n(3\text{-}21)$, are enumerated by the Fibonacci numbers. We demonstrate that the classical reverse-complement map on permutations restricts to a natural bijection between these two sets of maximal elements, revealing a deep symmetry between their underlying poset structures.

Classical discrepancy quantifies the irregularity of the distribution of a sequence in the unit interval. In this paper, we study the analogous notion for sequences in the ring of $p$-adic integers with a focus on the dynamically generated sequences. We prove that the orbits of ergodic $1$-Lipschitz self-maps of $\mathbb{Z}_p^d$ attain the optimal order of discrepancy and hence form low-discrepancy sequences. We also obtain bounds on the growth of the size of orbits of polynomial self-maps of $f: \mathbb{Z}_p^d \to \mathbb{Z}_p^d$ modulo $p^n$ for $d>1$. As a consequence, we show that orbit closures of $f$ have box dimension either zero or one. Our approach relies on the introduction of strong fixed points for such maps, together with several decomposition results for matrices over $\mathbb{Z}_p$.

Intermittency in turbulence manifests as intense vortices and sharp peaks of dissipation. Causing the breakdown of Kolmogorov's simple self-similar theory, it leads to anomalous scaling, multifractality and so far remains beyond the scope of a complete theoretical description. How intermittent flow structures influence these different measurements is not known quantitatively. With a simple filtering procedure-thresholding vorticity and inverting the Biot-Savart law to generate filtered velocity fields-we show the effects of intermittent flow structures can be disentangled. As extreme vorticity contributions to the velocity field are filtered out, the energy spectrum scaling persists, while the bottleneck is flattened, and structure function scalings tend towards their Kolmogorov values. The approach is more rapid for transverse exponents, revealing the selective importance of intensely swirling flow regions. Similarly, the extent of multifractality reduces as intermittency is filtered, shrinking the range of roughness singularity exponents. The residual fields are curiously more multifractal, but their structure begins to break away from an underlying turbulence skeleton. The effects on vortex stretching and strain self-amplification are quantified. Our work shows that a Biot-Savart approach can selectively remove the effects of intermittency from turbulence, and hence from its scalings.

We prove that the full group \(\SL(3,\R)\) occurs as the set of one-particle deformation gradients of periodic, unforced, single-shell solutions of both the three-dimensional Euler and Navier–Stokes equations. More precisely, given a particle label \(p\in\T^3\), a time \(T>0\), and \(F_*\in\SL(3,\R)\), every sufficiently large odd integer \(N\) admits a real-analytic curl eigenfield \(W_N\) with \(\operatorname{curl}W_N=NW_N\). With an explicit scalar amplitude, \(W_N\) yields a steady Euler solution; with an explicit exponentially decaying amplitude, it yields a Navier–Stokes solution for any positive viscosity, and in both cases \(\nabla_aX(p,T)=F_*\). The lifted particle trajectory is an embedded analytic arc with nowhere-vanishing velocity. The construction combines global trace-free symmetric-matrix control on \(\SL(3,\R)\), a Beltrami Cauchy problem along the controlled arc, Runge approximation, inverse localization on the torus, and a finite-dimensional endpoint correction. We also classify finite material-direction systems that determine every trace-free strain after arbitrary volume-preserving deformation. In dimension \(n\), this congruence-robust property holds exactly when the associated rank-one projectors span \(\Sym(n)\); hence the sharp number of scalar channels is \(n(n+1)/2\). In dimension three, among minimal systems the undeformed outer-product lower bound is maximized exactly by the six axes of a regular icosahedron. The realization theorem shows that the full deformation-group quantifier in this sensing result is dynamically attained within the rigid class above.

Many real-life systems can be found as examples of stochastic matching on hypergraphs, such as production lines or assemble-to-order systems. Two common features are the number of items required may vary between matchings, and there may intermediary items which exist as a combination of other items and not of external arrivals. Both of these phenomena can be modelled by considering the weighted variant of stochastic matching. In this work, we formalise the notion of stochastic weighted matching on hypergraphs. We also allow batch arrivals, meaning multiple items of multiple classes may arrive at the same time, and in particular, the arrivals can be correlated between classes. Unlike the classical setting where items arrive at discrete time $t \in \mathbb{N}$, we allow arrival processes to take place in continuous time $t \in \mathbb{R}_{\geq 0}$. We then extend the results from Nguyen and Bušić (2026) to overcome the intricacies brought up by this new setting. This allows us to derive necessary and sufficient criteria as direct generalisations of those in the unweighted setting, which depend only on the per-class arrival rates. As such, the correlation between classes bear no differences. The constructive proofs also give a maximally stable, periodic-review, size-based, arrival-rate agnostic policy.

Let $M$ be a smooth compact manifold, and $g$ a smooth non-degenerate symmetric bilinear form on $TM$. We prove that every smooth vector field can be written as a finite linear combination of iterated Lie brackets of gradient vector fields.

An important problem in neuro-engineering is the stabilization of neural fields. In applications, it is often assumed that the actuator placement can be chosen arbitrarily. In this work, we investigate the stabilizability of controlled Amari-type neural fields where the control input is prescribed to act only on a fixed subset of the neural field. We show that the linearized neural field is open-loop stabilizable under suitable assumptions on the interaction strength and a mild relative density assumption on the control set. Our geometric assumption requires that the volume of each cube intersected with the control set must be bounded below. As a consequence, we derive closed-loop stabilizability of the neural fields, under sensor/actuator placement constraints. Numerical simulations are used to illustrate the results and obtain empirical estimates on the control cost.

We construct a pressure-consistent operator-and-readout interface for Born, adjoint, and Gauss–Newton actions in constant-density acoustic full-waveform inversion (FWI) using Schrödingerised propagation. The energy variables $\pi=c^{-1}\partial_t u$ and $q=\nabla u$ yield an auxiliary-space Hamiltonian, while physical pressure $p=c\pi$ depends explicitly on wavespeed. Its derivative $D(c\pi)[c_0](\delta c)=c_0\delta\pi+\delta c\,\pi_0$ combines propagated wavefield sensitivity with a direct receiver-calibration term. Duhamel and receiver-row differentiation retain both contributions in the Born map, its adjoint, and the Gauss–Newton normal action. We prove a conditional consistency estimate with a periodic second-order finite-difference specialization and give a resource model for state preparation, normalization, quadrature, and selected-output measurement. A compiled nine-qubit instance realizes structured preparation, product-formula propagation, a derivative-LCU block, and calibrated pressure-overlap measurements. Bernoulli samples from ideal-circuit probabilities drive a four-parameter hybrid inversion. A two-qubit VQLS circuit represents the normalized update direction, while normal-system assembly, line search, and model refresh remain classical. Finite differences, tangent and reverse-adjoint recurrences, autodiff JVP/VJP evaluations, and explicit Jacobians verify the discrete Born, adjoint, and normal actions. Smooth periodic refinement confirms second-order convergence, whereas omitting receiver calibration leaves an order-one Born error and substantially changes the regularized Gauss–Newton direction. All ten predeclared finite-shot runs reduce the initial model error. These results specify the physical-pressure derivative and selected-output measurements needed to connect Schrödingerised propagation to a local FWI update.

The Heil-Ramanathan-Topiwala conjecture asserted that every finite collection of distinct time-frequency shifts of a nonzero square-integrable function is linearly independent. Recent counterexamples with Schwartz windows leave open the classification of configurations and windows for which independence holds. We prove linear independence for every nonzero window in $L^2(\mathbb{R})$ and every configuration consisting of finitely many commensurably spaced points on one line together with two distinct points on a parallel line. As a consequence, we obtain independence for every nonzero real-valued window and every four point configuration $\{(0,0),(0,1),(s,0),(a,b)\}$ with $abs\ne0$. The configurations underlying the known four-point counterexamples can be put in this form by translation and dilation, so they cannot yield dependence with nonzero real-valued windows. The proof combines factorization of Laurent polynomials with product estimates along orbits of irrational rotations. Finally, for a fixed nonzero window and a fixed three-point system, we show that the set of time-frequency parameters producing a shift in its span is compact, has Lebesgue measure zero, has Hausdorff dimension at most one, and is totally disconnected.

Let $\mathbf p=(p_1,\ldots,p_n)$ and $\mathbf q=(q_1,\ldots,q_n)$ belong to $(0,1/2]^n$, and let $\mu_{\mathbf p}$ and $\mu_{\mathbf q}$ be the associated measures on $2^{[n]}$. Suppose that $p_1q_1=\max_{i\in[n]}p_iq_i$. We prove that every pair of cross-intersecting families $\mathcal A,\mathcal B\subseteq2^{[n]}$ satisfies the sharp inequality $\mu_{\mathbf p}(\mathcal A)\mu_{\mathbf q}(\mathcal B)\leq p_1q_1$. This confirms a conjecture of Suda, Tanaka and Tokushige [Math. Program. 166 (2017) 113–130]. We also determine all equality cases. When $p_1q_1<1/4$, equality is attained only when both families consist of all subsets containing the same product-maximizing coordinate. At the endpoint $p_1q_1=1/4$, we identify precisely the additional extremal pairs, which are induced by half-sized increasing families on the coordinates satisfying $p_i=q_i=1/2$. We further resolve the remaining conjecture from the same paper by proving a dimension-free stability theorem. Assume that the first coordinate has maximum probability under both measures and that $p_1,q_1<1/2$. If $\mu_{\mathbf p}(\mathcal A)\mu_{\mathbf q}(\mathcal B)\geq(1-\varepsilon)p_1q_1$, then there exists a coordinate $j$ such that both $\mathcal A$ and $\mathcal B$ are within $c(p_1,q_1)\varepsilon$, in their respective measures, of the family of all subsets containing $j$. This improves the conjectured $O(\sqrt{\varepsilon})$ bound to a linear one. The main new ingredient in the sharp measure theorem is a log-odds interpolation combined with induction on coordinate sections, while stability follows from a semidefinite estimate and a one-coordinate approximation theorem.

We introduce a class of weighted Dirichlet spaces on the unit disk, called Douglas weighted Dirichlet spaces, characterized by the existence of a Douglas type boundary representation of the weighted Dirichlet integral. This representation naturally induces a nonlocal Dirichlet form on the unit circle in the sense of Beurling Deny and Fukushima. Our main result shows that the reproducing kernel of the weighted Dirichlet space associated with a regular Douglas weighted Dirichlet form is the image of the Szego kernel under the corresponding 1 resolvent. This resolvent representation is new for superharmonic weights. We also develop the potential theory associated with Douglas weighted Dirichlet spaces. We characterize the capacity in terms of reproducing kernels. Finally, as an application of the fact that every Douglas weight induces a regular nonlocal Dirichlet form, together with the general theory of Dirichlet forms, we obtain an associated symmetric pure jump Hunt process. In the classical Dirichlet case, this process is the wrapped Cauchy process.

The approximation of high-dimensional functions is a challenging task due to the often appearing curse of dimensionality. In this paper, we combine sparse grid with anchored projection techniques to derive sampling inequalities for Sobolev functions of a dominating mixed regularity which are effectively low dimensional. To this end, we derive new sampling inequalities for sparse grids and combine these with recently investigated regression processes of non-matching sampling processes.

LLM agents following the ReAct paradigm are promising enablers of complex multi-step tasks, including multi-hop question answering, code generation, and control of physical AI systems. Yet, when deployed at the edge, they must tightly manage their reasoning budget while remaining reliable and deferring to a cloud-side model only when local uncertainty is too high to act safely. We propose Think Short, Defer Smart (TSDS), a framework that synergistically integrates a lightweight convergence probe, which halts on-device reasoning once the intended action has stabilized, with a perplexity-based deferral rule that escalates uncertain actions to a cloud-side model. Both mechanisms are jointly calibrated on end-to-end episode trajectories via a multi-objective Learn-Then-Test (LTT) procedure, providing simultaneous finite-sample guarantees on expected episode reward and cloud-call rate. We evaluate TSDS on four ReAct benchmarks spanning arithmetic reasoning (GSM8K), multi-hop question answering (HotpotQA), code generation (MBPP), and multi-step embodied planning (household robot), and compare against thought-calibration-only and calibrated-deferral-only standalone baselines. TSDS reduces per-episode thinking compute by 43%-65% over deferral-only baselines across HotpotQA, MBPP, and the household robot task, while maintaining certified reward and cloud-call rate guarantees.

Plug and Play methods combine classical variational models with learned denoisers and have achieved strong results in imaging inverse problems. Their convergence has been widely studied for Gaussian data, whereas Poisson models require additional care because of the nonquadratic Kullback-Leibler fidelity. This work introduces GSDSplit+ , a splitting method built upon PnPSplit+, in which the generic denoising block is replaced by a relaxed Gradient Step Denoiser. The resulting method retains explicit Poisson fidelity updates. The denoiser is defined through the gradient of a learned convex potential parameterized by an Input Convex Neural Network. Its architecture and training promote both blind denoising capability and a smoothness condition sufficient for firm nonexpansiveness. Empirical smoothness estimates are used to select a relaxation parameter compatible with the convergence assumptions of PnPSplit+. Numerical experiments provide evidence that the trained denoiser operates in an FNE-compatible regime on the considered validation data. GSDSplit+ also yields stable reconstructions outside the empirically supported range and shows substantially lower sensitivity to the ADMM parameter, while remaining effective for different blur operators and Poisson noise levels.

Pivoted QR and pivoted LU decompositions are greedy algorithms used to compute low-rank approximations of matrices from selected columns, or selected rows and columns. Despite their practical robustness, general worst-case bounds comparing their errors with those of the best corresponding low-rank approximations contain exponentially growing factors and do not explain their behavior under modest singular value decay. We prove that under approximate greedy pivoting, their error is controlled by the determinant of a submatrix, which is bounded by the geometric mean of the leading singular values. Using this bound, we establish convergence rates under algebraic and geometric singular value decay. We also extend the LU analysis to functions of two variables. By bounding the determinants of arbitrary sampled submatrices, we obtain algebraic convergence rates under differentiability assumptions and geometric convergence under analyticity.

We study the qualitative behavior of solutions of Grad-Shafranov type equations arising in plasma physics with general differential operators and general nonlinearities. In particular, we extend recent estimates about threshold values for uniqueness, monotonicity and non-existence of the free boundary. The argument is based on a refined spectral analysis for weighted non-local problems together with comparison techniques and level set analysis.

Via the method of marked permutation tables presented in this paper, we generalize the formula for the sixth moment of a random determinant to account for entries with arbitrary distribution. That is, let $f_6(n) = \mathbb{E}(\det A)^6$, where $A$ is an $n$ by $n$ random matrix with independent and identically distributed entries. We show that the exponential generating function $F_6(t) = \sum_{n=0}^\infty f_6(n)t^n/(n!)^2$ is D-finite and we present it in a closed form. Our method relies on carefully decomposing marked permutation tables into a shell, a core, and a floating component, each of which has a separate contribution to $F_6(t)$. After this decomposition, it is sufficient to enumerate over a finite number of possible shells, which we did using a highly intricate computer program. We verified our result up to $n = 7$ in the general case and up to $n = 9$ for random matrices whose entries only take two values by using a different method for computing $f_6(n)$ for these cases.

This paper introduces a Coulomb-corrected labeled energy for smooth compactly supported solutions of the three-dimensional repulsive Vlasov–Poisson system. The resulting identity replaces the direct electric-field term along a characteristic by a weighted singular integral, which can be estimated more effectively. Thus this structure yields improved bounds on the velocity support and the decay of the electric field.

We study general position sets in strong products involving paths and cycles. For every connected graph $H$ and every $s\ge 2$, we prove that gp$(P_s \boxtimes H)=2$gp$(H)$. We also determine the corresponding values when the path is replaced by $C_4$, $C_5$, or $C_6$, and establish a general upper bound for gp$(C_s \boxtimes H)$. These results are then applied to strong products of two cycles. We determine several exact values, construct infinite families attaining the general upper bound, and provide counterexamples to the conjectured multiplicativity of the general position number under the strong product.

We study automorphism groups in five extremal families of polyhedral graphs. For every $n\ge14$, we prove that every minimum-order $3$-polytopal graph containing a vertex of each degree $3,4,\ldots,n$ is asymmetric. The proof uses an exact planar defect decomposition, a complete description of the high-degree tail, and a saturation theorem for the subgraph induced by the uniquely high-degree vertices. Duality gives the corresponding asymmetry result for minimum-face polyhedra containing faces of every size $3,4,\ldots,n$. For the three polyhedral graphs whose complements are also polyhedral, we determine the ordinary and extended automorphism groups and identify the extended group \[ \mathsf{Aut}^{\pm}(G_{13})\cong (C_2\times C_2)\rtimes C_4. \] Next, we classify automorphism groups of radius-one polyhedra. In the unique-dominating-vertex case they are cyclic or dihedral, and in the triangulated case the possibilities are \[ 1,\qquad C_2,\qquad C_3,\qquad C_2\times C_2,\qquad S_3. \] For polyhedra that are unigraphic among the class of self-dual, we show that their automorphism group is either $1$ or $C_2$. Finally, we consider polyhedra that are products of graphs, for each of the four standard graph products, and we classify them according to their automorphism group.

This paper establishes a critical space–time density estimate for the three-dimensional Vlasov–Poisson system in the whole space and on the torus. This estimate has two consequences. First, every nonnegative weak solution with finite initial kinetic energy in the bounded finite-energy weak-solution class conserves the total energy. This gives a positive answer, within this class, to the energy-conservation question raised by Lions and Perthame. Second, in the periodic repulsive case, it yields propagation of every velocity moment of order \(k>2\), thereby closing the previously unresolved interval \(2<k\leq3\).

The statistical analysis of object oriented data in non-Euclidean spaces heavily relies on generalized conditional Fréchet means, notably in the context of Fréchet regression. However, establishing the uniform convergence of these estimators presents several theoretical challenges. The difficulties are caused primarily by the absence of linear structures in general metric spaces, rendering standard techniques for verifying the asymptotic uniform equicontinuity of the estimator largely intractable. To overcome this limitation, this paper introduces an alternative theoretical framework for establishing uniform convergence that bypasses the need to verify uniform equicontinuity, under a novel structural condition on the empirical cost function of the generalized conditional Fréchet means. We demonstrate that this analytical condition is satisfied by various prominent Fréchet regression models across broad classes of metric spaces. Leveraging these foundational uniform convergence guarantees, we subsequently extend two widely used frameworks from Euclidean to non-Euclidean spaces: (i) a weighted Fréchet aggregation framework that facilitates both distributed regression and robust median-of-means regression; and (ii) an exceedance set estimation framework to identify critical covariate regions where the conditional generalized Fréchet mean surpasses a prescribed threshold, alongside a metric to quantify the aggregate magnitude of the exceedance. The theoretical properties of these proposed methods are empirically validated through Monte Carlo simulations and an application to dynamic transportation networks in New York City.

We establish sharp estimates for the discrete optimal constant of the fractional Hardy Inequality in dimension $N\geq 1$, with fractional exponent $s\in \left(0,\min\left\{1,\frac{N}{2}\right\}\right)$. The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the computations are carried out in a bounded, convex and smooth domain containing the origin, for which we employ a quasi-uniform and regular mesh.