Papers

In 1984, Frankl and Füredi asked for the maximum density of an $n$-vertex $r$-graph in which every $(r+1)$-set of vertices spans $0$ or $2$ edges. They gave a construction with asymptotic density $2^{1-r}$. We significantly improve this bound by constructing such $r$-graphs with density $\Omega(r^{-3})$, thereby improving the dependence on $r$ from exponential to polynomial. We also obtain lower bounds for the more general problem in which every $(r+1)$-set spans an even number of edges from $\{0,2,\ldots,2k\}$.

The extended complex plane is a fundamental object in complex analysis, hyperbolic geometry, and mathematical physics. Its geometry is governed by Möbius transformations, with the cross ratio serving as a central invariant. We present a formalization of these concepts in the Lean4 theorem prover. The extended complex plane is represented using Mathlib's Option type over $\mathbb{C}$, where the additional element represents the point at infinity. On this foundation, we define Möbius transformations, their action on the extended complex plane, and the cross ratio. We formalize several basic properties of Möbius transformations, including their group structure, and identify them with a projective general linear group. We also prove the uniqueness of a Möbius transformation mapping any three distinct points to any other three distinct points, and the invariance of the cross ratio. All proofs are machine-checked in Lean 4. The complete development comprises approximately 6,000 lines of Lean code, including about 40 definitions and 150 lemmas and theorems. This work provides a verified foundation for future formalizations of conformal geometry, hyperbolic models, modular forms, and applications in mathematical physics.

In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.

$q$-Matroids, a $q$-analogue of classical matroids have attracted a lot of attention over the last decade, yet their enumeration remains largely unexplored. In this paper, we study the number of $q$-matroids, paving and sparse-paving $q$-matroids defined on a fixed ground space and with prescribed rank. We derive new lower bounds using constructions from constant-dimension codes and improve existing estimates. On the upper bound side, we develop two approaches: a combinatorial method based on controlling the number of dependent hyperplanes for paving $q$-matroids, and an entropy-based counting argument applicable to classes of $q$-matroids closed under contraction. These techniques yield explicit upper bounds on the logarithmic number of $q$-matroids with fixed rank and ground space. Finally, we analyze the asymptotic behavior of these bounds, and identify gaps between lower and upper estimates, leading to conjectures on the true asymptotic growth.

In this note we investigate the Fitting ideals associated to the Tjurina ideal of a non quasi-homogeneous plane curve singularity. Special properties occur when the difference between Milnor number and Tjurina number is at most 2.

In this article, we introduce a novel data-dependent Shepard interpolation method inspired by the adaptive strategies proposed in [2]. In this case, as Shepard interpolation does not produce oscillations, our approach has the core objective of reducing the smearing near jump discontinuities in the data in one and two dimensions. While the original work in [2] focuses in on Radial Basis Function (RBF) interpolation, we extend these ideas to the Shepard framework by incorporating a data-dependent adaptation mechanism. Specifically, we modify the classical Shepard interpolation by adaptively adjusting the influence weights based on local smoothness indicators that modify the shape parameter. These indicators, similar to those used in [2], are designed to detect discontinuities: for grid-based data, we use squared undivided second-order differences, and for scattered data, we employ squared least-squares approximations of the Laplacian scaled by the square of the mean local separation of stencil points. The resulting data-dependent weighting scheme forces the kernels close to a discontinuity to behave like a local delta function, effectively reducing the smearing of the discontinuities introduced by the classical Shepard approach. We establish the theoretical foundation of the method, including the properties of the new interpolation and we theoretically prove that the reduction of the smearing of discontinuities is possible. Numerical experiments in one and two dimensions confirm that the proposed data-dependent Shepard interpolation significantly reduces the smearing of jump discontinuities while maintaining high accuracy in smooth regions.

Classical approximation theorems ask for a new neural network whenever the target accuracy is improved. This paper studies the opposite possibility: can the network be chosen once and for all, and can accuracy be bought only by letting it run longer? We prove that this is possible for every continuous function on [-1,1]. More precisely, each such function is uniformly approximated by the time evolution of a single ReLU recurrent neural network with fixed weights and fixed hidden dimension. The mechanism behind the construction is a new intermediate model, the Turing machine with neural units (TMNU). This model retains the algorithmic freedom needed to implement polynomial approximation schemes, while remaining rigid enough to be simulated by RNNs with explicit bounds on hidden dimension and weight magnitude. The resulting convergence rates reflect the underlying polynomial approximation rates. We complement the construction with minimax lower bounds showing that runtime is not merely a proof artifact, but an unavoidable resource in this fixed-network approximation paradigm.

We continue recent work of Chao, Gabel, Larson, and Nasr in using cyclotomic polynomials for dice relabeling. In their work, one idea they expand on is finding pairs of dice with different number of sides which maintain the sum frequency of two normal dice. We continue this idea in this paper by studying pairs of dice where the number of sides of each is a different perfect square (which we call "square-sided" dice). We additionally provide conjectures offering ideas for future exploration.

In this paper, based on recent results for the prescribed Hermitian-Yang-Mills (HYM) tensor and its twisted variants by Fan-Wang-Yang-Yau, we provide a dynamical construction of Hermitian-Einstein metrics on stable holomorphic vector bundles and its extension to Higgs bundles. Additionally, in the appendix, we use the heat flow method to give a new proof of the existence and uniqueness of solutions to the twisted prescribed HYM tensor equation, as well as its generalization to Higgs bundles.

We consider large-scale noisy derivative-free optimization (DFO) problems in which only function values are available and gradient or subgradient information cannot be reliably estimated. Matrix-adaptation evolution strategies (MAES) and their limited-memory variants are among the most robust DFO methods under noise; however, their performance may deteriorate when the noise level is large. In such regimes, sorting and selection may misidentify informative sampled points, making the recombination step less reliable and weakening the scaling information used by affine or matrix-adaptation mechanisms. This can substantially reduce the efficiency of MAES-type methods, especially in high-dimensional settings. To address this limitation, we propose a DFO method that replaces the full affine-scaling matrix with a diagonal approximation constructed from conjugacy-type conditions. The proposed mechanism does not attempt to estimate gradients, subgradients, or interpolation models, nor does it learn dense covariance information from noisy rankings. Instead, it uses consecutive normalized recombination displacements in a conservative diagonal update, thereby limiting the influence of unreliable selection information while preserving the derivative-free structure of the underlying evolutionary framework. As a result, the method is computationally cheaper than full matrix-adaptation schemes and limited-memory affine-scaling variants, while providing a stable scaling mechanism in noisy environments. Numerical experiments on noisy benchmark problems show that the proposed method is competitive with, and often more efficient than, MAES-type baselines, particularly when the noise level is large and ranking-based selection becomes unreliable.

The toughness of a graph is a crucial parameter for characterizing its structural properties. The toughness of a non-complete graph $G$ is defined as $\tau(G) = \min \{ \dfrac{|S|}{c(G - S)} : S \subseteq V(G), c(G-S) > 1 \}$, where $c(G)$ denotes the number of components of $G$. We define $\tau(K_n) = \infty$. A graph $G$ is said to be $\tau$-tough if $|S| \ge \tau \cdot c(G-S)$ for every vertex cut $S$ of $G$. Let $k \ge 3$ be an integer. For $\frac{1}{k-\eta}$-tough graphs with $\eta \in \{0, 1\}$, Liu, Fan and Shu \cite{a34} derived sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Jia and Lu \cite{a24}, for the case $\frac{1}{k-1} \leq \tau(G) < \frac{1}{k-2}$, established sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Motivated by these results, in this paper, we further investigate sufficient conditions for the existence of a spanning $k$-tree when $\frac{1}{k} \leq \tau(G) < \frac{1}{k-1}$. Specifically, for a connected $\frac{t}{t(k-1)+1}$-tough graph of sufficiently large order $n$ (where $t \ge 1$ is an integer), we provide sufficient conditions for the existence of a spanning $k$-tree in terms of the spectral radius and the signless Laplacian spectral radius. Furthermore, we establish a lower bound on the size (number of edges) to guarantee the existence of a spanning $k$-tree.

We prove a Littlewood-Paley formula for the Hardy space of Dirichlet series $\mathscr{H}^p$ with $1\leq p<\infty$ in terms of almost every vertical limit function. This significantly strengthens previous results, which hold either only as an average over the vertical limit functions or under additional assumptions of uniform convergence. As part of our approach, we obtain a Hardy-Stein identity for the derivative of the $p$-mean of almost every vertical limit. We further show that the mean counting function exists for any $f$ in $\mathscr{H}^p$ in terms of almost all of its vertical limit functions. This is done by establishing a version of Jensen's formula in this setting. In the process, we also deduce ergodic versions of Fatou's lemma and the monotone and dominated convergence theorems for the Kronecker flow.

The Fourier-Helgason (FH) transform for a noncompact symmetric space $G/K$ establishes the direct integral decomposition of the unitary representation of $G$ on $L^2(G/K)$ into irreducible principal series representations. By applying techniques of geometric quantization to the symplectic manifold $T^*(G/K),$ Lisiecki in 1987 gave a geometric interpretation of the FH transform in the case when $G$ is complex. He defined for general $G$ a ”horizontal” polarization on $T^*(G/K)$ and showed that, for complex $G$, the Blattner-Kostant-Sternberg (BKS) pairing between the Schrödinger vertical polarization Hilbert space, $L^2(G/K)$, and the Hilbert space of horizontally polarized functions coincides with the FH transform. However, in the same paper, Lisiecki showed that for noncomplex Lie groups the BKS pairing is nonequivalent to the FH transform and nonunitary in general. In the present paper, we resolve this discrepancy between the FH transform and geometric quantization in the case when $G$ is not complex. First, we show that the horizontal polarization is the infinite-time limit of the push-forward of the vertical polarization with respect to the geodesic flow for a $G$-invariant Riemannian metric. Then we lift the geodesic flow to an intertwining unitary parallel transport on the quantum bundle that we call quantum geodesic transform (QGT). Finally we show that the QGT has a well-defined limit, as the geodesic time goes to infinity, and that it is equal, up to the phase of the Harish-Chandra $c$-function and an irrelevant multiplicative constant, to the FH transform.

We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(\Omega;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(\Omega)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $\Omega=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $\Omega=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $\Omega$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

We study unit groups of weighted Leavitt path algebras. Let $K$ be a field of characteristic $0$ and let $(E,\omega)$ be a finite connected weighted graph. We prove that $L_K(E,\omega)^\times$ is abelian if and only if $L_K(E,\omega)$ is a domain. Equivalently, $L_K(E,\omega)^\times$ contains no non-cyclic free subgroup if and only if $L_K(E,\omega)$ is a domain.

We study high-frequency electromagnetic propagation in the rotating García–Díaz solution of Einstein gravity coupled to NLED. In this system, light is not governed only by the null cone of the spacetime metric, because the NLED field also behaves as an optical medium whose constitutive response determines the physical optical cones. Starting from the mixed electromagnetic potentials, we project the field $F$ and the excitation $P$ on a principal tetrad and obtain the aligned scalars $E$, $B$, $D$ and $H$. These scalars allow us to reconstruct the regular local constitutive branch connected with Maxwell theory through the map $(D,B)\mapsto(E,H)$. We then insert the resulting response matrix into the Fresnel characteristic problem. At the perturbative order considered here, the Fresnel quartic factorizes into two quadratic branches, each defining an effective optical metric. Both optical metrics admit Carter-type separation of the Hamilton–Jacobi equation and possess their own radial and angular potentials, critical constants and unstable critical families. By projecting these families onto the celestial sphere of a finite-distance observer, we obtain two critical contours, $\Gamma_+$ and $\Gamma_-$, which coincide in the Maxwell limit and split when the nonlinear constitutive response is active. We quantify this birefringent splitting through the maximum angular separation, the relative diameter shift and the normalized birefringent width. Numerical scans over the nonlinear coupling, spin and observer inclination show that the splitting is generated by the constitutive response, redistributed by rotation and stable under local projection changes within the perturbative domain. This provides a direct geometrical link between the local NLED response and a polarization-dependent critical structure on the observer screen.

We study the cohomology of a quasimap space that categorifies the constant term of the geometric Eisenstein series for the mirabolic parabolic subgroup of $GL$ over the function field $\mathbb{F}_q(C)$ of a smooth projective curve $C$. This cohomology carries a natural action of an algebra of correspondences whose commutative subalgebra is the ring of regular functions on the Coulomb branch, which here is the $A_{n}$-surface singularity. A choice of rank-one local system on $C$ induces an action of the étale fundamental group on the Coulomb branch; the scheme-theoretic fixed locus carries a natural vector bundle. Our main result identifies the cohomology of the quasimap space with the local cohomology of this vector bundle, for a generic range of parameters.

Mean curvature flow is a fundamental geometric evolution equation in which a submanifold moves in the normal direction with velocity equal to its mean curvature vector. Self-shrinkers arise naturally as self-similar solutions to the mean curvature flow and play an important role as models for finite-time singularities. Among nontrivial examples of compact embedded self-shrinkers, the rotationally symmetric self-shrinking torus constructed by Angenent is one of the most important. However, the uniqueness of the Angenent torus remains a major open problem. In this paper, we study rotationally symmetric self-shrinkers of type $\mathbb{S}^{1}\times \mathbb{S}^{n-1}$ from the point of view of ordinary differential equations. We analyze the profile curves of rotationally symmetric self-shrinkers, focusing on the behavior of their vertical points and the curves traced out by these points as the initial height varies. We give a new proof of the existence of the Angenent torus by showing that two families of vertical-point trajectories must intersect. We further derive the linearized equation associated with the rotationally symmetric self-shrinker equation and apply a Sturm-type comparison theorem to obtain sufficient conditions for the monotonicity of horizontal-point trajectories. In particular, we prove a comparison theorem for solutions near the spherical self-shrinker $x^{2}+r^{2}=2n$, and establish partial monotonicity results for the curves of horizontal points. These results provide a possible approach to the uniqueness problem for the Angenent torus.

We investigate the spectral stability of small-amplitude, periodic, traveling-wave solutions of the modified Camassa-Holm equation with cubic nonlinearities. More precisely, we analyze the $L^2(\mr)$-spectrum of the associated linearized operator in a neighborhood of the origin in the spectral plane. Inspired by a recently novel method based on Kato's perturbation theory [Berti et al, Full description of Benjamin-Feir instability of Stokes waves in deep water, Invent. Math., 230 (2022), 651-711.], we provide a complete description of the spectrum near the origin of the linearized operator–an integro-differential operator with periodic coefficients–and thus prove that such waves are not subject to modulational instability. Moreover, a spectral analysis reveals a remarkable threshold phenomenon: such waves with wave number $k^2\leq 3$ exhibit spectral stability, while instability emerges when $k^2>3$.

In 1929, O. Zariski found a pair of complex plane algebraic curves of the same degree and with the same collection of singularities, but embedded into the plane in a topologically different way. Accordingly, such curves belong to different components of the equisingular family. This phenomenon has been intensively studied till now. In this note, we propose a different insight on this subject: Two curves $C',C''\subset\PP^2$ form an anti-Zariski pair, if $(\PP^2,C')$ and $(\PP^2,C'')$ are homeomorhic, but $C'$ and $C''$ belong to different components of the equisingular family. We exhibit examples of anti-Zariski pairs and discuss related issues.

In this paper, we investigate the tree-independence number of graph classes that do not contain $K_{1,d}$ as an induced subgraph. Dallard et al. conjectured that for any positive integer $d$ and any planar graph $H$, the class of all $K_{1,d}$-free graphs without $H$ as an induced minor has bounded tree-independence number. Our main contribution towards this conjecture is showing that the conjecture holds for outerstring graphs. Additionally we give linear and quadratic bounds for the tree-independence number of various $K_{1,d}$-free graph classes, sharpening previous bounds. Finally, we bound the tree-independence number of $K_{2,d}$-free graphs additionally forbidding holes of length at least $5$.

A fiber bundle over a small category is a locally constant family of categories whose transition functors encode how a fixed fiber is transported over the base. Its Grothendieck construction assembles this data into a category over the base, while its global behavior is governed by monodromy. For finite acyclic categories, relative beat objects provide reductions of the total category over the fixed base, leading to relative cores that exist and are unique up to isomorphism. Monodromy classifies categorical fiber bundles by non-abelian cohomology and describes their strict gauge groups and sections. We also prove a fundamental-groupoid version of Quillen's Theorem~A, which gives reductions of the base that are more general than beat reductions and preserve the classification of bundles with fixed fiber. The results are illustrated by explicit finite examples.

We take up the study, initiated by Fred Cohen, of the Cartan–Leray spectral sequence for Euclidean configuration spaces, establishing a decomposition as a direct sum of atomic spectral sequences. As an immediate consequence, we recover a difficult theorem of Arone–Mahowald on the vanishing of Goodwillie derivatives of the identity.

Born-Infeld New Massive Gravity (BINMG) completes New Massive Gravity to all orders in curvature through the determinant of the metric shifted by the Einstein tensor. We recast it with an independent auxiliary metric $q_{\mu\nu}$, whose algebraic equation of motion $q_{\mu\nu}=g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)$ recovers the determinant action exactly on the regular branch and resums the infinite curvature series into a single relation. In the densitized variable $P^{\mu\nu}=\sqrt{-q}\,q^{\mu\nu}$ the three-dimensional action is polynomial, with all derivative dependence carried by the coupling $P^{\mu\nu}G_{\mu\nu}(g)$. The formulation makes known properties follow with substantially less algebra: the unique vacuum follows in one line, and the quadratic action yields a single Pauli-Fierz massive spin-2 field with the Fierz-Pauli tuning generated rather than imposed. On locally AdS backgrounds the conserved charges, BTZ mass and angular momentum, central charge, and entropy reduce to the Einstein results times a common factor. The formulation also isolates the nonlinear degree-of-freedom problem in the right variables, leaving the full Dirac count to separate work.

Drones are increasingly used in agriculture, where tight margins demand efficient planning. Current optimization tools suffer from exponential runtimes as problem sizes grow, necessitating practical heuristics for daily operations. This paper presents an operational framework and benchmarking analysis for drone spraying operations. We evaluate the trade-offs between facility siting methods and tiered routing parameters. For facility siting, comparing a Mixed-Integer Program (MIP) baseline against a $p$-Median heuristic shows that the heuristic reduces runtime by three orders of magnitude, from over 97 seconds to under 1.2 seconds, with only a 4% reduction in serviced field area. For route planning, a tiered problem decomposition approach partitioning the target area into 6 to 8 spatial clusters reduces computation time by an order of magnitude with minimal degradation in serviced area. This framework achieves minute-scale planning on commodity hardware, demonstrating operational relevance. Future research will incorporate weather modeling, integrated optimization of facility location and routing, and validation across diverse field geometries.

This work concerns regularity properties of weak solutions to elliptic equations in divergence form -div(a$\nabla$u) = div F , under low regularity assumptions on both the coefficient a and the source term F . We introduce generalized Morrey and Campanato spaces extending the classical definitions by replacing uniform boundedness requirements with suitable integrability conditions. Within this framework, we establish regularity estimates for the gradient of weak solutions in these generalized spaces. As applications, we recover classical Hölder and Lebesgue estimates and derive fractional Sobolev regularity results. In particular, the proposed approach yields fractional Sobolev estimates in situations where the coefficient may be discontinuous and the gradient of the solution is not expected to be locally bounded.

We propose a conservative adaptive rank method for the 1D1V Wigner-Poisson system. The method targets a central challenge in deterministic quantum kinetic simulations: reducing the cost of phase-space evolution while preserving the macroscopic invariants needed for physical fidelity. The scheme combines a sampling-based adaptive rank Wigner-Poisson update [7] with a conservative macroscopic correction. A conservative density-momentum solve provides local macroscopic updates, a Fermi-Dirac-type reconstruction transfers them to the kinetic solution, and a global quadratic moment correction enforces the discrete total energy constraint at the kinetic level. Unlike Maxwell-Boltzmann-type corrections commonly used in classical kinetic settings, the reconstruction uses a Fermi-Dirac-type form motivated by the model's quantum-statistical structure. The corrected state is incorporated into an ACA SVD representation, allowing the numerical rank to adapt to the phase-space complexity generated by the nonlocal Wigner operator and self-consistent Poisson field. Numerical experiments for the two-stream instability, strong Landau damping, and bump-on tail instability show that the method captures benchmark Wigner-Poisson dynamics for several values of the quantum parameter H, maintains bounded adaptive ranks, and preserves the specified global discrete invariants with conservation errors near machine precision. We also compare this formulation, which uses local density-momentum correction plus global total energy correction, with a related globally conservative formulation for mass, momentum, and energy [8]. The two approaches produce nearly identical phase-space and diagnostic results for the periodic benchmark test considered here, indicating that both correction strategies are compatible with adaptive rank compression for Wigner-Poisson dynamics in the tested 1D1V periodic setting.

Can intelligence be measured? We propose that intelligence can be defined as the lawful amplification of rare but valid futures: a system increases the probability of outcomes that would be unlikely under passive dynamics but remain admissible under the constraints of the domain. We start with the premise that an intelligent system must model the world and its own place within it. Because the system is part of the world it models, this leads naturally to recursive self-simulation: the system represents futures in which its own actions are part of the trajectory. Our central results give a necessity statement and a conditional near-sufficiency statement connecting this architecture to a precise thermodynamic measure of lawful amplification of rare-valid futures: high rare-valid lift is impossible unless the internal simulation identifies rare-valid futures with high fidelity; conversely, when rare-valid fidelity is high and the simulation contains an effective policy, the achievable lift approaches the actuation-limited optimum. Thus recursive self-simulation is not merely a plausible feature of intelligence but, under the stated assumptions, is necessary and nearly sufficient for high thermodynamic intelligence. The resulting framework makes intelligence measurable on a universal scale, from passive matter and feedback controllers, large language models, and humans as text generators to Maxwell-demon-like information engines.

Expanding products of modal logics are bimodal logics obtained from the combination of a `horizontal component' logic and a `vertical component' logic, lying between the fusion and the Cartesian product of the two logics. Gabelaia et al. showed that expanding products are often decidable when the first component is Noetherian, although their methods are semantical and do not yield complete axiomatisations. They do, however, propose a candidate, dubbed the expanding commutator of the two logics and known to be complete in many `non-Noetherian' cases. In this paper, we consider various expanding products of modal logics whose vertical component is $\sf GL$. We show that the standard axiomatisation is complete when the horizontal component is either $ {\sf K4}$ or $ {\sf GL} $, but incomplete when it is ${\sf Grz}$ or any logic between ${\sf K4.3}$ and ${\sf Grz.3}$, thus yielding a partial solution to a question posed by Gabelaia et al. more than two decades ago.

Inspired by the model theory of difference fields in characteristic zero, a class of automorphisms of an algebraic variety, here called compound fundamental isotrivial, is introduced. These are algebraic dynamical systems that are built up via a finite sequence of equivariant fibrations from (possibly nonautonomous) algebraic dynamics which trivialise after base extension over themselves. Every wild automorphism of an abelian variety is compound fundamental isotrivial. Conversely, it is shown that the only irreducible projective varieties admitting a wild automorphism that is compound fundamental isotrivial are the abelian varieties. That is, the wild automorphism conjecture of Reichstein, Rogalski, and Zhang is here proven for compound fundamental isotrivial dynamics. Along the way, a counterexample to the naive generalisation of the conjecture to the nonautonomous setting of $\sigma$-varieties is provided.