Papers

Evaluating layer potentials with rapidly decaying kernels becomes computationally challenging in the presence of boundary layers, as occurs for Helmholtz layer potentials with large imaginary wavenumbers. We introduce Quadrature by Matched Asymptotic Expansion (QBMAX), a high-order method that extends the Quadrature by Expansion (QBX) framework by incorporating the kernel's asymptotic decay behavior directly into the Taylor expansion of the potential. Numerical results in both 2D and 3D show improvements of up to four digits over QBX for the tested large-parameter cases, while the methods can perform similarly for smaller parameters or more strongly curved geometries. A symbolic weighted operation-count model indicates comparable costs for forming the two expansion kernels. The Taylor expansions have the same formal order, while QBMAX reduces the truncation-error constants in the flat-boundary model and generally exhibits less convergence degradation in the reported experiments.

For every $n\geq2$, we consider a Gaussian zonoid of revolution $Z_n$ arising from works of Vitale and Mathis. We prove that $Z_n^*$ is also a zonoid and compute the Banach–Mazur distance from $Z_n$ to the Euclidean ball. This distance is independent of the dimension and is approximately $1.10$. Consequently, the supremal Banach–Mazur distance among zonoids whose polars are zonoids does not converge to $1$. The construction also produces a separable real Banach space $X$, not isometric to a Hilbert space, such that both $X$ and $X^*$ embed linearly isometrically into $L_1$.

We develop a high-order, fully explicit operator-splitting method for 2.5D ideal magnetohydrodynamics on Cartesian meshes. The hydrodynamic subflow is advanced by an entropy-stable discontinuous Galerkin spectral element method equipped with stagewise oscillation elimination and a positivity-preserving limiter. The magnetic–velocity subflow uses compatible finite elements and mass-lumped reconstructions of electric field and current density. Its discrete-curl update exactly preserves the global $H(\mathrm{div})$ divergence-free subspace, while the ideal semidiscretization balances kinetic, magnetic, and internal energy. Curl-form artificial resistivity and a direction-resolved velocity filter return removed magnetic and kinetic energy to internal energy; consequently, the stabilized magnetic stage preserves positive internal energy and satisfies a discrete entropy inequality. The two solvers are composed by a second-order hydrodynamic–magnetic–hydrodynamic Strang splitting, yielding a matrix-free scheme that retains global mass, nodal admissibility, and magnetic divergence on accepted steps. Smooth Alfvén-wave and advected-vortex tests reveal an even–odd convergence pattern in the magnetic polynomial degree, verify second-order temporal accuracy, and show that stabilization preserves high-order accuracy. Field-loop, Orszag–Tang, rotor, MHD blast-wave, and Kelvin–Helmholtz calculations demonstrate robust performance for nonsmooth multidimensional flows and show that artificial resistivity suppresses grid-scale magnetic oscillations while retaining the resolved structures.

We study the dynamics of collisions and near-collisions in the system of three point vortices in the plane. The dynamics is known to be completely integrable and can be reduced to the study of Hamiltonian systems on two-dimensional symplectic leaves. This still allows for interesting behavior of the solutions, various aspects of which have been investigated in a large body of work by many authors. Our focus will be on collisions and the possibility of their regularization through perturbations of suitable parameters of the system. We will use the reduced coordinate given by $\zeta=(z_2-z_1)/(z_3-z_1)$, where $z_1,z_2,z_3$ are the complex numbers representing the positions of the vortices. This choice seems very natural, although we have not come across it in the existing literature on the topic. Some of our results concerning the near-collision behavior appear to be new, and we also prove inequalities between critical energy values that we have not found in the literature. The calculations we do to analyze the collisions and near-collisions also reproduce various classical results that were previously obtained by other authors using different coordinates, and parts of the paper therefore have an expository character.

Let $A^2(\mathbb D)$ be the unweighted Bergman space and write $Q_m(S)=T_{B_m(S)}$ for the map induced by the $m$th higher-order Berezin transform. Suárez asked in 2005 whether $Q_m(S)$ converges to $S$ in operator norm for every $S$ in the full Bergman Toeplitz algebra. We answer this question negatively in a strong form: there is a compact operator $S$ such that \[ \sup_{m\geq 0}\|Q_m(S)\|=\infty. \] In particular, along a strictly increasing sequence $(m_n)$ one has \[ \|Q_{m_n}(S)-S\|\longrightarrow\infty. \] The obstruction is a moving matrix edge. The proof uses a moving family of rank-one test operators, each supported in the $m$th matrix column. Near the corresponding edge, these test operators are sent to weighted Hankel matrices, and the limiting coefficients form an explicit Pascal kernel. We prove a standalone Pascal–Hankel theorem showing that the associated weighted Hankel transformation fails to map $\ell^2$ boundedly into trace class. Finite-section convergence, trace duality, and the Uniform Boundedness Principle then transfer this instability to the exact Bergman maps.

We study the particle number distribution of locally interacting bosonic quasi- free states, which arise in various areas of mathematical physics. We show that the particle number decomposes into an infinite sum of independent geometrically distributed random variables, confirming predictions from the physics literature. This representation yields exponential tail bounds for the particle number, a law of small numbers together with a large deviation principle at logarithmic speed when the lattice spacing diverges, and a central limit theorem when it vanishes. As an application, we obtain a detailed description of the statistics of the quantum depletion in Bose–Einstein condensates, with the constants explicit in terms of the scattering length of the interaction potential.

Will an attack on a network affect its structure in a major way? In this work we study this question by investigating a dynamic preferential attachment-type graph in which an attacker interferes with the growth mechanism. We show that our model of an attack exhibits a phase transition. If the skill of the attacker is low, the network preserves its scale-free structure inherited from the underlying preferential attachment structure (even though the tail of the degree distribution becomes lighter due to the attack). When the skill of the attacker increases and crosses a critical boundary, the scale-free property is destroyed. The skill of the attacker is expressed as the probability that, at a given step, the attacker succeeds in preventing adding an edge to the network and, instead, causes a deletion (in a preferential manner) of an edge. In all cases we describe the tail behaviour of the degree distribution.

In this paper, the log-conformation reformulation of a Giesekus fluid near a planar stick-slip singularity is investigated analytically. In what is, to our knowledge, the first use of the log-conformation formulation beyond its numerical purpose, we present results that match the asymptotic stress analysis of Evans [JNNFM 222 (2015) 24-33] for the stick, slip, and core regions under the assumption of a given Newtonian velocity field. Furthermore, the logarithmic reformulation allows us to extend the existing asymptotic theory substantially, showing, e.g., that the determinant of the conformation tensor is asymptotically constant, as well as thoroughly characterizing the transition behavior between the boundary layers and the core region. Combining these results, we obtain a composite asymptotic solution of the log-conformation equation that is valid uniformly throughout a close neighborhood of the singularity.

We characterize the von Neumann algebras that are isomorphic, as $C^*$-algebras, to reduced twisted $C^*$-algebras of locally compact Hausdorff groupoids equipped with continuous Haar systems of full support. They are exactly the subhomogeneous von Neumann algebras, equivalently finite products of matrix algebras over abelian von Neumann algebras. The groupoid can always be chosen compact, principal, and étale, with trivial twist and counting Haar system. We also prove that, for a groupoid in this class, the full or reduced twisted algebra is unital if and only if the groupoid is étale with compact unit space. This criterion reduces the classification for general Haar systems to the étale case. A further obstruction comes from controlled propagation, defined through faithful representations into the norm closure of uniformly sparse matrices. Every reduced twisted étale groupoid algebra and its Borel completion have controlled propagation, including for non-Hausdorff groupoids with locally compact Hausdorff unit space. For every infinite-dimensional Hilbert space $H$, every $*$-homomorphism from a nonzero quotient of $B(H)$ into an algebra with controlled propagation is zero. In particular, neither $B(H)$ nor the Calkin algebra embeds into any of these reduced or Borel groupoid algebras. We also prove that every von Neumann algebra with controlled propagation is finite. No separability or countability assumptions are required.

The main purpose of this article is to provide a fractional counterpart of the well-known Maz'ya inequality on the half-space, that is $$ \int_{\mathbb{R}^{d}_{+}}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dy\,dx\ge\mathcal{D}_{d,s,p}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)|^p}{x_d^{sp}}dx+C_{d,s,p,\tau}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)|^p}{x_{d}^{sp-\tau}\left(x_{d-1}^2+x_d^2\right)^{\tau/2}}dx, $$ where $\mathcal{D}_{d,s,p}$ stands for the sharp constant in the fractional Hardy inequality on a half-space $\mathbb{R}^{d}_{+}$. We also obtain a similar result in the setting of Sobolev–Bregman forms.

The goal of this paper is to present a one dimensional heat equation with a singular drift for which solutions of the Cauchy problem with smooth initial data are bounded and are discontinuous at an interior point (common to all solutions).

We present a phenomenological framework for fully-heavy tetraquark spectroscopy based on spectral graph theory. The four valence partons are the vertices of \(K_4\), with edge weights fixed by the colour-Casimir factors of the two colour-singlet diquark–antidiquark channels, \(\bar3\otimes3\) and \(6\otimes\bar6\); a physical state is a coherent mixture of the two, controlled by one mixing angle. Unlike an earlier version, which passed the colour-weighted adjacency matrix through an absolute-degree graph Laplacian, here the spectrum is obtained by directly diagonalising the adjacency matrix; we show analytically that the Laplacian step inverts the physical correspondence between colour attraction and mass ordering. Fixing the model's three parameters (mixing angle, energy scale, effective charm mass) against \(X(6900)\), \(X(7100)\), and the more tentative \(X(7200)\) gives \(\alpha=0.740\), \(\gamma=23.2\,\mathrm{MeV}\), \(m_c=1.77\,\mathrm{GeV}\); with three parameters fixed by three inputs this has zero residual degrees of freedom and is not itself a statistical test. The framework's value rests instead on two genuinely predictive, zero-additional-parameter applications: the all-bottom ground state falls at \(18.7\)–\(18.9\,\mathrm{GeV}\) for standard \(m_b\), with no retuning; and for \(T_{cc}^+\) the same mechanism under-binds by \(\approx93\,\mathrm{MeV}\), a quantitative diagnostic of missing long-range (molecular) dynamics. We discuss the sensitivity to the tentative status of \(X(7200)\), the tension between this static colour-configuration picture and the diquark radial-excitation interpretation favoured by recent CMS data, and what further data would make the model falsifiable in a stronger sense.

We study the maximal existence time $T^{*}(\varrho)$ of the solution of the semilinear wave equation $u_{tt}-\Delta u=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ in a bounded domain, with Dirichlet boundary condition and initial data $(\varrho f,\varrho g)$, where $1<q<p$ and the amplitude $\varrho$ is large. For nontrivial $f$ and sufficiently large $\varrho$, the concavity method gives $T^{*}(\varrho)\leq C\varrho^{-(p-1)/2}$ for $q\leq2p/(p+1)$ and $T^{*}(\varrho)\leq C\varrho^{-(p-q)/q}$ for $q>2p/(p+1)$, whereas the energy method gives a lower bound of order $\varrho^{1-p}$ only. We prove lower bounds with the same exponents as the upper ones, so that $T^{*}(\varrho)\asymp\varrho^{-\vartheta(p,q)}$ with $\vartheta(p,q)=\min\{(p-1)/2,\,(p-q)/q\}$. The proof rests on a hyperbolic rescaling which converts the large amplitude into a dilation of the domain and a coefficient $\varrho^{(q(p+1)-2p)/2}$ in front of the damping term, on a local existence theory in uniformly local energy norms whose existence time does not depend on that coefficient, and on a quantitative use of the dissipation when the coefficient is large. The threshold $2p/(p+1)$ is the value of $q$ at which the damping term is invariant under the rescaling. No attempt is made to optimize the constants: sharpness is meant throughout at the level of the exponent.

A method for constrained computation guidance and control is described. The method employs guidance updates by periodically re-solving a constrained optimal control problem. Feasibility of the optimal control problem is improved by preventing constraint violations in the perturbed system. Constraint violation is prevented by augmenting the optimal control problem with a penalty term intended to increase the margin between a constraint and its limit when the perturbed system is within certain proximity of the constraint limit. The additional penalty term employs a smooth maximum approximation to increase the margin between the constraint and its limit. The method developed in this paper is demonstrated on the maximum cross-range reusable launch vehicle (RLV) entry problem, for which the reference solution contains a long-duration constrained arc. The results demonstrate that the method developed in this paper is viable for path-constrained optimal guidance and control.

Anti-de Sitter (AdS) spacetimes are not globally hyperbolic, so boundary conditions are generally needed for well-defined dynamics. Ishibashi and Wald (IW) determined boundary conditions yielding well-posed dynamics with conserved positive energy for scalar, electromagnetic, and linearized gravitational fields, using nonlocal potentials and individual spherical harmonic modes. In this paper, we determine which IW conditions are local in the primary fields and which are AdS-invariant. In four spacetime dimensions, the standard AdS-invariant boundary conditions set the rescaled magnetic field to zero for electromagnetism and the rescaled magnetic Weyl tensor to zero for Einstein gravity on the conformal boundary. We generalize these standard boundary conditions by setting arbitrary linear combinations of the electric and magnetic fields, or the corresponding Weyl tensors, to zero, and we conjecture that these exhaust the local, AdS-invariant, conservative boundary conditions in AdS$_{1,3}$. Boundary conditions with non-AdS-invariance and in different spacetime dimensions are also studied. The AdS-invariant conditions in AdS$_{1,3}$ extend to Yang-Mills theory and nonlinear gravity in a general Fefferman-Graham setting. We construct phase spaces for these general boundary conditions. The covariant phase spaces for general conditions are naturally associated with Lagrangians containing Pontryagin terms and, in gravity, a Gauss-Bonnet term. In gravity, except for the standard boundary condition, no nontrivial asymptotic symmetries exist and all conserved charges such as ADM mass vanish identically, as in a closed universe. Spacetimes with Killing fields including AdS then exhibit linearization instability in the sense of Fischer and Marsden and Moncrief. For nonstandard boundary conditions, the modified symplectic structure yields new expressions for the entropy formula of black holes.

We develop a time–frequency framework for lattice GKP codes in which ideal codewords are realized in the modulation space $M^\infty$ and identified, through a vector-valued Zak transform, with a finite logical fibre over the continuous syndrome torus. Multi-window Gabor analysis then represents the logical vector by a finite block of adjoint-lattice coefficients. We prove that the normalized block map is an isometry, obtain an explicit recovering projection, and derive stable logical reconstruction. We further construct normalizable GKP approximants as lattice-envelope Gabor multipliers and establish weak-$*$ convergence and asymptotically isometric encoding. Finally, we recover displacement syndromes from phase relations between translated coefficient blocks and quantify their stability under additive perturbations.

Water distribution networks (WDNs) are susceptible to various failures, including but not limited to human errors, cyber-attacks, and network modifications, necessitating the vulnerability analysis of WDNs. Graph-theoretic centrality measures—as a main class of centrality measures—aim to rank the network components solely based on their influence (i.e., criticality) on the WDN topology in the case of input changes, while overlooking the WDN dynamics. To overcome such a limitation, this paper uses a control-theoretic centrality measure to identify the network's most and least influential pipes in WDNs, by simultaneously incorporating the dynamics and topology of the WDN. First, given a WDN modeled by nonlinear differential-algebraic equations (NDAEs) consisting of transient flow dynamics as differential equation (DE) and conservation of water mass as algebraic equation (AE) and considering the pipe flow rates as states of the state-space (SS) representation, we extract a linearized system modeled by linear ordinary differential equations (LODEs) around the equilibrium (i.e., steady) flow rate vector. Second, treating pipe flow rates as SS nodes (states), we introduce a node centrality-based measure, namely vulnerability vector (VV), to rank the network pipes based on their influence on the dynamics and topology of the WDN in the case of input changes. In particular, the network's most and least influential pipes can be identified through such a centrality-based approach. This enables water engineers to understand the WDN's vulnerability better and effectively prioritize the maintenance and operational efforts on the most influential pipes within the WDN.

We present a general proof of the Fong-Tsui conjecture for bounded operators on arbitrary complex Hilbert spaces. Specifically, we show that $|T|\leq|\operatorname{Re}T|$ implies that $T$ is self-adjoint. The argument combines a positive inverse of a Sylvester map with a spectral cutoff determined by the norm of the positive defect $|\operatorname{Re}T|-|T|$. A local vanishing lemma reduces the analysis to the classical squared self-adjointness criterion, while positivity of the defect yields a global norm contradiction. We formulate the argument as an abstract four-operator vanishing principle, without compactness, trace, or separability assumptions. We also establish a quantitative stability estimate: if $|T|\leq|\operatorname{Re}T|+\varepsilon I$ and $0\leq\varepsilon\leq|T|$, then $|\operatorname{Im}T|\leq6|T|^{7/8}\varepsilon^{1/8}$. The constant is independent of the dimension, and the exponent is not claimed to be optimal. Large language models (LLMs) were used to assist with proof development, algebraic calculations, numerical checks, and auditing of the arguments.

The total domination number $\gamma_t(G)$ of a graph $G$ is the minimum cardinality of a set $D\subseteq V(G)$ such that every vertex of $G$ has a neighbor in $D$. The annihilation number $a(G)$ is the largest integer $k$ for which the sum of the $k$ smallest degrees of $G$ is at most $|E(G)|$. A well-known conjecture, originating from Graffiti.pc and later formulated explicitly by Desormeaux, Haynes, and Henning, asserts that $\gamma_t(G)\le a(G)+1$ for every connected nontrivial graph $G$. The conjecture is known for graphs of minimum degree at least three and for several classes of graphs having vertices of degree one or two. In this paper we settle the minimum-degree-two case. More precisely, we prove $\gamma_t(G)\le a(G)+1$ for every connected graph $G$ with $\delta(G)=2$. The proof combines two sharp bounds on the total domination number with an estimate for the annihilation number. Moreover, in some specific cases, the stronger inequality $\gamma_t(G)\le a(G)$ holds.

We develop a machine-checked construction of non-Gaussian functional integrals using the Henstock–Kurzweil gauge integral and Chernoff product approximations. The central object is a finite family of bosonic modes with action S(phi) = (1/2) phi^T A phi + lambda * sum_i phi_i^4, where A is positive definite. We prove that the one-mode integral I(omega, j, lambda) is finite, strictly positive, monotone and infinitely differentiable in the coupling lambda on [0, infinity). Its derivatives are given by convergent integrals of phi^{4k} with the same weight, not by the divergent perturbative series. The M-mode influence functional factorises into one-mode integrals and is bounded by its Gaussian value. A Chernoff / Lie–Trotter splitting handles the non-commutativity of the free and non-Gaussian generators. All statements are formalised in Lean 4 with Mathlib; the accompanying file HkNonGaussian.lean is free of sorry and uses only the standard axioms propext, Classical.choice, Quot.sound. Four illustrative applications are worked out at the level of explicit formulas: the Duffing oscillator, local volatility (CEV) in finance, Wilson–Cowan neural fields, and non-Gaussian quantum reservoirs. The construction is completely direct and does not use Wick rotation, Wiener measure, zeta-regularisation or analytic continuation back from imaginary time.

This is a survey on an old open problem in combinatorics called the unit distance problem, and the field of mathematics around it, called incidence geometry. What do we know about the problem? Why is it difficult? How does it connect with other parts of math?

We prove that the Nieuwland number of the Octahedron is $3\sqrt{2}/4$. Along the way, we provide an elementary proof that Rupert's problem for the Octahedron admits no solution in which the outer projection is a hexagon.

Baste, Furst, Henning, Mohr, and Rautenbach conjectured that every finite regular graph of positive degree satisfies \(\gamma(G) \leq \gamma_e(G)\), where \(\gamma\) is the domination number and \(\gamma_e\) is the edge domination number, equivalently the minimum cardinality of a maximal matching. We show that the conjecture is false already for cubic graphs. The counterexample is a previously public 50-vertex cubic graph that had been used to refute the stronger independent-domination inequality \(i(G) \leq \gamma_e(G)\). For this graph we prove \(\gamma(G) = 16 > 15 = \gamma_e(G)\). The equality \(\gamma_e(G) = 15\) has a short counting proof, and a dominating set of order 16 is displayed explicitly. For the lower bound \(\gamma(G) \geq 16\), we give a self-contained exact reduction: after fixing which of the 20 clause vertices lie in a putative dominating set, the remaining problem is a finite set-cover problem on the 30 literal vertices. We enumerate all \(2^{20} = 1,048,576\) clause subsets, derive two explicit lower bounds, and solve exactly the 5,931 residual cases by a recurrence stated in the paper. The complete case counts and minima are displayed, and a short standard-library Python implementation is included in an appendix. A separate 893,049-node proof-tree certificate and a direct graph search provide independent verification. Thus the regular-graph conjecture is disproved. Combined with Gupta's recent theorem that every cubic graph on at most 48 vertices satisfies the conjectured inequality, the example is order-minimal among cubic counterexamples.

Let $\gamma\subset\mathbb R^{m},\,m\geq2,$ be a closed curve of length $2\pi$ with its curvature $\kappa$, parametrized by arc length, and let $\lambda_\gamma$ be the first eigenvalue of the periodic curvature Schrödinger operator $-d^2/d s^2+\kappa(s)^2$. We obtain \[ \lambda_\gamma\geq \frac{\sqrt{\pi}}{2} \left(\frac{\Gamma(7/6)}{\Gamma(5/3)}\right)^3. \] This is a near-sharp lower bound for the Ovals problem. Our proof introduces a new geometric approach. We derive a convolution identity from the closure condition and combine it with projection averaging over tangent directions and sharp Poincaré inequalities on antipodal arcs. As applications, we provide an improved two-state kinetic Lieb-Thirring inequality and the corresponding two-eigenvalue constant.

Suppose that L is a quantum logic (= an orthomodular poset) and A, B are Boolean subalgebras of L. Suppose further that s1(resp. s2) is a state on A (resp. on B). We ask when both s1 and s2 allow for a simultaneous extension over L. Under a natural necessity condition on A, B, s1, s2 we find a sufficient condition for the existence of the extension. This condition is the requirement of an abundance of two-valued states on L. We then show by the Greechie pasting technique that this condition is far from necessary even for state-space-rich logics.

This paper is a sequel to the authors' paper on rigidity of higher dimensional contact Anosov flows~[GRH]. At the time the authors were unaware of much earlier work of Hamenstädt~[Ham] devoted to the same problem. The methods of~[GRH] and~[Ham] are overlapping but not entirely the same. In this paper we strengthen some the results of Hamenstädt, by further bootstrapping the regularity of the conjugacy of 2-pinched contact Anosov flows. We also introduce a notion of symplecto-rigidity for symplectomorphisms and prove that some Anosov diffeomorphisms are symplecto-rigid. We further combine symplecto-rigidity with various rigidity techniques to establish a number of smooth rigidity results of Anosov symplectomorphisms. Some new phenomena are present in the realm of Anosov symplectomorphism. For example, while the well-known de la Llave example on the 4-torus demonstrates absence of rigidity in the space of smooth Anosov diffeomorphisms, however, we prove that it is rigid in the space of Anosov sympelctomorphisms.

We prove a Torelli-type theorem for a product of the moduli spaces of semistable vector bundles over smooth projective curves of genus $g\,\ge\, 4$. A similar result is proved for the product of the moduli stacks of semistable vector bundles. This is proved using a decomposition theorem for the product of the normal projective varieties with Picard rank one and discrete Picard group. As an application, we compute the automorphism group of the product of the moduli spaces and moduli stacks.

We prove that the full discrete Hardy-Littlewood maximal operator associated with Euclidean balls satisfies dimension-free bounds on $\ell^p(\mathbb Z^d)$ for every $1<p<\infty$. We also establish analogous dimension-free bounds for the full discrete spherical maximal operator when $d\geq 5$ and $2\leq p<\infty$. The main new idea is to approximate the relevant Fourier multipliers by finite linear combinations of normalized discrete Gaussian multipliers and their translates. We obtain these approximations uniformly in frequency and with uniformly bounded coefficients through a refined saddle-point analysis. The ball result resolves a question of E.M. Stein, while the spherical result gives, in the range $p\geq 2$, a dimension-free strengthening of the theorem of Magyar, Stein, and Wainger.

The twistor correspondence provides a duality between curves in $S^4$ and ruled surfaces in $\mathbb{CP}^3$ in which the conformal properties of the former are reflected in the projective properties of the latter. We use this to characterise a $14$–dimensional family of homogeneous ruled surfaces in $\mathbb{CP}^3$ which correspond to loxodromes in $S^4$.

In this article we study the antitonicity property of the Moore-Penrose inverse in the class of selfadjoint operators, with respect to the Löwner order. For this purpose, we employ different positive decompositions that selfadjoint operators admit. In addition, we relate a weak version of the antitonicity property with Thompson components.