Papers

We construct an explicit embedding of the rank two Racah algebra $\mathfrak{R}_2$ into the rank two Jacobi algebra $\mathfrak{J}_2$. Working in the differential model of $\mathfrak{J}_2$ on the triangle, we recall that the subalgebra structure of $\mathfrak{J}_2$ is organized by a pentagon whose four upper edges carry rank one Jacobi algebras and whose base carries a rank one Racah algebra. Promoting the two multiplication operators to Racah type generators by tridiagonalization turns every edge of the pentagon into a rank one Racah algebra and realizes $\mathfrak{R}_2$ inside $\mathfrak{J}_2$. We determine the common eigenfunctions of different commuting pairs explicitly as products of Gauss hypergeometric functions and Jacobi polynomials. Different overlap coefficients are computed involving univariate Wilson polynomials as well as, for one pair of bases, bivariate Racah polynomials of Tratnik type. We also recall how $\mathfrak{J}_2$ arises as a contraction of the rank two Racah algebra $\mathfrak{R}_2$.

In this paper we generalise two of the methods and corresponding results from our previous paper “On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators” [J. Spectr. Theory 14 (2024), 719–804] from tridiagonal operators on $\ell^2(\Z)$ to band operators $A$ on $\ell^2(G,Y)$ with a countable Abelian group $G$ and a Hilbert space $Y$. Again, we cover the pseudospectra of $A$, with error-control, via a union of pseudospectra of finite and moderately sized “local patches” of $A$. While a major application is to understand the case $G=\Z^d$ that is immanent in many physical problems, our new approach to the so-called $\tau$ and $\tau_1$ methods immediately extends to countable Abelian groups $G$.

Let $\delta$ denote the primitive act of distinction, formally realized as the one-step extension $r\mapsto Sr$ of a finite record. We prove that the $\delta$-orbit is initial among $\delta$-algebras: every $\delta$-algebra admits a unique structure-preserving map from the orbit. The map is injective when the successor operation is injective and the base point is not a successor. If the $\delta$-algebra also satisfies induction for all predicates, the map is bijective and gives the unique isomorphism with the generated $\delta$-orbit. The $\delta$-calculus is an intuitionistic first-order proof system over the signature $\{0,S,+,\cdot\}$. Each derivation carries a ledger recording the use of the law of excluded middle, the limited principle of omniscience, Markov's principle, and induction on quantified formulas. Every closed formula derivable in the forced fragment is true in the standard model. Starting from $\delta$, we construct the choice-free number tower $\delta \leadsto \mathbb{N}_\delta \hookrightarrow \mathbb{Z}_\delta \hookrightarrow \mathbb{Q}_\delta$. The metatheoretic systems $\mathbb{N}$, $\mathbb{Z}$, and $\mathbb{Q}$ each admit an explicit injection into $\mathbb{N}_\delta$. Under the law of excluded middle, every recognizer is either injective or has kernel congruence $\equiv_{i,p}$ for a unique pair $i\geq0$, $p\geq1$. In the noninjective case, the quotient is a finite monogenic monoid $M(i,p)$. A decidable congruence together with an explicit pair of distinct related elements implies the classification without additional nonconstructive principles. For a decidable congruence different from equality, Markov's principle is needed. For an arbitrary congruence, the dichotomy requires the law of excluded middle. The reverse implications show that the last two prices are exact. They are distinct from the syntactic ledger of derivations in the $\delta$-calculus.

We study the Poisson operator on the interacting Fock space associated with a discrete-time quantum walk, which we call the QW-Poisson operator. First, we investigate the spectral properties of the QW-Poisson distribution. In particular, we establish a relation between the spectral distributions of the Poisson operator and the reversed Poisson operator on a general interacting Fock space via a size-biased transform. Next, we study the edge behavior of the density of the QW-Poisson distribution. We show that a phase transition occurs at the left endpoint of the support: depending on the parameter, the density either decays to $0$ or blows up to $+\infty$. Moreover, this phase transition coincides with the transition in the number of atoms of the QW-Poisson distribution, equivalently, in the point spectrum of the QW-Poisson operator, and with whether $0$ belongs to the spectrum of the QW-Poisson operator. Finally, we study a connection between the interacting Fock space associated with a discrete-time quantum walk and noncommutative probability theory. More precisely, we compute the moment-generating function and moments of the QW-Poisson operator, and obtain a limit theorem for the Konno distribution via a Poisson approximation. We also investigate the Boolean self-decomposability of the Konno distribution and the shifted reversed QW-Poisson distribution.

Given $r \in \mathbb{N}$ and an edge-coloured complete graph such that the average number of colours incident with a vertex is at most $r$, Conlon and Stein asked whether there is a vertex partition into a bounded number of monochromatic cycles, and proved this for $r=2$. We settle the first open case by proving the corresponding statement for $r=3$.

Computing braid monodromy is a key tool for studying the topology of algebraic curves in the complex projective plane $\mathbb{P}^2$. We present an algorithm, named LA-MOKA, to compute this invariant in the specific case of complex line arrangements. To safely manage the problem of floating-point approximations when working over number fields, we translate the continuous geometry into a discrete combinatorial structure. We establish explicit conditions on this discretization that guarantee the topological correctness of the computed braid monodromy.

We are concerned with the existence of normalized solutions to the $L^2$-supercritical nonlinear Schrödinger equation on a noncompact metric graph $G$, \[ \begin{cases} -u''+W(x)u+\lambda u=\chi(x)|u|^{p-2}u, & \text{on every edge } e \text{ of } G,\\[2mm] \displaystyle\sum_{e\succ v}u'_e(v)=0, & \text{at every vertex } v\in V, \end{cases} \] under the mass constraint $\int_G |u|^2\,dx=\mu>0$, where $\lambda$ arises as a Lagrange multiplier. Here $p>6$, $\chi$ is the characteristic function of the compact core $\mathcal K$, so that the nonlinearity is localized, and the potential $W$ is bounded, nonnegative and vanishing at infinity along every unbounded edge, with $\int_{1}^{\infty}xW(x)\, dx<\infty$ on at least one of them. For every $\mu>0$ we obtain a positive solution with $\lambda>0$, arising as a constrained critical point at a strictly positive energy level. Our approach combines a uniform mountain-pass geometry for a family of approximating functionals, the monotonicity trick with Morse index type information, and a blow-up analysis ruling out the divergence of the Lagrange multipliers. A key point is the strict positivity of the multiplier, for which we exhibit a potential of inverse-square decay admitting a zero-energy $L^2$ solution on a half-line. To the best of our knowledge, this is the first existence result for normalized solutions of the $L^2$-supercritical NLS equation on a noncompact metric graph in the presence of an external potential.

We study large deviations of the maximum of the generalized TAP free energy introduced by Chen, Panchenko, and Subag for the Ising mixed $p$-spin model. The upper bound uses a new Guerra-type interpolation that may also be useful for other spin-glass models. The lower bound follows the strategy of Huang and Sellke. We show that supersymmetric critical points have no competitor at positive overlap on their own sphere. As such, they form a spherical code, which turns the annealed count into a lower bound on the probability of existence. This identifies the supersymmetric formula proposed by physicists with the large-deviation exponent for the existence of TAP maxima, rather than with the ordinary annealed complexity. The resulting rate function is the Legendre transform of a constrained Parisi value in the bottom-atom mass. The same argument, iterated in bands above TAP ancestors, would give a constructive proof of the Parisi formula for the Ising model under a technical strict stability assumption.

Let $b\ge 2$ be an integer and let $p\ge 3$ be a prime. We say that $p$ is a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime,} if $b^{p-1}\equiv 1 \pmod{p^2}$. When $b=2$, $p$ is also known simply as a Wieferich prime. Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N\ge 2$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Recently, the third author proved that $x^{2p}+2x^p+2$ is monogenic if and only if $p$ is not a Wieferich prime. In this article, we generalize this result to $x^{2n}+bx^n+b$ with certain restrictions on $b\ge 2$ and $n\ge 3$.

In this paper, we study distributions of two random variables \begin{gather*} \tau = \tau_1 g_{1-\tau_1} + \sum_{k=2}^\infty \tau_k g_{1-\tau_k} \prod_{i=1}^{k-1} g_{\tau_i} \equiv \Delta^{G_2}_{\tau_1\tau_2...\tau_n...}, \xi = \xi_1 g_{1-\xi_1} + \sum_{k=2}^\infty \xi_k g_{1-\xi_k} \prod_{i=1}^{k-1} g_{\xi_i} \equiv \Delta^{G_2}_{\xi_1\xi_2...\xi_n...}, \end{gather*} where $g_0$ is a given number belonging to interval $[\frac{1}{2}; 1)$, $g_1\equiv g_0 - 1$, $(\tau_n)$ and $(\xi_n)$ are sequences of random variables taking the values $0$ and $1$, and $(\tau_n)$ is a sequence of random variables that form a Markov chain with positive initial probabilities $p_{0}$, $p_{1}$ and matrix of transition probabilities $ \begin{pmatrix} p_{00} & p_{01} p_{10} & p_{11} \end{pmatrix},$ $(\xi_n)$ is a sequence of independent random variables taking the specified values with probabilities $p_{0n}$ and $p_{1n}$, respectively ($p_{0n}+p_{1n}=1$). We study structural, spectral, and fractal properties of distributions of $\tau$ and $\xi$. For random variable $\tau$, point spectrum (the set of atoms) and continuous spectrum (minimal closed support) of its distribution are studied exhaustively. We prove a theorem on the Lebesgue purity of distribution of random variable $\xi$ (an analog of the Jessen–Wintner theorem), i.e., conditions for the distribution to belong to one of the types: pure discrete, pure absolutely continuous, and pure singular.

For the quadratic family $f_c(z)=z^2+c$, the only parameters in the Mandelbrot set $\cal M$ for which the Julia set $\cal J_c$ has Hausdorff dimension $1$ are $c=0$ and $c=-2$. Near $c=0$, Ruelle's theory gives a real-analytic expansion of the dimension. The tip $c=-2$ of $\cal M$, however, is a non-hyperbolic parameter and the dimension function $c\mapsto \mathrm{dim_H}(\cal J_c)$ is highly discontinuous there. We prove the sharp first-order asymptotic for the lower envelope of the Hausdorff dimension at the tip: If $c\in \cal M $ then $\mathrm{dim_H}(\cal J_c)$ lies asymptotically above $1+ \Omega \sqrt{|c+2|}$ with the Jaksztas constant $\Omega =\sqrt{\frac{2}{3}}\frac{1}{\pi\log 2}$. This is a surprisingly precise contribution to the Yoccoz problem about unfolding attractors. The proof develops a thermodynamic formalism for degenerating families of box mappings. At each scale, for parameters $c\to -2$, the induced dynamics exhibit a uniform property of exponential tails, generating improved control of their pressure functions.

We introduce an interface multicontinuum homogenization method (I-MCHM) for high-contrast elliptic problems whose subdomains may have distinct microscopic patterns and unequal numbers of continua. Multicontinuum models upscale microscopic fields to macroscopic continuum quantities; here the continua are the high- and low-permeability regions, and the macroscopic variables are the corresponding local averages of the fine-scale solution on observing cells of size $H_\epsilon$. Although the fine-scale solution is continuous across a material interface, these macroscopic continuum fields need not coincide on the interface. Moreover, when adjacent subdomains retain different numbers of continua, a one-to-one coarse transmission condition cannot be defined. Standard subdomain-wise multicontinuum homogenization therefore determines the bulk equations but leaves the interaction among these coarse variables unspecified. I-MCHM closes this gap by constructing two-sided constrained local bases on interface neighborhoods and assembling a coarse bilinear form with bulk cut-cell integrals and interface-segment corrections over the subdomain adjacency graph, without prescribing a pointwise transmission law. Numerical experiments on straight, curved, and triple-junction geometries, including continuum-coupling ablations and unequal continuum counts, demonstrate the accuracy of the resulting upscaled model.

In this paper, we find all regular Borel subalgebras of Lie(Aut($\mathbb{A}^2$)), i.e., maximal solvable subalgebras generated by homogeneous derivations with respect to the standard $\mathbb{Z}^2$-grading. It follows that a regular Borel subalgebra has derived length $2$ or $3$. We also describe isomorphism classes of such subalgebras. On the way, we give a combinatorial description of the regular Abelian subalgebras and the regular solvable subalgebras of Lie(Aut($\mathbb{A}^2$)).

In this paper, only Carleman estimates are used, without energy estimates, we derive observability inequality. The main tool consists in the use of a new Carleman estimate.

The independence polynomial of a simple graph $G$ is given by \( I_G(z) = i_0 + i_1 z + i_2 z^2 + \cdots + i_\alpha z^\alpha \), where \( i_\alpha \) denotes the size of a maximum independent set, also called the independence number of the graph. The independence polynomial has the notable feature of being essentially closed under graph composition (lexicographic product). In this paper, we determine the independence polynomials of size five. For a disconnected graph $G$, we exploit the fact that $I_G(z)$ factors as the product of the independence polynomials of the connected components of $G$. Furthermore, we classify all independence polynomials that can occur for such a disconnected graph $G$ and, by examining their component structures, we characterize the disconnected configurations that may arise.

Let $S$ be a nonempty finite set of positive integers, let $q=\max S$, and let $B_n(S)$, $n>2q$, be the normalized product sequence associated with the Chebyshev-type polynomial of the cyclic circulant graph $G_n(S)$. When $G_n(S)$ is connected, $B_n(S)$ is both the normalized spanning-tree number and a normalized special value of the reduced spectral function, a determinant-type function constructed from the non-trivial adjacency spectrum. Chebyshev root representations and the existence of linear recurrences for fixed-step circulant spanning-tree sequences are known. Starting from these representations, we explicitly construct a monic annihilating polynomial $\mathcal H_S(X)\in\mathbb Z[X]$ of degree $3^{q-1}$, which yields a general upper bound for the recurrence order of $B_n(S)$. By collecting coincident exponential bases and accounting for possible cancellations, we determine the minimal annihilating polynomial and give a sufficient condition under which the minimal recurrence order is exactly $3^{q-1}$. For $S=\{1,2,3\}$ and $S=\{1,3\}$, we explicitly derive the corresponding ninth-degree annihilating polynomials and prove their minimality.

We prove a resolvent estimate for the generator of the frame flow on hyperbolic manifolds away from vertical lines of resonances. A byproduct of the proof is an optimal essential spectral gap property for the generator, hence giving another proof of exponential mixing of frame flows with respect to the volume measure of the frame bundle. This extends the result of [https://arxiv.org/abs/2005.08387v2] in dimension 3 to any dimension. We make extensive use of the Borel-Weil calculus developed in [https://arxiv.org/abs/2405.14846] to overcome difficulties of this higher-dimensional case.

We investigate nonnegative mild solutions of $\partial_t u+(-\Delta)^{\ln}u=f(u)$ in $(0,T)\times\mathbb R^N$, with initial datum $u(0,\cdot)=\mu u_0$, $\mu>0$. Unlike the classical and fractional heat semigroups, the positive logarithmic heat kernel exists only for $0<t<N/2$, and the corresponding linear evolution may become singular at its terminal time, with lifespan and growth depending on the spatial decay of $u_0$. The behavior of $f$ near zero determines local solvability: if $\int_{0^+} d\sigma/f(\sigma)<\infty$, then no finite nonnegative solution exists on any positive time interval. Under suitable assumptions on $u_0$ and $f$, we establish well-posedness for the nonintegrable logarithmic heat kernel. If $f$ has at most global linear growth, the nonlinear solution attains the full linear lifespan, while the Osgood condition at infinity implies that the maximal existence time tends to zero as $\mu\to\infty$. We further distinguish slow-decay, fast-decay, and critical-tail initial data. In the noncritical regimes, a weighted Osgood tail condition yields blow-up strictly before the linear terminal time; if it fails, an amplitude threshold occurs under additional assumptions on $f$. In the critical regime, the dividing power is $3/2$: a square-root weighted Osgood condition yields premature blow-up, while its failure again leads to an amplitude threshold. Finally, we obtain terminal-time blow-up estimates and sharp rates for power nonlinearities.

We study branching of regular abnormal geodesics in sub-Riemannian manifolds of rank two. As an application, we provide examples of two new phenomena: continuous families of strictly abnormal branching geodesics, and geodesics with an accumulation of countably many branching points.

The study of Rota-Baxter operators on the polynomial algebra in one variable was initiated by S.H. Zheng, L. Guo, and M. Rosenkranz (2015). Monomial Rota-Baxter operators on $F[x]$ were classified in 2016, while monomial averaging operators on $F[x]$ and $F_0[x]$ were classified in 2024. In the current work, we continue the study of these operators on Laurent polynomial algebra $F[x, x^{-1}]$ over a field of arbitrary characteristic.

Let $(R, \mathfrak m)$ be a Noetherian local ring of dimension $d \geq 1$ with $\mathrm{depth} R \geq d-1,$ and let $I$ be an $\mathfrak m$-primary ideal. In this paper, we study bounds on the second Hilbert coefficient of $I$, denoted by $e_{2}(I)$. Under the assumption that the associated graded ring $G(I)$ has depth at least $d-1,$ we first establish a lower bound for $e_{2}(I).$ We then extend several known results from the Cohen-Macaulay case to this general setting and obtain upper bounds for $e_{2}(I)$ in terms of the sectional genus denoted by $\mathrm{g}_{s}(I)$ and the Hilbert coefficients of $I$ and those of a minimal reduction $Q$ of $I$. We further analyze the extremal case when $e_{2}(I)$ attains this bound and relate it to the depth of $G(I)$. In addition, for Buchsbaum local rings, we establish a sharp upper bound for $e_{2}(\mathfrak m)$ using the technique of $S_{2}$-fication. Finally, in the Cohen-Macaulay case, we give sufficient conditions to ensure good properties on the depth of $G(I)$ and of $G(I^n)$ under the assumption that $e_{2}(I)=0$.

For a system of $s$ homogeneous polynomials of degree $d$ in $n$ variables, say ${\bf{f}} = 0$, we consider the problem of constructing resultant systems. A resultant system is a finite set of polynomials in the coefficients of the input polynomials, the vanishing of which characterizes the systems $\bf{f}$ with a common non-zero solution. The classical approaches for constructing resultant systems rely either on maximal minors of large coefficient matrices or on the coefficients of a resultant of generic linear combinations of the input polynomials. Typically, they produce resultant systems containing a very large number of polynomials. We develop new constructions based on taking resultants of linear combinations of the input polynomials; this results in resultant systems of small cardinality. Our main results are: 1) We prove that a resultant system with ${d+n-1 \choose n-1} s-n^2+1$ polynomials exists; each polynomial is the resultant of $n$ linear combinations of the input polynomials. This improves the previously known upper bounds, even for systems of bivariate homogeneous polynomials. 2) Under the assumption that the input polynomials are non-zero, we construct explicit resultant systems with cardinality $\mathrm{poly}(s,d)$, when $n$ is fixed.

We prove Pólya's conjecture for the Neumann eigenvalues of the Laplacian on Euclidean balls in dimensions three and higher. The proof further develops the approach introduced in our earlier work on the two-dimensional case and on Dirichlet eigenvalues in arbitrary dimensions. The main difficulty in the higher dimensional Neumann case is that one has to estimate zeros of the derivatives of ultraspherical Bessel functions, rather than of the usual Bessel functions. For low-lying eigenvalues, we use variational estimates involving dimension-dependent test functions, which is a novel ingredient allowing us to control a larger dimension-scaled frequency range. Other components of the proof include phase-function bounds, lattice-point counting techniques, and computer-assisted arguments.

Local-in-time well-posedness is established for a recently proposed diffuse interface model describing incompressible two-phase flows. The result constitutes the first analytical study of a model introduced by ten Eikelder et al. for the motion of a binary mixture of macroscopically immiscible, viscous, incompressible fluids with unmatched densities. In contrast to classic diffuse interface models based on a single mean velocity, this model is derived within the framework of mixture theory, assigning each phase its own momentum and mass balance, which results in a system of two coupled Navier–Stokes equations and two mass transport equations. The proof of the well-posedness result uses a fixed-point strategy, where the main difficulty lies in the analysis of the principal part of the associated linearized system.

The aim of this paper is threefold: (i) We study the complex Green's function (the analytic extension of the real Green's function) for multiply connected domains with some symmetry and transform it to a simple form with the help of Walsh's conformal map onto lemniscatic domains. (ii) For the complement of the union of two real intervals, we represent the complex Green's function with the help of Jacobi's elliptic and theta functions. (iii) Using this representation, we explicitly obtain all parameters of the lemniscatic domain corresponding to the complement of the two intervals. In addition, using an equality between the corresponding complex Green's functions, we obtain a numerical method for computing the conformal map from the complement of two intervals onto a lemniscatic domain which yields more accurate results than a previous method from the literature.

Batteryless overlays couple passive throughput to Wi-Fi sounding overhead and channel state information (CSI) aging. This paper investigates channel sounding for ultra-high reliability (UHR) operation in a Wi-Fi 8/IEEE 802.11bn-inspired downlink multi-user multiple-input multiple-output (MU-MIMO) system with a batteryless passive overlay. We optimize the post-sounding transmission interval to maximize the aggregate throughput of the active Wi-Fi and passive links, while jointly accounting for sounding overhead, CSI aging, modulation and coding scheme (MCS), passive attenuation, and passive data rate. A packet-level cross-layer model evaluates the cycle-average throughput, and a data-driven search identifies the optimal interval under different operating conditions. Simulations demonstrate that passive overlay reshapes the conventional sounding tradeoff: depending on the MCS and passive-link configuration, the additional passive throughput may or may not compensate for the associated Wi-Fi reliability loss, causing the optimal interval to shift. The results provide design guidance for reliable and low-power MU-MIMO WLANs.

The Möbius transformation that generates the spherical Cauchy distribution from the uniform is the tangent-normal lift of a one-dimensional optimal transport: the monotone rearrangement of the cosine about the location axis. This identifies the probabilistic nature of the Möbius transformation and suggests a generalization: replacing the Cauchy target by any rotationally symmetric law, for instance the Poisson kernel or spherical cardioid, yields a generalized Möbius transformation. Möbius transport of a von Mises-Fisher base gives tractable anisotropic distributions on the sphere, with closed-form densities that inherit the base normalizing constant and allow immediate simulation. The Möbius-von Mises-Fisher and isotropic scaled von Mises-Fisher distributions, the latter also arising from a Möbius transport, are illustrated on paleomagnetic directions and short-period comet orbits, where they outperform classical and recently proposed alternatives.

By extending the Rokhlin-Sinai machinery relating to the entropy and countable Lebesgue component in the spectrum, we establish that the dynamical Möbius-Sarnak process has a countable Lebesgue component. Inspired by recent work of M. Lin and the author, we extend the notion of spectral measure to all operators on Banach spaces. This generalization is further motivated by the Bellow-Losert extension of Wiener's notion of the spectral measure of sequences. Furthermore, we establish unconditionally that the topological entropy of the Möbius flow is given by $\frac{6}{\pi^2}\log 3$. Among other consequences, we recover a recent result by el Abdalaoui-Nerurkar which asserts that for any quasi-generic measure for the Möbius function, the Möbius flow equipped with this measure has a countable Lebesgue component in its spectrum. It follows that the Sarnak Möbius orthogonality conjecture holds for any topological dynamical system with singular spectrum. We further show that all the potential spectral measures of he Möbius function are absolutely continuous with respect to Lebesgue measure.

A role colouring of a graph $G$ is an assignment of colours to the vertices of $G$ such that two vertices of the same colour have identical sets of colours in their neighbourhoods. This model is used to capture the idea of vertices having roles in a contact network, consistent with the set of roles of their contacts. We define an extension of the role colouring problem to temporal graphs. Temporal roles are defined via an automaton with states and transitions capturing both the current colour of a vertex and information about its current and past adjacencies. We show, by a reduction from the static problem, that the temporal role colouring problem is NP-complete. To contend with this intractability, we explore several parameterisations. We give fixed-parameter tractability results with respect to the number of states of the automaton combined with either the vertex-interval-membership width or the tree-interval-membership width of the temporal graph. We further show the problem is in FPT parameterised by the treewidth of the underlying graph, the lifetime of the temporal graph and the number of colours combined.

We study iterative methods for computing the Moore–Penrose inverse of split-quaternion matrices. We first establish a consistent framework based on a \(2\times2\) real representation and the associated \(i\)-conjugate transpose. This representation gives a direct definition of the Moore–Penrose inverse and clarifies the treatment of nonzero zero divisors. We then analyze Newton–Schulz iterations for rectangular and rank-deficient matrices. Using a thin singular value decomposition of the real representative, we derive the convergence conditions, the evolution of the projector residuals, and an exact relation between the residual and the error. The resulting analysis applies both to an embedded real implementation and to a native split-quaternion iteration, which are shown to be equivalent under the representation. We also propose a low-degree polynomial initialization inspired by Souriau's inverse recursion. The polynomial is obtained by a least-squares approximation of the inverse Gram factor and is combined with a spectral acceptance test and a safe fallback initialization. Finally, we apply the proposed methods to cross and CUR approximations of split-quaternion matrices. We characterize the optimal middle factor for fixed sampled rows and columns and give conditions under which the cheaper cross factor gives an exact reconstruction. Numerical experiments illustrate the accuracy of the methods and the practical benefit of the polynomial initialization.