We introduce a new class of semidefinite programming (SDP) relaxations for sparse box-constrained quadratic programs, obtained by a novel integration of the Reformulation Linearization Technique into standard SDP relaxations while explicitly exploiting the sparsity of the problem. The resulting relaxations are not implied by the existing LP and SDP relaxations for this class of optimization problems. We establish a sufficient condition under which the convex hull of the feasible region of the lifted quadratic program is SDP-representable; the proof is constructive and yields an explicit extended formulation. Although the resulting SDP may be of exponential size in general, we further identify additional structural conditions on the sparsity of the optimization problem that guarantee the existence of a polynomial-size SDP-representable formulation, which can be constructed in polynomial time.

We present deep, high spatial resolution interferometric observations of 0.88 mm continuum emission from the TYC 5709-354-1 system, hereafter WISPIT 2, obtained with the goal of detecting circumplanetary emission in the vicinity of the newly discovered WISPIT 2b planet. Observations with the most extended baseline configuration offered by ALMA, achieving an angular resolution of $25 \times 17$ mas ($3.3 \times 2.2$ au), revealed a single, narrow ring with a deprojected radius of 144.4 au and width of 7.2 au, and no evidence of circumplanetary emission within the cavity. Injection and recovery tests demonstrate that these observations can rule out point-like emission at the location of WISPIT 2b brighter than $\approx 45$ $μ$Jy at the $3σ$ level. While these data can rule out PDS 70c like circumplanetary emission, the upper limit is consistent with empirical mass-flux relationships extrapolated from the stellar regime. Visibility modeling of the continuum ring confirms that WISPIT 2b lies significantly interior to the mm dust ring, raising doubts about the ability of WISPIT 2b to be the only driver of the dust structure. Possible solutions include either another lower mass companion, residing between WISPIT 2b and the cavity edge, likely in the gap seen by SPHERE at $\sim130$ au, or that WISPIT 2b is either substantially more massive than IR-photometry based estimates ($\sim15$ $M_{\rm Jup}$) or on a moderately eccentric orbit. The combination of observations sensitive to the gas and dust distributions on larger spatial scales and dedicated hydrodynamical modeling will help differentiate between scenarios.

Table learning, which lies at the intersection of machine learning and modern database systems, has recently attracted growing attention. However, existing table learning frameworks typically require explicit data export and extensive feature engineering, creating a high barrier for database practitioners. We present TLSQL (Table Learning Structured Query Language), a system that enables table learning directly over relational databases via SQL-like declarative specifications. TLSQL is implemented as a lightweight Python library that translates these specifications into standard SQL queries and structured learning task descriptions. The generated SQL queries are executed natively by the database engine, while the task descriptions are consumed by downstream table learning frameworks. This design allows users to focus on modeling and analysis rather than low-level data preparation and pipeline orchestration. Experiments on real-world datasets demonstrate that TLSQL effectively lowers the barrier to integrating machine learning into database-centric workflows. Our code is available at https://github.com/rllm-project/tlsql/.

The answer to the title-question is affirmative. The analysis of the geometry of continuous and differentiable curves in three-dimensional Euclidean space suggests that the point represents the location of the center of charge of the electron, satisfies a system of ordinary differential equations of fourth order, and moves at the speed of light. The center of mass of the electron is a different point and will be determined by the evolution of the center of charge. It is the relative motion of the center of charge around the center of mass that gives rise to the spin and magnetic properties. The invariance of the mass and the absolute value of the spin for the center of mass observer imply that in the interaction of the electron with an external electromagnetic field the particle has to radiate. The analysis of a Poincaré invariant interaction of two electrons implies that the only relevant parameter that characterizes the interaction in a natural system of units, is the fine structure constant. The fundamentals of General Relativity have to be revisited.

We present a multiwavelength analysis of 850 $μ$m-selected SMGs (deblended S$_{\rm 850}\gtrsim$ 1mJy) in the SSA22 field, where our deepest JCMT/SCUBA-2 observations reach a sensitivity of $σ_{850}\sim$ 0.80mJy beam$^{-1}$. Using multiple identification methods, we have identified 248 deblended SMG candidates for 192 SCUBA-2 sources. The average multiplicity of SCUBA-2 sources is $\sim$26%, with brighter sources exhibiting higher multiplicity. After applying quality cuts based on SED fitting reliability, our final sample comprises 221 SMGs associated with 186 SCUBA-2 sources. The SSA22 SMGs have a median infrared luminosity of (2.25$\pm$0.25) $\times$10$^{12}$ L$_{\odot}$, with $\sim$ 63% ($\sim$ 8%) of the sample classified as ULIRGs (HLIRGs). The median redshift of the sample is $z = 2.00 \pm 0.08$, while optically faint galaxies exhibit higher median redshift ($\sim 2.20 \pm 0.17$). The comoving volume density of SMGs increases by a factor of $\sim 6$ at $z \lesssim 4$, plateauing at $\sim$ 1.78-3.16 $\times$ 10$^{-5}$ cMpc$^{-3}$ over $z \sim$ 1-3 (including the overdensity). The significant overdensity of SMGs within large-scale structures demonstrates their reliability as tracers of cosmic structure formation at high redshift. The median stellar mass and SFR of our SMG sample are $(1.55 \pm 0.22) \times 10^{11}$ M$_\odot$ and $166 \pm 25$ M$_\odot$ yr$^{-1}$, respectively. We observe a clear ``downsizing" signature: after cosmic noon ($z \lesssim 2$), massive SMGs exhaust their gas reservoirs and transition to quiescence, while lower-mass SMGs continue forming stars and dominate the cosmic SFR density. The sample has a median dust mass of (1.95 $\pm$ 0.14) $\times$ 10$^{9}$ M$_{\odot}$. The dust fraction ($ M_{\text{dust}}/M_{\text{star}}$) has a median value of (1.4 $\pm$ 0.2) $\times$ 10$^{-2}$. The median $A_V$ of SMGs is 3.09$\pm$0.07mag.

We analyze the spectral energy distributions (SEDs), star formation histories (SFHs), and infrared-radio correlation (IRRC) of 221 850 $μ$m-selected submillimeter galaxies (SMGs) in the SSA22 deep field. The median mass-weighted age is 567 Myr. Most galaxies in our sample began forming $\sim$ 1.68 Gyr after the Big Bang, entered the `SMG phase' after $\sim$ 1 Gyr of evolution -- when they are predominantly observed -- and largely transitioned out of the `SMG phase' to become quiescent within an additional $\sim$ 0.2 Gyr. A subset of massive galaxies shows rapid early assembly with high star formation efficiencies ($\sim$0.2-0.8). The majority of SMGs reside at the high-mass end of the star-forming main sequence, with a characteristic stellar mass of $M_{star} \sim 10^{11}$ M$_\odot$, above which galaxies are predominantly either on the main sequence or already quenched. We observe a downsizing trend: more massive galaxies tend to ``mature" earlier, completing their major episodes of star formation at higher redshifts compared to lower-mass systems. Our sample contributes $\sim$ 21% (28%) to the cosmic star formation rate density (stellar mass density), including the overdensity, with its relative contribution peaking at 50-60% in the redshift range $z=2.5-3.5$. The median infrared-radio correlation parameter $q_{IR}$ is 2.37, evolving as $(1+z)^{-0.11}$, likely due to AGN contributions at high redshift and intrinsic differences between low- and high-redshift populations.

We develop a one-out-of-two oblivious transfer protocol over the binary-input additive white Gaussian noise (BI-AWGN) channel using polar codes. The scheme uses two decoder views linked by automorphisms of the polar transform and publicly draws the encoder at random from the corresponding automorphism group. This yields perfect secrecy for Bob at any blocklength. Secrecy for Alice is obtained asymptotically via channel polarization combined with privacy amplification. Because the construction deliberately injects randomness into selected bad bit-channels, we derive a relaxed reliability criterion, which is empirically certified via Monte-Carlo simulations. We also evaluate finite-blocklength performance. Finally, we characterize the polar-transform automorphisms as bit-level permutations of bit-channel indices, and exploit this structure to derive and optimize an achievable finite-blocklength rate.

Self-similar quasicrystals (like the famous Penrose and Ammann-Beenker tilings) are exceptional geometric structures in which long-range order, quasiperiodicity, non-crystallographic orientational symmetry, and discrete scale invariance are tightly interwoven in a beautiful way. In this paper, we show how such structures may be generalized from Euclidean space to Minkowski spacetime. We construct the first examples of such Lorentzian quasicrystals (the spacetime analogues of the Penrose or Ammann-Beenker tilings), and point out key novel features of these structures (compared to their Euclidean cousins). We end with some (speculative) ideas about how such spacetime quasicrystals might relate to reality. This includes an intriguing scenario in which our infinite $(3+1)$D universe is embedded (like one of our spacetime quasicrystal examples) in a particularly symmetric $(9+1)$D torus $T^{9,1}$ (which was previously found to yield the most symmetric toroidal compactification of the superstring). We suggest how this picture might help explain the mysterious seesaw relationship $M_{\rm Pl}M_{\rm vac}\approx M_{\rm EW}^{2}$ between the Planck, vacuum energy, and electroweak scales ($M_{\rm Pl}$, $M_{\rm vac}$, $M_{\rm EW}$).

The basis number of a graph $G$ is the minimum $k$ such that the cycle space of $G$ is generated by a family of cycles using each edge at most $k$ times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis number at most 2, and more generally, graphs embedded on a fixed surface of bounded genus are known to have bounded basis number. Generalising this, we prove that graphs excluding a fixed minor $H$ have bounded basis number. Our proof uses the Graph Minor Structure Theorem, which requires us to understand how basis number behaves in tree-decompositions. In particular, we prove that graphs of treewidth $k$ have basis number bounded by some function of $k$. We handle tree-decompositions using the proof framework developed by Bojańczyk and Pilipczuk in their proof of Courcelle's conjecture. Combining our approach with independent results of Miraftab, Morin and Yuditsky (2025) on basis number and path-decompositions, one can moreover improve our upper bound to a polynomial one: there exists an absolute constant $c>0$ such that every $H$-minor free graph has basis number $O(|H|^c)$.

Ultametrics are an important class of distances used in applications such as phylogenetics, clustering and classification theory. Ultrametrics are essentially distances that can be represented by an edge-weighted rooted tree so that all of the distances in the tree from the root to any leaf of the tree are equal. In this paper, we introduce a generalization of ultrametrics called arboreal ultrametrics which have applications in phylogenetics and also arise in the theory of distance-hereditary graphs. These are partial distances, that is distances that are not necessarily defined for every pair of elements in the groundset, that can be represented by an ultrametric arboreal network, that is, an edge-weighted rooted network whose underlying graph is a tree. As with ultrametrics all of the distances in the ultrametric arboreal network from any root to any leaf below it are are equal but, in contrast, the network may have more than one root. In our two main results we characterize when a partial distance is an arboreal ultrametric as well as proving that, somewhat surprisingly, given any unrooted edge-weighted phylogenetic tree there is a necessarily unique way to insert roots into this tree so as to obtain an arboreal ultrametric.

Despite much research, hard weighted problems still resist super-polynomial improvements over their textbook solution. On the other hand, the unweighted versions of these problems have recently witnessed the sought-after speedups. Currently, the only way to repurpose the algorithm of the unweighted version for the weighted version is to employ a polynomial embedding of the input weights. This, however, introduces a pseudo-polynomial factor into the running time, which becomes impractical for arbitrarily weighted instances. In this paper, we introduce a new way to repurpose the algorithm of the unweighted problem. Specifically, we show that the time complexity of several well-known NP-hard problems operating over the $(\min, +)$ and $(\max, +)$ semirings, such as TSP, Weighted Max-Cut, and Edge-Weighted $k$-Clique, is proportional to that of their unweighted versions when the set of input weights has small doubling. We achieve this by a meta-algorithm that converts the input weights into polynomially bounded integers using the recent constructive Freiman's theorem by Randolph and Węgrzycki [ESA 2024] before applying the polynomial embedding.

We present a quantum algorithm for the simulation of the linear advection-diffusion equation based on block encodings of high order finite-difference operators and the quantum singular value transform. Our complexity analysis shows that the higher order methods significantly reduce the number of gates and qubits required to reach a given accuracy. The theoretical results are supported by numerical simulations of one- and two-dimensional benchmarks.

In this series of papers, we investigate properties of a finite group which are determined by its low degree irreducible representations over a number field $F$, i.e. its representations on matrix rings $\operatorname{M}_n(D)$ with $n \leq 2$. In particular we focus on representations on $\operatorname{M}_2(D)$ where $D$ is a division algebra having an order $\mathcal{O}$ such that $\mathcal{O}$ has finitely many units, i.e. such that $\operatorname{SL}_2(\mathcal{O})$ has arithmetic rank $1$. In this first part, the focus is on two aspects. One aspect concerns characterisations of such representing spaces in terms of Serre's homological goodness property, small virtual cohomological dimension and higher Kleinian-type embeddings. As an application, we obtain several characterisations of the finite groups $G$ whose irreducible representations are of the mentioned form. In particular, such groups $G$ are precisely those such that $\mathcal{U}(R G)$, with $R$ the ring of integers of $F$, can be constructed from groups which virtually map onto a non-abelian free group. Along the way we investigate the latter property for congruence subgroups of higher modular groups and its implications for the congruence kernel. This is used to obtain new information on the congruence kernel of the unit group of a group ring. The second aspect concerns the conjugacy classes of the images of finite subgroups of $\mathcal{U}(R G)$ under the irreducible representations of $G$. More precisely, we initiate the study of a blockwise variant of the Zassenhaus conjectures and the subgroup isomorphism problem. Moreover, we contribute to them for the low rank representations above.

We study the computational complexity of scheduling jobs on a single speed-scalable processor with the objective of capturing the trade-off between the (weighted) flow time and the energy consumption. This trade-off has been extensively explored in the literature through a number of problem formulations that differ in the specific job characteristics and the precise objective function. Nevertheless, the computational complexity of four important problem variants has remained unresolved and was explicitly identified as an open question in prior work. In this paper, we settle the complexity of these variants. More specifically, we prove that the problem of minimizing the objective of total (weighted) flow time plus energy is NP-hard for the cases of (i) unit-weight jobs with arbitrary sizes, and (ii)~arbitrary-weight jobs with unit sizes. These results extend to the objective of minimizing the total (weighted) flow time subject to an energy budget and hold even when the schedule is required to adhere to a given priority ordering. In contrast, we show that when a completion-time ordering is provided, the same problem variants become polynomial-time solvable. The latter result highlights the subtle differences between priority and completion orderings for the problem.

We investigate the interplay of nonreciprocity and nonlinearity in a one-dimensional nonlinear Klein-Gordon chain of classical oscillators coupled by asymmetric springs, akin to a mechanical analogue of the Hatano-Nelson model with onsite nonlinearity. Using multiple-scale analysis, we show that families of nonlinear skin breathing modes -- time-periodic, boundary-localized oscillations -- emerge from their linear counterparts at any nonreciprocal strength. We derive an explicit nonlinear frequency shift for these families of nonlinear breathing modes, showing its dependence on amplitude, nonlinearity type, lattice size, and nonreciprocity, and we predict the emergence of genuine skin end breathers at the boundary once their oscillation frequency and higher harmonics enter the spectral gaps of the linear spectrum. Numerical pseudo-arclength continuation confirms full families of solutions for both hardening and softening nonlinearities. Furthermore, the Floquet analysis shows that these modes can be either linearly stable or unstable, with Floquet eigenvectors exhibiting skin localization inherited from the asymmetric couplings. Our results extend the nonlinear non-Hermitian skin effect from stationary modes to intrinsically time-periodic excitations, providing a pathway to engineer and control breathing modes in nonreciprocal mechanical metamaterials.

Gravitational wave oscillations arise from the exchange of energy between the metric perturbations and additional tensor modes. This phenomenon can occur even when the extra degrees of freedom consist of a triplet of massive Abelian vector fields, as in Multi-Proca dark energy models. In this work, we study gravitational wave oscillations in this class of models minimally coupled to gravity with a general potential, allowing also for a kinetic coupling between the vector field and dark matter that can, in principle, enhance the modulation of gravitational wave amplitudes. After consistently solving the background dynamics, requiring the model parameters to reproduce a phase of late-time accelerated expansion, we assess the accuracy of commonly used analytical approximations and quantify the impact of gravitational wave amplitude modulation for current detectors (LIGO--Virgo) and future missions such as LISA. Although oscillations are present in these scenarios, we find that the effective mass scale (the mixing mass) governing the phenomenon is $m_g \sim μ_A$, where $μ_A$ is the (time-dependent) effective mass of the vector dark-energy field. Detectability of gravitational wave oscillations, however, requires $m_g \gg H_0$, which is in tension with the ultra-light masses typically needed to drive accelerated expansion $μ_A \sim H_0 \sim 10^{-33}\,\mathrm{eV}$. Therefore, if gravitational wave oscillations were to be detected at the corresponding frequencies, they could not be attributed to these classes of dark-energy models.

Given pointed cellular spaces $X$ and $Y$, $X$ compact, and an integer $r\ge0$, we define a relation $\overset r\approx$ on $[X,Y]$ and argue for the conjecture that it always coincides with the $r$-similarity $\overset r\sim$.

The proposed fixed-target experiment, MUonE, at CERN will aim to measure the hadronic contribution to the running of the QED coupling by analysing the scattering of muons on electrons. Here we present state-of-the-art predictions for the process $μ^{\pm}e^-\rightarrowμ^{\pm}e^-$, where for the first time an all-order resummation of soft and soft-collinear logarithms has been performed. Further, we match this resummation with the complete next-to-leading and the dominant next-to-next-to-leading higher-order corrections. We find that the resummation has a dominant effect in the signal region, while the systematic matching significantly reduces the perturbative uncertainty.

Classical spectral graph theory and graph signal processing rely on a symmetry principle: undirected graphs induce symmetric (self-adjoint) adjacency/Laplacian operators, yielding orthogonal eigenbases and energy-preserving Fourier expansions. Real-world networks are typically directed and hence asymmetric, producing non-self-adjoint and frequently non-normal operators for which orthogonality fails and spectral coordinates can be ill-conditioned. In this paper we develop an original harmonic-analysis framework for directed networks centered on the \emph{adjacency} operator. We propose a \emph{Biorthogonal Graph Fourier Transform} (BGFT) built from left/right eigenvectors, formulate directed ``frequency'' and filtering in the non-Hermitian setting, and quantify how asymmetry and non-normality affect stability via condition numbers and a departure-from-normality functional. We prove exact synthesis/analysis identities under diagonalizability, establish sampling-and-reconstruction guarantees for BGFT-bandlimited signals, and derive perturbation/stability bounds that explain why naive orthogonal-GFT assumptions break down on non-normal directed graphs. A simulation protocol compares undirected versus directed cycles (asymmetry without non-normality) and a perturbed directed cycle (genuine non-normality), demonstrating that BGFT yields coherent reconstruction and filtering across asymmetric regimes.

We study the finite temperature Casimir effect for parallel plates in the $κ$-Minkowski space-time. Using the Matsubara formalism and imposing the Dirichlet boundary conditions on a massless $κ$-scalar field, we compute the $κ$-deformed corrections to thermal Casimir free energy, pressure, entropy, and internal energy. Our results demonstrate that space-time non-commutativity enhances the attractive nature of the thermal Casimir force while preserving thermodynamic consistency; the system satisfies the Nernst theorem and laws of thermodynamics remain intact in $κ$-deformed space-time. Our analysis yields an upper bound on the deformation parameter as $a\leq10^{-18}m$. Furthermore, our results indicate that non-commutative effects become experimentally observable in Casimir effect studies when the ratio of the non-commutative scale to plate separation satisfies $a/L\leq 10^{-12}$. We also obtain the expression for Stefan-Boltzmann's law in $κ$-Minkowski space-time.

We analyze entanglement generated by the Schwinger effect using a mode-by-mode formalism for scalar and spinor QED in constant backgrounds. Starting from thermal initial states, we derive compact, closed-form results for bipartite entanglement between particle-antiparticle partners in terms of the Bogoliubov coefficients. For bosons, thermal fluctuations enhance production but suppress quantum correlations: the logarithmic negativity is nonzero only below a (mode-dependent) critical temperature $T_c$. At fixed $T$, entanglement appears only above a critical field $E_{\text{entang}}$. For fermions, we observe a qualitatively different pattern: the fermionic logarithmic negativity is non-vanishing at finite temperature, and is monotonically suppressed by thermal noise. As a function of the electric field, it is non-monotonic, featuring a temperature-independent optimal field strength $E_*$ and decreasing on both sides of the maximum. We give quantitative estimates for analog experiments, where our entanglement criteria convert directly into concrete temperature and electric field constraints. These findings identify realistic regimes where the quantum character of Schwinger physics may be tested in the laboratory.

We study L-equivalence in the Grothendieck ring of varieties and its interaction with categorical invariants of cubic fourfolds. Assuming a Derived Torelli-type criterion for Kuznetsov components and a mild condition on the discriminant of the transcendental lattice, we prove a counting formula for Fourier--Mukai partners of such cubic fourfolds. As an application, we exhibit cubic fourfolds with a fixed algebraic lattice admitting a unique non-trivial Fourier--Mukai partner, which is trivially L-equivalent to the original. Finally, we show that L-equivalence classes of cubic fourfolds are finite.

Interstellar detections of several cyano derivatives of large polycyclic aromatic hydrocarbons (PAHs) have now been achieved, enabled by accurate laboratory measurements of their microwave rotational spectra. These results highlight the continued promise of other N-containing unsaturated PAHs, such as aromatic imines, as candidates for future laboratory studies and astronomical searches. In this work, we present broadband spectroscopic measurements of 9-iminofluorene and benzophenone imine in the 6-18\,GHz band. These measurements yield accurate rotational, centrifugal distortion, and $^{14}$N quadrupole coupling constants for both molecules. Using these experimentally-derived constants, we attempted a search for both molecules in the cold molecular cloud TMC-1 using observations from the Green Bank Telescope (GBT). Neither of these two ketimines was detected above the current noise level, establishing upper limits for their column densities of $5.1\times10^{12}\,\text{cm}^{-2}$ for 9-iminofluorene and $1.3\times10^{13}\,\text{cm}^{-2}$ for benzophenone imine. We also attempted a search for phenylmethanimine (both E/Z isomers) as the simplest aromatic aldimine, but neither was detected in TMC-1. To provide insight into these non-detections, we propose and evaluate different formation pathways using respective potential energy surfaces as determined by high-precision quantum chemical calculations. The result suggests the presence of an entrance barrier to forming the intermediate species, potentially explaining the low abundance.

We consider a spacetime singularity at $t = 0$ arising in a Kasner-type metric that solves the gravitational equations modified by quantum effects of a conformal field theory (CFT). The resulting constraints can be solved efficiently when expressed in terms of symmetric polynomials. Focusing first on the trace part of the modified gravitational equation, we determine the corresponding solution surfaces in Kasner-parameter space. The geometry of these surfaces depends sensitively on the ratio $η= A/C$, the quotient of the conformal charges characterizing the underlying CFT. We then fully integrate the conformal anomaly near the singularity for a generic Kasner-type metric and obtain the corresponding stress-energy tensor. Its components are expressed in terms of three symmetric polynomials (of degrees $2$, $3$ and $4$) and depend on seven arbitrary constants, which may be interpreted as parameterizing different choices of the quantum state at the singularity. By imposing a set of constraints we reduce this parameter space to a single free constant. Subsequently, we solve, at leading order near the singularity, the modified gravitational equations. Among the admissible solutions, we identify, in particular, those that develop a curvature singularity while remaining geodesically complete.

Box Decoding is a sort-free tree-search MIMO detector whose complexity is independent of the QAM order, achieved by searching a fixed candidate box around a zero-forcing (ZF) estimate. However, without pruning, the number of visited nodes grows exponentially with the MIMO dimension, limiting scalability. This work proposes two deterministic, low-complexity, sort-free pruning strategies to control node growth. By exploiting the geometric symmetry of the QAM grid and the relative displacement between the ZF estimate and nearby constellation points, the proposed methods eliminate unnecessary metric evaluations while preserving QAM-order independence. The resulting detector achieves substantial complexity reduction with negligible error-rate degradation and enables fully parallel, hardware-efficient implementations for large-scale MIMO and higher-order QAM systems.

Let $φ$ be a plurisubharmonic function defined in a neighborhood of the origin in $\mathbb C^n$. For each real number $t>-n$, we associate to $φ$ the weighted log canonical threshold \[ c_t(φ):=\sup\Bigl\{c\geq 0:\|z\|^{2t}e^{-2cφ}\in L^1_{\mathrm{loc}} \text{ near }0\Bigr\}. \] In this paper, we prove a sharp slope inequality showing that all difference quotients of the function $t\mapsto c_t(φ)$ are uniformly controlled by the Lelong number $ν_φ(0)$. Moreover, we derive explicit lower bounds for the growth of $c_t(φ)$ in terms of the complex Monge-Ampère mass of $φ$ at the origin. Our arguments combine weighted integrability estimates, restrictions to complex lines, and techniques from pluripotential theory.

We use the measurable Hall's theorem due to Cieśla and Sabok to prove that (i) if two measurable sets $A,B \subset \mathbb{R}^d$ of the same measure are bounded remainder sets with respect to a given irrational $d$-dimensional vector $α$, then $A, B$ are equidecomposable with measurable pieces using translations from $\mathbb{Z} α+ \mathbb{Z}^d$; and (ii) given a lattice $Γ\subset \mathbb{R}^m \times \mathbb{R}^n$ with projections $p_1$ and $p_2$ onto $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, if two cut-and-project sets in $\mathbb{R}^m$ obtained from Riemann measurable windows $W, W' \subset \mathbb{R}^n$ are bounded distance equivalent, then $W, W'$ are equidecomposable with measurable pieces using translations from $p_2(Γ)$. We also prove by a different method that for one-dimensional cut-and-project sets, if the windows $W, W' \subset \mathbb{R}^n$ are polytopes then the pieces can also be chosen to be polytopes; this fails in dimensions two and higher.

Tracking maneuvering targets requires estimators that are both responsive and robust. Interacting Multiple Model (IMM) filters are a standard tracking approach, but fusing models via Gaussian mixtures can lag during maneuvers. Recent winnertakes-all (WTA) approaches react quickly but may produce discontinuities. We propose SAFE-IMM, a lightweight IMM variant for tracking on mobile and resource-limited platforms with a safe covariance-aware gate that permits WTA only when the implied jump from the mixture to the winner is provably bounded. In simulations and on nuScenes front-radar data, SAFE-IMM achieves high accuracy at real-time rates, reducing ID switches while maintaining competitive performance. The method is simple to integrate, numerically stable, and clutter-robust, offering a practical balance between responsiveness and smoothness.

In this paper, we introduce the 0-smoothing invariant $\mathcal{F}$ of virtual knotoids constructed from local modification at classical crossings, which take values in a free $\mathbb Z$-module generated by non-oriented flat virtual knotoids. We prove that $\mathcal{F}$ is a Vassiliev invariant of order one. It was observed by Henrich that smoothing invariant she constructed for virtual knots provides less information than the gluing invariant. We demonstrate the same property for the 0-smoothing invariant of virtual knotoids: $\mathcal{F}$ provides less information than the gluing invariant introduced by Petit. To prove this result, we use the extension of the singular based matrix invariant originally introduced by Turaev for singular virtual strings.

We consider a Cox--Ingersoll--Ross (CIR) type short rate model driven by a mixed fractional Brownian motion. Let $M=B+B^H$ be a one-dimensional mixed fractional Brownian motion with Hurst index $H>1/2$, and let $\mathbf{M}=(M,\mathbb{M}^{\mathrm{It\hat{o}}})$ denote its canonical Itô rough path lift. We study the rough differential equation \begin{equation}\label{eqn1} \dd r_t = k(θ-r_t)\,\dd t + σ\sqrt{r_t}\,\dd\mathbf{M}_t,\qquad r_0>0, \end{equation} and prove that, under the Feller condition $2kθ>σ^2$, the unique rough path solution is almost surely strictly positive for all times. The proof relies on an Itô type formula for rough paths, together with refined pathwise estimates for the mixed fractional Brownian motion, including Lévy's modulus of continuity for the Brownian part and a law of the iterated logarithm for the fractional component. As a consequence, the positivity property of the classical CIR model extends to this non-Markovian rough path setting. We also establish the convergence of an implicit Euler scheme for the associated singular equation obtained by a square-root transformation.