Papers

Low Earth Orbit (LEO) satellite communications (SATCOM) has emerged as a key enabler of global connectivity for 6G networks. To overcome the significant path loss of space-to-ground links, high-gain directional beamforming (BF) is indispensable. As LEO systems evolve toward wider bandwidths to support data-intensive applications, however, they encounter a fundamental physical limitation known as the beam-squint effect, which induces frequency-dependent beam misalignment. Conventionally, the beam-squint effect has been treated as a critical performance impairment that must be mitigated. This article introduces a paradigm shift in wideband LEO satellite systems by redefining beam-squint as a valuable source of frequency-spatial diversity and presents the principles of rainbow BF. Rather than mitigating beam squint, rainbow BF deliberately exploits it to generate frequency-dependent beams, enabling different frequency components to illuminate distinct spatial directions using only a single or a small number of radio frequency chains. By supporting dynamic frequency-spatial beam allocation, rainbow BF offers enhanced flexibility and scalability for wideband LEO SATCOM. We further illustrate the benefits of rainbow BF through three representative LEO SATCOM applications: i) massive multiple access to overcome the latency and throughput bottlenecks of conventional beam hopping; ii) integrated sensing and communications for simultaneous target detection and data transmission; and iii) rapid satellite acquisition to reduce search overhead and improve link reliability. Finally, we discuss key implementation challenges and outline promising future research directions for rainbow BF in wideband LEO SATCOM.

We study outward meromorphic continuation from circular boundary data in the unit disk. The unknown function is holomorphic in the unit disk and admits a meromorphic continuation to a larger disk, where finitely many exterior simple poles are superposed on an unknown holomorphic background. The positive Fourier coefficients of the boundary trace are Taylor coefficients at the origin, and exterior poles generate a finite exponential-sum component in these coefficients. We introduce shifted determinant characteristics and prove that, in the pure finite-pole model, the determinant for the correct order factors exactly into a nonzero constant times the polynomial whose zeros are the reciprocals of the exterior poles. The same zero set is obtained for noiseless equispaced discrete Fourier coefficients; sampling changes only the amplitudes through an aliasing factor. For data containing a holomorphic background, discretization effects, and noise, roots of a single empirical determinant are only candidate reciprocal poles. We therefore propose a root-propose and contour-certify procedure: determinant roots generate candidate regions, while local argument-principle counts, contour moments, empirical margins, and persistence over determinant orders and shifts certify visible poles. A Rouché-type perturbation analysis gives sufficient conditions for stable local zero counts and explains how residues, pole separation, distance to the target annulus boundary, shifts, and noise affect visibility. Numerical experiments verify the pure-pole identity, demonstrate certification under background and noise, and show that high noise, weak residues, boundary-near poles, and close poles naturally lead to partial recovery of contour-certified visible poles.

We formulate compactifications of continuous maps naturally. We consider real analytic functions mainly. We are interested in topological properties and combinatorial ones of explicit resulting maps. For understanding them, we use their Reeb spaces, being quotient spaces of the spaces of the domains of the functions and defined by the equivalence relation identifying two points in same components of their level sets. They are known to be $0$- or $1$-dimensional (metrizable) cell-complexes, in our situations or more general certain tame cases. Reeb spaces have been important in understanding topological properties and combinatorial ones of functions and spaces roughly, since the last century. These compactifications have been explicitly studied by the author previously and recently. We have obtained real algebraic functions whose Reeb spaces are not so complicated and which seem to be of most natural and simplest. We present new discussions and examples.

This paper studies a finite-horizon multi-item capacitated lot-sizing problem in which demand quantities are deterministic, while demand-arrival periods are stochastic. Each demand occurs once within a known time window and must be satisfied no later than its deadline. The proposed model makes production and allocation decisions at the demand level, allowing it to represent capacity competition, demand-specific backlog, and allocation-dependent inventory dynamics. The stochastic problem is formulated as a discrete-time Markov decision process (DTMDP), including the state space, feasible actions, transition kernel, and one-period cost function. To isolate the computational effect of stochastic timing, each stochastic instance is first compared with a deterministic counterpart in which each arrival distribution is replaced by its most likely arrival period. This comparison shows that stochastic timing substantially increases the number of states, the number of transitions, solution time, and memory pressure. A genetic algorithm (GA) is then proposed for the stochastic-timing problem. The GA searches over feasible state-feedback policies and evaluates each policy exactly under the DTMDP transition model. Computational experiments on 330 benchmark instances show that the GA remains close to the exact stochastic solution whenever the latter is available, with an average optimality gap of about $3.44\%$. On the difficult benchmark instances, comprising 90 test cases, the GA remains below the $5\%$ optimality-gap threshold and achieves an average optimization speedup of $6.89 \pm 1.41$ at the $95\%$ confidence level. For instances that cannot be solved exactly on the available hardware, an empirical Bellman-time regression is used to estimate the missing exact resolution time and extrapolate the expected GA speedup.

We extend Toën's derived Brauer theory to the open site of Spivak's derived smooth manifolds. For a compact derived smooth manifold $X$, we define a spectralized structure sheaf $\mathbb{O}_{X}$ and prove that every Picard twist can be represented by a derived Azumaya algebra, whose Morita equivalence class is uniquely determined by the twist. In the open locally Morita trivial setting, this gives a natural bijection \begin{align*} \Phi_{X}:\mathbf{dBr}^{\mathrm{open}}(X) \xrightarrow{\cong} H^{1}(|X|;\underline{\mathbb {Z}}) \times \pi_0\Gamma(|X|,\mathrm B^2GL_1(\mathbb O_X)). \end{align*} If we consider $C^\infty_X(\mathbb C)$ on ordinary smooth manifolds, the second factor is $H^3(|X|;\underline{\mathbb Z})$, which recovers the Dixmier-Douady invariant. If we consider $C^\infty_X(\mathbb R)$, the second factor is $H^2(|X|;\underline{\mathbb Z/2})$, which recovers the corresponding invariant of Donovan and Karoubi.

In this paper, we prove that any closed minimal hypersurface $M^4$ of $\mathbb{S}^5(1)$ with constant scalar curvature and constant Gauss-Kronecker curvature must be isoparametric. Specifically, $M^4$ is either an equatorial 4-sphere, a Clifford torus $\mathbb{S}^2\left(\frac{\sqrt{2}}{2}\right)\times \mathbb{S}^2\left(\frac{\sqrt{2}}{2}\right)$ or $\mathbb{S}^1\left(\frac{1}{2}\right)\times \mathbb{S}^3\left(\frac{\sqrt{3}}{2}\right)$, or a Cartan's minimal hypersurface. Consequently, the squared norm of the second fundamental form $S$ can only take the values 0, 4, 12. This result provides strong support for Chern's Conjecture.

We construct three nonconstant polynomials $a,b,c\in \mathbb{R}[X]$ such that \[ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 \] are all squares in $\mathbb{R}[X]$. The entries in the construction are quadratic, so the construction has the smallest possible positive degrees. By composition with nonconstant polynomials, this gives infinitely many examples over $\mathbb{R}[X]$ with all entries nonconstant.

Deep neural networks often contain substantial hidden-state redundancy, but most compression methods operate directly on weights, neurons, or quantised representations without explicitly characterising the dynamical role of internal states. This paper proposes a controllability-observability framework for empirical state-order reduction of deep neural networks. By viewing a trained network as a depth-indexed nonlinear dynamical system, we construct data-driven reachability, observability, and balanced Gramians from hidden-state snapshots and output Jacobians. The resulting A/B/C tests estimate layer-wise reachable, observable, and jointly reachable–observable ranks. These ranks are then used not only as diagnostic measures of hidden-state redundancy, but also as actual compressed layer widths for realised reduced networks. Experiments on MNIST and CIFAR-10 compare the proposed balanced realisation against projection-based reduction, unstructured pruning, structured pruning, low-rank SVD, dynamic INT8 quantisation, and linear baselines. On MNIST, a four-layer SiLU DNN is reduced from state order 1024 to 277, giving 72.95% state compression and 73.48% parameter compression, while maintaining 95.45% accuracy compared with 96.60% for the full model. On CIFAR-10, a larger SiLU DNN is reduced from state order 4608 to 1339, giving 70.94% state compression and 83.09% parameter compression, while preserving accuracy from 54.45% to 54.44% and reducing CUDA inference latency by approximately 3X. The results show that balanced reachable-observable ranks provide a principled empirical minimal-realisation criterion for designing compact neural architectures with little or no loss in accuracy.

In algebraic geometry, it is well known that Hochschild cohomology and, in particular, the algebraic structure it carries, plays a central role in studying the infinitesimal noncommutative deformations of geometric spaces. This thesis provides, for the first time, an explicit interface between J. Lurie's work on higher centers and the Hochschild cohomology of an algebraic $\mathbb{k}$-scheme within the framework of formal deformation quantization. Our motivation stems from the mysterious appearance of the square root of the Todd genus in Kontsevich's formality theorem for algebraic varieties, as well as the conjectural relationships between these objects and the motivic Galois group. Our main result is a canonical solution to Deligne's conjecture for Hochschild cochains in the affine and global cases, even for singular schemes, by exhibiting the Hochschild complex as an $\infty$-operadic center. We show that this equips the Hochschild complex with a universal $\mathbb{E}_2$-algebra structure that precisely agrees with the classical Gerstenhaber bracket and cup product on cohomology in the affine and smooth cases. Finally, we interpret this universal $\mathbb{E}_2$-algebra structure in terms of operadic formal moduli problems by exhibiting a relationship between $\infty$-operadic centers and formal automorphism groups.

Continuous Ising solvers embed a discrete optimization problem into a continuous dynamical system and recover the spin configuration by sign readout, but dense interaction evaluation gives an $O(N^2)$-per-step cost. We show that this cost is not intrinsic: during late-stage simulated bifurcation the trajectory collapses onto a lower-dimensional active subspace, and saturated coordinates can be eliminated exactly by a variational frozen-set identity whose couplings fold into an induced field on the unresolved subsystem. We prove large-parameter recovery for the external-field quartic model, the hard-box limit of ballistic confinement, and a robust-margin freezing criterion. The resulting algorithm, iSTAR (Ising Stable-set Tail-Aware Reduction), exploits this collapse by detecting stabilized coordinates and continuing only on the active tail. An online certified implementation on the G-set benchmark preserves the same-seed baseline in all runs and removes on average 64.4% of the dense interaction work.

We determine some sufficient conditions for the split extension of two free finitely generated non-abelian Lie algebras $L = F_1 \leftthreetimes F_2$ over an infinite field $K$ to be incoherent.

We prove the following two statements: (1) If $X$ is a compact Kähler fourfold with pseudo-effective canonical bundle and no subvarieties of codimension one or two, then $K_X$ is a torsion line bundle; (2) If $X$ is a compact Kähler fourfold with no subvarieties of codimension one or two, then $X$ is not uniruled. Using the Beauville-Bogomolov decomposition and a characterization of Uniruled manifolds by Wenhao Ou, we get that if $X$ is a compact Kähler fourfold with no subvarieties of codimension one or two, then either $X$ is a torus or an irreducible holomorphic symplectic manifold.

This paper investigates the higher-order commutators generated by fractional maximal operators with rough, spatially dependent kernels in the framework of variable exponent Morrey spaces. Under minimal log-Hölder continuity assumptions on the variable exponent profiles and suitable geometric constraints on the Morrey-type weights, we establish comprehensive strong interior boundedness results between the appropriate spaces. We further analyze the critical endpoint boundary configurations where classical strong-type boundedness fails due to Luxemburg norm blow-up, proving that a corresponding sharp weak-type estimate remains structurally valid and revealing the underlying transitions between different boundedness regimes. In addition, an abstract real interpolation framework of Grafakos–Martell type is developed to bridge these endpoint and interior strong estimates, thereby recovering a full continuous scale of intermediate regularity properties. These results extend the classical harmonic analysis scheme to a broader nonhomogeneous setting and provide new insights into the continuous interplay between rough kernels, variable exponents, and local weighted growth conditions.

We prove a sparse embedding theorem for induced embeddings of bounded-degree graphs. The theorem applies to pairs $G\subseteq \Gamma$: the graph $G$ supplies the positive edges of the target graph, while the ambient graph $\Gamma$ supplies the induced constraints which must be avoided. Its main feature is that these two types of constraints are kept separate throughout the embedding process. As an application, we show that, for every fixed $\Delta,r\ge2$, there are constants $A,C>0$ such that every $n$-vertex graph $H$ with maximum degree at most $\Delta$ satisfies $r_{\text{ind}}(H;r)\le C n^{\Delta+2}(\log n)^A$. This improves the exponent in the polynomial bound of Conlon, Fox and Zhao for bounded-degree induced Ramsey numbers. The proof combines the aforementioned embedding theorem with a sparse random transference argument, in which the random host is used only to certify robust deterministic hypotheses for every colour class.

Low-precision neural networks are attractive for resource-constrained hardware, but fixed-point arithmetic introduces failure modes that are often hidden by idealised quantisation models. In particular, two's-complement overflow wrapping can corrupt hidden activations by changing both their magnitude and sign, leading to unstable numerical error propagation and severe accuracy degradation. This paper proposes a Lyapunov-stabilised quantisation framework for low-precision neural networks operating under hardware-style wrapping arithmetic. The hidden-state energy is monitored through a layerwise Lyapunov function, and a monotone projection is applied to enforce bounded and non-increasing state evolution across depth. The method is evaluated on MNIST using a compact patch-based transformer under post-training quantisation and quantisation-aware training with fixed-point bit-widths from 4 to 16 bits. Monte Carlo results show that unconstrained wrapped quantisation-aware training collapses to near-chance accuracy across 6-16 bits, with activation overflow rates exceeding 11%. In contrast, the proposed monotone Lyapunov projection suppresses activation overflow to below 0.012% and restores stable low-precision learning, achieving 86.55% accuracy at 12 bits. These results demonstrate that Lyapunov-based state control can act as a hardware-aware stabilisation mechanism for reliable fixed-point neural inference and training.

We strengthen a result of Liu from his papers arXiv:2011.09412, arXiv:2105.01022, by proving that profinite isomorphisms of hyperbolic 3-manifolds are regular in the sense of Boileau and Friedl arXiv:1505.07799

Lie-Poisson electrodynamics (LPE) is a non-Abelian and nonlinear deformation of usual electrodynamics, where the gauge algebra is defined through a Lie-algebra-type Poisson bracket on space-time. This field-theoretical model arises from noncommutative U(1) gauge theory in the semiclassical limit. We focus on the geometric approach to LPE in the absence of charged matter. We establish a non-trivial field redefinition which, under mild technical assumptions, maps the LPE dynamics to that of Maxwell theory. We emphasise that despite the dynamical equivalence, these two theories are not physically equivalent, since the corresponding local gauge-invariant observables are different. We show that our field redefinition is actually a Seiberg-Witten map between LPE and Maxwell theory. Using this map, for any symmetry of the Maxwell action, we construct generators of LPE symmetries and the corresponding conserved currents. In particular, we obtain deformed Poincaré transformations, thereby solving the problem of constructing space-time symmetries in LPE. The deformed symmetries are constructed as active transformations of the gauge field. In the outlook, we propose a natural quantization prescription for LPE based on our field redefinition.

The hyperbolic potential $$ V(r)=-\frac{\alpha}{(R^2-r^2)^2} $$ was identified by Olshanii as the negative-curvature counterpart of Newton's off-center circular-orbit problem. We first show that this form is not an isolated ansatz. For a planar radial natural Hamiltonian at fixed energy, if the Jacobi metric is regular at the force center and has nonzero constant Gaussian curvature, then, up to radial and overall scales, $$ E-V(r)=\frac{\alpha}{(R^2\pm r^2)^2}. $$ Thus the spherical and hyperbolic off-center models are the two nonflat regular radial constant-curvature branches. We then complete the dynamics of the negative-curvature branch. At zero energy its Jacobi metric is a constant multiple of the Poincare disk metric. An explicit Runge–Lenz-type vector and the angular momentum form an on-shell $\mathfrak{so}(2,1)$ moment map; their algebra gives the supporting-circle equation without integrating the equations of motion. The cotangent lift of circular inversion preserves the moment map and intertwines the exterior and punctured-interior flows after a positive time change. The singular circle is at infinite Jacobi distance but is reached in finite Newtonian time. Finally, requiring the magnetic two-form to preserve the full orientation-preserving hyperbolic isometry group uniquely forces it to be a constant multiple of hyperbolic area, hence $B(r)=-Q/(R^2-r^2)^2$ in disk coordinates. The resulting shifted $\mathfrak{so}(2,1)$ Casimir gives the standard circle–horocycle–hypercycle transition at $Q^2=8m\alpha R^2$, now as an explicit coupling statement in the off-center Newtonian system.

In this paper, we study the two-dimensional Schrodinger oscillator within a dynamical noncommutative (DNC) space. By leveraging perturbation theory, we derive the energy eigenvalues and eigenvectors and systematically analyze the effects of both dynamical and non-dynamical noncommutative settings. First-order corrections to the eigensystem are obtained, revealing that the energy shift explicitly depends on the DNC parameter tau. Furthermore, we explore the thermal properties of the system by employing the partition function. Numerical results are presented to provide a comprehensive analysis of the system behavior under the considered effects. Notably, in the DNC framework, the commutation relations and the deformation parameter are position-dependent. Using the two-dimensional Bopp-shift, we effectively map the noncommutative problem to its commutative counterpart.

We study the small-noise asymptotics of Euclidean heat regularizations of probability measures supported on manifolds with corners. Near a boundary or corner stratum, the relevant regime is a conical boundary layer in which the observation point approaches the stratum at the same scale as the Gaussian smoothing parameter. After rescaling this layer, the support is replaced to leading order by its inward tangent cone. We prove a two-term expansion for the heat-regularized density in this regime. The leading coefficient is the Gaussian mass of the linearized cone, weighted by the density on the support and by the adapted corner Jacobian; the first correction records the variation of the density, the Jacobian, and the quadratic geometry of the embedding. A localization argument then yields the corresponding expansion for the full heat regularization, with the nonlocal contribution exponentially small. From this density expansion we derive logarithmic asymptotics and uniform expansions for the score, the log-Hessian, and the scale derivative of the score. These formulas show how lower-dimensional support, boundary faces, corners, and curvature are encoded in the singular differential structure of small-noise Gaussian regularizations.

Flow matching is a powerful tool for generative modeling, but emerging applications in robotics, planning, and control require inference-time constraints on generated outputs. Such constraints are often complex and highly nonlinear. As a result, methods designed for linear constraints like image inpainting are rarely sufficient, and projection or optimization-based alternatives can be prohibitively expensive. In this paper, we introduce Lagrangian Dual Flows, a new family of constrained generation techniques based on Lagrangian dual dynamics. By flowing a dual co-state alongside generated samples, we can guarantee nonlinear constraint satisfaction without expensive optimization subproblems, pseudoinverses, or projection steps during the denoising process. The resulting constrained generation algorithms are simple, effective, and open new theoretical connections between flow matching and primal-dual methods in numerical optimization.

We consider compactly supported RBFs having algebraically decaying Fourier transforms. Here we focus especially on generalized Wendland RBFs and their modification by making them smoother away from zero, a process we call eccentric smoothing. Specifically, we consider mapping properties of the integral operators (sometimes known as covariance operator) for these novel type RBFs. Moreover, we show that eccentric smoothing makes wavelet-inspired compression technique for the kernel matrix feasible.

Enterprise security teams commonly summarize remediation through mean time to remediate (MTTR), SLA compliance, dwell time, or detection delay. These metrics are useful, but they can hide how fixes actually reach the estate: continuously, through scheduled maintenance windows, in deployment rings, or through emergency bypass paths. This paper introduces a remediation-cadence audit for enterprise vulnerability management. The audit records routine mean lag, release period, release fraction, cohort geometry, emergency/routine split, non-fielding delay, local residual-pressure evidence, and declared rate scenario. It compares a continuous same-mean shortcut with the recorded release calendar and reports a local capacity verdict plus a calendar discount: the fraction of mean-only local capacity consumed by calendarized fielding. Worked notional packets with the same 30-day mean lag show why this matters. Under the normalized screening scenario, a two-month release train consumes 17.4% of mean-only capacity, a monthly train 5.2%, and a two-week screen 1.3%; across a 16-fold attacker-adjustment rate band, the two-month discount remains at least about 12% and the monthly discount stays in the resolution-sensitive 3–8% range. The audit therefore turns cadence assessment into an evidence-resolution question: when the discount is material relative to residual-pressure uncertainty or claimed headroom, MTTR/SLA should not be used alone as fielding evidence. Release-geometry checks show that deployment rings do not automatically recover the continuous benchmark, and cohort staggering can help or hurt near capacity. The result is a reproducible governance diagnostic, not a breach predictor or CVE prioritizer.

We revisit the hybridizable discontinuous Galerkin method of Cui and Zhang for the Helmholtz equation with first-order absorbing boundary condition. Their analysis proves stability without imposing a mesh constraint coupling $h$ and $\kappa$, but the explicit stability bound still contains negative powers of the mesh size. We prove that this mesh-dependent blow-up is not intrinsic to the HDG discretization. Under the same geometric and mesh framework as in the reference analysis, for fixed polynomial degree and for stabilization parameters uniformly bounded above and below, we establish a generalized stability estimate whose constant is independent of the mesh size. For every prescribed compact interval $0<\kappa_0\le\kappa\le\kappa_1<\infty$, the stability constant can be chosen locally uniformly with respect to $\kappa$. The proof replaces the Rellich-identity and inverse-estimate argument by a compactness argument based on the HDG discrete distributional gradient and the Crouzeix–Raviart lifting mechanism. As consequences, we obtain well-posedness, convergence for minimal $L^2$ data, and projection-based error estimates in which the negative powers of the mesh size appearing in the previous theory are removed. Numerical experiments confirm the predicted mesh-uniform behavior and show a clear contrast with the mesh-dependent growth suggested by the previous

We introduce the class of cooperative conic games, a new family of transferable utility games whose characteristic function can be computed by solving a conic optimization problem. This framework unifies a broad range of optimization-based cooperative games within a common mathematical formulation, encompassing linear, second-order cone, semidefinite, and other convex nonlinear optimization problems admitting conic representations. Exploiting the structural properties of conic programs and conic duality theory, we derive general conditions ensuring the existence of core payoff allocations (or cost shares) and show how such allocations can be computed directly from optimal dual solutions. In particular, the proposed methodology replaces combinatorial verification procedures for core nonemptiness with tractable optimization-based criteria. The expressive power of the framework is illustrated through several representative applications. We show that it models cooperation in multi-agent optimal control, generalizes linear production games to nonlinear production settings such as Cobb–Douglas models, introduces cooperative formulations for portfolio selection under Markowitz, conditional value-at-risk, and entropic risk measures, and unifies a broad class of combinatorial optimization games through exact conic reformulations. These examples involve the most relevant cones in modern convex optimization, highlighting both the versatility of the proposed framework and its potential to strengthen the interaction between mathematical programming and cooperative game theory.

We present an open, reproducible branch-and-cut (B&C) algorithm for the capacitated profitable tour problem (CPTP) and its open s-t path variant, the capacity-constrained elementary shortest-path problem. The solver re-implements the formulation and cut families of Jepsen et al. (2014) on a fully open mixed-integer programming stack (HiGHS; Huangfu and Hall, 2018), and adds bound-based preprocessing, domain propagation, and reduced-cost variable fixing. We claim no new method; the contribution is twofold. First, an open, reproducible artifact: to our knowledge the first branch-and-cut for this problem class on a fully open stack, with the formulation, every separator, and all benchmark scripts released, so the results below can be rerun and the solver reused and extended as a baseline. Second, a component study on this common modern stack, benchmarked against a dynamic-programming/labelling reference, that decomposes which components pay off and where the running time goes. We find that the capacity-class cuts account for essentially the entire benefit (adding them to a connectivity-only baseline lifts the number of instances solved from 52 to 64 of 76 and shrinks the search tree more than tenfold), while comb and rounded generalized-large-multistar cuts, reduced-cost fixing, and bound-based propagation add nothing measurable. We also report a negative result: the shortest-path-incompatibility (SPI) cut, a variant of the node-precedence inequalities of García (2009), finds no violated inequality on any instance. The solver and all experiments are released as open, reproducible software (Spoorendonk, 2026).

We show that among partially hyperbolic diffeomorphisms with one dimensional center there is a $C^1$-open and dense subset for which either there is a proper quasi-attractor or both strong foliations are minimal. The same result holds among volume-preserving partially hyperbolic diffeomorphisms and has consequences about robust transitivity beyond the conservative setting. The proofs involve a careful study of minimal $\mathcal{W}^u$-saturated sets and interact with recent results in the subject.

The Alon–Tarsi number $AT(G)$ of a graph $G$, defined via the graph polynomial, is a strengthening of the list chromatic number $\chi_{\ell}(G)$. We study the Alon–Tarsi number of squares of planar graphs. The square of a graph $G$ is the graph obtained by joining every pair of vertices whose distance in $G$ is at most $2$. Recently, Kim and Luo (2026) proved that $\chi_{\ell}(G^2)\le 6$ for every subcubic planar graph containing no $k$-cycles for $4\le k\le 8$. We strengthen this result by proving that $AT(G^2)\le 6$ for every such graph $G$.

Comparing whether two dynamical systems implement the same computation despite differences in coordinates or measurements is a central problem in neuroscience and machine learning. Dynamical Similarity Analysis [DSA; Ostrow et al., 2023] addresses this problem by aligning finite-dimensional Koopman approximations through an orthogonal similarity transformation. Here we show that orthogonal alignment is neither necessary nor sufficient for topological conjugacy: conjugate systems may require a non-orthogonal basis-transfer matrix that DSA cannot capture, while non-conjugate systems may have orthogonally equivalent Koopman operators that DSA fails to distinguish. We use this observation to formulate Conjugacy-based Similarity Analysis (CSA), which restricts alignments to those induced by candidate state-space bijections rather than arbitrary orthogonal matrices. We prove that CSA's fitted alignment is the finite-data projection of the composition operator associated with the candidate bijection, and use controlled examples to show why this distinction matters when observable dictionaries are chosen explicitly or implicitly from data. These results clarify what Koopman-based similarity measures must ensure to support claims of identifying conjugacies between computational systems.

We study irregular double-phase parabolic equations with variable exponents and non-divergence data, \[ u_t-\operatorname{div} \left(\mathcal{F}(z,\nabla u)\nabla u \right)=f(z),\quad z=(x,t)\in Q_T:=\Omega\times (0,T), \] under the homogeneous Dirichlet boundary conditions. Here, $\Omega \subset \mathbb{R}^N$, $N \geq 2$, is a bounded domain, $T>0$, \[ \mathcal{F}(z,\nabla u)=a(z)|\nabla u|^{p(z)-2} + b(z) |\nabla u |^{q(z)-2} \] with given Lipschitz-continuous exponents $p,q$ that satisfy a suitable balance condition. The nonnegative coefficients $a(z), b(z)$ satisfy the inequality $a(z)+b(z)>0$ in $Q_T$, the space and time derivatives of $a$ and $b$ belong to $L^d(Q_T)$ with some $d$ depending on the data. If \[ f\in L^\sigma(Q_T) \quad \text{for} \ \sigma \in (2, N+2] \quad \text{and} \quad \mathcal{F}((\cdot,0),\nabla u_0)\,|\nabla u_0|^{r+2}\in L^1(\Omega), \] where \(0\le r\le K(N,\sigma,p,q)\) if \(\sigma<N+2\), while \(r\ge0\) is arbitrary if \(\sigma=N+2\), then the problem has a unique strong solution, for which we prove the global transfer of integrability from the initial data and the forcing term to the double-phase flux in the spirit of Calderón-Zygmund theory, higher integrability of the gradient, and the second-order space regularity: \[ \begin{split} & \text{$\mathcal{F}((\cdot,t),\nabla u(\cdot,t))|\nabla u(\cdot,t)|^{r+2}\in L^1(\Omega)$ for a.e. $t\in (0,T)$}, \\ & \text{$|\nabla u|^{2(\min\{p(z),q(z)\}-1)+r+s}\in L^1(Q_T)$ for every $s\in\left(0,\frac{4}{N+2}\right)$}, \\ & \mathcal{F}(z,\nabla u)|\nabla u|^{\frac{r+2}{2}} \in L^2(0,T;W^{1,2}(\Omega)). \end{split} \] The results improve and complement the results in \cite{Arora-Shmarev-JGA-2026} and extend them to the full range $r \geq 0$.