Papers

Unsteady thin-airfoil theory (UTAT) coupled with discrete-vortex methods has been widely employed in reduced-order aerodynamic modeling. Due to the non-uniqueness of potential-flow solutions, the Kutta condition is imposed to determine the circulation around the airfoil. Although the unsteady Kutta condition is commonly associated with the zero-loading condition at the trailing edge, its implications for the continuity of the vortex-sheet strength remain comparatively underexplored. In particular, the classical series expansion employed for the bound vorticity in unsteady thin-airfoil theory is not uniformly convergent at the trailing edge, leading to mathematical inconsistencies in the vortex-sheet and pressure distributions. In this context, the present work seeks to advance the mathematical framework of unsteady thin-airfoil theory through a physically consistent formulation of the bound vortex-sheet strength for the Wagner problem. A recurrence relation is derived for the Wagner coefficients, allowing the construction of a uniformly convergent series expansion for the bound vorticity. The proposed formulation ensures continuity between the bound and wake vortex sheets while simultaneously recovering zero pressure loading at the trailing edge, thereby providing a mathematically consistent representation of the unsteady Kutta condition. To investigate the implications of the modified framework, a discrete-vortex method based on UTAT is developed and compared with the classical formulation. The results demonstrate that the proposed approach eliminates spurious oscillatory behavior near the trailing edge and significantly improves the regularity of the computed vorticity and pressure distributions.

We consider the problem of finding stationary points for stochastic convex optimization problems. Rather than surrogates to stationarity, such as a proximity-to-stationarity guarantee or small gradient of the Moreau envelope, we ask for a stronger notion: that the subdifferential of the objective actually contains a small element. This criterion is non-trivial, because subdifferentials of convex functions fail to converge uniformly, even in arbitrarily small neighborhoods of the optimum. Our convergence guarantees rely on dimension theory to decompose the graph of the subdifferential of a convex function, showing how stochastic sampling preserves "pieces" of these graphs, and allowing effective application of proximal-point-like methods.

A multiple double is an unevaluated sum of doubles. An NVIDIA tensor core is a specialized high performance compute core for matrix multiplication. The Ampere A100, released in 2020, introduced tensor cores capable of 64-bit floating-point arithmetic. Every multiple double arithmetical operation requires renormalization, which involves branching, for which tensor cores are unsuited. To solve this problem caused by renormalization, we apply a solution similar to the Ozaki scheme [Ozaki et al, Numerical Algorithms, 2012]. Our software is available under the GPU GPL license on github.

We prove a Gaussian comparison inequality for closed convex sets with reference probability at least 1/2. For centered Gaussian vectors whose covariance matrices are ordered in the Loewner sense, the smaller covariance law assigns at least as much probability as the larger covariance law to every closed convex set with measure at least 1/2 under the larger covariance. More generally, we show that convolving a nondegenerate Gaussian law with any independent centrally symmetric distribution cannot increase the probability of a closed convex set when the convolved probability is at least 1/2. This provides a one-sided analogue of Anderson's Theorem for Gaussian measures. As a statistical application, the result justifies one-sided and order-restricted inference using conservative covariance estimators at significance levels below 1/2 for test statistics whose acceptance regions are closed and convex but not necessarily symmetric.

We show that for every $k \in \mathbb{N}$, every graph $G$ contains $k$ vertex-disjoint cycles of different lengths, or there exists a set $X \subseteq V(G)$ with $|X| \in \mathcal{O}(k^6\mathsf{polylog}(k))$ such that $G-X$ has at most $k-1$ cycle lengths. We also prove analogous results for facial lengths of embedded graphs. Let $G$ be a graph with a closed 2-cell embedding $\psi$ on a surface $\Sigma$ of Euler genus $g$, let $c$ be a colouring of the faces $\mathcal{F}(\psi)$ of $\psi$, and let $R(G,\psi)$ be the radial graph of $(G, \psi)$. Then there exist $k$ faces $F_1, \ldots , F_k \in \mathcal{F}(\psi)$ that are given pairwise distinct colours by $c$ and are pairwise at distance at least $d$ in $\psi$, or there exists a set $X \subseteq V(G)$ of order at most $\mathcal{O}(k^2dg)$ such that $|\{ c(F) \mid F \in \mathcal{F}(\psi) \text{ and } V(F) \cap \bigcup_{x \in X} N^d_{R(G,\psi)}(x) = \emptyset \}| \leq k(k+2)$. Finally, using a result from additive combinatorics, we show that there are subdivided ladders with only a small number of cycle lengths. This suggests that it may be difficult to improve our bounds.

This paper addresses the problem of non-adaptive global robust disturbance rejection for two distinct classes of nonlinear systems. The first class, denoted by C1, consists of nonlinear systems in strict-feedback form, with linear and Hurwitz zero-dynamics (whose states are unavailable for feedback), and enhanced - within this work - by forcing, unmatched, additive disturbances. Nonlinear tools are herein employed to demonstrate that the proposed control architecture - based on the high-gain paradigm - achieves closed-loop input-to-state stability with respect to the forcing disturbances, along with global asymptotic convergence towards an attractor which can be rendered as small as desired. Then, owing to the established input-to-state stability property, a uniformly bounded control action is additionally embedded within the control architecture. The additional unit, designed following the sliding-mode paradigm, is aimed at improving the disturbance rejection task, by potentially lowering the required high-gain control expenditure. The second class of systems, denoted by C2, is constituted by minimum-phase, uncertain, nonlinear systems with relative degree greater than one, featuring possibly unbounded, with possibly unbounded derivatives, output-dependent nonlinearities, with matched additive forcing disturbances. To solve the problem of output-feedback, non-adaptive, global robust disturbance rejection for systems within C2, first, an open-loop observer is employed in lieu of a classic dynamic extension adopted in earlier works, as the latter is no longer implementable due to the presence of unknown forcing disturbances. Subsequently, the results derived for C1 are then adapted to C2, to yield an output-feedback dynamic controller providing closed-loop global uniform boundedness, along with asymptotic regulation towards an attractor which can be rendered as small as desired.

Each WSI slide contains thousands of candidate tissue patches, while supervision is usually available only at slide level. Existing bag-construction strategies like Uniform extraction and handcrafted heuristics do not control redundancy while attention-based multiple-instance models couple patch importance to a particular downstream classifier, and coreset methods optimise embedding-space coverage without modelling task-relevant patch quality. We introduce InfoDPP-PAC, a principled patch-selection framework that combines teacher-seeded Gaussian process relevance modelling, determinantal log-determinant diversity,submodular greedy optimisation, and a concentration-based adaptive stopping rule. The main theoretical result shows that the log-determinant diversity term used in DPP-style selection is the Gaussian process mutual information between a selected subset and the latent relevance function. We further derive a PAC-style certificate for residual information gain, allowing the number of retained patches to vary by slide rather than being fixed a priori. The empirical study evaluates whether the selected subset is diverse, spatially and morphologically covering, non-redundant, and enriched for the teacher-derived relevance signal. It does not claim end-to-end diagnostic improvement after retraining a downstream MIL model. On 202 HISTAI gastrointestinal whole-slide images, the adaptive rule uses 83.7% fewer patches on average than a fixed full budget while retaining 97.9% of full-budget composite selection quality. At a matched budget, InfoDPP-PAC achieves the highest mean teacher-derived relevance score among fourteen baselines, with diversity and composite scores close to the strongest coreset methods. The results support InfoDPP-PAC as a controlled quality-diversity-cardinality selection framework, rather than as a downstream clinical predictor.

We consider observables $X$ whose realizations in a sampled dataset are restricted, for example by measurement resolution, to a finite set of distinguishable categories within their possibly infinite theoretical domain. Given the probabilities of observable categories, we study the coarse-grained probability mass of families of possible datasets generated through a forward sampling process. The combinatorial construction induces an intrinsically discrete $p$-value defined directly from the sampling process rather than through additional probabilistic structure on observables. Specifying the sample means of $d+k$ arbitrary functions $g_\alpha(X)$ defines a linear family of datasets whose probability mass is obtained as a weighted sum over integer lattice points contained within the associated polyhedron. To overcome the intractable large-$N$ combinatorics, we derive via saddle-point techniques a density approximating these probability masses in the continuum limit of forward sampling within the multinomial universality class. As a demonstration, we consider conditional sampling, where $d$ structural means are fixed while $k$ means vary over admissible datasets. The information geometry emerging from the saddle-point density, together with the spherical symmetry arising at large $N$ from the intrinsic $p$-value construction, enables efficient computation of the $p$-value in the Laplace approximation via the $\chi^2_k$ distribution. The resulting statistic is given by the semi-analytic expression $2N$ times the Kullback-Leibler divergence between the information projections associated with the corresponding structural and observed linear families. These projections can be computed efficiently via standard numerical routines converging for sufficiently well-behaved sample means.

In computing, there is a need for number representation schemes that provide large dynamic range with low error. Many applications, including embedded systems and edge machine learning, have stringent memory constraints yet require large dynamic range for data representation. We present Weighted Integer (WINT), a simple and configurable mantissa exponent number format with user-selectable mantissa (m) and exponent (e) bit allocations (also referred to as configurations) that enables application-specific precision versus range tradeoffs at design time. We develop a complete analytical framework for computing Mean Relative Error (MRE), the primary metric for characterizing WINT's error. Since exact MRE calculations grow exponentially with mantissa size, we introduce harmonic and Taylor series approximation methods that achieve O(1) time complexity regardless of configuration. The Taylor series and harmonic approximations demonstrate significant speedups over the exact method while maintaining accuracy within 0.2% for the configurations presented. Our experiments across 8 to 32-bit configurations show that allocating 2 exponent bits consistently yields both lower MRE by 12-33% and 2X greater range than the integer baseline for bit widths of 12 and above. Allocating 3 exponent bits extends range by 16X while reducing MRE by 15-50% for bit widths of 16 and above

A cheap swarm of unreliable agents can be steered to a correct consensus by a few strong, expensive "oracle" correctors. We ask how much one must spend, and where to place the oracles. We model the swarm as a consensus on a graph in which each oracle pins one node toward the truth at a cost-coupled, concave strength, and measure quality by the coherence H(R)=tr M(R)^{-1}. Our first result is that H stays submodular (each added oracle helps less than the last) even when the oracles differ in strength, so a cost-benefit greedy comes within 1-1/e of the best placement at any budget. Inverting the budget gives the budget-correctness frontier B*(eps), the least spend that guarantees an eps-correct consensus: closed-form on the complete graph, and a minimal oracle count k* when oracles cost the same. Whether a budget then buys a few strong oracles or many medium onese curvature of the cost-quality law: diminishing returns favour spreadsharply increasion. Measured onthe Qwen3 ladder (0.6-32B), the law is concave for math verificatio convex foremergent code tracing, so the verdict is genuinely task-dependent.https://github.com/YehudaItkin/budgeted-oracle-placemen

In a closed fibered hyperbolic 3-manifold M ,the inclusion of a fiber S, with S and M lifted to the universal covers, gives an exponentially distorted embedding of the hyperbolic plane into hyperbolic 3-space. Nevertheless, Cannon and Thurston showed that there is a map from the circle at infinity of the hyperbolic plane to the 2-sphere at infinity of hyperbolic 3-space. The Cannon-Thurston map is surjective, finite-to-one, and gives a space-filling curve. Here we use properties of geodesics to prove that many natural measures on the circle when pushed forward by the Cannon-Thurston map become singular with respect to many natural measures on the 2-sphere. The circle measures we consider are the Lebesgue measure and stationary measures that arise from fully supported random walks on the surface group. The measures on the sphere we consider are the Lebesgue measure and stationary measures that arise from geometric random walks on the 3- manifold group. We obtain the singularity of measures from the following properties of typical geodesics. We prove that a hyperbolic geodesic sampled with respect to a pushforward measure asymptotically spends a definite proportion of its time close to a fiber. On the other hand, we show that a hyperbolic geodesic sampled with respect to a natural measure on the sphere spends an asymptotically negligible proportion of its time close to a fiber. For a more restricted class of circle measures, namely the Lebesgue measure and stationary measures from geometric random walks on the surface group, we also prove an effective result for the proportion of time spent close to a fiber.

Sepsis is a leading cause of mortality, yet optimal treatment policies remain contested. Existing reinforcement learning (RL) approaches learn fixed strategies for sepsis treatment, limiting adaptability to changing clinical objectives during inference. We propose EHRMPC, a framework that decouples learning patient dynamics from optimizing treatment by training a patient digital twin in the form of a generative electronic health record (EHR) model. The digital twin predicts clinical trajectories under interventions and enables model predictive control (MPC) to optimize treatments via inference-time planning over simulations. We evaluate EHR-MPC on a multicenter ICU sepsis cohort spanning 8 hospitals in the Mass General Brigham health system using both off-policy importance sampling and on-policy simulation-based evaluation. Relative to RL baselines, EHR-MPC achieves comparable off-policy performance and improved simulation performance. Unlike RL, this work frames sepsis treatment optimization as inference-time control over learned patient dynamics, establishing a general framework for decision making with generative clinical models.

We study cocentral algebra-split abelian Hopf algebra extensions over an algebraically closed field of characteristic zero for a fixed action of a group on a finite abelian group. We describe the extension classes in terms of crossed families of normalized 2-cocycles and construct the obstruction to representing crossed families of cohomology classes by cocycles satisfying the crossed identity. A bicharacter obstruction maps to the strict lifting obstruction and may remain nonzero even when its image vanishes. For permutation modules, we obtain an explicit description of the corresponding cocentral extension groups. We also study finite reductions of geometric representations of Coxeter groups and compute the first cohomology of the associated linear coefficient modules for finite dihedral groups and the infinite dihedral group.

We propose a Jacobi-like relative value iteration (RVI) algorithm and a Gauss-Seidel-like implementation for the ergodic risk-sensitive control (ERSC) problem of a controlled discrete time Markov chain (DTMC) on a finite state space. Under the assumption that the DTMC is irreducible and recurrent under every stationary Markov policy, we prove that the iterates of the proposed RVI algorithms converge at a geometric rate. The main challenge stems from the multiplicative structure of the ERSC cost criterion and the associated Bellman-like operators, which prevents us from adapting the analogous global contraction and bi-Lipschitz continuity properties that underlie the proof of convergence in the average cost setting. We overcome this by establishing local contraction properties for the risk-sensitive Bellman-like operators and a local bi-Lipschitz continuity property for their fixed points, and use these properties to show the iterates converge geometrically. We conclude by implementing our proposed RVI algorithms on two examples: service effort control for a single-server queue of finite capacity, and maximizing the exit rate from a finite domain (on a graph).

Let $k$ be a field, and let $X$ be a smooth projective surface over $k$. Fix a Lefschetz pencil on $X$, and let $C$ be its fibre at the generic point of $\mathbb P^1$. The closed immersion of $C$ in to $X_{k(t)}$ induces the Gysin homomorphism from the Jacobian $A$ of the curve $C$ to the Chow group $A_0(X_{k(t)})$ of $0$-cycles of degree $0$ on $X_{k(t)}$. Embedding $k(t)$ in to an uncountable universal domain $\Bbbk $, we obtain the corresponding homomorphism from $A(\Bbbk )$ to $A_0(X_{\Bbbk })$, whose kernel is either countable or the union of translates of a certain abelian subvariety inside $A_{\Bbbk }$, due to the Deligne-Katz irreducibility of monodromy action on vanishing cycles. We prove three dichotomy theorems on the structure of the kernel of Gysin homomorphism on $0$-cycles, in terms of étale monodromy action, and working over a field of arbitrary characteristic.

We study a class of 3-wave kinetic equations arising in wave turbulence theory, with regularized kernels. For radial, nonnegative initial data, we construct an exact global-in-time strong solution which remains nonnegative and is analytic with respect to time. The proof combines a careful analysis of the resonant interaction surfaces with a time power-series construction and a continuation argument based on the conservation of the energy moment.

We study exact support recovery by best subset selection in linear regression under a fixed design. For a known support size $s$ and ambient dimension $d$, we derive a non-asymptotic upper bound on the probability that best subset selection fails to recover the true support. The bound is expressed through a deterministic subset-separability parameter, which measures how well the true support can be distinguished from competing supports after projection. The result holds for all sample sizes $n$ exceeding a certain sufficient threshold which we state explicitly in terms of the signal-to-noise ratio, the subset separability of the realized design, and a logarithmic factor of order $\ln s + \ln(d - s)$. In contrast to random-design analyses, no full log-combinatorial term over the candidate support class appears. We discuss how such terms may reappear when the design is random and the separability parameter must be controlled uniformly over many competing subsets. The fixed-design formulation and the proof strategy also indicate settings in which the effective complexity of best subset selection may be reduced, for instance, under structured designs or restricted candidate subset classes.

Various situations in the theory and applications of double categories, ranging from a loose Yoneda theory and loose compact closure to double-operadic systems theory, require a notion of double copresheaf in which the action is by loose morphisms rather than tight ones. In this paper, we develop and compare several models for loose copresheaves on double categories. First, we introduce a new notion of morphism between double categories, called twisted double functors, which send tight morphisms to loose morphisms and vice versa, and use these to define twisted copresheaves. We exhibit numerous examples of twisted double functors, starting with the twisted Hom functor and the twisted representables on a double category. Corresponding to this functorial notion of loose copresheaf is a fibrational one, an internal version of a discrete opfibration that we call a loosely discrete opfibration. We prove that twisted copresheaves and cloven loosely discrete opfibrations are equivalent via an elements construction. Finally, we compare twisted bimodules with double categories over the walking loose arrow, or double barrels, via a collage construction.

Throughout, all graphs are simple, finite and have vertex sets of the form $\{ 0, 1, 2, \ldots, n - 1 \}$ for some $n \in \mathbb{N}$. For graphs $G$ and $H$, and a permutation group $\Gamma$ on the vertex set of $H$, we say that $H$ is $\Gamma$-embeddable in $G$ if there exists a graph homomorphism from $H$ to $G$ of the form $\psi \circ \varphi$, where $\varphi \in \Gamma$ and $\psi$ is an increasing injection. Recently, standard and ordered Ramsey numbers of graphs were unified through the introduction of permutational Ramsey numbers, defined as follows. For graphs $H_1, H_2, \ldots, H_k$ and permutation groups $\Gamma_1, \Gamma_2, \ldots, \Gamma_k$ on their respective vertex sets, the permutational Ramsey number $R(H_1^{\Gamma_1}, H_2^{\Gamma_2}, \ldots, H_k^{\Gamma_k})$ is the minimum $n \in \mathbb{N}$ such that for every $k$-edge-coloring of a complete graph on $n$ vertices, there exists some $j \in \{1, 2, \ldots, k\}$ for which $H_j$ is $\Gamma_j$-embeddable in the spanning subgraph of the complete graph comprising the edges of color $j$. Here, we consider reflective (resp. dihedral) Ramsey numbers, which are a specific class of permutational Ramsey numbers in which each group $\Gamma_j$ is the reflection group (resp. dihedral group) on the naturally ordered vertex set of $H_j$. Focusing on the two-color case, we apply the SAT-based approach originally proposed by Poljak for ordered Ramsey numbers and recently extended to cyclic Ramsey numbers. We utilize the Kissat SAT solver to obtain exact values and lower bounds for small reflective and dihedral Ramsey numbers whose two arguments belong to the following graph classes: monotone and alternating paths, monotone cycles, start-central stars, complete graphs and nested matchings. We also derive several general results and formulate conjectures based on the computational findings.

Quick-commerce dark stores dispatch e-grocery orders within 15 to 30 minutes, so operators such as Getir, Glovo, and GoPuff must commit drivers before orders arrive. Demand follows a latent regime that persists across hours, while unfulfilled orders spill forward as a compounding backlog. This decision binds whether drivers are employed on fixed shifts or drawn from a gig platform whose incentives are set ahead of the hour, yet existing models do not learn the regime as orders arrive. We formulate the single-store problem as a partially observable Markov decision process in which the firm infers the regime from realized orders. We show that optimal staffing rises with backlog and prove the single-store problem is indexable, a property open for multi-action partially observed problems in general. We then extend the framework to a driver pool shared across stores through a Lagrangian relaxation that decouples the network into per-store subproblems. The result is a two-level allocation policy that prices the value of tracking demand in real time and ranks stores by a provably valid priority index. On 2021 to 2022 data from the European firm SuperGlovo, which staffs full-time drivers to comply with Spain's Rider's Law, belief updating adds 10.9%, about \$188K across 27 stores, over the same program with the belief held fixed. The gain concentrates in the stores whose regimes are most persistent, observable before deployment. Online allocation reduces to a table lookup and a ranked list with a cutoff price.

We show fast mixing of Glauber dynamics for certain quadratic Gibbs measures with large external fields. The main ingredient is an overlap condition that allows us to control correlation matrices uniformly over all pinnings, by controlling norms of small submatrices of the interaction matrix. Using stochastic localization, we then obtain a lower bound on the spectral gap and, consequently, polynomial-time mixing of Glauber dynamics. As a direct application, we consider the Sherrington-Kirkpatrick model, whose interaction matrix is a scaled GOE matrix. For this model, we show that for any fixed finite inverse temperature $\beta$, there exists a strength of external field $\theta$, not depending on the size of the system, for which Glauber dynamics mixes in polynomial time (with high probability on the draw of the interaction matrix).

Biological and physical systems ranging from Fermi surfaces and skeletal structures to reaction–diffusion patterns and cosmological models may be viewed as binary mixtures in which a smooth interface separates two complementary phases. While the interface is often directly observable, the topology of one of the phases may remain hidden. To study such systems, we introduce tubular tilings, a geometric framework for discretizing binary mixtures on smooth manifolds of arbitrary dimension and topology. We prove that tubular tilings satisfy global Euler balance laws relating the topology of the ambient manifold, the discretized phases, and their interfaces. These balance laws provide a practical inference principle: topological information about a hidden phase can be recovered from the observable phase and the geometry of the separating interface. We further show that, in dimensions $d>2$, tubular tilings form a subclass of soft tilings, the recently discovered class of corner-free tessellations. Applications to Fermi surfaces and cosmological shell decompositions illustrate how the theory can be used to extract otherwise inaccessible topological information about complex geometric structures.

We revisit Talagrand's CH compactum as a test object for the two-disjoint-copies property and for pointwise quotient questions. The two-disjoint-copies property, or 2DCP, is a topological sufficient condition for the existence of infinite-dimensional metrisable quotients of spaces $C_{\operatorname{p}}(X)$; recent work asks whether Talagrand's compactum has this property. Assuming $\diamondsuit(S)$ for a stationary co-stationary $S\subseteq\omega_1$, we carry out Talagrand's inverse-limit construction with additional diagonalisation. The resulting compactum $T$ keeps Talagrand's conclusions: $C(T)$ is Grothendieck, the weak-star compact ball $M_1(T)$ contains no copy of $\beta\omega$, and $T$ has no non-trivial convergent sequences. At the same time, no two disjoint non-metrisable closed subspaces of $T$ are homeomorphic; hence $T$ has no 2DCP and is not locally homogeneous. We also give a ZFC example of a perfect compact space with 2DCP which is not locally homogeneous and contains neither $\beta\omega$ nor $2^\omega$. Finally, we isolate a general locally convex observation, in the spirit of the Banakh–Gabriyelyan theory of the Josefson–Nissenzweig property, showing that pointwise quotients onto $(\ell_p)_{\operatorname{p}}$, $1\leqslant p<\infty$, force the Josefson–Nissenzweig property. Consequently Talagrand compacta have no classical pointwise sequence quotients $(c_0)_{\operatorname{p}}$, $(\ell_p)_{\operatorname{p}}$, or $(\ell_\infty)_{\operatorname{p}}$. The full metrisable quotient problem for these $C_{\operatorname{p}}$-spaces remains open. Several open problems are included.

In 1942, Freudenthal showed that a simplex in Euclidean space can be subdivided such that the quality (well-shapedness of the simplex, quantified in terms of e.g. fatness) of the simplices in the subdivision is lower bounded. This answered a question of Brouwer. Recently, Brunck discussed the same problem for simplices in two-dimensional spaces of constant curvature and provided a closely related construction. In this paper we generalize Brunck's result to arbitrary dimensional spaces of constant curvature by combining Freudenthal's construction and radial projection. We contrast this approach with Brunck's construction.

We examine the kinematic foundations of relativity by considering two inertial frames, $S$ and $S'$, in a standard configuration, where $S'$ moves along the common spatial $x$-axis at a constant velocity $v$. By relaxing Einstein's second postulate regarding the universal invariance of the one-way speed of light, we adopt an operational framework grounded strictly on the directly observable two-way (round-trip) speed of light, evaluated alongside the principles of spacetime homogeneity, linearity, and reciprocity. Within this setting, we demonstrate that: (i) the classical Lorentz transformation is recovered exactly along the coordinate axes under a generalized two-way synchronization scheme without requiring Einstein's second postulate; (ii) Langevin's light-clock argument fundamentally implies that the longitudinal scale factor $b$ matches the standard Lorentz factor $\gamma$; and (iii) transverse lengths remain strictly invariant ($b_z = 1$). Crucially, to resolve the ontic paradox of multiple co-existing wavefront centers, we introduce a kinematically symmetric model relative to an absolute cosmological rest frame (or geometric anchor) $K$, wherein $S$ and $S'$ move with equal and opposite velocities ($u$ and $-u$, respectively). Within this generalized framework, allowing for anisotropic one-way light propagation via Reichenbach-type parameters ($\varepsilon$ or $\kappa$) yields a consistent, linear velocity-addition law, a generalized Doppler effect, and a flat but oblique spacetime metric. Finally, we prove that all round-trip observables remain strictly invariant under synchronization gauge transformations (reflecting the Tangherlini-Edwards perspective) and demonstrate that the resulting non-diagonal metric structure is fully consistent withthe Sagnac effect over closed spatial loops.

In this article, we study the edge-to-edge dihedral tilings of the sphere by kites and regular $m$-gons with $m\ge 4$. All such tilings have been identified and fully classified, using various combinatorial and geometric tools. New phenomena are observed among the tilings.

We study the algebraic structure underlying Sp-orientations in the $\eta$-periodic motivic stable homotopy category $SH(k)[\eta^{-1}]$. Borel classes determine a geometric formal ternary law, but the HW-Hurewicz map shows that its universal coefficients generate a proper $W(k)$-subalgebra $\Lambda\subsetneq (MSp[\eta^{-1}])_*$, although $\Lambda[1/2]=(MSp[\eta^{-1}])_*[1/2]$. Thus the failure of generation is purely 2-primary. To capture part of the missing information, we introduce framed involutions. The spectrum $MSp[\eta^{-1}]$ carries a canonical framed involution, yielding a Quillen-type idempotent with telescope $MSL[\eta^{-1}]$ and a canonical splitting $(MSp[\eta^{-1}])_* \cong \mathcal{R}_{fr} \otimes_{\mathbb{Z}} (MSL[\eta^{-1}])_*$, where $\mathcal{R}_{fr}$ is the universal ring of framed involutions. We then axiomatize formal ternary laws, construct the universal Walter ring $\mathcal{W}^{\eta}$, and prove that $\mathcal{W}^{\eta}$ is isomorphic to the Lazard ring $L$ after inverting 2. If $W(k)\cong \mathbb{Z}$, the universal geometric formal ternary law together with the canonical framed involution induces a classifying map $\phi:\mathcal{W}^{\eta}\to (MSp[\eta^{-1}])_*$ that is injective and becomes an isomorphism after inverting 2. Integrally, however, additional secondary power series are needed to recover the full cobordism coefficient ring.

In this paper we consider explicit Runge–Kutta (RK) methods for the numerical solution of Initial Value Problems (IVPs) in differential equations in which the last function evaluation in a step is reused, substituting the first evaluation for the next step. This requirement implies that, except for the first step, the computational cost of an $s$–stage explicit RK method reduces from $s$ to $(s-1)$ function evaluations per step. It will be seen that, in general, if a RK method has order $p$, when introducing this reused stage the new method has in general order $<p$. In this context we study the conditions on the coefficients of an explicit RK method with order $p$ such that the global error of the method reusing the last stage has the same order $p$ of accuracy. A detailed study on the conditions that the coefficients of an $s$ stage method with order $p$ must satisfy for the method reusing the last stage has also order $p$. In particular, for the families of $s$–stage explicit RK methods with the highest order $p=s$ we identify those methods reusing the last stage which retain the order of of the original method. The results of some numerical experiments are presented to confirm the theoretical results.

Let $A=(a_{ij})$ be an $n\times n$ real-valued random matrix with independent, mean-zero, variance-one entries whose fourth moments are uniformly at most $K$. Suppose that there exists $\kappa \in (0, 1)$ such that the entries of $A$ satisfy $$ \max_{i,j}\sup_{u \in \mathbb{R}} \mathbb{P}(\lvert a_{ij} - u\rvert < 1) \le \kappa. $$ We prove that there are constants $c,C>0$, depending only on $K$ and $\kappa$, such that for every fixed invertible $n\times n$ matrix $M$ and every $\varepsilon\ge0$, $$ \mathbb{P}!\left(s_{\min}(MA) \le \frac{\varepsilon}{\lVert M^{-1}\rVert_{\mathrm{HS}}}\right) \le C\varepsilon + e^{-cn}. $$ In the Gaussian case, we also show that the above estimate is sharp in the sense that $\mathbb{E}[s_{\min}(MA)]\asymp \lVert M^{-1}\rVert_{\mathrm{HS}}^{-1}.$

Continuous variable quantum key distribution protocols (CV-QKD) with discrete modulation have been intensively investigated to bridge the gap between ideal Gaussian modulation and modern coherent optical communication systems. To mitigate the penalty of discrete modulation, probabilistic constellation shaping (PCS) is applied to the modulation format and is typically performed by distribution matching (DM) algorithms. In this paper, we address the application of DM algorithms to perform PCS in CV-QKD protocols. We investigate the impact of approximating optimized Maxwell-Boltzman distributions with DM algorithms based on Huffman (HDM) and constant composition (CCDM) codes on the protocol's secret key rate (SKR) and tolerance to excess noise. Our results show that specifically symbol-by-symbol HDM degrades the SKR by at least 30%, whereas CCDM matches the optimal SKR with code length of $10^3$ or more symbols. Furthermore, we also provide a statistical analysis of symbol dependence for both approaches, showing that CCDM must operate with blocks of at least $10^5$ symbols for the correlations become negligible. Finally, we propose an algorithm to generate independent symbols following near-optimal distributions.