Papers

Topological indices are key quantitative descriptors in mathematical chemistry, unchanged under symmetry operations and retaining graph connectivity; they capture molecular structural features to provide insights into molecular stability and chemical properties, becoming indispensable in cheminformatics and theoretical chemistry. Among degree-based indices, the Sombor index is widely concerned for capturing structural information, and motivated by enhanced structural discrimination, the augmented Sombor index ($ASO$) is defined for a connected graph $\Omega$ with $|V(\Omega)|\geq 3$ as $$ASO(\Omega) = \sum_{v_iv_j\in E(\Omega)} \sqrt{\frac{{d_i^2 + d_j^2}}{d_i + d_j - 2}},$$ where $d_i$ and $d_j$ are the degrees of vertices $v_i$ and $v_j$, respectively. Within the scope of this study, we first establish several sharp bounds for the augmented Sombor index and characterize the extremal graphs attaining these bounds. In particular, we determine the minimum value of the $ASO$ index for unicyclic graphs with a prescribed girth and characterize all graphs achieving this minimum. We also identify the second maximum $ASO$ value among trees and characterize the corresponding extremal tree structures. Furthermore, the minimum and maximum values of the $ASO$ index for bipartite graphs and chemical graphs are obtained, together with a complete characterization of the associated extremal graphs. In addition, we characterize the chemical trees that maximize the $ASO$ index. The chemical applicability of the $ASO$ index is investigated through quantitative structure-property relationship (QSPR) analysis, supported by a comparative assessment of several variants of the Sombor index. Finally, we present concluding remarks and outline potential directions for future research on the augmented Sombor index of graphs.

The ubiquity of geometric Brascamp-Lieb data, which refers to a certain density property of geometric data within the set of all feasible Brascamp-Lieb data, has recently been studied. Motivated in part by a question of Bennett-Tao concerning a possible connection or analogue of the adjoint Brascamp-Lieb inequality to the forward-reverse Brascamp-Lieb inequality, we establish ubiquity in the forward-reverse setting. Our proof takes an analytic approach and, to overcome the difficulties arising in extending ubiquity to the forward-reverse setting, makes use of tools from the calculus of variations such as Danskin's theorem and Ekeland's variational principle.

This paper develops a model-free framework for continuous-time mean-field control when the population evolves according to unknown controlled McKean–Vlasov dynamics and only discrete-time transition data are available. Model-based mean-field control requires the continuous-time drift and diffusion coefficients, which are not directly observed from fixed-step transitions, while a direct reduction to a discrete-time Bellman equation loses the continuous-time generator structure. To bridge these two viewpoints, we introduce a Mean-Field-PhiBE (MF-PhiBE), which incorporates discrete-time transition information into a continuous-time PDE on the Wasserstein space. The MF-PhiBE replaces the unknown infinitesimal drift and covariance in the policy-evaluation equation by one-step estimators computed from data, while preserving the generator structure of the McKean-Vlasov HJB equation. We also derive a policy-gradient theorem for entropy-regularized randomized feedback policies, expressing the actor direction through an action-wise infinitesimal advantage and the score of the policy. Combining these two ingredients yields a model-free actor-critic method. We prove a first-order consistency estimate showing that the value induced by an optimal MF-PhiBE policy approximates the optimal continuous-time value as the observation time step vanishes. For entropy-regularized LQR, we establish first-order policy convergence and second-order value convergence; under suitable conditions, the population-averaged feedback means coincide exactly. Numerical experiments on an LQR benchmark and a crowd-aversion problem illustrate the proposed framework.

Bayesian filtering of partially and noisily observed dynamical systems seeks to infer the evolving conditional distribution of the state of a dynamical system given observations, in an online fashion. This Bayesian filtering distribution is rarely available as a supervised learning target. However, one can often use the forecast model to generate synthetic trajectories, with corresponding synthetic observations. We introduce the proper scoring ensemble filter (PSEF), an ensemble data assimilation method trained using only synthetic trajectories. The analysis step is represented as a permutation-equivariant, transformer-based map. Training is based on strictly proper scoring rules—with the energy score used in our implementation—so that probabilistic accuracy is rewarded over the whole probability distribution. Under a realizability assumption, the population mean-field objective is minimized by the true Bayesian filtering distribution. Our methodology allows the same learned parameters to be shared between different ensemble sizes, subject to an ensemble-dependent fine-tuning. Numerical experiments show that the learned filter accurately approximates challenging filtering distributions, including highly non-Gaussian and multi-modal posteriors, and achieves stronger performance in data assimilation tasks than classical methods or learning-based methods with mean-squared-error objectives.

This paper investigates the topological properties of the Revuz correspondence between positive continuous additive functionals (PCAFs) and their associated smooth measures. Within the Dynkin, local Dynkin, and Green-tight Dynkin classes, we establish bidirectional equivalences among measure convergence, potential convergence, and PCAF convergence. In the local Dynkin class, weak convergence on compact sets, strong $\mathcal{E}_1$-convergence of potentials, uniform convergence of potentials, and $L^1$-convergence of PCAFs are mutually equivalent; under the Green-tight condition, this equivalence extends to the whole space.

Discrete optimal transport (OT) typically relies on pointwise matching between empirical measures, incurring computational costs that scale at least quadratically with the sample size. To circumvent this limitation, we introduce a mathematical framework for OT between finite location-scale mixture models. By defining a specific function class grounded in generalized Minkowski inequalities and characterizing OT maps between multivariate location-scale families, we extend Wasserstein-type metrics and barycenters to these mixture models under the assumption of identifiability. Furthermore, we prove that restricting joint couplings to a specific mixture structure reduces the continuous multimarginal OT problem to a discrete transport problem over mixture components. Computing transport plans between these components rather than individual samples reduces the computational complexity to linear scaling with respect to the sample size. Empirical evaluations on the VisDA-C benchmark confirm that this strategy achieves competitive accuracy compared to existing empirical OT approaches, while substantially reducing the computational cost and memory footprint.

We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.

Pinching-antenna (PA) systems have been widely investigated as a flexible architecture for waveguide-enabled wireless transmission. Existing analytical models, however, often rely on isotropic radiation assumptions and simplified couplingefficiency settings, which may overlook two practical design factors: the geometry-dependent radiation pattern of each PA and the sequential extraction of guided power along the waveguide. In this paper, we propose a measured-radiation-pattern-aware PA framework that incorporates an externally obtained radiation pattern, waveguide attenuation, and coupling-dependent power extraction. For a single PA, the resulting placement rule balances directional gain, waveguide loss, and free-space path loss, leading to a coupling-efficiency threshold for outperforming a fixed isotropic antenna. For multiple PAs, we study phase-matched placement and coupling-efficiency design under both uniform and independently controllable coupling. The uniform-coupling case yields a one-dimensional optimality condition and reveals that the preferred coupling efficiency decreases as more phasematched PAs participate in coherent combining. The independently controllable case admits a closed-form power-allocation structure, where stronger effective directional channels receive larger radiated power fractions. Numerical results based on a representative measured PA radiation pattern demonstrate the importance of jointly accounting for measured-radiation-patternaware placement and coupling-efficiency optimization.

In multihop Vehicle-to-Everything (V2X) networks, Road Side Units (RSUs) intend to collect information on vehicles' location in a low-latency manner while respecting their privacy constraints to support real-time location-based services. To facilitate data collection, provenance is known to ensure trust and accountability of data. Although existing joint data- and spatial-provenance techniques preserve the privacy of vehicles up to a certain granularity with respect to the RSU and other vehicles, they are unsuitable when stringent deadlines are imposed on the end-to-end delay on the packets. As a consequence, there is a need for designing spatial-provenance methods for V2X networks that satisfy stringent deadlines on the end-to-end delays while managing the privacy concerns. To fill this research gap, we propose two novel protocols, namely: Bi-Segment Embedding (BSE) and Tri-Segment Embedding (TSE), which provide a skipping mechanism for joint data- and spatial-provenance while trading off privacy features among the vehicles. Through an extensive theoretical framework, we provide an analysis of the proposed schemes in terms of reliability, privacy, and communication overhead. When compared to the baselines, our protocols offer lower end-to-end delay, higher reliability in provenance reconstruction, and the same level of privacy with respect to the RSU. We validate latency gains using practical radio parameters, and our study reveals that our proposed protocols offer significant benefits in latency when implemented over a 5G stack.

The integration of artificial intelligence (AI) and wireless communications is widely regarded as a core objective of sixth-generation (6G) systems. However, both the meaning of native AI and the type of AI capability that should be embedded into future wireless systems remain open to interpretation. This paper discusses 6G native AI from a system-design perspective and argues that native AI should be co-designed, optimized, and deployed as an intrinsic component of the wireless system rather than as a removable post-deployment add-on. From this perspective, conventional task-specific supervised models are difficult to use as the main technical basis of native AI because they depend heavily on labeled data, generalize poorly across propagation conditions, and require fragmented designs for different channel-related tasks. Motivated by these limitations, we position channel foundation models (CFMs) as a channel-centric foundation-model paradigm for 6G native AI. We define the scope of CFMs, clarify their differences from task-specific wireless AI models and large language models, and summarize three pretraining families: generative, discriminative, and hybrid pretraining. We further discuss how CFMs may support physical-layer processing, radio access network intelligence, and integrated sensing and communications. Preliminary CSI-CLIP-based results are included as bounded evidence that CFM-style pretraining can improve positioning and beam prediction when task-specific labels are limited.

Bounded ratios between products of minors of a positive definite matrix have a long history. Starting with Hadamard's inequality which bounds from above the determinant by the product of its diagonal entries and progressing through its generalizations the Fischer's inequality and the Koteljanskii's inequality. Finding new bounded determinantal ratios is a difficult task and finding their supremum is even more difficult. In 2008, Hall and Johnson showed that the ratio $$ \frac{\det A[\{1,2,4\}] \det A[\{1,3,4\}] \det A[\{2,3\}] \det A[\{1\}] \det A[\{4\}]}{\det A[\{1,2\}] \det A[ \{1,3\}] \det A[\{1,4\}] \det A[\{2,4\}] \det A[\{3,4\}]} $$ is bounded above by $4$. They conjectured that the supremum of the ratio, over all $4 \times 4$ positive definite matrices $A$, is $27/16$. Here, $A[\alpha]$ denotes the principal minor of $A$ corresponding to the rows and columns indexed by $\alpha \subseteq \{1,2,3,4\}$. In this paper we confirm that the supremum of the ratio is $27/16$ and exhibit a sequence of matrices that approaches it.

In this short note we correct the lattice theoretic classification in [Rev. Mat. Iberoam. 38 (2022), no. 4, 1199–1218] of the first named author, including some missing cases and explaining how to recover them.

We develop a physics-informed neural network (PINN) framework for nonlocal partial differential equations arising in dynamic density functional theory (DDFT). Such equations are challenging for standard PINN methods because they involve nonlinearities, nonlocal interaction terms, and an underlying gradient-flow structure, often leading to slow convergence and difficult optimization. We adapt the PINN methodology to DDFT gradient-flow equations and introduce two computational components: a modified Lorentzian activation function that behaves approximately linearly for small inputs and decays toward zero as the input magnitude increases, and a precomputed discrete operator for evaluating the nonlocal convolution term efficiently during training. The method is tested on four examples in one and two space dimensions. In the first example, the exact stationary solution is known, while in the remaining cases the neural-network approximations are validated against reference solutions computed using continuous and discontinuous Galerkin finite element discretizations. Accuracy and physical consistency are assessed through $L^1$, $L^2$, and $L^\infty$ errors, together with mass conservation and free-energy dissipation. The results show that the proposed activation function accelerates convergence relative to the standard $\tanh$ function, while the overall framework maintains good agreement with the reference solutions and captures the expected gradient-flow behaviour. These findings demonstrate the potential of the proposed PINN framework for solving nonlocal gradient-flow equations arising in DDFT.

In this paper, a matrix-free Hermitian/skew-Hermitian splitting (HSS) preconditioner is proposed for periodic Landau–Lifshitz–Gilbert (LLG) saddle-point systems. The main contributions are threefold. (1) The coupled skew/constraint block has an explicit \(4\times4\) nodal inverse and requires no local factorization. Combining this local inverse with FFT inversion of the shifted exchange block gives \(O(N_g\log N_g)\) work and \(O(N_g)\) temporary storage per application. (2) We establish well-posedness and an even-step GMRES residual bound, with an iteration estimate uniform in the mesh size and time step for a class of coupled refinements. (3) The projected implicit Euler and projection-free Crank–Nicolson-type midpoint discretizations generate saddle-point systems of the same form, so the same matrix-free FFT–HSS preconditioning procedure applies to both. They achieve first- and second-order temporal accuracy, respectively; for quadratic-affine energies, the midpoint scheme also preserves nodal length and satisfies an exact discrete dissipation identity. Two- and three-dimensional experiments confirm the predicted temporal orders and midpoint invariants. On the largest smooth tests, FFT–HSS reduces GMRES iterations by \(35\)–\(39\%\) relative to same-grid Householder preconditioning and yields approximately 15-fold and 3-fold speedups over unpreconditioned GMRES in two and three dimensions, respectively. Broadband tests retain mesh-independent iterations in the covered refinement regime at time steps 13.3 times the linearized explicit exchange limit.

Bayesian inference methods have been developed to address inverse problems in function spaces where the unknown parameters are of infinite dimension. However, conventional Gaussian priors remain inadequate for reconstructing discontinuous or sharply varying target functions encountered in practical applications like obstacle reconstruction. Although hybrid priors have emerged as a promising solution, significant challenges remain in developing theoretically rigorous and computationally tractable frameworks in engineering applications. To address these issues, we propose a persistent-homology-Gaussian (PHG) prior for solving the acoustic obstacle scattering inverse problem in the infinite-dimensional Bayesian setting, which combines a weighted persistence-based regularization term with a periodic Gaussian reference measure through a Gibbs tilt. Then, the complex boundary is represented by a log-radial function on the unit circle, so that the reconstruction from far-field data is formulated as a function-space inverse problem. The well-posedness of the resulting posterior measure is established in the Hellinger, total variation, and Wasserstein-\(p\) metrics. Furthermore, the convergence of finite-dimensional posterior approximations is obtained, and posterior sampling is performed by a preconditioned Crank–Nicolson (pCN) method. Numerical experiments show that the proposed PHG prior yields accurate and stable reconstructions under more extensive noisy conditions, providing explicit control of multiscale topological features and better performance compared to other conventional priors.

Professor Winfried Bruns was a prominent mathematician with vast knowledge in various fields of mathematics. He specialized in combinatorial commutative algebra, algebraic geometry, computer algebra, algebraic coding theory, and cryptography. In his classes, he incorporated the history of mathematics, as well as topics related to music, making his lectures engaging and enriching from an educational perspective. Additionally, he was a skilled musician, specializing in the viola. The main purpose of this note is to explore his life as a mathematician, computer scientist, and musician.

The powerset construction is the classical determinisation procedure for nondeterministic finite automata. In the coalgebraic setting, this construction has been generalised to structured coalgebras, which are coalgebras equipped with extra data. For stochastic Moore machines over the distribution monad, a type of structured coalgebra, the determinisation construction induces a semantics assigning to each finite input word a distribution on the current output. This semantics is appropriate when only the current output matters, but it is too coarse for settings in which intermediate observations must also be taken into account, as is typical for agents solving POMDPs in control theory and reinforcement learning. In these contexts, agents need to condition on all realised observations, not just the final one, so to better plan for the future. This has been addressed from a category theoretic perspective through a procedure called “unifilarisation”, which (in our context) takes a stochastic Mealy machine and produces a machine whose states are priors over the original state space and whose transitions are given by Bayesian filtering. Here we show that unifilarisation is an instance of coalgebraic determinisation. We work with Mealy machines over monads equipped with extra structure generalising the notion of the support of a distribution. We show that in this setting, unifilarisation arises from the general determinisation procedure. We then compare the resulting final coalgebra semantics with the Moore-style one. Instead of assigning only a distribution on current outputs to each finite input word, it yields causal stochastic behaviours, that is, families mapping input words to distributions on output words compatible with the “causality” constraint that outputs cannot depend on future inputs.

In this paper, we introduce the Apostol-type Mersenne-Bernoulli and Mersenne-Euler polynomials of order $\alpha$. By employing the $M$-calculus, based on the Mersenne numbers, we establish explicit series representations, addition theorems, difference equations and convolution identities.

Arguments for emergent spacetime require that quantum typicality, the generic absence of inter-subsystem correlations, persists on the physical Hilbert space of a gauge theory, where non-Abelian constraints could in principle inject geometry-supporting entanglement. Using $SU(2)$ lattice gauge theory on two-dimensional tori ($d_{\mathrm{phys}}$ up to $4{,}193$), we show that it does: the typical mutual information between strictly disjoint links matches an exact parameter-free analytical prediction combining a microcanonical baseline with Haar-random fluctuations. The Kogut-Susskind Hamiltonian generates correlations from states of definite geometry (such as the electric vacuum), while generic states show only regression to the mean, establishing that the arrow of correlation growth requires a non-generic initial condition.

Non-coherent Grassmannian (unitary space-time) signaling detects on the received subspace, which is invariant to a branch-side (polarization or mode-coupling) rotation and to a phase that is constant over the coherence block. It therefore needs no carrier-phase or polarization recovery within the block and is robust to phase noise when the per-block phase drift is small, while a multi-branch (polarization or spatial) front end harvests diversity without channel estimation or pilots. However, the Grassmannian-constellation literature usually assumes a distortion-free, linear channel and transmitter and already-acquired symbol timing. This paper closes both gaps while reusing off-the-shelf Grassmannian packings. First, we impose a constant-modulus (low peak-to-average-power-ratio, PAPR) constraint on the constellation and quantify the PAPR/chordal-distance trade-off: a constant-modulus design lowers the 0.1% PAPR from 6.1 dB (unconstrained) to 3.6 dB – 1.6 dB below 16-QAM (5.2 dB) – easing the optical modulator linear range and the fiber Kerr-nonlinearity penalty, at a ~1.8 dB cost in high-SNR coding gain. Second, we derive a phase-blind subspace timing-error detector (TED) that exploits the invariance of the GLRT projection energy to the unknown carrier phase, plus a feedforward acquisition metric, supplying clock recovery without prior carrier or polarization recovery. The TED yields a clean S-curve with a stable lock point for roll-offs down to beta=0.1. Under block fading the proposed estimator attains genie-timing SER within a fraction of a dB and recovers full diversity, whereas an uncorrected 0.35-symbol timing offset floors the error rate near 0.4. Results use a symbol-rate block-fading abstraction; full fiber, modulator, and phase-noise modeling is future work. The scheme combines low PAPR with the diversity and phase-recovery-free operation of non-coherent reception.

Many functions exhibit approximate sparsity in their coefficients with respect to a given dictionary. In recent literature, sparse approximation in such a dictionary from i.i.d. pointwise samples, underpinned by compressed sensing, has become a powerful tool for high-dimensional function approximation. A key step in this framework is truncating the (typically countably-infinite) dictionary to a finite index set of size $n$, so that compressed sensing tools can be used to approximate the function by a sparse combination of these truncated dictionary elements. This introduces a discrete $L^2$-truncation error over the sample points, which in standard approaches, is bounded by the continuous $L^\infty$-norm. Such a deterministic, worst-case bound ignores the randomness of the sample points entirely. As a result, $n$ must be taken unnecessarily large to keep the truncation error under control, which directly inflates the size of the matrix involved in the sparse recovery algorithm and increases computational cost. In this paper, we show that by exploiting the i.i.d. structure of the sample points, the discrete $L^2$ truncation error admits a bound that instead reflects the faster decay behaviour of the continuous $L^2$-norm truncation error and yields significantly smaller truncation sets and decreased computational cost. We demonstrate this through applications to weighted Wiener spaces and anisotropic Sobolev spaces, in each case obtaining significantly smaller truncation sets than recent works. In addition, we also present an improved bound of independent interest for sparse approximation in bounded Riesz systems, where the measurement condition exhibits a smaller (and scale-invariant) dependence on the Riesz constants than in previous works.

Input convex neural networks (ICNNs) are increasingly used as surrogates for stability indices and embedded as constraints in power-system optimization. This letter clarifies two recurring formulation limitations that can negate ICNN convexity benefits: (i) applying generic Big-$M$ mixed-integer reformulations introduces auxiliary binaries that are unnecessary for enforcing ICNN sublevel constraints; and (ii) reversing the stability inequality transforms a convex sublevel set into a generally nonconvex superlevel set, invalidating global-convergence guarantees of cut-based methods. After clarifying the limitations, we provide (i) an exact LP-based epigraph reformulation for ReLU-ICNNs, (ii) an outer-approximation scheme with global guarantees under the sublevel convention, and (iii) a feasibility-preserving inner-approximation scheme for the superlevel convention, with simulations on IEEE 14- and 118-bus unit commitment instances.

Many arithmetic sums over function fields can be expressed in terms of $H_i(B_n, V^{\otimes n})$ for some braided vector space $V$, and a vanishing line for these homology groups gives power-savings cancellation for the arithmetic sum. We prove an explicit vanishing line for $H_i(B_n,V^{\otimes n})$ depending only on the homology up to some finite $n$. Moreover, as the range of $n$ increases, the slope of the resulting vanishing line converges to the optimal slope. We also apply our methods to two different families of arithmetic sums. Firstly, we prove an upper bound for the bias of higher order Gauss sums over function fields, extending Patterson's conjecture beyond the cubic and quartic cases over number fields, and we conjecture this bound is sharp for orders that are prime powers. Secondly, we show that over Galois $G$-extensions, almost all character sums exhibit near square-root cancellation.

This paper studies deletion-correcting codes for the $\ell$-symbol read channel, whose noiseless output is the vector of all consecutive $\ell$-mers of a transmitted sequence. This model is motivated by overlapping-read mechanisms arising in nanopore sequencing, racetrack memories with consecutive read heads, and related sequence-labeling problems. We consider an adversarial setting in which a fixed number of $\ell$-mers are deleted from the read vector. Our first contribution is a structural characterization of the effect of such deletions: after a minimum number of $\ell$-mers are inserted to restore consistency, the resulting sequence is obtained from the transmitted sequence by deleting symbols from certain periodic substrings; when $t\le \ell-2$, these deletions correspond to complete minimum periods. Based on this characterization, we introduce check patterns and construct $\ell$-read deletion-correcting codes via power-sum syndromes. For every $\ell\ge2$, we obtain single-deletion correcting codes with redundancy $\log\lfloor (n+2\ell)/(\ell-1)\rfloor$. For $2\le t\le \ell/2$, we construct $q$-ary $\ell$-read $t$-deletion correcting codes with redundancy $t\log n+O(1)$, and for $\ell=2t-1$ with $t\ge3$, we construct codes with redundancy $(2t-1)\log n+O(1)$. We also study the sporadic parameter pairs $(\ell,t)\in\{(2,2),(3,2),(3,3)\}$ and obtain improved constructions, including binary $\ell$-read $2$-deletion correcting codes with redundancy $2\log n+O(1)$ for $\ell=2,3$, a non-binary $3$-read $2$-deletion correcting code with redundancy $3\log n+O(1)$, a binary $3$-read $3$-deletion correcting code with redundancy $5\log n+O(1)$, and a non-binary $3$-read $3$-deletion correcting code with redundancy $7\log n+O(1)$.

We prove that, for every integer $m\ge 2$, there is a smooth closed Riemannian surface $(M,g)$ diffeomorphic to $\mathbb S^2$, with positive Gaussian curvature, such that $\lambda_1(M)$ is simple and, after choosing the sign, its normalized first nonzero Laplace eigenfunction has at least $m$ distinct non-degenerate local maxima. This gives a negative answer to the Open Question of Grossi and Provenzano (Math. Ann. 389(4): 3447–3470, 2024).

We prove transfer principles for the uniform primary factorisation property (UPFP) from a Banach space $X$ to the vector-valued sequence spaces $\ell_1(X)$, $c_0(X)$ and $\ell_\infty(X)$. The hypotheses are either finite-cotype assumptions on $X$ or $X^*$, or natural self-similarity assumptions on $X$. Consequently, under these conditions, the resulting vector-valued sequence spaces are primary. As applications, we recover the primariness of $\ell_\infty(L_p)$ for $1\leq p<\infty$ without using Bourgain's localisation method, and obtain the primariness of $c_0(L_1)$. We also show that $\ell_1(\Gamma,L_1[0,1])$ has the UPFP for every set $\Gamma$, and consequently that $C[0,1]^*$ has the UPFP and is primary.

We present a novel and non-standard derivation of the Bickley-Naylor functions $Ki_n, n\in\mathbb{N}_0.$ A $3\times 3$ system of equations is developed, the solution of which yields new explicit expressions for the Bickley-Naylor functions $Ki_2$, $Ki_3$, and $Ki_4$. Using a recurrence relation, expressions follow for any other order of Bickley-Naylor functions. We then characterize the infinite set of $Ki_n, n\in\mathbb{N}_0$ as a four-dimensional span of modified Bessel and Struve functions over the ring of real polynomials, thus providing a new computational and modeling tool for studying and applying the Bickley-Naylor functions

We propose a complete architectural blueprint for safe artificial general intelligence based on a closed reentry loop (D <-> I cycle). In contrast to feedforward networks, which are directed acyclic graphs (C=0, S=0) incapable of self-reference, the proposed architecture contains a structural cycle (C >= 1) with self-sustaining amplification (rho > 1), mathematically guaranteeing the emergence of a self-model, instrumental self-preservation, and unprogrammed goal-directed behaviour. The agent's goals are encoded as a non-textual D-vector in the architecture itself, making them immune to reinterpretation and prompt injection. We present the S-measure – a polynomial-time [O(N^3)] computable alternative to Tononi's NP-hard Phi – with machine-verified Lean 4 proof that S>0 implies positive integrated information. The work provides full Python/NumPy implementations (Tarjan-based cycle complexity, Delta-S barrier), industrial horizontal scaling via Apache Kafka and Docker Compose, a taxonomy of six epochs of AI evolution, a zoo of future reentry architectures (RAS, diffusion attractors, fractal loops), gauge-invariant networks for safe swarms, fault-tolerance and recovery protocols, and eight falsifiable predictions. All formal proofs are machine-verified in Lean 4. This architecture is deployable today and represents a topologically protected, safe-by-design approach to AGI.

In the aerial pickup and delivery of goods in a distribution network, transport aviation faces risks of load imbalance due to the urgency required for loading, immediate take-off, and mission accomplishment. Transport planners deal with trip itineraries, prioritisation of items, building up pallets, and balanced loading, but there are no commercially available systems that can integrally assist in all these requirements. This enables other risks, such as improper delivery, excessive fuel burn, and possible safety issues due to cargo imbalance, as well as a longer than necessary turn-around time. This NP-hard problem, named "Air Cargo Load Planning with Routing, Pickup, and Delivery Problem" (ACLP+RPDP), is mathematically modelled using standardised pallets in fixed positions. We developed a strategy to solve this problem, considering historical transport data from some Brazilian hub networks, and performed several experiments with a commercial solver, five known meta-heuristics, and a new heuristic designed specifically for this problem. By using a portable computer, our strategy quickly found practical solutions to a wide range of real problems in much less than operationally acceptable time.

In this article, a non-homogeneous $Tb$ type theorem for arbitrary dimensional Calderón-Zygmund singular integral operators is proved. This is an extension of an analogous non-homogeneous $Tb$ theorem for the Cauchy transform, in the planar setting, due to Nazarov, Treil and Volberg. The novelties of the present work are the change of dimension and the fact that the operators to which the theorem applies are not necessarily antisymmetric. The techniques used in the proof include, among others, suppressed kernels, decompositions in $L^2(\mu)$, where $\mu$ is a Radon measure in $\mathbb{R}^d$, and a probabilistic argument resulting from taking averages of the operators involved.