Papers

This paper is concerned with a class of the long time Kähler-Ricci flow on a compact Kähler manifold. It is shown that the uniform $\mu$-entropy or uniform Sobolev inequality along the normalized Kähler-Ricci flow with semiample canonical bundle. As a consequence, we prove that the scalar curvature of the Kähler metrics along the normalized Kähler-Ricci flow converge to negative Kodaira dimension of the compact Kähler manifold.

A central operator schedules each vehicle's passage time through a bottleneck to achieve a dynamic system optimum (DSO). The assignment depends on each vehicle's preferred arrival time, which is private and must be elicited from each vehicle. Mechanisms that elicit exact preferences, such as the Vickrey-Clarke-Groves (VCG) mechanism, can achieve strategyproofness but involve relatively complex rules and a computational burden on the operator. We focus instead on coarse reporting, in which each vehicle selects from a finite menu of time slots of a common width. This discrete interface already structures reservation and appointment systems in practice, including managed lanes for automated vehicles, airport slot allocation, and delivery appointment windows. We design a slot-based DSO mechanism on this coarse interface, in which the operator implements DSO assignment based on the reported slots and charges a capacity shadow price as a toll, and evaluate its performance. We prove that both the worst-case misreporting gain and the expected efficiency loss decrease quadratically in the slot width. The efficiency loss decays in this way under binding capacity, while the worst-case misreporting gain requires an additional condition on the preferred arrival time distribution and the schedule cost function. Analyzing the no-toll case, we find that the misreporting incentive persists, however finely the slots are refined, indicating that the toll also serves to elicit truthful reports. Numerical experiments support these theoretical results and show that they continue to hold in parameter regions outside the sufficient conditions.

We use perfect matchings and labelled Fano planes to construct and study the $2025$ orthogonal bases of positive roots in the $E_8$ root system. The set of these bases forms a highly structured, Bruhat-like graded poset $(\Omega, \leq_Q)$ whose rank function can be computed from the cardinalities of so-called generalized Rothe diagrams. We give combinatorial characterizations of these diagrams in terms of matchings and Fano planes, and we explain how to compute the ranks of the elements of $\Omega$ using suitable combinatorial statistics such as the weights of perfect matchings. We establish simple formulas for the rank generating functions of $\Omega$ and of its 50 congruence classes under a natural order congruence relation. Our derivation of the generating functions contains some intermediate results on general perfect matchings and labelled Fano planes that can be stated without mentioning root systems and may be of independent interest.

In this paper we consider the friendship-bias of the vertices in an infinite rooted Galton–Watson tree. The friendship-bias of a vertex is the difference between the average degree of the neighbours of the vertex and the degree of the vertex itself. A vertex is said to be of type $\chi \in S$, with $S = \{-,0,+\}$, when its friendship-bias is, respectively, strictly negative, zero or strictly positive. We consider the fractions $f_l^\chi$ of vertices of type $\chi \in S$ along a random downward path up to branching depth $l \in \mathbb{N}$ and derive a large deviation principle (LDP) for the triple $(f_l^\chi)_{\chi \in S}$ as $l\to\infty$. The branching depth of a vertex counts the number of branchings that occur along the path that connects the vertex to the root of the tree. The rate in the LDP is $l$, while the rate function in the LDP is identified in terms of a variational formula minimising a relative entropy under a linear constraint. We focus on the case of binary branching, for which the rate function is already quite involved. We identify the qualitative properties of the rate function and show how it can be computed numerically. We briefly indicate how to proceed for more general branching and for vertex types along a tree consisting of a finite number of random downward paths. Our paper is the first to consider large deviations of vertex types.

To address computational intensity and crisp weight limitations in the alpha-cut intervals based Fuzzy Best-Worst Method (alpha-FBWM), this research proposes the Linear alpha-cut intervals based Fuzzy Best-Worst Method (Linear alpha-FBWM). By reformulating the non-linear optimization into a single linear programming model, the framework directly determines criteria weights as Triangular Fuzzy Numbers (TFNs) to retain uncertainty. To evaluate weight alignment with decision-makers' initial fuzzy preferences, an Ordinal Preference Violation (OPV) metric is introduced based on the prominence effect. Numerical examples demonstrate that Linear alpha-FBWM minimizes logical violations and matches or outperforms the original alpha-FBWM. To efficiently handle large-scale datasets, we extend this into the Linear alpha-cut intervals based Parsimonious Fuzzy Best-Worst Method (Linear alpha-PFBWM), embedding the linear formulation. This model allows initial alternative ratings as fuzzy numbers and computes non-reference alternative priorities via fuzzy interpolation without early defuzzification. The framework is validated using a literature example and a real-world warehouse selection case study involving 20 alternatives across Gujarat for a multinational paint firm. The integrated approach reduced the required expert evaluations from 185 to 20 initial ratings and 35 pairwise comparisons. This 81.08% reduction in pairwise comparisons proves the framework to be an efficient, reliable, and scalable tool for complex industrial decision-making.

Integrated sensing and communication (ISAC) in cell-free (CF) massive multi-user multiple-input multiple-output (MU-MIMO) system is a promising architecture for high-rate communications and high-accuracy multi-target sensing. However, centralized coordination among distributed access points (APs) incurs substantial fronthaul overhead and computation complexity. This paper proposes a low-complexity hybrid precoding framework for CF massive MU-MIMO ISAC systems with partially-connected architectures at the APs. By applying hybrid architecture at the APs, the proposed framework converts the original high-dimensional channel information into a low-dimensional effective channel, enabling digital precoding over the compressed channel domain and thereby substantially reducing both fronthaul overhead and baseband computational complexity. We formulate the joint hybrid precoding design as an ergodic sum-rate (ESR) maximization problem with position error bound (PEB) constraints to ensure multi-target sensing accuracy. An efficient alternating optimization (AO)-based solver is then developed, where the PEB constraint is reformulated into tractable convex constraints, while the digital-domain optimization is carried out over the reduced-dimensional effective channel and the analog precoding is refined on the constant-modulus manifold. For dynamic user topology, we further propose multi-branch (MB) rate-splitting (RS) minimum mean-square-error Tomlinson-Harashima precoding (MMSE-THP) update algorithm that combines multi-branch ordering with recursive MMSE-THP matrix updates, enabling common and private digital precodings to be refreshed without repeated full matrix recomputation. Simulation results demonstrate that the proposed scheme achieves high ESR and accurate multi-target sensing while reducing computational complexity by 87.02% compared with conventional baselines.

Let $G$ be a connected graph, and let $b\geq2$ be an even integer. The distance spectral radius of $G$ is denoted by $\mu(G)$. An $H_b$-factor of $G$ is a spanning subgraph $F$ of $G$ with $d_F(v)\in\{1,3,5,\ldots,b-1,b\}$ for any $v\in V(G)$, where $d_F(v)$ is the degree of $v$ in $F$. Lu and Wang provided a sufficient condition with respect to the number of odd components in $G-S$ for a connected graph $G$ of even order to contain an $H_b$-factor, where $S$ is a vertex subset of $G$ [H. Lu, D. Wang, On Cui-Kano's characterization problem on graph factors, J. Graph Theory 74 (2013) 335–343]. In this paper, motivated by Lu and Wang's above result, we establish an upper bound on the distance spectral radius $\mu(G)$ of a connected graph $G$ to guarantee that $G$ contains an $H_b$-factor.

The essence of the density topology lies in the family of Lebesgue measurable sets where each point of a set is a density point of that set. The motivation of this work is to investigate the family of measurable sets for which, at every point within a set belonging to this family, the upper density of that set is positive. We obtain a strong generalized topology, and its essential properties are demonstrated in comparison with those of the classical density topology.

We study the space of splines $\mathcal{S}^{\mathbf{r}}(\Sigma^\mathscr{A})$ where ${\mathbf{r}}$ denotes a smoothness distribution and $\Sigma^\mathscr{A}$ is the fan of a central hyperplane arrangement $\mathscr{A}$ in $\mathbb{R}^3$. This is the first step in the analysis of splines on three-dimensional cross-cut partitions, which naturally generalize planar cross-cut partitions. We show that the Hilbert function of $\mathcal{S}^{\mathbf{r}}(\Sigma^\mathscr{A})$ is bounded by an expression that involves the dimensions of specific Koszul homology modules constructed from the defining equations of the hyperplane arrangement $\mathscr{A}$ and the smoothness distribution function. By exploiting this connection with Koszul homology, we are able to: 1) compute the dimension of the spline space in high degrees, 2) compute all values of the dimension of the spline space if $\mathscr{A}$ is generic with five or fewer hyperplanes, and 3) compute the Hilbert function of the spline space if $\mathscr{A}$ is a generic arrangement with sufficiently many hyperplanes and ${\mathbf{r}}$ is a constant distribution. As an application of our methods, we compute $\dim \mathcal{S}^0_d(\Sigma^\mathscr{A})$ and $\dim \mathcal{S}^1_d(\Sigma^\mathscr{A})$ for all values of $d$ when $\mathscr{A}$ is a generic arrangement.

This paper presents an axiomatization of Ludwig von Mises' praxeology in many-sorted first-order logic, isolating the foundational layer. We introduce a formal language with five sorts (Actors, Actions, Ends, Things, Times) and six primitive relations (Acts, Avail, EndOf, Use, a preference order, and a time order), together with a base axiom system organized into three layers: the structure of action itself, the actor's preference order together with its demonstration in choice, and material scarcity. The base system captures purposeful action in its bare praxeological form. Working entirely within the base system we derive the core classical Misesian propositions as Hilbert-style theorems: the asymmetry of demonstrated preference, the existence of opportunity cost, the structural scarcity of time, the subjectivity of opportunity cost, the law of diminishing marginal utility, and the increasing marginal disutility of labor. Where a theorem requires structure beyond the praxeological core, as with diminishing marginal utility, the additional premises are made explicit; identifying these hidden premises is one of the methodological payoffs of the approach. A self-contained Lean 4 companion encodes the language as Lean 4 type classes and constructs a concrete infinite-time Robinson Crusoe model whose acceptance by the type-checker is a constructive consistency proof of the full base theory.

Relative symplectic cohomology is an invariant of compact subsets of a closed symplectic manifold, introduced by Varolgunes. There are many examples of computations of this invariant over the Novikov field, but the collection of computed examples over the Novikov ring is still quite limited. One reason for this is that such computations require determining the relevant Floer complexes for Hamiltonians that are not necessarily $C^2$-small Morse functions. In this work, we present a computation of relative symplectic cohomology over the Novikov ring for balls and their complements in $\mathbb{C}P^n$. Our computation relies on explicit descriptions of Floer complexes, in the Morse–Bott setting with cascades, for J-shaped Hamiltonians on $\mathbb{C}P^n$. This allows us to deduce new estimates for the stable displacement energy of the boundaries of balls in $\mathbb{C}P^n$.

For datasets with unknown but stationary serial dependence, a robust long run variance estimator is essential to handle diverse scenarios. Spectral variance estimators are commonly used but tend to exhibit significant negative bias in the presence of positive correlation. To overcome this, zero lugsail estimators have been introduced, offering zero asymptotic bias regardless of the correlation structure. However, there are currently no guidelines for selecting the optimal bandwidth for lugsail estimators, a critical component in the estimation process. We propose an inference optimal bandwidth rule for lugsail estimators, based on nonstandard fixed-smoothing limiting distributions developed in our study. This approach significantly improves bias correction, accounts for variability, and provides an estimator optimized for robust inference. Our theoretical findings are supported by a simulation study.

We ask whether knotting can be recognised using persistent homology. Starting from a point-cloud representation of a curve, we compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score to these cycles. Motivated by proteins but tested more broadly, the method reveals systematic differences between knotted and unknotted structures in both protein families and synthetic examples. This suggests that knotting leaves a detectable persistent-homology-based signature.

Adaptive optimizers combining preconditioning, momentum, and weight decay (Adam and AdamW) are, under Polyak-Ruppert averaging, candidate engines for one-pass inference. Does the averaged iterate keep the classical Polyak-Ruppert central limit theorem (CLT), with sandwich covariance $H^{-1}SH^{-1}$ (Hessian $H$, gradient covariance $S$), under momentum and non-convergent preconditioning? The preconditioner-only analysis does not carry over: with momentum the canonical decomposition collapses to a tautology. Treating the augmented state (iterate, momentum buffer) as a time-varying linear stochastic approximation (SA), we prove (under local stabilization) positive drift stability, a non-autonomous Polyak-Ruppert CLT, and a projection identity. The upshot: the iterate-marginal covariance is exactly the plain stochastic gradient descent (SGD) sandwich $H^{-1}SH^{-1}$, so the adaptivity is asymptotically invisible. This holds for SA-Adam (sub-linearly vanishing momentum gain, $\gamma\in(\alpha,1)$; the sub-linear regime is essential), not constant-$\beta$ deployed Adam. Coupled $L_2$ weight decay yields the ridge-penalized sandwich, extending one-pass inference to regularized problems.

We develop a non-overlapping domain decomposition method for the numerical solution of the Stokes-Biot model of fluid-poroelastic structure interaction in a mixed form. The model is based on a velocity-pressure formulation for the free fluid, a three-field stress-displacement-rotation formulation with weakly symmetric stress for the solid deformation, and a Darcy velocity-pressure formulation for the fluid in the poroelastic media. Mass conservation, balance of stress, and the Beavers-Joseph-Saffman slip with friction condition are imposed on the interface. The interface conditions are incorporated through Lagrange multipliers modeling the traces of the displacement and the Darcy pressure. The system is discretized using stable mixed finite element spaces for Stokes flow, elasticity, and Darcy flow. The domain is decomposed into a union of subdomains of either Stokes or Biot type with three types of interfaces: Stokes-Stokes, Biot-Biot, and Stokes-Biot. On the Stokes-Stokes interfaces, a normal stress Lagrange multiplier is introduced to impose weakly velocity continuity, while the Biot-Biot and Stokes-Biot interfaces are equipped with displacement and pressure Lagrange multipliers to impose weakly continuity of normal stress and normal velocity, respectively. The global problem is reduced via Schur complement to an interface problem for the Lagrange multipliers, which is solved by GMRES. Each iteration requires the solution of local Stokes or Biot problems, which can be performed in parallel. We show that the resulting interface operator is positive definite and analyze the convergence of the GMRES iteration through fields-of-value analysis. Numerical experiments are presented to illustrate the performance of the method.

The stationary hydrogen atom has Coulomb degeneracy across orbital levels, whereas the Aufbau/Madelung ordering is an empirical, many-electron rule established in atomic physics. We examine the hydrogen atom through a regularized de Broglie–Bohm representation, in which stationary amplitude current constraints generate separable Sturm–Liouville branches. In this formulation, the radial, orbital, and magnetic sectors acquire canonical Langer-like inverse square corrections. The modified boundary value problems allow analytical solutions and produce a hydrogen-like spectrum with regularized radial and angular indices. Consequently, radial Coulomb quantization acquires an orbital dependent shift, lifting the Coulomb degeneracy and producing a spectral ordering that follows the Aufbau/Madelung sequence. On this basis, we construct the ordering of the regularized de Broglie–Bohm states and show that the spectral structure retains the standard degenerate Rydberg sequence in the l=0 sector. The separated amplitudes are represented by generalized special function branches, including the associated Laguerre, Legendre, and Bessel functions with non-integral parameters arising from regularized separation. Therefore, the treatment is intended as an analytical examination of spectral ordering in a regularized one center Coulomb problem rather than as a replacement for the many electron atomic structure theory. Keywords: de Broglie–Bohm representation; Coulomb spectrum; canonical regularization; Langer correction; Sturm–Liouville equations; Aufbau principle; Madelung ordering; associated Legendre functions; associated Laguerre functions; Bessel functions.

A general framework is developed for deriving sharp inequalities on simple graphs from majorization and Schur-convexity. After establishing majorization relations between the spectrum of an arbitrary graph and the spectra of the complete, complete bipartite, and matching graphs, it is shown that every positive Schur-convex spectral functional yields several sharp inequalities relating $\lambda_1$, $|\lambda_n|$, and $\|G\|_\ast$. This reduces the problem of proving graph inequalities to the choice of a suitable Schur-convex function. This optimization problem is then studied within the family of random vector norms, whose moment and cumulant expansions connect the framework to the numbers of closed walks. This yields new sharp results, recovers classical inequalities from a unified viewpoint, and produces further bounds in settings such as triangle-free and square-free graphs.

We consider mean curvature flow with free boundary through cylindrical or half-cylindrical singularities, namely singularities of the types $\mathbb{R}^k\times S^{n-k}$, $\mathbb{R}^k_+\times S^{n-k}$ or $\mathbb{R}^k\times S^{n-k}_+$. Using the foundational results for free boundary Brakke flows by Edelen and the first author, and the recent classification of ancient asymptotically cylindrical flows by Bamler-Lai, we prove that all these singularities have a mean-convex neighborhood. Moreover, generalizing work of Hershkovits-White to the free boundary setting we show that the free boundary level set flow is nonfattening provided all singularities have a mean-convex neighborhood. We conclude that free boundary flow through singularities is well-posed as long as all singularities are of cylindrical or half-cylindrical type.

We provide several new characterizations of Property A for bounded degree graphs. In particular, we show that $(X,d)$ has Property A if and only if there is a proper gauge $\omega$ such that the Lipschitz free space $\operatorname{LF}(X,\omega\circ d)$ is isomorphic to $\ell_1$. As a consequence, all finitely generated groups with Property A admit proper uniformly Lipschitz affine actions on $\ell_1$. Moreover, for groups with finite Nagata dimension, we obtain actions with compression exponent 1. This result applies to higher rank lattices, such as $\operatorname{SL}(3,\mathbb{Z})$. We also show that a countable discrete group coarsely embeds into $L_1$ if and only if it admits a proper uniformly Lipschitz affine action on a subspace of $L_1$.

Mazur's knot exterior admits a geometric description using a single regular ideal octahedron. The resulting hyperbolic structure is closely related to the Whitehead link exterior through Adams' theorem on thrice-punctured spheres. The same octahedral framework applies to the family of Jester manifolds introduced by Sparks. Using hyperbolic geometry, Thurston's hyperbolic Dehn filling theorem, and Mostow–Prasad rigidity, we prove that Mazur and Jester boundary 3-manifolds are pairwise distinct up to finite ambiguity. Using recent results on systolic geodesics, we remove the remaining finite ambiguity and prove that the boundaries of all Mazur and Jester manifolds are pairwise nonhomeomorphic, regardless of orientation. Consequently, the corresponding compact, contractible $4$-manifolds are pairwise nonhomeomorphic.

We associate a profinite group to every group \(G\) acting on a set \(\Omega\) with finite orbits. For each finite \(G\)-stable subset \(A\subseteq\Omega\), let \(G_A\leq\operatorname{Sym}(A)\) be the induced finite permutation group. The groups \(G_A\), with the natural restriction maps, form an inverse system, and we define $\Gamma_\Omega:=\varprojlim_A G_A$. We show that \(\Gamma_\Omega\) acts naturally on \(\Omega\) and is canonically topologically isomorphic to the closure of the image of \(G\) in \(\operatorname{Sym}(\Omega)\), endowed with the topology of pointwise convergence. We introduce the finite-level exactness property \(\textup{FLEP}\), under which subgroups of \(\Gamma_\Omega\) are recovered up to closure from their fixed-point sets, and closed subgroups are recovered exactly. We prove several equivalent formulations of \(\textup{FLEP}\). Under this condition, the fixed-point set construction gives an inclusion-reversing bijection between closed subgroups of \(\Gamma_\Omega\) and the fixed subsets of \(\Omega\) arising from closed subgroups. We apply the theory in two directions. First, every profinite group \(\Gamma\) is recovered from its normal finite-quotient action on $\coprod_{N}\Gamma/N$, where \(N\) ranges over the open normal subgroups of \(\Gamma\). For this action, \(\textup{FLEP}\) holds precisely when every finite quotient \(\Gamma/N\), with \(N\) open and normal, is a Dedekind group. Second, if \(G\leq\Aut(E)\) acts on a field \(E\) with finite orbits and \(F=E^G\), then \(E/F\) is Galois and the construction yields a canonical topological isomorphism $\Gamma_E \cong_{\mathrm{top}} \operatorname{Gal}(E/F)$, where \(\operatorname{Gal}(E/F)\) has the Krull topology. Thus the Krull Galois group is recovered from finite-orbit data.

Motivated by the optimization of bounded binary black-box functions, we study the problem of learning polynomial surrogates over the Boolean hypercube. To ensure that optimizing the surrogate yields good solutions for the underlying objective, we require uniform $L_\infty$-error guarantees rather than the usual $L_2$-type guarantees. We characterize the minimax sample complexity of uniform estimation under subgaussian noise for two classes of bounded polynomials. First, for polynomials of degree at most $d$ on $n$ variables, the sample complexity scales as $n^{d+1}$. Second, for $s$-sparse Fourier-Walsh polynomials with $s \leq n$, it scales as $ns^2$. These rates differ structurally from the noiseless setting, where uniform exact recovery scales as $n^d$ and $ns$, respectively. Our lower bounds hold even for arbitrary adaptive learners, showing that the additional factors are intrinsic to the noisy cases. Standard Fourier-analysis tools for the $L_2$-norm do not naturally extend to the $L_\infty$-setting in a way that yields uniform guarantees. Our proofs overcome this difficulty by relying on suitably chosen auxiliary norms that serve as proxies for controlling the $L_\infty$-error. Together, our results provide a tight characterization of the sample complexity of learning optimization-safe polynomial surrogates.

We study Schubert varieties of the affine Grassmannian for the general linear supergroup $GL_{n|m}$. An explicit computational study is conducted in low-dimensional cases, namely for dimensions $n|m = 1|1$ and $2|1$. We describe the supervariety structures that arise in these settings, providing coordinate descriptions, equations, and morphisms.

Autonomous capture of a tumbling object with a space manipulator requires trajectory generation under real-time onboard constraints. However, coupled bus-arm dynamics, rotating safety geometry, and visibility requirements make the underlying optimization severely nonconvex, hence expensive to solve and sensitive to initialization. This paper develops a causal-transformer warm-start for the computationally dominant attitude-manipulator stage of a two-stage sequential convex programming (SCP) framework. Across 300 held-out scenarios evaluated on embedded hardware, the learned warm-start reduces the iterations of the second-stage SCP by up to 32% and runtime by 16% while maintaining numerical feasibility and matching the final control cost. Furthermore, linear and flow matching action decoders are compared across action-chunk lengths and training dataset sizes, where the flow matching decoder exhibits stronger training-data efficiency than the linear decoder. When SCP is terminated based on feasibility, the runtime is further reduced while the learned prior suppresses the severe high-cost tail observed with heuristic initialization. These results demonstrate that learned warm-starts can reduce onboard optimization cost while steering SCP toward basins containing feasible, low-cost solutions.

In this paper we study the structure of the $\infty$-category of spectral Lie algebras. We show that this $\infty$-category admits an interesting symmetric monoidal structure, defined by an analog of the smash product of pointed spaces, and that the free Lie algebra functor $\mathrm{Sp} \to \mathrm{Lie}(\mathrm{Sp})$ is symmetric monoidal with respect to it. Moreover, this property of the free functor essentially characterizes the spectral Lie operad (among nonunital operads in spectra). This result may be thought of as Koszul dual to the more familiar fact that the free commutative algebra functor takes direct sums to tensor products. One of the key ideas is that the $\infty$-category of spectral Lie algebras behaves in many ways like the $\infty$-category of pointed spaces. More precisely, we deduce structural facts about spectral Lie algebras from familiar statements about spaces by differentiating, in the sense of Goodwillie calculus. The tool to do this is the highly structured generalization of Arone-Ching's chain rule established by Blans-Blom. Numerous other features of spectral Lie algebras follow as well, such as a version of Mather's second cube lemma, the relation between the James construction and loop-suspensions, the Hilton-Milnor splitting, and a version of the EHP sequence.

In this paper we study the class of graphs without cycles of size 4 and independent sets of size 4 as induced subgraphs: $\mathop{Free}\{C_4, 4K_1\}$. This is one of the three minimal minimal open cases for the complexity of the colouring problem when restricted to classes defined by excluding induced subgraphs of order 4. We investigate the clique width of some subclasses of $\mathop{Free}\{C_4, 4K_1\}$. We introduce a new framework: the $(k,l,m)$-decomposition and prove that if all the graphs of a class $\cal G$ are $(k,l,m)$-decomposable, then graphs in $\cal G$ have bounded clique width. We give a few examples of such class, found with the help of a program we designed. We also show, for any graph $G \in \mathop{Free}\{C_4, 4K_1\}$ that is 3 cliques coverable, an infinite family in $\mathop{Free}\{C_4, 4K_1\}$ of supergraphs of $G$ which have unbounded clique width.

The irrationality of e+pi remains open, despite the separate transcendence of e and pi. This paper studies the problem from the viewpoint of finite irrationality certificates and gives a bounded no-go audit for low-complexity Apéry-type proof mechanisms. First, we prove exact equivalences between the hypothesis e+pi in Q and eventual factorial-arithmetic phenomena: a ceiling recurrence, a factorial-Cantor digit condition, and a divisibility criterion. These criteria identify what rationality would force, while showing why tail conditions are not finite obstructions. Second, we formulate an Apéry-type certificate framework based on integer linear forms L_n = A_n(e+pi)+B_n with A_n,B_n in Z, L_n nonzero, and |L_n| tending to zero. A mixed integration-by-parts identity produces such forms from integer polynomials. We then audit several low-complexity constructions, including mixed Padé approximation, crossed separate approximations to e and pi, simple J-fractions, holonomic ansatzes, Rodrigues-type families, and an integer kernel-lattice search. The main contribution is a rigid boundary probe: no-go filters marking a tested zone where analytic smallness is destroyed by denominator clearing, coefficient growth, primitive reduction, or continued-fraction shadows. In the final kernel-lattice audit, 145 raw candidates reduce to 133 primitive records; the best signals are dominated by continued-fraction shadows, while non-CF candidates do not form a degree-continuing family. Thus, within the tested low-complexity families, no non-circular Apéry-type mechanism for e+pi is found.

In this article, we investigate the construction of linear codes over a finite ring $\mathcal{S}$, where $\mathcal{S}$ is taken to be an extension of a commutative non-unital ring $I$ of order $p^2$. Our approach is based on the defining set method. The defining sets considered in this work are derived from general simplicial complexes that may contain multiple maximal elements. We determine the parameters of these codes over $\mathcal{S}$ and study their Gray images. We also study the corresponding subfield-like codes. We show that these Gray image codes and subfield-like codes produce several families of divisible codes. Furthermore, we establish sufficient conditions under which these codes are minimal, optimal, and self-orthogonal. As applications of our results, we obtain several families of projective few-weight codes, and locally recoverable codes with small locality. We also study the minimal access structures of secret-sharing schemes associated with the duals of these minimal codes. Moreover, we construct several families of strongly regular graphs from projective two-weight codes and determine their parameters explicitly.

In survival analysis, dependent censoring poses significant challenges in accurately estimating model parameters and survival functions. This study introduces a novel framework leveraging Extended Generalized Marshall-Olkin (EGMO) models to address dependent censoring mechanisms. Geometric optimization techniques are employed to develop efficient estimation procedures that capture dependencies between failure and censoring times. We establish their asymptotic properties. Simulation studies and real data applications illustrate the method's robustness and effectiveness.

Perfect resource placements in dense Eisenstein–Jacobi (EJ) networks partition the network into hexagonal radius-$t$ service cells. This paper studies local repair of such placements after resource failures. For one failed resource, we prove that one replacement cannot cover the failed hexagon and two always suffice, giving $\rho_{\mathrm{EJ}}(t)=2$ for all $t\ge1$. Among minimum-size repairs, the sharp minimum-overlap formula $\Omega_{\mathrm{EJ}}(t)=t^2$ follows from the three-strip geometry of EJ balls. For two failed resources, independent repair gives a four-replacement upper bound, but unlike the Gaussian case EJ repair is not always additive: two infinite neighboring displacement families admit three-replacement repairs, proved optimal by a two-ball impossibility argument. Additive behavior is established algebraically via endpoint-rigidity and diagonal-corridor theorems. For $q$ failed resources, independent canonical repair gives a universal $2q$ upper bound, exact when failed cells are pairwise more than $4t$ apart. Dense cluster subadditivity is proved for infinite four-fault and six-fault families with exact repair numbers four and five, giving savings of four and seven over independent repair. An exact inclusion–exclusion identity governs repeated coverage for arbitrary multi-fault repairs. An audit over 19,400 instances confirms widespread subadditivity. EJ local repair is structurally distinct from the Gaussian case: the one-fault overlap is quadratic, two-fault repair can be non-additive, and clustered repairs reuse replacement balls across multiple failed cells.