Papers

Two families of polytopes underlie the combinatorics of associative operations. Associahedra and multiplihedra respectively capture the information in the operation itself and in the morphisms that respect that operation. The first applications of these polytopes, from Stasheff, were for modeling homotopy associative spaces and their homotopy homomorphisms. Later, lax and weak higher categories used both as the shapes of commuting diagrams. More recently, the associahedra have been generalized to versions based on graphs, and then to acyclic versions based on posets: the acyclonestohedra. Meanwhile, the associated multiplihedra have also been generalized to graph associahedra and generalized permutohedra. Here we complete that picture: we use the graph multiplihedra to define and realize new acyclic poset multiplihedra for the acyclic poset associahedra. Quotients of these are also studied, concluding with the conjectured interval polytopes of the order polytopes of posets.

Distributed learning systems typically assume that local data is already available at clients with fixed quality, while in practice, data is sensed through imperfect physical processes whose quality depends on modality, resolution, sensing power, and sample size. We model sensing noise as a structured, modality-dependent covariance and derive a non-convex learning convergence bound whose irreducible sensing floor is governed by the alignment between the modality noise covariance and the loss-sensitivity geometry. Thus, the optimal modality minimizes this noise-gradient alignment rather than total noise power alone. The analysis further yields a sensor-hardware achievability bound for epsilon-stationarity and a hardware-saturation threshold on the accumulated dataset size. We jointly optimize modality, resolution, power, and sample count and demonstrate the performance gain through simulations.

Thin current sheets are central small-scale structures in electron magnetohydrodynamics (EMHD), closely associated with energy dissipation and fast magnetic reconnection at electron scales. We study their formation numerically in a $2\frac{1}{2}$-dimensional EMHD system on a periodic domain with split fractional dissipation, where the magnetic potential and the vertical magnetic component are damped separately. The local theory is governed by a symmetric combined damping balance, but the numerical onset of small-scale growth need not follow this symmetry. A scaling analysis identifies the out-of-plane current as the primary concentration observable, since it is regularized only through the magnetic-potential equation. Using a validated Fourier pseudospectral exponential time-differencing solver with resolution-controlled diagnostics, we find a clear decay/concentration dichotomy. The onset boundary is markedly asymmetric: current-sheet formation appears to be controlled mainly by damping of the magnetic potential, rather than by the combined damping strength. The analyticity strip collapses to the grid scale, the concentration sharpens under grid refinement, and the observed growth is consistent with an energy-critical self-similar rate, with exponent near three. These experiments indicate that magnetic-potential damping is the apparent binding constraint for current-sheet concentration, refining the symmetric sum picture.

Barry introduced a central transform of integer sequences and proposed four conjectures concerning the Hankel transforms of central transform of four rational families. We prove these four conjectures. The proofs are unified within a common algebraic framework: we interpret the Hankel determinants as Gram determinants and use a basis of shifted monic Chebyshev polynomials to reveal the finite-band structure of the associated Gram matrices.

For a (molecular) graph $G$ and any real number $\alpha\ne 0$ , the zero-order general Randić index , denote by $^0R_\alpha$, is defined by the following equation: \begin{align*} {^0R_\alpha} (G) =\sum_{v\in G}d_G (v) ^{\alpha} (\alpha \in \mathbb{R}-\left\{0\right\}) . \end{align*} The deficiency of $G$, denoted by $def(G)$, is equal to the cardinality of vertices which are not covered by a maximum matching in $G$. A graph G is called $l$-deficient if $def(G)\le l$. In this paper, we use this index to give sufficient conditions for a connected graph, bipartite graph and a balanced bipartite graph $G$ to satisfy the $l$-deficient property, and show that none of these conditions can be dropped. We will also use these results to enhance and generalise the results that already obtained by M. An and K. C. Das in 2018 and G. Su et al. in 2022.

In this article we establish global a priori estimates on the solution of a generic rough differential equation driven by an $\alpha$-Hölder path covering the full range of regularity $\alpha\in(0,1)$, under the hypothesis of Lipschitz continuity (but neither boundedness nor coercivity) of specific combinations of the coefficients and their derivatives, called elementary differentials. Our main tool is a closed formula for the remainder of a generalised Taylor expansion in terms of products of gradients of the elementary differentials, which allows to take advantage of cancellations in an optimal way.

Minimum-variance portfolios are well known to be highly sensitive to covariance estimation error. In this paper, we show that by imposing a block diagonal correlation structure, we can derive closed-form expressions for long-only minimum-variance portfolios that make this fragility explicit. These analytical solutions reveal that fragility is driven by threshold effects arising from the interaction between correlation structure and the assets' volatilities. Motivated by the latter insight, we propose robustification approaches that can be interpreted as structured shrinkage schemes that selectively attenuate unstable coupling while preserving the dominant risk structure. Unlike global shrinkage techniques, the proposed corrections are analytically grounded, require minimal tuning, and remain closely aligned with the minimum-variance solution. The framework extends naturally to general covariance matrices through clustering-based approximations. Empirical results on controlled simulations highlight a clear regime dependence. In homogeneous volatility settings, correlation-oblivious shrinkage achieves the best trade-off between risk and stability. In contrast, under heterogeneous volatility, correlation-aware shrinkage performs best by inducing sparsity and avoiding exposure to high-risk assets. Across regimes, the proposed methods consistently reduce out-of-sample variance and turnover relative to classical and clustering-based benchmarks, providing a principled and practical approach to robust portfolio construction.

Let $f$ be the density of the Chernoff random variable $\mathrm{argmax}_{t\in\mathbb{R}}\{W(t)-t^2\}$, where $W$ is a two-sided Brownian motion. This note proves the conjecture of Balabdaoui and Wellner (2014) that $f$ is strongly log-concave. The proof was generated in its entirety by GPT-5.6 Sol.

The deep integration of diverse services in sixth-generation (6G) networks poses significant challenges to conventional task-agnostic channels, often resulting in performance conflicts. To resolve these bottlenecks, this article introduces a physical-layer computing paradigm enabled by stacked intelligent metasurfaces (SIMs), transforming the wireless environment from a passive medium into a programmable signal processor. Specifically, we establish a unified framework to map high-level service requirements directly to wave-domain synthesis. We then investigate the fusion of diverse services, demonstrating how the deep computational architecture of SIMs resolves resource conflicts in integrated sensing and communication (ISAC) and integrated communication and computation (ICC) scenarios. Furthermore, we critically analyze fundamental challenges, including diffractive channel modeling and inverse task-to-phase mapping, while validating through numerical results that this approach elevates the system from simple coexistence to true service symbiosis. Finally, we discuss key research directions to pave the way for service-native 6G architectures.

We study nonlinear parabolic equations with delayed diffusion terms governed by finite signed measure kernels. The atom of the kernel at the origin is absorbed into the present-time operator, while the remaining part is treated as a residual delay kernel. Under structural assumptions on the effective present-time operators and a pathwise coercivity condition for the total memory operator, we prove the existence and uniqueness of weak solutions and their stability under weak-star convergence of the kernels. The stability result covers collapsing delayed atoms, whose mass is transferred to the present-time diffusion coefficient in the limit. We verify the assumptions for p-Laplacian type examples, including separated kernels and a regularized class of kernels reaching the origin.

The paper defines a certain function \(\chi(n)\) on the set of odd integers \(n \ge 3\). The integrality property of this function is investigated. The main result establishes that the function is integer-valued if and only if \(n\) is a prime number or an odd Giuga number. Thus, the integrality of \(\chi(n)\) serves as a single criterion unifying three classes: prime numbers, Carmichael numbers, and odd Giuga numbers. The proof relies exclusively on the von Staudt–Clausen theorem on the denominators of Bernoulli numbers and Korselt's criterion for Carmichael numbers, which makes the exposition elementary. A connection is proven between prime numbers \(p\) for which \(\chi(p)=p-1\) and Novák–Carmichael numbers. It is proven that the primes satisfying this criterion form a sequence with the identifier A337119 in the Online Encyclopedia of Integer Sequences. For the classes \(P_\mu\) of primes with a fixed value \(\chi(p)=\mu\), a characterization is obtained in terms of the exponents of the prime divisors of the numerator of the corresponding Bernoulli number \(B_{p-1}\). For some \(\mu\), explicit congruences are provided. The work is accessible to a wide readership due to its minimal use of advanced mathematical apparatus.

Let $G$ be a reductive affine algebraic group over $\mathbb C$ with toral identity component $T$ and finite component group $K$. We describe finite indecomposable semisimple module categories over $\Rep(G)$ using equivariant data that incorporate the extension class. We prove that every fiber functor is classical and determine the image of its Hopf $2$-cocycle support in $K$ and its intersection with $T$. This yields a minimality criterion involving a finite stabilizer cocycle and an alternating bicharacter on a sublattice of $\widehat T$ of index at most $|K|$. When $\dim T=1$, every Hopf $2$-cocycle for $G$ has finite support of order at most $|K|^2$. We compute all gauge classes for the split and nonsplit extensions with component group $\mathbb{Z}/2\mathbb{Z}$ and inversion action, and compare the support formulas with the Künneth coordinates for commutative direct products. Finally, for an arbitrary reductive group $G$, a minimal Hopf $2$-cocycle in a regular deformation of the trivial Hopf $2$-cocycle forces $G$ to be a torus. This uses locality and rigidity of algebraic matrix coefficients on the semisimple quotient of $G$.

A Schubert variety $X_0$ on a rational homogenous space $X=G/P$ is said to be homologically rigid, if any subvariety $Z$ on $X$ representing the same homology class with $X_0$ must satisfy $Z=g\cdot X_0$ for some $g\in{\rm Aut_0}(X)$. We say $X_0$ is Schur rigid, if furthermore any subvariety $Z$ on $X$ whose homology class is a multiple $r$ of that of $X_0$ must satisfy $Z=g_1\cdot X_0+\cdots+g_r\cdot X_0$ for some $g_1,\cdots ,g_r\in{\rm Aut_0}(X)$. Homological rigidity and Schur rigidity of Schubert varieties in rational homogeneous spaces of Picard number one have been well studied in extensive literature. In this paper, we study both rigidity problems of Schubert varieties in rational homogeneous spaces of higher Picard numbers. We show that in the long root cases, including all cases when $G$ is of type $ADE$, smooth Schubert varieties have homological rigidity. Besides, we give the complete list of Schubert varieties of subdiagram type with/without homological rigidity. Furthermore, for a Schubert variety $X_0$ of subdiagram type, we show that it has Schur rigidity in long root cases unless $X_0$ admits a fiber bundle structure over the projective space.

Let $F_0,\dots,F_{n-1}$ be Borel functions on a standard Borel space $X$ and let $G_F$ be the graph they generate. We prove that for every Borel probability measure $\mu$ on $X$ there are a forward-invariant $\mu$-conull Borel set $A \subseteq X$ and a proper Borel $(2n+1)$-coloring of the restriction of $G_F$ to $A$; consequently $\chi_M(G_F) \le 2n+1$ in the total sense of Kechris and Marks. This answers positively, for every $n$ and with the optimal constant, the measure half of Problem 5.14 of the survey "Descriptive graph combinatorics" of Kechris and Marks – the measurable version of a question raised by Kechris, Solecki, and Todorcevic (Adv. Math. 141, 1999). No invariance, no local finiteness, and no finiteness of the Borel chromatic number are assumed. The argument is elementary and specific to measure; the corresponding Borel problem and the Baire-measurable half remain open.

Existing side-channel theories treat leakage as a holistic quantity $I(X;Y)$, without characterizing its internal structure. This paper studies the structural decomposability of encrypted-traffic side-channel leakage. Via the structural causal model $X\!\to\!Y_{\mathrm{size}}\!\to\!Y_{\mathrm{dir}}\!\to\!Y_{\mathrm{time}}$ and the mutual-information chain rule, total leakage is decomposed into three sequential increments for packet size, direction, and timing. Defenses are formalized as mechanism replacement by a strategy variable~$D$; coupling information $C_{\mathrm{size,dir}}=I(Y_{\mathrm{size}};Y_{\mathrm{dir}}\!\mid\!X)$ measures inter-dimensional dependence, and the Markov residual gives a testable condition for a single-dimension defense to sever downstream leakage. Causal efficacy~$\eta_d$ quantifies per-unit-cost suppression, and a Fisher-geometric approximation $I(X;Y)\approx\frac{1}{2\ln 2}\mathrm{Tr}(G\Sigma_\theta)$ holds under small perturbations. On the Wang dataset (95 websites), $Y_{\mathrm{dir}}$ dominates undefended leakage (0.637\,bits), while Tor's fixed 512-byte cells make $Y_{\mathrm{size}}$ degenerate; FRONT suppresses the direction term by 63%, yet its Markov residual of 0.021\,bits (95% CI $[0.016,0.027]$) shows it cannot sever timing leakage; $\eta_{\mathrm{dir}}=0.97$ vs. $\eta_{\mathrm{time}}\approx 0$ confirms FRONT's design intent. This yields a computable, structured leakage-accounting method for multi-dimensional joint defense design.

Guo, Kang and Zwaneveld conjectured that colouring the \((d+1)\)-clique blowup of a graph with defect \(d\) requires as many colours as colouring the graph properly. Norin and Steiner refuted this, proved that the two chromatic numbers differ by at most a factor of two, and asked for the exact constant; the best separation known was \(11/10\). We prove that the constant is exactly two for every fixed defect \(d\ge 2\). Every rational ratio in \([1,2)\) is attained, by graphs whose clique number and fractional chromatic number equal the defective parameter: the separation is invisible to the standard lower bounds. The engine is an exact transfer theorem: a weighted list-colouring obstruction spread over an arbitrary auxiliary graph yields prescribed defective and ordinary chromatic numbers; Kneser graphs then give sharpness. A 29-vertex graph with defective parameter four and chromatic number five answers Norin's Barbados question at the smallest possible defect and palette.

Black-hole thermodynamics has been reproduced with remarkable success from the BFSS matrix model, but the corresponding test in the closely related IKKT matrix model has awaited its recently proposed finite-temperature formulation, within which we investigate this correspondence. Introducing a cubic Myers term, we compute the complete one-loop thermodynamics – free energy, internal energy, entropy, and heat capacity – of a fuzzy $S^2$ background, and compare against the D0 and D2 black holes as the closest available reference points, since no gravitational dual is known for this specific background. At fixed Myers coupling, the leading entropy exhibits an $O(N^3)$ scaling rather than the $O(N^2)$ behavior characteristic of conventional black holes. Holding the physical radius fixed instead yields a genuine $O(N^2)$ scaling, showing that the power of $N$ depends on the large-$N$ prescription and is therefore not by itself an unambiguous test of a black-hole interpretation. We further find no intrinsic Hawking temperature – only a characteristic scale – and a heat capacity that is negative at low temperature but positive at high temperature. The absence of an intrinsic Hawking temperature, tied to the missing gravitational dual rather than to the choice of large-$N$ prescription, persists in either scaling limit. Rather than establishing a black-hole interpretation of the finite-temperature IKKT matrix model, these results provide a quantitative benchmark against which future analytical, numerical, and holographic investigations can be tested.

Given Hilbert space commuting operators $T, S \in \mcl(H)$, such that $T$ is $w$-hyponormal with $\ker T \subseteq \ker T^*$ and $S$ is normal operator. Let $\phi_{T, S} \in \mcl(\mcl(H))$ be the elementary operator defined by $\phi_{T, S} (X) = T X S^*-S X T^*$. In this paper, we show firstly that (1) $\ker \phi_{T, S} \subset \ker \phi_{T^*, S^*}$. (2) The range of $\phi_{T, S}$ is orthogonal to the kernel of $\phi_{T, S}$ ( $ \mcr(\phi_{T, S}) \perp \ker \phi_{T, S} $ ) if and only if $\ker T \cap \ker S=\{0\}$. Secondly, we will extend these results to the elementary operator $\Phi \in \mcl(\mcl(H))$ defined by $\;\Phi(X)=A X D-C X B$ where $[A, C]=[B, D]= 0$. Related orthogonality results for the elementary operator $\Phi$ are also given.

This expository article serves as a methodical guide, presenting a rigorous mathematical formulation of the D'Alembert-Lagrange principle and the derivation of the Lagrange equations for mechanical systems subject to ideal constraints. Designed for educational purposes, the paper establishes a clear geometric framework for the extended phase space, defining the space of virtual displacements and constraint reactions independently of their specific analytical representations. A central focus of this instructional text is the transition from the general equation of dynamics to the Lagrange equations of the second kind for holonomic systems. Specifically, we demonstrate that the Lagrange equations can be naturally and elegantly derived from considerations of the covariance of the variational derivative under the embedding mapping of the configuration manifold.

Let $\mathcal{X}$ be a finite set, let $p$ and $Q$ be strictly positive probability distributions on $\mathcal{X}$, and let $u\in\mathbb{R}^{\mathcal{X}}$ be a log-weight. Define $\Psi_p(u)=\log\sum_x p(x)e^{u(x)}$ and $\pi_{p,u}(x)=p(x)e^{u(x)-\Psi_p(u)}$. The variational free energy $F(Q;p,u)=D(Q\|p)-\mathbb{E}_Q[u]$ satisfies the exact identity $\Psi_p(u)=-F(Q;p,u)+D(Q\|\pi_{p,u})$, whose residual is the canonical divergence of the dually flat simplex. We prove a converse. Suppose $G(Q;p,u)=A(Q,p)+B(Q,u)$ is additively separable and its gap $\Psi_p(u)+G(Q;p,u)$ is nonnegative and vanishes whenever $Q=\pi_{p,u}$. Then $G=F$ and the gap equals $D(Q\|\pi_{p,u})$; $A$ and $B$ are unique up to a $Q$-dependent additive gauge. The gap is not assumed to be a divergence or to depend only on the target, and no continuity, measurability, differentiability, convexity, or membership in a prescribed divergence family is assumed. The proof bounds increments of a corrected accuracy term by centered log-moment-generating costs of order $k^{-2}$ per step. Telescoping $k$ equal exponential tilts gives an $O(k^{-1})$ bound and forces every increment to vanish. Thus target factorization and strictness are conclusions. Counterexamples show that separability, nonnegativity, and tightness are each necessary. Within this class, no $\alpha$-divergence distinct from Kullback–Leibler and no Rényi divergence of order other than one can occur as the residual.

In this paper, we propose a parameterized framework for vector spaces in which each vector is represented by a family of parameter-dependent realizations.

Bönisch, Duhr, and Maggio introduced three meromorphic modular forms \(C_4,C_{6a},C_{6b}\) on \(\Gamma_0(2)\), arising from a hypergeometric K3 family, and conjectured that they are magnetic of depths \(1,2,2\). Writing \[ C_4=\sum_{n\ge1}c_4(n)q^n,\qquad C_{6a}=\sum_{n\ge1}c_{6a}(n)q^n,\qquad C_{6b}=\sum_{n\ge1}c_{6b}(n)q^n, \] we prove the stronger denominator-one statements \[ \frac{c_4(n)}n,\qquad \frac{c_{6a}(n)}{n^2},\qquad \frac{c_{6b}(n)}{n^2}\in\mathbb Z \qquad(n\ge1). \] The weight-four case is reduced to a termwise binomial divisibility by a hypergeometric change of Hauptmodul. For weight six we identify the two forms with canonical level-two CM forms of discriminants \(-8\) and \(-4\): \[ f_{3,-8,0,1,1}=-64C_{6a},\qquad f_{3,-4,2,1,1}=32C_{6b}. \] An explicit pair of vector-valued weakly holomorphic forms of weight \(-3/2\) then gives the full odd-prime divisibility through the higher-level theta-lift coefficient formula of Löbrich–Schwagenscheidt. The prime \(2\) is treated independently. If \(t=(\eta(2\tau)/\eta(\tau))^{24}\), \(H=\eta(\tau)^4/\eta(2\tau)^2\), \(J=2E_2(2\tau)-E_2(\tau)\), \(u=64t\), and \(\mathcal T=H^4J\), we prove the infinite-family contraction \[ U_2\bigl(\mathcal Tt\,\mathbb Z_2[[u]]\bigr) \subseteq 2^5\mathcal Tt\,\mathbb Z_2[[u]]. \] Consequently \(v_2(c_{6\bullet}(2^rm))\ge5r\), which is stronger than the slope \(2r\) required for double magneticity. Thus the complete level-two K3 packet is magnetic with global denominator one.

WAVES 2026, 17th International Conference on Mathematical and Numerical Aspects of Wave Propagation Concordia University (John Molson Building), Montreal, Canada, June 22-26, 2026 WAVES 2026 is the seventeenth meeting in a long-running biennial series that has, throughout its history, alternated between Europe and North America to advance the mathematical and numerical study of wave propagation. Hosted at the John Molson Building of Concordia University in Montreal from June 22 to 26, 2026, the conference continues this tradition as a leading international forum where theory, computation, and application meet. The scientific program spans the full breadth of mathematical and numerical techniques for wave phenomena, from the modeling and analysis of the governing partial differential equations to the design, analysis, and implementation of efficient computational methods. Representative themes include acoustic, electromagnetic, elastic, and seismic wave propagation; scattering and inverse problems; high-frequency and asymptotic methods; finite element, boundary integral, and time-domain discretizations; and absorbing boundary conditions and domain truncation, with applications reaching across the physical sciences and engineering.

We present a corrected version of a theorem from the paper "N-free posets and orthomodularity" published in Order 43(1) (2026).

We discuss the problem of limit cycles in an aggregate-type models of liquidity growth proposed and studied both theoretically and numerically by W. Semmler and M. Sieveking in \cite{SemmlerSieveking}. Their model is locally of predator-prey type with cannibalism in predator (profit) and logistic bound on prey's population (liquidity). We modify the model to include the weak Allee effect in liquidity and to study the local bifurcations of equilibria and limit cycles by means of the Hopf-Andronov theory. The main motivation for this study follows the economic theory of the so-called corridors of stability introduced by Leijonhuvud in \cite{Leijonh}. Similar importance of these effects is, however, found also in the originally developed dynamical models in biology, especially in mathematical ecology.

We construct a smooth projective variety in positive characteristic whose $\mathbb{Q}_{\ell}$-coefficient etale cohomology ring is not the scalar extension of any graded $\mathbb{Q}$-algebra, providing an example of a new type of obstruction to characteristic zero liftability.

Let $G$ be a finite group and let $V$ be a finite-dimensional $G$-module over a field $k$. We construct explicit counterexamples in characteristic $2$ to several questions and conjectures of Wehlau concerning Noether numbers. For the $2$-group $G=D_8$, we exhibit a submodule $U\subseteq V$, with $\dim_kU=5$ and $\dim_kV=6$, such that $\beta\bigl(k[U]^G\bigr)=6>5=\beta\bigl(k[V]^G\bigr)$, thereby disproving submodule monotonicity. Writing $X=V^*$ and $Q=U^*$, the corresponding nonsplit exact sequence $0\longrightarrow k\longrightarrow X\longrightarrow Q\longrightarrow0$ also satisfies $\beta\bigl(k[Q]^G\bigr)=8>6=\beta\bigl(k[Q^*]^G\bigr)$ and $\beta\bigl(k[Q]^G\bigr)=8>5=\beta\bigl(k[X]^G\bigr)$. Thus the modular Noether number need not be invariant under duality, and quotient monotonicity also fails. Notably, the basic counterexamples already occur for $2$-groups in defining characteristic. The constructions remain valid over every field of characteristic $2$, and the $D_8$ conclusions propagate by inflation to every finite group admitting $D_8$ as a quotient.

We study the inference-time behavior of deep linear encoder-only transformers through the lens of interacting particle systems. In this perspective, tokens are modeled as particles that interact dynamically through successive linear self-attention layers. We show that in embedding dimension two, for any key, query, and value matrices, the dynamics can be reformulated as a generalized Kuramoto-type model with pure second-harmonic coupling. This formulation is amenable to Watanabe–Strogatz theory which reveals the dynamics are intrinsically low-dimensional regardless of the parameter matrices. For a class of token initializations associated with the Ott–Antonsen (OA) manifold, we show that the parameter matrices induce a diverse variety of long-time behaviors in linear transformers, including clustering, oscillations, and bifurcations. The oscillations and bifurcations are characterized by uncovering a hidden Hamiltonian structure in the dynamics. By establishing a structural stability result, we further show that dynamics initialized near the OA manifold exhibit the same long-time behavior as those initialized exactly on the manifold. Motivated by our theory in dimension two, we conduct numerical experiments for analogous parameter regimes in higher-dimensional transformers. Our numerical experiments suggest that the long-time behaviors characterized in our theoretical results persist in higher dimensions.

This paper investigates the geometry of foliations on smooth algebraic varieties over an algebraically closed field of arbitrary characteristic $p \ge 0$. We address several specific features of foliations in positive characteristic, aiming to highlight both similarities and differences with the characteristic zero case. First, we provide a complete classification of regular foliations on minimal rational and del Pezzo surfaces, establishing that in positive characteristic, most of them are $p$-closed. However, there are regular foliations on Hirzebruch surfaces over arbitrary characteristic $p>0$, and on weak del Pezzo surfaces, in characteristic $p=2$, which are not $p$-closed. We extend the Camacho-Sad index and its associated sum formula to arbitrary characteristic, using it to study the behavior of foliations and distributions with small singular set. For foliations on projective spaces, we prove that the existence of an invariant hypersurface with a sufficiently small singular set forces the foliation to be $p$-closed and imposes strict divisibility conditions on the degree of its normal bundle. Finally, we establish versions of the Bott vanishing theorem in both Hodge cohomology and the Chow ring for $p$-closed foliations, providing a positive characteristic analogue to classical vanishing results in complex geometry.

Building on our previous general computational solution to Bellman's Lost-in-a-Forest Problem, we present a new approach and analytical formulas for the previously well-known escape path for the infinite unit-strip forest and unit broadworm by Zalgaller. Earlier studies addressed these problems exclusively through geometric methods. We reformulated the problem as an interval-cover problem and then formulated it as a constrained functional minimization problem. This constrained functional minimization can be directly discretized and subsequently solved as a convex optimization. Furthermore, we extend the analysis of various line segment. Finally, we show that, in the case of a closed escape path for the unit strip, the optimal solution is a curve of constant unit width.