Papers

In this work, we introduce $t$-colorable $k$-plane drawings, that is, drawings of graphs with a $t$-edge-coloring where every edge is crossed by at most $k$ edges of each color. We give tight upper bounds on the edge- and crossing density for small values of $t$ and $k$ and show that the recognition of such drawings is NP-complete if $t\geq 2$ and $k\geq1$.

Let $S$ be a semilattice-ordered semigroup and let $\mathrm{cl}$ be a closure operator on $S$. We consider the space $$X := \{A \in \mathcal{P}(S) \mid A^\mathrm{cl} = A\}$$ of all $\mathrm{cl}$-closed subsets of $S$, endowed with the subspace topology induced by the hull-kernel topology on $\mathcal{P}(S)$. We prove that $X$ is a spectral space and a retrocompact subset of $\mathcal{P}(S)$ if and only if $\mathrm{cl}$ is algebraic. Assuming that $\mathrm{cl}$ is algebraic, we then investigate the operation $$A \star B := (AB)^\mathrm{cl}$$ induced on $X$ by the multiplication on $S$. We obtain finite characterizations of spectrality of the map $\star: X \times X \to X$ in terms of finite subsets of $S$ and finitely generated $\mathrm{cl}$-closed subsets. Analogous criteria are established for left and right translations. We prove that if the underlying poset $(S, \leqslant)$ is well-quasi-ordered and the closure operator $\mathrm{cl}$ is algebraic and order-compatible, then the induced operation $\star$ is spectral. For the identity closure operator, we characterize spectrality of $\star$ by the finite decomposition property. We also show that spectrality of $\star$ implies spectrality of all left and right translations, whereas the converse fails in general, even for an algebraic multiplicative closure operator. Finally, we introduce three closure operators naturally associated with semilattice-ordered semigroups, study their algebraic and multiplicative properties, and apply the general results to the corresponding spectral spaces and induced operations.

This paper is devoted to short quartic Weyl sums in an intermediate range of rational approximations and to their application to Waring's problem with almost equal summands. We obtain a pointwise estimate for such sums and use it, together with the Hardy–Littlewood circle method, to establish an asymptotic formula for the number of representations of a sufficiently large natural number as a sum of seventeen fourth powers of integers lying in a short interval about a common centre. The asymptotic formula is valid for $N^{6/25+\varepsilon}\leq H\leq N^{1/4-\varepsilon}$ and improves the previously known admissible lower bound for $H$.

Identifying cost-effective indigenous building materials that minimise heat penetration through walls is critical for indoor thermal comfort in low-income rural housing in hot-dry climates, where summer temperatures routinely exceed 45 C. We present a two-stage computational framework for thermal ranking of five low-cost indigenous wall materials: mud brick, clay-straw adobe, lime-stabilised bamboo panel, fired clay brick, and lime-mud composite. First, a validated Crank-Nicolson finite difference method (FDM) solves the one-dimensional transient heat equation with Robin boundary conditions under diurnal solar and outdoor air-temperature forcing, generating 1500 periodic-day solutions across a nine-dimensional parameter space by Latin Hypercube sampling. Second, a Physics-Informed Neural Operator (PINO) with a Fourier Neural Operator (FNO) backbone learns the parameter-to-solution operator mu -> T(x,t), enforcing both data fidelity and PDE consistency. The trained PINO attains a relative L2 field error of 5.14e-4 and a 0.201 K mean absolute error on the peak inner surface temperature, preserving the FDM material ranking exactly; PINO trained on 150 FDM samples matches a data-only FNO trained on twice as many, so the physics loss is most valuable when data are scarce. The periodic-day formulation also yields the ISO 13786 time lag and decrement factor, reproduced to within 0.99 h and 0.010. At nominal hot-dry summer conditions, clay-straw adobe achieves the best cost-performance index among widely available materials. A climate sweep, confirmed by FDM spot checks, reveals a regime boundary: under sub-ambient outdoor conditions the ranking inverts to conductive fired clay brick, delineating heat-exclusion and heat-rejection regimes. The framework supports evidence-based material selection for post-flood reconstruction in hot-dry regions.

The key theorem is a connection between motivic superpolynomials of plane curve singularities in any ranks with superpolynomials of the corresponding instanton slices, Nekrasov-type instanton sums with conductors. In this case, instanton slices are related to Compactified Jacobians, but they form a much wider class and, generally, have no connection to plane curve singularities. Thus, we extend significantly theory of affine Springer fibers in type A. For instance, we obtain a motivic interpretation (counting $\mathbb{F}_q$-points of some stacks) of superpolynomials for hyperbolic knots K12n242, K12n725, among many others. New formulas for Nekrasov's instanton sums as limits of superpolynomials of families of plane curve singularities are another application, with an impressive connection to nonsymmetric Macdonald polynomials. The main conjecture states that instanton superpolynomials for arbitrary Young diagrams coincide with the corresponding (new) DAHA superpolynomials if and only if the latter are positive or (equivalently) the former are superdual. Only very few such diagrams come from plane curve singularities. Our DAHA superpolynomials are parallel to the EHA ones, but we used the nonsymmetric approach. The connection with Khovanov-Rozansky polynomials of the corresponding Coxeter knots is expected. Among applications: the Volume Conjecture reformulated in terms of instanton slices, topological invariance of unibranch motivic superpolynomials for semigroups with up to 4 generators, a mixed-characteristic conjecture, upgraded Weak Riemann Hypothesis, and superpolynomials for Gorenstein curve singularities and multibranch ones.

Robust generalised Bayesian methods protect parameter inference from contamination but do not generally infer its proportion or identify individual outliers. We propose Hölder–Bayes, which applies the Hölder divergence to a scaled model to construct a joint generalised posterior over model parameters and the contamination proportion, without specifying an outlier distribution. Each posterior draw flags a corresponding number of observations least compatible with the fitted model; averaging selections propagates joint uncertainty to outlier-detection scores. Under fixed, non-negligible contamination, we establish uniform boundedness of the posterior influence function, a finite-sample $n^{-1/2}$ bound on posterior-averaged excess risk, and a Bernstein–von Mises theorem. Tail overlap between the outlier distribution and the fitted model controls the population risk minimiser's bias and the associated change in limiting posterior covariance. Interpreting the posterior temperature as an affine volume scaling yields a closed-form, data-driven choice. Simulations and real-data experiments demonstrate robust estimation, contamination-proportion recovery, and posterior-guided data cleaning.

Machine learning demand forecasts optimize statistical accuracy yet leave excess operational volatility that inflates safety stock and amplifies the Bullwhip effect. We introduce Contextual Deconvolution (CD), a two-stage estimator that reframes demand sensing as a convex decomposition: a kernel-modulated banded operator separates transient promotion-driven shocks from a smooth structural baseline, and hierarchical partial pooling enables catalog-scale deployment without per-SKU training. The operator is data-derived, not imposed—it reduces to the identity wherever the promotional response is impulsive (most of M5, all of Favorita) and contributes only where genuine multi-day carryover exists, so the gains rest on the structural decomposition itself. Evaluating strictly out-of-sample on 30,490 M5 SKUs and 2,845 Favorita items, with calendar-aware baselines given CD's identical future calendar, we anchor the contribution on a full inventory-cost accounting: CD lowers safety stock, holding cost, and order variance but under-provisions event spikes, reducing total cost only when holding costs exceed $\sim$20% of stockout costs (95% CI $[17\%,25\%]$); otherwise it is an operational-stability and inventory-capital layer, not an expected-cost minimizer. Its accuracy contribution is reliability rather than central tendency: across eleven baselines, CD attains the lowest cross-sectional dispersion of per-SKU error and mis-forecasts by more than 200% on 0.8% of SKUs versus 9.9–20.6% for every baseline, ranking first on both in all four M5 draws. Because the Variance Ratio and std-based safety stock are minimized by any sufficiently smooth forecast, we treat them as diagnostics, not objectives. A supporting analysis shows the learned demand operators are non-normal, yet CD's compact parametric kernel matches their operational performance interpretably.

We discuss the structure constants of spacelike $\mathcal{N}=2$ Liouville theory on the two-sphere. Due to the absence of a particular $b \leftrightarrow b^{-1}$ self-dual symmetry, where $b$ is the theory's coupling, the standard analytic bootstrap toolkit that was used to solve the bosonic and $\mathcal{N}=1$ theories cannot be implemented in a straightforward way. Our approach, instead, relies on the fact that $\mathcal{N}=2$ Liouville theory is dual by mirror symmetry to the $\mathrm{SL}(2)_k/\mathrm{U}(1)$ supercoset, whose target space is the $2$d fermionic black hole. Leveraging this duality, we obtain explicit expressions for the winding number preserving and violating structure constants on the supercoset side, and test them on the Liouville side through a semiclassical analysis finding agreement up to one-loop order. We also discuss the chiral rings of the theory and evaluate correlators of $\frac 12$-BPS operators.

Data-driven model predictive control (MPC) using Koopman operator theory is a promising approach for constrained control of unknown nonlinear systems. While linear Koopman realizations are commonly used due to their simplicity, bilinear Koopman realizations can provide significantly higher approximation accuracy for nonlinear control systems. However, robust MPC (RMPC) formulations that account for modeling errors in bilinear Koopman realizations remain limited. This paper proposes a RMPC framework for unknown nonlinear systems with general nonlinear constraints based on data-driven bilinear Koopman realizations. A central difficulty is that finite-dimensional Koopman predictors need not preserve the manifold of valid lifted states, so multi-step prediction in lifted coordinates may leave the region where one-step error certificates apply. We address this issue by reprojecting each predicted lifted state back onto the manifold, thereby obtaining an error-aware discrete-time control-affine predictor in the original state space without impractical assumptions. For this predictor, we develop a discrete-time robust control contraction metric based homothetic tube construction, and then formulate a tube-based RMPC problem with terminal ingredients. Under the proposed formulation, we prove robust satisfaction of the original nonlinear constraints by the true closed-loop trajectory, recursive feasibility, and convergence to a neighborhood of the target state. Numerical experiments demonstrate robust stabilization of nonlinear systems and the advantages of the proposed method over existing Koopman-based RMPC approaches in terms of performance.

In this article, we generalize the notion of continuous superpotentials on compact Kähler manifolds to arbitrary complex manifolds in terms of local potential functionals and study related properties. In particular, we study the associativity of the Dinh-Sibony product and also a sufficient condition for the continuity of the Dinh-Sibony product.

We show that the Levi-Civita connection for the Taub-NUT metric is singular on Misner strings and generates an Einstein tensor containing the squares of the delta function. This can be improved by introducing torsion, in which case the Einstein tensor may be interpreted as that of a cosmic string with negative variable tension. Komar flows in Misner strings are generated by torsion, which is therefore responsible for the asymptotic Komar charges. The Taub-NUT metric then appears as a soliton supported by two spin-fluid beams.

Let $A^2$ be the Bergman space of the open unit disc $\mathbb{D}$ in the complex plane consisting of holomorphic functions that are square integrable with respect to the area measure in the disc, and $H^\infty$ be the Hardy algebra of bounded and holomorphic functions on $\mathbb{D}$. With pointwise operations, $A^2$ is an $H^\infty$-module. It is shown that the $H^\infty$-module $A^2$ is not flat. It is also shown that there exists a maximal ideal $\mathfrak{m}$ in $H^\infty$ such that $\mathfrak{m}A^2=A^2$.

Conserved gravitational charges are commonly expressed as surface integrals of the metric perturbation and its first derivatives. We show that, in the usual asymptotically AdS and asymptotically flat settings, the standard charges admit equivalent representatives as fluxes of linearized curvature. The construction uses a divergence-free rank-four tensor whose trace is proportional to the cosmological Einstein tensor. On a maximally symmetric AdS background, it reproduces the known curvature representation of the Abbott-Deser charges. The asymptotically flat construction is not obtained by taking a naive $\Lambda \rightarrow 0$ limit, since the Killing two-form vanishes for translations. We instead introduce an antisymmetric Poincaré Killing potential. A representative Killing potential adapted to the algebraic Bianchi identity converts the linearized Einstein current into a total divergence and yields a single curvature-flux formula. Its translation sector reproduces the ADM energy–momentum, while in four dimensions its Lorentz sector reproduces the angular momentum and boost/center-of-mass charges under the standard Regge-Teitelboim falloff and parity conditions. The normalization is checked explicitly for Schwarzschild, boosted Schwarzschild, and Kerr data. On non-maximally symmetric Einstein backgrounds, background Weyl curvature generates additional terms, so the maximally symmetric construction does not directly extend to a pure codimension-two curvature-flux formula.

Under small-ball bounds for the Malliavin determinants of two $\mathbb R^k$-valued Sobolev mappings on a Gaussian space, we estimate the total variation distance between their laws both in terms of the Kantorovich–Rubinstein distance and in terms of the distance between the mappings in the corresponding Sobolev space. In particular, our results yield new total variation distance estimates for distributions of random vectors whose components belong to finite sums of Wiener chaoses, with exponents improved by an asymptotic factor of two. The proof is based on fractional regularity estimates for distributions of Sobolev mappings. Namely, we show that if an $\mathbb R^k$-valued mapping has components in $W^{2,p}(\gamma)$ and the determinant of the corresponding Malliavin matrix satisfies a small-ball bound of order $\varkappa\in(0,1]$, then the law of the mapping has fractional regularity of order \[ \frac{\varkappa}{1+(2k-1)\varkappa p^{-1}}. \] In particular, for large $p$, this gives regularity of order $\varkappa$ up to an $O(p^{-1})$ loss.

We study the metric geometry of finite-time singularities of volume-noncollapsed Kähler-Ricci flows and the geometry at infinity of Ricci shrinkers. For a compact volume-noncollapsed Kähler-Ricci flow approaching its first singular time, we prove that the time slices converge to a unique Gromov-Hausdorff limit, canonically identified with the time-zero slice of the Ricci-flow spacetime completion. Its regular part is identified with the complement of the null locus, and we show that the singular set has Hausdorff codimension at least four. For Kähler-Ricci shrinkers, we relate the geometry at infinity to the associated Fano fibration. We prove local smooth convergence to a Kähler cone metric on the locus where the fibration is biholomorphic, and characterize asymptotic conicality by biholomorphicity outside a compact set. For Ricci shrinkers with bounded scalar curvature, we show that local compactness of the time-zero slice implies pointed Gromov-Hausdorff convergence of the full self-similar flow to that slice. As applications, we prove uniqueness of the tangent space at infinity for Kähler-Ricci shrinkers with bounded scalar curvature and Euclidean volume growth whose soliton vector field generates an $S^1$-action, and for all four-dimensional Ricci shrinkers with bounded scalar curvature.

The edge-girth of an edge $e$ in a simple connected graph is the length of a shortest cycle containing $e$, with $g_e = \infty$ if no such cycle exists, and the edge-girth sequence of a graph is the nondecreasing sequence of the edge-girths of its edges. We characterize the sequences that are realizable as the edge-girth sequence of a simple connected graph. A constant sequence $(g^{(m)})$ with $g$ finite and $m \geq 1$ is realizable if and only if $m = g$ or $m \geq \lceil 3g/2 \rceil$. Otherwise, writing $S = S_0 \uplus (g^{(m)})$, where $g$ is the maximum value of $S$ and $m$ its multiplicity, we prove that $S$ is realizable if and only if $S_0$ is realizable and $m$ lies in an explicit set determined by $g$ and by the maximum diameters of the graphs realizing $S_0$, alone or extended by fewer than $g$ edges of edge-girth $g$. The proof rests on a chaining lemma, which turns a shortest cycle of a realization into a lower bound on the maximum diameter of the realizations of a shorter sequence.

The Erdős-Selfridge odd covering problem (Erdős problem #7) asks whether a covering system of $\mathbb{Z}$ exists whose moduli are all odd, distinct, and greater than 1. The problem is open. We present a Lean 4 formalization, checked end to end by the proof kernel, of the exclusion: any covering of $\mathbb{Z}$ by finitely many congruence classes with distinct odd moduli > 1 has lcm of the moduli exceeding 10000. The proof composes a formalized density argument (a covering by divisors of $N$ exceeding 1 forces $2N \le \sigma_1(N)$, so the lcm is abundant or perfect), a kernel-checked abundancy floor (no odd $N < 945$ qualifies), a family of Chinese-Remainder capacity certificates – decidable per-$N$ arithmetic inequalities each refuting every covering with distinct moduli > 1 dividing that $N$ – for all 23 odd abundant numbers below $10^4$, and a kernel-checked enumeration establishing that those 23 are the only odd non-deficient candidates. The result is transported to the official StrictCoveringSystem $\mathbb{Z}$ formulation of Erdős #7 in google-deepmind/formal-conjectures, with a bidirectional periodicity bridge between coverings of $\mathbb{Z}$ and finite checks over $\mathbb{Z}/N\mathbb{Z}$ suitable for consuming future SAT-style search output. All 63 published theorems depend on exactly propext, Classical.choice, and Quot.sound: no sorry, no native_decide, no solver in the trusted base. The mathematical content is known – the density argument is folklore, and far larger uncertified classifications of covering numbers exist – so the contribution is epistemic rather than mathematical: these exclusions are theorems of the Lean kernel, with an axiom gate enforced mechanically in continuous integration.

We extend the celebrated result of Alberti, stating that Borel vector fields coincide with gradients of $C^1$-functions outside of a set of arbitrary small measure. We prove that a similar statement holds true in the setting of fractional gradients and $C^{0,s}-functions. Furthermore, we establish analogous results for general nonlocal gradients with appropriate function spaces. Both statements are proven by means of the translation method, demonstrating its use for the purposes of geometric measure theory. In particular, this article illustrates how the translation method transfers rigidity phenomena from the classical gradient to a broad class of nonlocal gradients.

In this article, a comprehensive framework for goodness-of-fit testing for distributions on Stiefel manifolds is developed. The approach is based on integrals of the squared differences between empirical and theoretical characteristic functions, yielding test statistics that are consistent against all fixed alternatives. For the Fisher-Bingham family of distributions, explicit computable forms of the test statistic are derived. Simplified expressions for important special cases, including the matrix Fisher, matrix Bingham, and uniform distributions are provided. In the case of testing uniformity on hyperspheres, we obtain the complete asymptotic distribution of the test statistic, enabling computationally efficient asymptotic testing. For general Fisher-Bingham distributions, we establish theoretically justified Monte Carlo testing procedures for both simple and composite hypotheses. Simulation studies demonstrate accurate Type I error control and strong power across a wide range of alternatives. The practical relevance of the proposed methodology is illustrated by an application to data on the orbits of comets.

We prove a realization theorem for stealthy point processes based on lattice induction, together with a converse in dimension one. Every probability-preserving $\mathbb R^d$-action induced from a full-rank lattice admits a generating Delone cross-section whose Bartlett spectrum vanishes on a neighborhood of the origin. The construction can be chosen so that first-order linear statistics detect a nonzero part of the inducing spectral type. Bernoulli bases yield a nonzero absolutely continuous component, while weakly mixing bases of singular maximal spectral type yield a nonzero singular-continuous component. To our knowledge, these are the first rigorously constructed translation-invariant stealthy point processes on $\mathbb R^d$ with non-pure-point Bartlett spectrum. Conversely, let $\eta$ be an ergodic translation-invariant point process on $\mathbb R$ with positive intensity and local second moments. If \[ \int_{0<|\xi|<1}\frac{1}{\xi^2}\,d\sigma_\eta(\xi)<\infty, \] then its translation action has a nonzero eigenvalue and is lattice-induced. Consequently, an ergodic probability-preserving Borel $\mathbb R$-space is lattice-induced if and only if it admits a generating stealthy Delone cross-section. This should be compared with a theorem of Borichev, Sodin and Weiss stating that a translation-invariant point process on $\mathbb Z$ with proper spectral support is periodic.

According to the Singer conjecture, the $L^2$-Betti numbers of a closed aspherical manifold are expected to vanish outside the middle dimension. In this paper, we study a natural analog of the Singer conjecture concerning the vanishing of twisted $L^2$-Betti numbers outside the lower middle dimension. These invariants can be thought of as $L^2$-Betti numbers of an infinite cyclic cover, depending on a first cohomology class. This is related to a previous conjecture by the author, which connects the twisted $L^2$-Euler characteristic to the Thurston norm, and which we prove for a class of closed arithmetic hyperbolic manifolds.

For integers $n\ge s\ge2$, let $e(n,s)$ be the maximum size of a family $\mathcal F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erdős–Kleitman problem, is closely related to the well-known Erdős matching problem. Frankl and Kupavskii posed a meta-conjecture predicting that the maximum is always attained by a weighted family. Fix $m\ge3$, write $n=ms+c$ with $0\le c<s$, and set $\ell=s-c$. For $0\le k\le m$, let $a_k=ms-kc-1$. For $A\in\binom{[n]}{a_k}$, define \[ \mathcal H^k(m,s,\ell;A):= \{F\subseteq[n]: k|F|+|F\cap A|\ge m(k+1)\}. \] This defines a unified class of weighted families with matching number less than $s$. Among these families, $\mathcal H^0$, $\mathcal H^1$, and $\mathcal H^m$ were previously known to be extremal in different ranges of $c$. We show that for $1\le k\le m-1$, all families $\mathcal H^k$ are uniquely extremal in some ranges of $c$. More precisely, we prove that for every $m\ge3$ and every $1\le k\le m-1$, there exist constants $\alpha=\alpha(m,k)>0$, $\beta=\beta(m,k)>0$ and an integer $s_0=s_0(m,k)$ such that, for all integers $s\ge s_0$ and all integers $c$ with $0\le c<s$, the only extremal families for $e(n,s)$ are the families $\mathcal H^k(m,s,\ell;A)$ with $A\in\binom{[n]}{a_k}$ whenever $\beta s^{(k-1)/k}\le c\le \alpha s^{k/(k+1)}$. In particular, this result determines an infinite number of new extremal families for the Erdős–Kleitman problem and verifies the Frankl–Kupavskii meta-conjecture in these ranges. This also provides a quantitative extension of the result of Kupavskii and Sokolov on the extremality of $\mathcal H^1$.

We consider the classical analytic linearization problem for vector fields on the torus $\mathbb{T}^d$ close to a constant vector field $\omega$. Our goals are twofold. First, we provide a geometric framework in which the arithmetic condition governing analytic linearization arises naturally from the orbit of a unimodular lattice associated with $\omega$ under a diagonal flow on $\operatorname{SL}(d,\mathbb{Z})\backslash \operatorname{SL}(d,\mathbb{R})$. Within this framework, a summability condition emerges as the natural criterion for convergence. We prove that it is equivalent to several classical formulations of the Brjuno condition for linear forms, including those involving best approximation vectors and switching times of the diagonal flow. As a byproduct, we obtain a new quantitative linearization theorem with fully explicit estimates. In particular, the loss of analyticity of the conjugacy is controlled by a Brjuno function.

In this paper, we establish a necessary and sufficient condition for a group action on a bifoliated plane to arise from a topological Anosov flow supported by an orientable $3$-manifold. More precisely, we show that a group action on a non-singular bifoliated plane arises from such a flow if and only if it is an orientation preserving strong Markovian action, that is, an orientation preserving action of the plane preserving a family of rectangles satisfying suitable Markov-type properties. Furthermore, we prove that this family of rectangles can be lifted to a Markov partition of the associated flow.

This work develops well-posed homogenized strain-gradient models for linear elastostatics and elastodynamics in periodic media, with a primary focus on elastic wave propagation. Using the classical two-scale asymptotic expansion method, we carry out second-order periodic homogenization for media in $\mathbb{R}^d$ ($d = 2, 3$), with no restriction on the periodicity cell geometry or material distribution. Reciprocity identities applied to suitably chosen pairs of cell solutions provide alternative expressions for the effective stiffness and inertia tensors arising at the leading, first and second orders, substantially reducing the number of cell problems that must actually be solved. Since direct two-scale homogenization beyond leading order generically yields ill-posed effective operators, a Boussinesq-trick procedure is introduced, involving a tunable scalar weight, to recast the resulting fourth-order partial differential equation as a valid strain-gradient elasticity (SGE) model possessing the requisite symmetry, sign-definiteness and coercivity properties. These properties are then used, via the Hille-Yosida theorem, to establish the well-posedness of the corresponding transient initial-value and forced-response problems. Several practically relevant special cases are examined, including centrosymmetric cells, homogeneous mass density and homogeneous elasticity, each yielding simplified model structures. Numerical illustrations on three two-dimensional periodicity cells (square, hexagonal and a non-centrosymmetric chiral lattice) compare the resulting dispersion relations against reference Floquet-Bloch computations and assess transient wave propagation, demonstrating the model's capacity to capture anisotropic and dispersive effects beyond classical elasticity while preserving mathematical well-posedness.

This study develops a compartmental epidemic model for Hepatitis B virus (HBV) transmission incorporating vertical transmission, spontaneous recovery in acutely infected individuals, saturated treatment response for chronically infected individuals, and vaccination of susceptible individuals. The basic reproduction number, \(\mathcal{R}_0\), is derived using the next-generation matrix method, providing a threshold quantity for understanding disease dynamics. To identify the most influential parameters, sensitivity analysis is performed using Latin Hypercube Sampling (LHS) together with Partial Rank Correlation Coefficients (PRCCs). The qualitative behavior of the model is investigated through stability analysis of the disease-free and endemic equilibria. It is shown that the disease-free equilibrium is globally asymptotically stable when \(\mathcal{R}_0 < 1\), while the endemic equilibrium is globally stable when \(\mathcal{R}_0 > 1\). Pontryagin's Maximum Principle is applied to optimize public health interventions, leading to three optimal control strategies aimed at reducing the combined cost of treatment and vaccination. The results provide a rigorous theoretical framework for designing cost-effective interventions against HBV transmission.

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb F_q$, and let $\mathcal F\subsetneq\genfrac{[}{]}{0pt}{}{V}{k}$. The projective Ore-degree of $\mathcal F$ is the minimum, over all $k$-subspaces $S\notin\mathcal F$, of the sum of the $\mathcal F$-degrees of the projective points contained in $S$. We prove sharp projective Ore analogues of the vector-space Erdős–Ko–Rado and Hilton–Milner theorems. The Ore–Erdős–Ko–Rado theorem holds for $n\ge2k+1$, with equality only for a full point-star. For nontrivial intersecting families, we determine the sharp Ore–Hilton–Milner threshold, together with the complete equality classification, when $q\ge3$ and $n\ge2k+1$, or when $q\ge2$ and $n\ge2k+2$. We further determine a sharp projective Ore-degree threshold forcing a direct-sum matching of size $s$ when $s\ge3$ and $n\ge(2s-1)k-s+4$, and derive a multicolour Ramsey consequence.

The Borel classes generating the stable cohomology of the general linear group can be represented by invariant differential forms. It is known that pulling these forms back along a tropical Torelli map yields canonical convergent integrals associated to graphs, which are closely connected to the cohomology of $\mathrm{GL}_n$ and of graph complexes. A natural question is what numbers these graph integrals are. We answer this for primitive canonical integrals by showing that they coincide with a family of complex position-space integrals arising in deformation quantisation. As a consequence, canonical integrals of graphs evaluate to single-valued multiple zeta values. We further deduce that every single-valued multiple zeta value occurs as a rational linear combination of Feynman periods of graphs with massless propagators. Finally, in the commutative graph complex, our result implies that the two associated cocycles agree.

The conditional gradient method is attractive when linear minimization over the feasible region is substantially cheaper than projection. Its classical convergence theory, however, is formulated for compact feasible sets, whereas many natural convex feasible regions are closed and unbounded. This paper studies a simple compact-restriction principle for applying conditional gradient steps to unbounded feasible regions. The first scheme uses one compact convex set containing the initial objective sublevel set. The second scheme updates the restriction by intersecting compact convex sets generated along the iterations. In both cases the linear minimization oracle is solved only over compact subsets, but the resulting objective values converge to the global optimum of the original problem, provided the compact restrictions contain the corresponding objective sublevel sets. For smooth convex objectives we obtain the standard superlinear convergence rate of the objective. We also record constructive restrictions based on strong convexity, exact sublevel sets, and epigraph caps, and include a nonsmooth conditional subgradient extension with a sublinear convergence rate under a curvature assumption. Numerical experiments illustrate the behaviour of the fixed and dynamic restrictions on unbounded feasible regions.

In this paper, new generalizations of the hyper Bessel type differential operator called it as $\ell$-H derivative operator, are studied, and using these, some of the classical results of calculus are extended. Also, new generalization of a natural number is defined and then its combinatorial properties are extended. After this, the $\ell$-H integral is defined, and then its basic properties are assessed and classical results of calculus pertaining to the integral are extended.