Papers

Let $(\mathcal{H}\_t)_{t \geq 0}$ denote the standard Ornstein-Uhlenbeck semigroup over ambient space $\mathbb R^d$ equipped with Gaussian measure $d\gamma(x) = e^{-|x|^2} dx$. Further let $\mathcal{H}^*(f)(x) = \sup_{t > 0} \mathcal{H}_t(|f|)(x)$ denote the associated maximal operator. Since $(\mathcal{H}_t)_{t \geq 0}$ is a Markovian semigroup over $(\mathbb R^d,d\gamma(x))$, it is well known that $\mathcal{H}^*$ is bounded on $L^p(\mathbb R^d,d\gamma(x))$ for all $1 < p \leq \infty$. We show that $\mathcal{H}^*$ is also bounded on the weighted space $L^p(\mathbb{R}^d,\omega(x)e^{-\frac{p}{2}|x|^2}dx)$ for $1 < p <\infty$, for weights $\omega\,: \mathbb R^d \to (0,\infty)$ belonging to a certain Muckenhoupt type class $A^\alpha_p$, where $\alpha \in [0,1)$. Essentially, a weight $\omega$ belongs to $A^\alpha_p$ if and only if $\omega(Q) \omega^{-\frac{1}{p-1}}(Q)^{p-1} \leq C |Q|^p$ for all cubes $Q$ contained in any reference cube $N_\alpha(R_x) \subseteq \mathbb R^d$, where the well chosen family $(N_\alpha(R_x))_{x \in \mathbb R^d}$ covers $\mathbb R^d$, $N_\alpha(R_x)$ contains $x$ and the side length of $N_\alpha(R_x)$ is comparable to $1/\max(1,|x|)^\alpha$, with constants depending only on $d$. The class $A^\alpha_p$ contains the usual Muckenhoupt class. We also use the weight class $A_p^{loc}$ from \cite[Definition 2.2]{B} defined in a similar way as $A^\alpha_p$ but with dyadic subcubes contained in some $N(R)$ built over the Gaussian dyadic grid (see below), and we show that the local part $\sup_{t >0}\mathcal{H}_t(|f|\chi_{N(R_x)})(x)$ is bounded on $L^p(\mathbb{R}^d,\omega(x)e^{-\frac{p}{2}|x|^2}dx)$ if and only if $\omega \in A^{loc}_p$.

In Proposition 3.5 of our paper [J. Langemets and K. Pirk, Stability of diametral diameter two properties, RACSAM (2021)], we claimed that the convex DLD2P lifts from an $M$-ideal to its superspace. The proof contains a quantifier error. We show that its valid part yields the following replacement: for every $\varepsilon>0$, every slice of the unit ball of the superspace contains a $(2-\varepsilon)$ $\Delta$-point of norm arbitrarily close to one. In particular, the unit ball is the closed convex hull of these points at each fixed scale, and the superspace has the local diameter two property. Finally, we prove that the original claim was false, we construct Banach spaces $Y$ and $X$ such that $Y$ has the convex DLD2P, $Y$ is an $M$-ideal in $X$, yet $X$ fails the convex DLD2P.

We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in $\mathbb{Z}[\sqrt{-3}]$. A pigeonhole argument of Thue represents the relevant primes by $X^2+3Y^2$, and explicit division in $\mathbb{Z}[\sqrt{-3}]$ lifts primitive odd-power norms to powers in that order. For cubes, the resulting coordinate identities transform $2K(K^2+3M^2)=C^3$ into another equation of the same signed form with a strictly smaller non-zero integer height. The construction gives $|C'|\le |C|/39$. Half-sums and half-differences of the two odd entries place the sum and difference parity configurations in this single descent.

For a fixed integer $k\ge 2$, let $W_{2k+1}=K_1\vee C_{2k}$ be an odd wheel graph. The fixed-size spectral extremal problem aims to determine \[ \operatorname{spex}(m,W_{2k+1}):=\max\{\rho(G): e(G)=m,\ G \text{ is } W_{2k+1}\text{-free}\}, \] where $\rho(G)$ denotes the adjacency spectral radius. Based on this problem, Yu, Li, and Peng [12] proposed the following conjecture: When $m-\binom{k}{2}$ is divisible by $k$ and $m$ is large, every $W_{2k+1}$-free graph of size $m$ satisfies \( \rho(G)^2-(k-1)\rho(G)\le m-\binom{k}{2} \) with equality precisely for $K_k\vee qK_1$. For nonzero residue class, Yu, Zhang, and Zhang [13] proposed the following conjecture: Let $r$ be a nonzero remainder when $m-\binom{k}{2}$ is divided by $k$ and $m$ is large. Then $S_{k,m}$ is the unique graph among $W_{2k+1}$-free graphs of size $m$ having maximum spectral radius, where $S_{k,m}$ is obtained from $K_k\vee qK_1$ by adding a vertex $z$ and joining it to exactly $r$ vertices of the $K_k$. Very recently, Fang, Zhai and Zhang [4] confirmed the Yu–Li–Peng conjecture for $k\ge 2$. Chen, Gao and Li [2] confirmed Yu-Zhang-Zhang conjecture for $k\ge 3$. When $k=2$, then $W_5=K_1\vee C_4$. For large odd $m$, determining $\operatorname{spex}(m,W_5)$ is still open. In this paper we address the odd-size problem. Our result disproved Yu-Zhang-Zhang conjecture for $k= 2$. In our proof, a universal defect bound shows that only $O(1)$ edges can lie outside the dense core. Perron localization then reduces this to at most one edge. A discrete defect inequality forces the complete bipartite crossing and quantizes the two matching deficiencies. Exact quotient-polynomial comparisons eliminate the remaining cross-edge and odd–odd candidates.

Lecture notes from a graduate course given by Michael Krivelevich at the School of Mathematical Sciences of Tel Aviv University in the spring semester of 2026. Topics covered include: phase transition and the giant component in $G(n,p)$; long paths and cycles in supercritical and sparse random graphs; thresholds for connectedness and perfect matching; general model of a random subgraph of a finite graph; phase transition and the giant component in the random hypercube; polynomial diameter of the giant component; perfect matchings in the random hypercube.

Graphical discreteness is not preserved under quasi-isometry in general. We prove that, for every finite thick generalized $m$-gon $\Gamma$ with $m\geq 3$, graphical discreteness is nevertheless constant on the class of finitely generated groups quasi-isometric to the right-angled Coxeter group $W_\Gamma$: such a group $\Lambda$ is graphically discrete if and only if $\Gamma$ is nonflexible, meaning that no nontrivial graph automorphism of $\Gamma$ fixes a closed star pointwise. The proof combines quasi-isometric rigidity of Fuchsian buildings with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. For finite-rank Coxeter systems, we also show that the automorphism group of the standard Cayley graph is compact-by-discrete if and only if it is discrete, equivalently, if and only if the defining diagram is nonflexible. We further obtain obstructions to graphical discreteness from one-vertex doublings, classify compact-by-discreteness for standard chamber graphs of graph products of finite groups, and derive applications to finite projective planes and to right-angled Coxeter groups with Menger curve boundary. Our methods do not settle whether nonflexible finite thick generalized polygons exist.

We extend the Besov rough path framework of Friz and Seeger to paths lying in Besov spaces that are Muckenhoupt-weighted in time. We establish a weighted Besov sewing lemma, construct Young and rough integrals, and prove local Lipschitz continuity of the Itô-Lyons map in weighted Besov norms. The resulting estimates deliver "location-sensitive" control of errors in RDE solution increments. We illustrate the framework with a diffusion whose coefficient is singular at a specified time.

Let $p$ be a prime and $\mathbb{F}_q/\mathbb{F}_r$ a finite field extension with $q=p^m$ and $r=p^s$. For any $k\mid q-1$, we consider $r$-ary irreducible cyclic codes (ICC) of the form $C(k,q/r) = \{(Tr_{q/r}(\gamma \omega^{ik})_{i=0}^{n-1})\}_{\gamma \in \mathbb{F}_r}$, with $\omega$ a primitive element of $\mathbb{F}_q$ and $ n= \tfrac{q-1}{k}$, and generalized Paley (GP) graphs $\Gamma(k,q) = Cay(\mathbb{F}_q, \{ x^k : x \in \mathbb{F}_q^* \})$. We show that there is a simple closed formula relating the weight distribution of $C(k,q/r)$ with the spectrum of $\Gamma(k_r,q)$, where $k_r=\gcd(k, \frac{q-1}{r-1})$. Then, we give $Spec(\Gamma(k,q))$ explicitly for those graphs associated with irreducible 2-weight cyclic codes in the semiprimitive and exceptional cases. Finally, we give the weight enumerators of irreducible cyclic codes associated with Hamming GP-graphs.

The Girard–Hutchinson estimator provides an extremely simple randomized estimate of the Frobenius norm of a matrix $A$ that can only be accessed implicitly via matrix-vector products. In particular, if $\Omega$ is a random Gaussian matrix with $r = O(1/\varepsilon^2)$ columns, than $\frac{1}{r}\|A\Omega\|_F^2$ provides a $(1\pm \varepsilon)$ multiplicative approximation to $\|A\|_F^2$ with high probability. In this work, we introduce a closely related estimator, given by \begin{align*} {\frac{1}{r}\|A\Omega\|_F^2 + \frac{1}{r}\|\Psi^T A\|_F^2 - \frac{1}{r^2}\|\Psi^T A\Omega\|_F^2}, \end{align*} where $\Psi$ is a second, independent random Gaussian matrix with $r$ columns. We prove that this estimator yields a $(1\pm\varepsilon)$ multiplicative approximation to $\|A\|_F^2$ when $r = O(1/\varepsilon)$, a quadratic improvement over Girard–Hutchinson. This dependence on $\varepsilon$ is optimal. Our method, which we call Hutch# (pronounced “Hutch sharp”), matches the complexity of the Hutch++ algorithm [Meyer, Musco, Musco, Woodruff, 2021]. However, unlike Hutch++, Hutch# uses only non-adaptive matrix-vector products with $A$ and $A^T$ and requires no orthogonalization or other advanced linear algebra steps. Thus, Hutch# combines the simplicity of the Girard–Hutchinson estimator and the optimal query complexity of Hutch++.

We give an algebraic proof in characteristic zero of the Miyaoka–Mori criterion: every point of a curve of negative canonical degree on a smooth projective variety lies on a rational curve. Our jet technique gives, in addition, an effective numerical decomposition of the original curve class. For each prescribed point, the decomposition contains a rational curve through that point, with a positive coefficient independent of the point and with anticanonical degree at most $\dim X+1$. Together with BDPP cone duality, this recovers the projective uniruledness criterion over $\mathbb C$. The main result of this paper was obtained using the Pharos system. A detailed report on the use of Pharos and on the Lean~4 formalization of the main arguments is given in the Appendix B, written by Bin Dong, Guoxiong Gao, Zeming Sun, and Bin Wu.

The Baringhaus-Henze-Epps-Pulley (BHEP) tests for multivariate normality are affine-invariant goodness-of-fit tests based on a Gaussian-weighted $L^2$ distance between empirical and Gaussian characteristic functions. In 1990, Henze and Zirkler expressed the limiting null distribution through the eigenvalues of an integral operator on the standard Gaussian space. In 1997, Henze and Wagner obtained a simpler covariance kernel and raised the problem of calculating the eigenvalues of the resulting operator on a Gaussian-weighted space. Although subsequent work treated the univariate case and numerical approximations in a few low dimensions, the complete all-dimensional spectral problem remained open. This paper determines both complete spectra for every dimension $d \in \mathbb{N}$ and every smoothing parameter $\beta > 0$. The two operators are shown to have the forms $\mathcal{X}_{\beta,d}^*\mathcal{X}_{\beta,d}$ and $\mathcal{X}_{\beta,d}\mathcal{X}_{\beta,d}^*$ for the same Hilbert-Schmidt operator $\mathcal{X}_{\beta,d}$. Consequently, their nonzero eigenvalues agree, including multiplicities, while the null space of the Henze-Zirkler operator is identified exactly. The Gaussian integral operator in the Henze-Wagner decomposition is diagonalized by Mehler's formula, and rotational symmetry confines the finite-rank correction to the sectors associated with spherical harmonics of degrees $0$, $1$, and $2$. The degree-$1$ and degree-$2$ eigenvalues are characterized by scalar transcendental equations, and the radial eigenvalues by an explicit pole-safe Fredholm determinant. The paper establishes nonnegativity, multiplicities, eigenfunction reconstruction, completeness, the trace identity, and a complete characterization of all exceptional pole cases.

We introduce Sharper Transductive Local Complexity (STLC), a localized complexity method for transductive learning under uniform sampling without replacement. The construction starts from a Bernstein-type concentration inequality for the supremum of the test–train empirical process. Its proof uses the modified log-Sobolev inequality for the swap walk and a two-parameter entropy closure. A peeling argument with a surrogate localization functional then gives excess-risk bounds with the same fixed-point and confidence terms as the classical inductive local Rademacher-complexity bounds, without the additional logarithmic confidence factor in earlier transductive results. For realizable learning over a binary class of VC dimension $\dVC$, with training size $m$, test size $u$, and $u\ge m\ge\dVC$, STLC yields $\cO\{\dVC\log(me/\dVC)/m\}$. This matches the standard inductive rate and, when $m\ge9$, is within a logarithmic factor of the transductive minimax lower bound of order $\dVC/m$. For transductive kernel learning, STLC gives a spectrum-adaptive excess-risk bound without the multiplicative imbalance factors appearing in the earlier local-complexity bound.

In this paper we determine the Ramsey number $R(3,3,k)$ up to a constant factor, showing that $$R(3,3,k) = \Theta\bigg( \frac{k^3}{(\log k)^2} \bigg).$$ The proof of the lower bound combines the Hefty-Horn-King-Pfender construction for $R(3,k)$ with the method of Alon and Rödl. Using the same proof, we also determine the $r$-colour Ramsey numbers $R_r(3,\ldots,3,k)$ up to a constant factor for every fixed $r \geqslant 3$.

In this paper, we prove that the defocusing mass-critical Hartree equation is global well-posed and scatters for any initial data $u_0\in L^2$, and prove the analogous result in the focusing case, provided the mass of $u_0$ is strictly less than that of the ground state $Q$. This result removes the radial restriction from [C. Miao, G. Xu and L. Zhao, J. Math. Pures Appl. (9), 91(2009), 49-79.], thereby proving the scattering conjecture for the Hartree equation at the scaling-critical regularity.

Let $n\ge2$, and let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector with $\mathrm{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standard Gaussians, and put $\overline{Z}=(Z_1+\cdots+Z_n)/n$. We prove $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\,\max_i(Z_i-\overline{Z})\le t\}$ for every $t\in\mathbb{R}$, and for each fixed $t>0$ equality holds only when $\mathrm{Cov}(X_i,X_j)=-1/(n-1)$ for all $i\ne j$. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance $1/(n-1)$, which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider $n$ equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least $n-1$. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.

We study dynamic pricing problems under inter-temporal price constraints. We have resources with limited capacities. At each time period, we decide which products to make available and what prices to charge for the available products. The sale probability for a product depends on its price. If we make a sale for a product, then we collect a revenue reflecting the price and consume the capacities of a combination of resources. We work with two types of inter-temporal constraints. In price monotonicity, the prices charged for a product at different time periods have to be monotone. In promotion fatigue, we can discount a product at most once over each time interval of a fixed length. Computing the optimal policy is intractable. We use fluid approximations to construct policies. Traditionally, policies from fluid approximations make randomized decisions at each time period by following an optimal solution to the fluid approximation, but such randomized decisions easily violate price monotonicity or promotion fatigue constraints. We develop policies that sample price paths according to an optimal solution to the fluid approximation, while satisfying the inter-temporal constraints. Letting $c_{\min}$ be the smallest initial capacity of a resource and $L$ be the maximum number of resources used by a product, our policies have a performance guarantee of $\max\Big\{ \frac{1}{8L}, \, \frac{1}{2} - \sqrt{\frac{\log c_{\min}}{2 \,c_{\min}}} - \frac{L}{c_{\min}}\Big\}$. Thus, under large resource capacities, our policies are guaranteed to obtain at least half of the optimal total expected revenue. The latter performance guarantee is tight in the sense that no policy can, in general, obtain more than half of the optimal objective value of the fluid approximation even under large resource capacities. We unify our approach to open the path for extensions to other inter-temporal price constraints.

For a convex body $K\subset\mathbb{R}^n$, let $w(K)$ denote the volume of its maximum-volume inscribed ellipsoid. We prove that every closed halfspace $H$ whose boundary passes through the center of the maximizing ellipsoid satisfies \[ w(K\cap H)\le\frac{\sqrt e}{2}\,w(K). \] The constant is optimal uniformly over all dimensions, as witnessed by a family of circular cones, thereby establishing Khachiyan's conjecture. The proof converts containment, maximality, and the central-cut condition into algebraic constraints on positive definite matrices. Two complementary spectral bounds from a diagonal model extend to arbitrary center displacements through a directional rank-one estimate for fractional trace powers. Concavity determines their joint optimum. We also derive finite-dimensional bounds and a necessary condition for near equality, with self-contained supporting proofs and an alternative resolvent argument. An AI language model discovered the proof in a human-directed research process. Lean 4 with mathlib verifies the main theorem, sharpness, and supporting results.

We propose and analyze a fully discrete finite element scheme for the Beris-Edwards system of nematic liquid crystal dynamics, in which the incompressible Navier-Stokes equations are coupled to a gradient flow for the Landau-de Gennes Q-tensor. The scheme combines a linearly implicit formally second-order accurate backward differentiation formula in time with an incremental Chorin projection step for the incompressibility constraint, conforming finite elements in space, and the invariant energy quadratization approach with mass lumping for the nonlinear bulk potential. Each time step requires the solution of one linear system and one Poisson problem. We show that the scheme is uniquely solvable, that it preserves the symmetry and trace-free structure of the discrete Q-tensor and molecular field, and that it satisfies a discrete energy law without any restriction on the time step. Our main result is that, as the mesh size $h$ and the time step $\Delta t$ tend to zero subject to $h^{2} = o(\Delta t)$, the approximations converge along a subsequence to a weak solution of the Beris-Edwards system. The convergence proof addresses two difficulties: the projection method produces two velocity approximations, only one of which is uniformly bounded in $L^2(0,T;H^1_0(\Omega))$, and the coupling term $\mathcal{H}\nabla Q$ requires strong convergence of $\nabla Q$, which we obtain from the structure of the equation for $\mathcal{H}$ rather than from any discrete $H^2$-bound. Numerical experiments in two dimensions exhibit approximately second-order convergence in space and time, and reproduce the splitting of a $+1$ point defect into two $+1/2$ defects and the transport and deformation of a skyrmion induced by a constant pressure gradient.

For every $\rho$ in (0, 1/2), we construct an entire function F of order $\rho$ whose logarithmic derivative has zero as a deficient value. This disproves an old conjecture of W. H. J. Fuchs.

A formal category theory is constructed (in the form of a proarrow equipment), encoding weak coherent enrichment over a monoidal model category $\mV$. We describe how basic categorical concepts formulated via the equipment translate back to enriched categories. We characterize Dwyer-Kan equivalences of enriched categories as $2$-categorical equivalences. Specializing to either the Kan-Quillen model structure on simplicial sets, or the Quillen-Serre model structure on topological spaces, we prove that the resulting formal category theory is equivalent to the one associated with the $\infty$-cosmos of quasicategories, thereby extending the formal approach to $(\infty,1)$-categories in the sense of Riehl-Verity to encompass both simplicial and topological categories. A notion of classifying object, formulated internally to the equipment of $\mV$-categories, leads to enriched versions of Quillen's Theorem A.

In this exposition based on a lecture series at the CIMPA School on Operator Algebras and K-theory, 2025, we highlight modern perspectives on noncommutative geometry using dualisable categories. To this end, we show that several categories naturally arising in functional analysis (derived categories of bornological and condensed modules), operator algebras (E-theory) and stable categories (noncommutative motives) are dualisable. This in particular leads to a definition of algebraic K-theory for analytic spaces. Furthermore, the dualisability of E-theory and motives leads us to pursue an analogy between topological and algebraic K-theory, inspired by assembly map (Baum-Connes and Farrell-Jones) conjectures.

We prove an averaging formula for the Ehrhart polynomial of a cyclic polytope whose vertices are given by integer parameters on the moment curve. More precisely, its Ehrhart polynomial is the average of the Ehrhart polynomials of an explicitly constructed family of lattice zonotopes. Since lattice zonotopes are magic positive, this formula implies magic positivity for these cyclic polytopes. Consequently, their $h^\ast$-polynomials are real-rooted, and their $h^\ast$-vectors are log-concave and unimodal.

In this paper, we introduce a novel second-order, positivity-preserving finite element method for the Vlasov equations using a convex limiting algorithm. The method employs strong-stability-preserving (SSP) Runge-Kutta time integration and a tensor-product construction of the phase-space mesh for efficient high-dimensional implementations. The convex limiting algorithm combines the robust first-order positivity-preserving graph viscosity approach with high-order residual-based viscosity stabilization to obtain a high-order positivity-preserving scheme. Both novel first-order and high-order methods applicable to high-dimensional problems such as the Vlasov system are presented. In addition, we propose a divergence-cleaning technique for Maxwell's equations to ensure that the divergence constraints of the electromagnetic fields are satisfied. Numerical experiments are provided to demonstrate the accuracy and robustness of the proposed methods.

We prove the existence, uniqueness, and partial regularity outside a proper analytic subset for the Kähler current solving the $J$-equation at the birational minimal slope. This confirms Datar–Mete–Song's conjecture 1.5 in \cite{DMS26}. We introduce an analytic threshold, and prove its equivalence to the birational threshold introduced by Datar–Mete–Song. One of the key tools is the approximation of subsolution by Bergman's kernels, which is motivated by the work of Demailly on the approximation of plurisubharmonic functions with analytic singularities. As an application of the Bergman kernel approximation, we combine the results of Fang–Ma \cite{FM26} to give an analytic characterization of the $J$-null locus of a semistable pair $(\alpha,\beta)$. This removes the technical assumptions in \cite{L26b}.

Under suitable relative homotopy and boundary admissibility assumptions, we establish a min-max theory in arbitrary codimension that produces nonconstant branched immersed disks or spheres in closed Riemannian manifolds, with controlled Morse index, where the prescribed mean curvature type tensor and the nonorthogonal, nonconstant contact angle condition of disks along a supporting submanifold are determined by the same differential 2-form. In Euclidean space, under suitable topological and convex barrier assumptions, we obtain such disks with free boundary on a closed supporting hypersurface and a Morse index bound depending only on the ambient dimension.

Chvátal conjectured that every downset has a maximum-size intersecting family which is a star, that is, consists of all members of the downset containing a fixed element. Recently, Chang, Liu and Liu gave a proof of this conjecture, which follows as a corollary of more general results such as Kleitman's conjecture and a version of Kahn's conjecture (which they also prove). We give a short, direct (spectral) proof of Chvátal's conjecture, and a proof of Kleitman's conjecture along the same lines. We also prove a strengthening of Chvátal's conjecture concerning the projection packing number. Finally, we propose two spectral Chvátal conjectures which are motivated by extensive numerical experiments.

Let $G$ be a finite solvable group such that, for any prime $p$ and any quotient $Q$ of $G$, there are two Sylow $p$-subgroups of $Q$ intersecting in $O_p(Q)$. Then, for every family $(P_i)_{i=1}^n$ of Sylow subgroups of $G$ for distinct primes $p_1,\ldots,p_n$, there exists $x \in G$ such that $P_i \cap P_i^x = O_{p_i}(G)$ for all $i$. This covers groups of odd order, partially settling a conjecture of the second and fourth authors and unifying old results of Bialostocki and Mann on the intersection of nilpotent subgroups. We also prove a result for all symmetric and alternating groups, completing the proof of the conjecture for simple groups as initiated by Burness and the first author.

We study finite-time blow-up for a semilinear shifted Tricomi equation with decreasing propagation speed and an oscillatory scale-invariant mass. We focus on the Strauss-type critical regime and prove that every weak solution with finite speed of propagation, arising from nonnegative nontrivial energy data satisfying a suitable localization condition, blows up in finite time. For sufficiently small initial data, we also obtain the corresponding critical exponential upper bound for the lifespan. The proof relies on a positive self-similar solution of the homogeneous adjoint equation represented by the Gauss hypergeometric function. Combined with a previously established lower bound for the nonlinear term and new estimates adapted to the critical case, this construction reduces the PDE problem to a nonlinear differential inequality. An ODE comparison argument then yields both finite-time blow-up and the lifespan estimate.

Let $A$ be an $\alpha$-Hölder cocycle over a transitive two-sided subshift of finite type, and assume that $A$ is strongly bunched. We prove that if there exist $k\in\{1,\ldots,d-1\}$ and $c>0$ such that \[ \lambda_k(p)-\lambda_{k+1}(p)\ge c \] for every periodic point $p$, then $A$ admits a dominated splitting of index $k$. Thus, under strong bunching, a uniform asymptotic gap in the Lyapunov spectrum of periodic measures implies a uniform geometric splitting along all orbit segments. This extends the periodic-gap criterion of Kassel–Potrie from locally constant cocycles to Hölder cocycles in arbitrary dimension, and extends the two-dimensional fiber-bunched result of Velozo to higher dimensions under the stronger bunching assumption.

Terminal intercept-angle control against a maneuvering target can substantially increase the required missile acceleration, potentially leading to saturation and interception failure unless acceleration limits are explicitly addressed. The engagement is therefore formulated as a linear-quadratic optimal-control problem with bounded acceleration commands. Polynomial approximations of the line-of-sight projection coefficients are used to better represent the nonlinear engagement geometry and estimate the time-to-go. The bounded optimal command is derived over saturated and unsaturated arcs, whose switching times are computed at each guidance step. The guidance law is derived for arbitrary linear missile dynamics and implemented for zero-order missile dynamics. For the zero-order model, the conditions under which the terminal demands can be met are derived in closed form, yielding the minimum and maximum reachable commanded terminal intercept angles. Performance is evaluated in nonlinear simulations. Compared with its unconstrained counterparts, the bounded formulation yields substantially smaller miss distances and terminal-angle errors when saturation is encountered. Unlike corresponding bounded miss-only guidance laws, the proposed law does not reduce to its unconstrained counterpart for minimum-phase missile dynamics because the acceleration command can saturate near the end of challenging engagements. The bounded law anticipates this saturation and compensates through earlier maneuvers.