Papers

Principal component analysis (PCA) is a fundamental tool for learning low-dimensional structure from high-dimensional data. When data are collected from multiple sources, the underlying task distributions may exhibit unknown degrees of similarity, with some tasks potentially arising from arbitrary distributions. We propose new multi-task PCA procedures that exploit similarity structure across tasks to improve eigenspace estimation while remaining robust to outlier tasks. We establish non-asymptotic convergence rates and show that the proposed procedures attain minimax optimal rates in a range of regimes. One of the procedures builds on the matrix-depth notion of Chen, Gao, and Ren (2018) and can achieve the optimal dependence of the estimation error on the proportion of outlier tasks, addressing a key challenge in robust multi-task learning. Extensive simulations and real-data analyses demonstrate the effectiveness of the proposed methods.

Motivated by questions and results of Erdős, Sárközy and Sós (1989), and Khalfalah and Szemerédi (2006), we introduce an ergodic approach to additive problems concerning solutions of equations of the form \[ x+y=\lfloor\alpha(n)\rfloor,\quad x,y\in A,n\in\mathbb{N} \] where $\alpha(t)$ is a sufficiently regular function, such as a polynomial or a logarithmico-exponential function of polynomial growth (e.g. $\alpha(t)=t^{5/2}+t^2\log t$), and $A$ is either a given set of positive upper density, or a cell of a prescribed finite partition of $\mathbb{N}$. We give a complete description of the Hardy field functions of polynomial growth for which these partition and density problems are always solvable. The ergodic approach also applies to the more general problem of finding $x\in A_1,y\in A_2$, where $A_1,A_2$ are two given sets of positive density. This in turn allows us to treat equations of the form $kx+ly=\lfloor\alpha(n)\rfloor$, where $k,l\in\mathbb{Z}$. Some of our results also hold true when $A$ is a subset of the primes with positive relative upper density.

For a zero-one matrix $P$, let $\operatorname{Ex}(n,P)$ denote the maximum number of ones in an $n\times n$ zero-one matrix avoiding $P$. We prove that $$\operatorname{Ex}(n,P) \leq n^{1+O_P(1/\log\log n)}$$ for every acyclic $P$, settling the weak form of the Pach-Tardos conjecture.

The article arXiv:2410.21454 constructed a braided $\mathrm{W^*}$-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category $\mathcal{B}$ in arXiv:2506.19969 is equivalent to $\mathsf{Hilb}(\mathcal{B})^{\rm op}$, the opposite of the unitary ind-completion of $\mathcal{B}$, as a braided $\mathrm{W^*}$-tensor category.

A longstanding conjecture attributed to Smith (1984) asserts that for every $k\ge2$, any two longest cycles in a $k$-connected graph share at least $k$ vertices. In this paper, we prove the first linear lower bound, showing that any two longest cycles in a $k$-connected graph share at least $k/600$ vertices. Departing from previous Turán-type extremal arguments, we develop a novel structural approach that also yields applications to related problems on longest cycles and paths.

Christensen and Pedersen proved that every properly infinite $\mathrm{AW}^*$-algebra is monotone sequentially complete, and Saitô and Wright developed a transfinite form of their dilation argument. We revisit the transfinite construction using normality of $\mathrm{AW}^*$-algebras. Normality simplifies the limit stages by turning suprema into compressions of joins, so the construction only needs a supply of fresh orthogonal projections large enough to contain the supports of the summands at successor stages. We use this simplified proof to show that a $*$-homomorphism between $\mathrm{AW}^*$-algebras that preserves only the joins needed to encode such a sum preserves the sum itself. We also use it to deduce order-continuity facts about $\kappa$-join-preserving $*$-homomorphisms. We also show that a finite $\mathrm{AW}^*$-algebra has suprema for all bounded positive families whose supports have bounded total center-valued dimension.

We give an FPRAS for the permanent of an $n\times n$ $0/1$ matrix with running time $\widetilde O(n^3)$. Our algorithm extends to a strongly polynomial FPRAS for arbitrary nonnegative matrices with the same running-time bound (but in the unit-cost arithmetic model). Jerrum, Sinclair, and Vigoda (2004) gave the first FPRAS for the permanent of a nonnegative matrix. The running time was subsequently improved to $\widetilde{O}(n^7)$ by Bezáková, Štefankovič, Vazirani, and Vigoda (2008), and recently to $\widetilde{O}(n^6)$ by Chen, Vigoda, and Yang (2026). We introduce a multicommodity-flow bound inspired by electrical flows, replacing the usual path-length factor by routing energy. For the classical JSV chain with perfect-matching weights multiplied by $n^2$, we prove a relaxation-time bound of $O(n^3\log n)$ and obtain comparable bounds for estimating the stationary probabilities of having a specified pair of unmatched vertices. We use a separate chain on assignments to generate a warm start at any given temperature, avoiding the long initialization of the JSV chain. Finally, we replace the JSV chain with a new chain that uses only $n$ hole weights and speeds up hole-weight learning by a factor of $n$.

A theorem of Kollár, Miyaoka, and Mori states that on a rationally connected variety, any finite collection of points lies on a rational curve. Motivated by this, it is natural to ask what one can say about families of curves passing through many general points of an arbitrary smooth projective variety X. When X has nonnegative Kodaira dimension, we establish a sharp linear lower bound for the genus of such curves in terms of the number of points and the dimension of X, generalizing a theorem of Arapura and Archava. By contrast, the least possible gonality of a connecting family eventually stabilizes as the number of points grows. We characterize its limiting value in terms of varieties dominating X that are generically finite covers of rationally connected varieties. As an illustration, we study these invariants for hypersurfaces of large degree, determining in particular the joint asymptotic behavior of the minimal connecting genus as the number of points and the degree vary. Finally, we briefly consider higher-dimensional connecting subvarieties, proving linear bounds for their canonical volumes and computing asymptotic results for hypersurfaces.

As a fundamental task across computational statistics, scientific computing and machine learning, sampling from high-dimensional probability distributions has received increasing attention in recent years. Numerous sampling algorithms have been proposed, among which underdamped Langevin Monte Carlo (ULMC) methods based on underdamped Langevin dynamics (ULD) have emerged as a class of efficient ones. In this work, we introduce a “universal" predictor-corrector formulation that bridges Euler-type, UBU-type and randomized schemes through different choices of method parameters. Notably, the “universal" integrator induces two novel classes of low-cost integrators, termed low-cost randomized integrators (LC-RIs) and low-cost UBU integrators (LC-UBUIs), as well as their exponential-free variants based on polynomial and rational approximations. The resulting new UBU-type and randomized schemes require only one gradient evaluation and two Gaussians per iteration, considerably reducing the number of gradient evaluations or Gaussians per iteration required by existing counterparts. Further, a general framework of long-time error analysis is developed for general discretization schemes in a probability metric. Under certain smoothness and non-log-concavity conditions, we rely on the unified framework to establish non-asymptotic $\mathcal{W}_1$-error bounds of both old and new schemes, revealing convergence rates of order $\mathcal{O}(d^{\frac{1}{2}}h)$ for Euler-type schemes, order $\mathcal{O}(d h^2)$ for UBU-type ones and order $\mathcal{O}(d^{\frac{1}{2}}h^{\frac{3}{2}})$ for randomized ones. In the strongly convex setting, the same non-asymptotic error bounds can be recovered in $\mathcal{W}_2$-distance. Numerical experiments corroborate the theoretical findings.

In this paper we construct a two parameter family of generalized Gaussian fields which interpolates between the Gaussian Free Field, conditioned Gaussian Free Fields, and Gaussian Free Fields with independent noise. Depending on the values of the parameters, we characterize these generalized Gaussian fields as partial conditionings of a Gaussian Free Field with perturbations of its first two Fourier modes. We show that this family arises as the fluctuations of the height function associated with the corner process of a two parameter variant of Wigner random matrices.

It is known that, for ordinary Hölder multifunctions with 0 < {\alpha} < 1, Hölder regularity alone does not guarantee the existence of a Hölder selection, and in general even a continuous selection may fail to exist. We ask how this situation changes when the parameter is measured through a Stieltjes clock. For 0 < {\alpha} < 1, we characterize the clocks for which every compact-valued g-Hölder multifunction admits a g-Hölder selection. The determining condition is uniform disconnectedness of the clock image Im(g). For left-continuous and nondecreasing clocks, this condition is equivalent to uniform jump dominance: every positive clock increment contains a jump carrying a fixed positive fraction of that increment. Under this condition, a selection can be chosen through any prescribed point of the graph, with explicit control of its Hölder regularity. Conversely, when the condition fails, there exists a compact-valued g-Hölder multifunction with no g-Hölder selection. The results show that the selection property is governed not simply by the presence of jumps, but by how their sizes are distributed across scales.

We consider a certain pair of families of Picard rank 1 Calabi–Yau threefolds, that have appeared earlier in mathematical literature in unrelated contexts: the family $\mathcal{X}$ of degree 33 threefolds in $G(2,6)$ (constructed by Miura) and the family $\mathcal{Y}$ of arithmetically Gorenstein degree 21 threefolds in $\mathbb{P}^8$ (constructed by Schenck–Stillman–Yuan). After establishing a natural geometric correspondence between their general members, we go on to show that any pair of corresponding threefolds in these families satisfy certain classical dualities. We moreover discover that their geometries may be related by a mathematical gauged linear sigma model, using which we prove that they are derived equivalent. This settles a conjecture of Miura, who predicted the existence of non-trivial Fourier–Mukai partners to members of $\mathcal{X}$, based on a study of its mirror moduli. This conjecture was also formulated later by Gerhardus–Jockers, in a physical context. In fact, starting from the other family $\mathcal{Y}$, we conjecturally arrive at the same mirror moduli. Thus, such a pair is a new, and quite possibly the last, addition to the small list of deformation families of non-birational, double-mirror Calabi–Yau threefolds having Picard rank 1.

Can one construct a graph $G$ on the set of integer points ${\mathbb Z}^2$ in the plane such that the length of the shortest path between any two vertices of $G$ differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant $c>0$ such that for every $i=1,2,\ldots,$ every $\rho$-fat plane convex set $S$ can be cut into $2^i$ convex pieces of equal area, each of which is at least $c\rho$-fat. (A convex set is $\rho$-fat if the ratio of its inradius to its circumradius is at least $\rho$.) We prove that there exists an (unweighted) spanning subgraph $G$ of an enlarged copy of ${\mathbb Z}^2$ such that, for every pair of vertices at Euclidean distance $d$, their shortest-path distance in $G$ lies between $d-O(1)$ and $d+o(d^{5/6})$. The same bound can be achieved by a planar graph with vertex set ${\mathbb Z}^2$, in which every edge joins two vertices at Euclidean distance at most 2.

We study efficient algorithms for realizing the first-order oracle complexity of optimization of $G$-Lipschitz convex functions with respect to the $\ell_{q}$-norm over an $\ell_{p}$-ball of radius $R$, where $1\leq p,q\leq \infty$. For $p<q$, we obtain error $\widetilde{O}_{p,q}(GR/T^{1/p-(1/q-1/2)_{+}})$ after $T$ oracle queries, efficiently realizing the nearly optimal rates of (MBG+26), thereby resolving the nonsmooth end of the COLT 2015 open problem (Guz15b). In particular, the rate is $\widetilde{O}(GR/T)$ for Euclidean Lipschitzness over an $\ell_1$-ball of radius $R$ ($p=1,q=2$). Our solution consists of reducing convex Lipschitz optimization to the chasing nested convex sets problem in sublevel sets of an evolving bundle (LNN95; BBE+20): at each query we either find a point with low function value or we produce a deep cut in the current sublevel of the bundle, that we chase. The dichotomy between stability of selectors and forced movement by deep cuts bounds the number of iterations of the algorithm near optimally. For nested subsets of $R B_{p}^{d}$, we introduce a novel notion of stable center whose movement is bounded by $\widetilde{O}_{p,q}(RT^{1-1/p+(1/q-1/2)_{+}})$ in the $\ell_{q}$-norm after $T$ steps, which we show is nearly optimal in high dimensions. A Monte Carlo average of the proposed selector achieves near-optimal rates with high probability and can be implemented in polynomial time for our optimization algorithm in the real-arithmetic model.

We study the free minimization problem for a nonlinear, nonlocal functional associated with Robin-type nonlocal exterior conditions depending on a positive parameter $\alpha$. We prove that, for every $\alpha>0$, the minimizer exists, is unique, and satisfies suitable decay properties at infinity. We also investigate the regularity of the maps $\alpha \mapsto u_\alpha$ and $\alpha \mapsto E_\alpha$, where~$u_\alpha$ denotes the minimizer and $E_\alpha$ the corresponding energy. Finally, we derive asymptotic expansions of the energy as $\alpha\to+\infty$ and as $\alpha\to0^+$. The latter regime requires distinguishing between two cases, depending on whether the forcing term has zero mass. In one case, the limiting energy possesses a minimizer and $E_\alpha$ converges to its energy, but in the other case $E_\alpha$ diverges to $-\infty$.

Godsil, Roberson, Šámal and Severini conjectured that the quantum chromatic number of the categorical product of two graphs equals the minimum of the quantum chromatic numbers of the factors. We disprove this conjecture: we construct explicit finite graphs $G,H$ with \[ \chi(G\times H) \leq 1538 < 1539 = \min(\chi_q(G),\chi_q(H)).\]The graphs are obtained from Zhu's counterexample to Hedetniemi's conjecture by using a base graph for which the Lovász theta number of the complement, and not only the fractional chromatic number, is large. The lower bound for the first factor is the theta bound. For the second factor we adapt Zhu's argument to projections that do not commute: the step that fixes the colors of a clique is replaced by identities between operators. Both lower bounds hold for colorings by projections in an arbitrary nonzero unital $C^*$-algebra. Hence the conjecture also fails for the spatial, approximate, commuting-operator and $C^*$-algebraic variants of the quantum chromatic number. We also give smaller counterexamples certified by exact integer data. The graph constructions, the certificates and the counterexample statements in the projective formulation are formalized in Lean~4.

We show that manifolds admitting Zoll families of minimal hypersurfaces are diffeomorphic to $\mathbb{S}^{n}$ or to $\mathbb{R}\mathbb{P}^{n}$, and we characterize their Zoll families in both cases. Then, we prove that a metric on $\mathbb{R}\mathbb{P}^3$ is surface Zoll if and only if its projective widths are all equal. Our last result asserts that a closed surface with the first four widths equal to $2\pi$ is isometric to $\mathbb{R}\mathbb{P}^2$ with its constant curvature one metric.

We study first-order black-box convex optimization over an $\ell_p$-ball for objectives Lipschitz in the $\ell_q$-norm, solving in the affirmative the nonsmooth version of the COLT open question (Guz15b) on whether the geometry of a smaller feasible set ($p < q$) can improve convergence rates in convex optimization, and matching prior lower bounds up to logarithmic factors. Our rates include \(\widetilde O(1/T)\) for convex Euclidean-Lipschitz optimization over the $\ell_1$-ball, improving on the $O(1/\sqrt{T})$ classical rate under general assumptions. The key technical device is a new online learning game, where the comparator is evaluated using the maximum of affine losses observed so far. We bound the value of this game above and below in terms of a combinatorial online learning quantity: the sequential fat-shattering dimension, which we characterize for the $\ell_p / \ell_q$ case. Our results generally apply when the feasible set $X$ and the set of possible subgradients $H$ are convex, centrally symmetric, and admit a type of minmax theorem, advancing on a fundamental question by Sridharan [Sri12, Section 10.1.2, Q3]. As a geometric consequence of our analysis, of independent interest, we obtain estimates for the expected distance of a convex hull of samples to their mean in several Banach geometries, a version of the celebrated Wendel's theorem (Wen62), but quantitative and for bounded general distributions as opposed to centrally symmetric ones.

We study the large-$N$ asymptotics of a constrained spherical integral for non-Hermitian random matrices, in which the norms and mutual scalar product of two vectors are fixed. In the delocalized regime, the asymptotics are governed by the non-Hermitian transforms $\mathcal R_1$ and $\mathcal R_2$. At saddle-point level, the constrained integral exhibits a transition to a localized regime controlled by the largest singular value of a shifted matrix and by the overlap of its associated left and right singular vectors. Motivated by the Hermitian spherical-integral mechanism and by the Coulomb-gas picture, we formulate conjectures for one-eigenvalue large deviations and boundary fluctuations. Throughout, our analysis is carried out in the spirit of mathematical physics.

We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.

We prove that solution maps of the Vlasov–Poisson equation are discontinuous at the zero initial datum in $X_p=L_x^pL_v^\infty$ for every $1\leq p<\infty$, in $\mathbb{R}^{d} \times \mathbb{R}^{d}$ with $d \le 3$, and for both attractive and repulsive interactions. For every $T>0$, the trajectory of the solution in $L^\infty([0,T];X_p)$ is discontinuous at zero, and for every sufficiently small fixed $t>0$, the fixed-time map with values in $X_p$ is also discontinuous at zero. The counterexamples are smooth, nonnegative, bounded by one, and supported in a common compact subset of phase space. Their mass and initial $X_p$ norm tend to zero, whereas their solution norm at the observation time is at least one. The construction places narrow spatial packets on a lattice. Free transport allows a different velocity to select a packet at each spatial point, while a smallness estimate on the field ensures that the nonlinear characteristics are close to that of the free transport. For the same families, the phase-space $L^q$ norms converge to zero uniformly in time for every $1\leq q<\infty$.

In 2020 M. Mucciconi and L. Petrov introduced a long-range deformation of the quantum open non-relativistic Toda system. We prove the integrability of the deformed Toda system by constructing a $2 \times 2$ Lax operator, which produces the commutative family of differential operators containing the Hamiltonian of the deformed Toda system. Moreover, we show that the same integrable deformation exists on both classical and quantum levels and can be applied to both non-relativistic and relativistic Toda systems. For the open non-relativistic deformed Toda systems we also present an $n \times n$ Lax matrix and prove that it produces the same family of Hamiltonians. We also show how to obtain the van Diejen-type deformed Toda system. Lastly, we show that on the quantum level the algebraic Bethe ansatz technique can be applied to the deformed Toda system.

We prove RE-completeness of the quantum homomorphism problem parameterised by families of graphs derived from the classic metric association schemes. These include Kneser graphs, $q$-Kneser graphs, and the complements of Johnson, Grassmann, and Hamming graphs. Our proof develops a spectral method for establishing non-contextuality of quantum polymorphisms. It combines an equality analysis of Roberson's bound on the projective packing number in terms of Schrijver's theta with a structural argument inspired by Erdős-Ko-Rado theory.

We consider a Coulomb gas on a Jordan curve $\gamma$ in an external potential $V$ at inverse temperature $\beta>0$ and obtain an asymptotic expansion of the free energy up to $o(1)$ and a central limit theorem for linear statistics. We focus on the one-cut regime, where the density of the weighted equilibrium measure of $\gamma$ in $V$ is strictly positive on $\gamma$. The constant term in the (normalized) expansion consists of two parts: the Fredholm determinant of a generalized Grunsky operator and the Dirichlet energy of the logarithm of the density of the weighted equilibrium measure of $\gamma$. The coefficient of the latter vanishes for $\beta=2$. The variance of the fluctuations of the linear statistics only depends on the Dirichlet energy of the test function and is therefore independent of $V$. Essential in our approach is that the generalized Grunsky operator and the accompanying equilibrium parametrization allow us to transport the particles on the curve in the external potential to a reference object in a way that preserves the equilibrium measure. In our setting, the unit circle is the natural reference object.

Let $m \ge 4$. We prove that there is a forward orbit of the 2D Euler system for scale-invariant $m$-fold symmetric vorticities, namely \begin{equation*} \partial_tg + 2G\partial_\theta g = 0, \quad 4G + \partial_{\theta\theta}G = g, \end{equation*} which is dense in $S :=\{g \in L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1): g\sin(m\theta) \ge 0 \text{ and }\|g\|_{L^\infty} \le 1\}$ equipped with the topology of weak$^*$ convergence on $L^\infty(\mathbb{S}^1)$, where $L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1)$ denotes the space of odd and $m$-fold symmetric functions in $L^\infty(\mathbb{S}^1)$. The system above was first derived by Elgindi and Jeong, who showed well-posedness in the $m$-fold symmetric class. In subsequent work by Elgindi, Murray and Said, it was shown that solutions with regulated vorticity relax to piece-wise constant steady states with finitely many jumps; our construction shows that this regularity assumption cannot be relaxed. The proof starts from a simple linear mechanism: by evolving prescribed step functions backwards from chosen times and superimposing the resulting data, one can arrange for a single linear orbit to approximate a countable dense family. We show that this mechanism persists for the nonlinear Euler coupling. The key stability estimates imply that the error introduced at each stage becomes arbitrarily small when the next approximation time is taken sufficiently far in the future, allowing the construction to be iterated indefinitely.

We prove that every compact operator $T$ on a separable infinite-dimensional complex Hilbert space is a commutator of two compact operators, thereby answering a question of Pearcy and Topping. Moreover, the compact factors $A$ and $B$ can be chosen such that $[A, B] = T$ and $\max\{\Vert{}A\Vert{},\Vert{}B\Vert{}\}\leq c\Vert{}T\Vert{}^{1/2}$ for a universal constant $c$.

We develop a complex-analytic approach to the model theory of the (real) unrestricted exponential. As a consequence we derive sharp forms of many of the foundational results for the structure ${\mathbb R}^\text{RE}_{\exp}$ (and more general structures). In particular we establish sharp o-minimality, a sharp form of Wilkie's theorem of the complement, a sharp form of Wilkie's conjecture and a sharp form of piecewise definability by terms. Our approach is based on a complexification of the LE-preparation theorem of Lion–Rolin. We also develop a parallel complex theory for ${\mathbb R}_\text{an,exp}$, proving for example that the rational points of height $H$ on a nowhere-dense definable set can be interpolated by an algebraic hypersurface of degree $\text{poly}(\log H)$. This generalizes a theorem of Cluckers–Pila–Wilkie who proved the same statement for ${\mathbb R}_\text{an}^\text{pow}$.

In this paper we propose a reconstruction of the theory of multiple generality in Stoic Logic using the Natural Term Logic (NTL) developped by the author in \cite{ntl}. Contrary to a frequent misconception, it can be shown conclusively, based on the available evidence, that Stoic logic was far more than a mere propositional logic (we further argue that even identifying the Stoic conditional with any type of propositional connective is misleading). Its more general and complex logical theory - and notable the treatment of multiple generality - was rather patterned after the logico-syntactic mechanisms of natural language (subject to a regimentation which recalls Polish notation) rather than the variable-based quantifier logic of the modern Fregean tradition. NTL is likewise is a variable-free formal framework which captures the core syntactic and logic mechanisms of natural language (and notably a large array of features involving intensionality and anaphoric constructions). Bobzien and Shogry \cite{bobzienshogry} have presented abundant evidence and arguments for the case that quantified conditionals and multiple generality were treated by the Stoics through a disambiguifying regimentation of syntax using indefinite pronouns and anaphoric constructions involving such pronouns. There are however some patent difficulties involving how exactly the equivalent of universal quantification was treated. In this paper we explore some possible building blocks for a formal reconstruction of the Stoic logic of multiple generality based on a variant of NTL. We focus on formalizing the more general form of Stoic modus ponens involving quantified conditions which was clearly attested by ancient authors. We also sketch, as a preliminary study for formalizing the grammatical theory deployed in Stoic logic, how NTL can be used to formalize adjectival and adverbial constructions.

In this paper, we investigate the stochastic two-dimensional anisotropic magnetohydrodynamics system on $\mathbb T^2$ with velocity dissipation acting only in the horizontal direction. Under multiplicative noise, we prove the existence of global martingale solutions and global probabilistically weak solutions, together with pathwise uniqueness.

We study multi-type critical Bienaymé trees through the lens of the skeleton-blocks structure, seeing them as a monotype skeleton on which we have added multi-type blocks. This decomposition enjoys a remarkably elegant compatibility with the concept of local convergence, allowing us to find simple proofs of traditional local limit theorems.