Papers

A new robustness metric $\nu$ was recently proposed as a substitute for the structured singular value $\mu$ to better capture robustness to sparse diagonal uncertainty, but its computation has remained an open problem. In this paper, we show that computing $\nu$ is equivalent to maximizing the spectral radius of a nonnegative matrix product. This equivalence allows us to transfer existing results on spectral-radius maximization to $\nu$, including a refined upper bound and conditions under which the bounds coincide. We then provide reformulations and structural results that enable an algorithm to solve the nonconvex optimization problem for nontrivial problems using global solvers. This also enables identification of the most fragile parts of the system. Our results do not yet provide a scalable solution for computing $\nu$, but they are an important step toward computability and interpretability.

In this paper, we investigate Hamiltonian stationary Lagrangian currents with single-valued phases in open sets of Ricci-flat Kähler manifolds. We first establish the relation between phase and mean curvature for a class of integral Lagrangian currents in Ricci-flat Kähler manifolds. Then we derive several fundamental properties of integral Lagrangian currents with single-valued harmonic phases, including an almost monotonicity formula, an Allard-type regularity theorem, and the structure of limits of currents in this class. Finally, we apply the above results to study the regularity and rigidity of Hamiltonian stationary Lagrangian graphs in complex Euclidean space under certain conditions.

This paper is concerned with the parabolic-elliptic chemotaxis system involving weakly singular sensitivity and a nonlocal source: $u_t=\Delta u-\chi\nabla\cdot\left(\frac{u}{v^k}\nabla v\right) +u^\alpha\left(r-\mu\int_\Omega u^\beta\,dx\right)$ and $0=\Delta v-v+u^\gamma$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb{R}^N\) with \(N\ge1\), where \(\chi,r,\mu,\gamma>0\), \(k\in(0,1)\) and \(\alpha,\beta\ge1\). In view of the enhanced aggregation induced by singular sensitivity and the suppression of excessive population growth by the nonlocal damping, we identify sufficient conditions under which solutions remain globally bounded. We show that classical solutions are globally bounded in the following cases: \par\smallskip \begin{center} \small \setlength{\tabcolsep}{5pt} \setlength{\arrayrulewidth}{0.3pt} \renewcommand{\arraystretch}{1.25} \begin{tabular}{c|c|c|c} \multirow{2}{*}{$\beta>1$} & \multirow{2}{*}{$\gamma=1$} & $N=1$ & $1\leq\alpha<1+2\beta$ \\ \cline{3-4} & & $N\geq2$ & $\begin{gathered} 1\leq\alpha<2,\quad \alpha+\beta>2+\tfrac{N}{2},\\[-0.5mm] \text{or}\quad 2\leq\alpha<1+\tfrac{2\beta}{N} \end{gathered}$ \\ \hline \multirow{2}{*}{$\beta=1$} & $\gamma=1$ & $N=1$ & \multirow{2}{*}{ $\begin{array}{l@{\quad}l@{\quad}l} \text{Case 1:} & 1\leq\alpha<1+\tfrac{2}{N} & \text{if}\quad m_0<\tfrac{r}{\mu},\\[4pt] \text{Case 2:} & \alpha\geq1 & \text{if}\quad m_0\geq\tfrac{r}{\mu} \end{array}$ } \\ \cline{2-3} & $0<\gamma<\tfrac{2}{N}$ & $N\geq2$ & \\ \end{tabular} \end{center} \par \smallskip \noindent Here \(m_0:=\int_\Omega u_0\) denotes the initial total mass. It is worth noting that both the uniform-in-time \(L^p\)-estimate and the \(L^\infty\)-bound obtained by Moser iteration are derived without first establishing a uniform positive lower bound for \(v\).

Let $f\colon X\to\Delta$ be a smooth Kähler morphism from complex manifold $X$ to the unit disc. We prove that the canonical bundle $K_{X_t}$ is nef for every fiber as soon as it is nef for one fiber. This answers, in arbitrary dimension and in the Kähler setting, the deformation-openness problem for non-nefness of the canonical bundle raised by Campana and Peternell.

We study the algebraic structure of principal ideal codes and CSS-type stabilizer constructions over the ring $ R_q=\mathbb Z_q[i],$ where every prime divisor of $q$ is congruent to $1$ modulo $4$. Although such ring-based constructions have recently been considered in the context of classical and quantum error correction, the precise relationship among principal ideal cardinalities, Hermitian duality, algebraic cosets, and stabilizer syndromes requires further structural analysis. We obtain cardinality formulas for principal ideals and their annihilators. For a length-one principal ideal code $C=\langle\alpha\rangle$, we show that its Hermitian dual is $ C^{\perp_H}=\operatorname{Ann}(\sigma(\alpha)). $ For nested codes satisfying $ C_2^{\perp_H}\subseteq C_1\subseteq C_2, $ we determine the corresponding CSS-type stabilizer and prove that the kernel of the physical $X$-error syndrome map is $ \ker(\operatorname{Syn}_X)=C_2. $ Consequently, $C_2/C_1$ parametrizes logical $X$-operator classes and $R_q^n/C_2$ parametrizes physical $X$-error syndrome classes. Using this syndrome quotient, we construct a transversal of $C_2$ in $R_q^n$ whose elements have pairwise distinct physical $X$-error syndromes. The corresponding $X$-type error family is correctable by stabilizer syndrome measurement, yielding a syndrome-based recovery procedure that is consistent with the stabilizer structure. Explicit examples illustrate the resulting duality, cardinality, and syndrome structure.

We present an explicit construction of examples showing that the estimate $ \|S(f)\|_{L^2(w)\to L^{2,\infty}(w)}\lesssim \big([w]_{A_2} \log(1 + [w]_{A_2})\big)^{1/2}$ for the dyadic square function is sharp in terms of the characteristic $[w]_{A_2}$.

Identical particles have dynamically invariant labels in all setups where their degrees of freedom can be partitioned into two parts, which we call internal and visible, with the visible part subject to unitary evolution, while the dynamically invariant internal part accounts for the partial distinguishability of particles. We give the explicit form of the visible state in terms of the indistinguishability function, generalizing the standard symmetrization/antisymmetrization postulate for bosons/fermions with internal modes. The indistinguishability function on the symmetric group accounts for the permutation symmetry of the dynamically invariant label state and gives a self-contained description of the symmetry spectrum of the visible state. It is shown that the indistinguishability of single particles emitted independently from a stable source is completely characterized by the projective measures on the symmetry spectrum. We reveal a hierarchy of successive upper bounds on the projective measure of indistinguishability in terms of marginal projective measures. We point out an experimentally feasible way to get the upper bounds on the indistinguishability of an arbitrarily large number of single photons by direct experimental readout of their marginal projective measures. We also analyze the indistinguishability of coherent superpositions and convex mixtures and resolve a recently posed problem.

In this paper, we study the decomposition, in the caterpillar or weak-coupling limit, of a spectral curve with two irregular poles of order two, namely a spectral curve of type $(2,2)$, into two spectral curves each having one regular singularity(order-$1$ pole) and one irregular pole of order two, namely spectral curves of type $(1,2)$ or $(2,1)$. Such spectral curves can be identified with a double Gelfand–Zeitlin (DGZ) system. We show that the Duistermaat–Heckman measure on the DGZ side can be identified with the Weyl measure on the gauge-theory side arises from gluing two quivers. Furthermore, the Fourier expansion of the isomonodromic tau function can be interpreted as a Peter–Weyl expansion in the real polarization of the DGZ system. On the other hand, we introduce the nodal caterpillar spectral network. Applying the caterpillar spectral network, we then obtain an explicit expression for the Stokes matrix up to a normalization factor. This expression agrees with the existing analytic formulas. Moreover, we identify the variation of endpoint normalization arising in the Stokes computation with the equivariant Euler class of the fluctuation complex over the Coulomb moduli space. Finally, using the gluing technique for spectral curves and the DGZ system, we conjecturally predict the perturbative coefficients of the higher-rank $P_{\mathrm{III}}$ isomonodromic tau function from the viewpoint of spectral geometry.

Recent advances in multimodal foundation models have intensified the need to understand how different modalities share, preserve, and complement information. Mutual Information (MI), the Information Bottleneck (IB), and Partial Information Decomposition (PID) provide complementary perspectives, yet existing studies often treat them as isolated tools. This survey presents an information-theoretic perspective connecting these principles as progressively refined views of multimodal information processing: MI characterizes inter-modal dependency, IB explains task-oriented information preservation under compression, and PID decomposes preserved information into redundancy, uniqueness, and synergy. We review 170 recent studies (2018–2026) and 12 foundational works, organizing multimodal learning around four challenges: cross-modal alignment, information-efficient fusion, interaction-type characterization, and scaling to multimodal foundation models. Rather than using application domains as primary taxonomy axes, we interpret healthcare, robotics, recommendation systems, affective computing, and wireless communications as empirical validations of these information principles. Beyond taxonomy, we organize existing multimodal paradigms within a single information-theoretic coordinate system – the Generalized Multimodal Information Lagrangian – in which they occupy exact or approximate parameter corners, and whose unoccupied regions name candidate method families the literature has not yet built. We further discuss how emerging multimodal foundation models instantiate these principles at scale and identify open challenges including scalable information estimation in high-dimensional settings, standardized evaluation across information-theoretic methods, combinatorial complexity of multimodal PID, and the transition from post-hoc information analysis toward information-aware multimodal learning.

We prove that the orbifold fundamental group of the moduli space of smooth complex cubic surfaces is a hierarchically hyperbolic group. This places it within a powerful geometric framework whose prototypical examples are mapping class groups. The proof uses a known incomplete complex hyperbolic metric on the moduli space, and applies in a broader setting which includes a number of other moduli spaces.

V. Fedorchuk, A. Chizogidze, and T. Banakh in 2003 and V. Bogachev in 2024 posed the following questions: (i) is it true that $P_\tau(X)$ is $C$-embedded in $P_\sigma(X)$; (ii) Is it true that $P_R(X)$ is $C$-embedded in $P_R(\beta X)$ if and only if $X$ is pseudocompact, where $P_\sigma$, $P_\tau$, and $P_R$ are the functors of probability $\sigma$-additive on the Baire $\sigma$-algebra, $\tau$-additive, and Radon measures on the space $X$? The answers to these questions are negative. However, if instead of probability measures we consider the corresponding alternating measures $M_\sigma$, $M_\tau$, and $M_R$, the situation changes. It is proved that (i) $M_\tau(X)$ is $C$-embedded in $M_\sigma(X)$; (ii) $M_R(X)$ is $C$-embedded in $M_R(\beta X)$ if and only if $X$ is pseudocompact. The question of $C$-embedding of measure spaces is an extension of the question of coincidence of measure spaces, which is a development of the classical concepts of universally measurable and universal measure zero sets. A general theorem is obtained, which leads to the mentioned results.

This paper investigates Anderson localization in subwavelength elastic periodic systems with random perturbations. For the unperturbed system, we use layer potential techniques to reformulate the eigenvalue problem as boundary integral equations, and derive asymptotic formulas for the subwavelength eigenvalues. For perturbed systems, we apply the Floquet transform to obtain a periodic formulation and derive equations determining the resonant frequencies under general perturbations. Numerical experiments for perturbed periodic monomers and dimers agree with the theoretical predictions. We further demonstrate Anderson localization by increasing the strength and number of random perturbations. These results provide a mathematical foundation for understanding subwavelength localization in elastic metamaterials.

Two families $\mathcal{F},\mathcal{G}\subseteq S_n$ are cross $t$-intersecting if every $\sigma\in\mathcal{F}$ and $\tau\in\mathcal{G}$ agree on at least $t$ points. A $t$-coset is a coset of the stabilizer of $t$ points. A subset of $S_n$ is non-trivial if it is not contained in any $t$-coset. Let $d_m$ denote the $m$-th derangement number. We prove that, for all $t\geq1$ and $n\geq400t$, every pair of cross $t$-intersecting families $\mathcal{F},\mathcal{G}\subseteq S_n$ satisfies the following: (i) $|\mathcal{F}||\mathcal{G}|\leq((n-t)!-d_{n-t}-d_{n-t-1})((n-t)!+t)$ if $\mathcal{F}\cup\mathcal{G}$ is non-trivial. (ii) $|\mathcal{F}||\mathcal{G}|\leq((n-t)!-d_{n-t}-d_{n-t-1}+t)^2$ if both $\mathcal{F}$ and $\mathcal{G}$ are non-trivial. (iii) $\min\{|\mathcal{F}\setminus\mathcal{C}|,|\mathcal{G}\setminus\mathcal{C}|\}\leq t((n-t-1)!-(n-t-2)!)$ for some $t$-coset $\mathcal{C}$ if $t\geq2$. We also characterize all extremal configurations. The first result extends a theorem of Ellis (2011) to an exponentially wider range and sharpens the stability theorem of Keller, Lifshitz, Minzer and Sheinfeld (2024); the second gives a product version of the classical Hilton–Milner–Frankl theorem for permutations; and the third settles the remaining cases $t\geq2$ of a conjecture of Ellis (2011) in a stronger form. In all three results, the linear dependence on $t$ is essentially optimal. Our proofs are based on the spread approximation method introduced by Kupavskii and Zakharov and on an approach to cross $t$-intersection problems developed by the present authors, with several essential refinements. As an application of our approach, we prove a product version of the Hilton–Milner–Frankl theorem for the alternating group.

In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation \[ \det(\nabla u)=f\mbox{ in }\mathbb{R}^2, \] for a class of data. For every $f\in C_c^1(\mathbb{R}^2)$ and $p>1$, we construct a solution $u\in \dot W^{1,2p}(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2)$. In particular, no sign condition or integral constraint is imposed on the compactly supported data. For $f\in C_c^\infty(\mathbb{R}^2)$, the construction yields a smooth solution. We also consider rapidly decaying data and prove the existence of global solutions with bounded gradient for every $f\in\mathcal{S}(\mathbb{R}^2)$. Finally, we extend the construction to compactly supported data that are measurable in one variable and $C^1$ in the other. Our proof relies on a similar idea as in [Moser, Trans. Amer. Math. Soc. 1965].

We revisit a multiterminal lossy source coding problem named successive refinement and derive the rate-distortion-perception region under the strong-sense perfect perception constraint in the presence of unlimited common randomness. Specifically, in successive refinement, one aims to compress a source sequence and allows two distinct decoders to recover the source sequence at different distortion levels. By imposing the strong-sense perfect perception constraint, our results refine the previous result by analyzing the impact of the perceptual quality. Our achievability proof is inspired by output constrained lossy source coding and our converse proof adapts the proof steps of the standard successive refinement problem. Furthermore, we provide a numerical example of the Bernoulli source to illustrate our result and show that the Bernoulli source under Hamming distortion is successively refinable even with the strong-sense perfect perception constraint.

We study $\partial$-Morin maps, a class of smooth maps from manifolds with boundary such that both the maps themselves and their restrictions to the boundary have only Morin singular points, and the singular point sets of the maps are disjoint from the boundary. We develop a gluing construction that provides a unified framework for studying maps from manifolds with boundary through corresponding maps from closed manifolds. Using this framework, we derive Euler characteristic formulas for $\partial$-Morin maps and a congruence relating the signature to the self-intersection number of the singular point set for $\partial$-fold maps from $4$-manifolds to $3$-manifolds. We also establish an inequality relating singular fibers of stable maps from a $3$-manifold to the plane to the simplicial volume of the source manifold. We further apply these results to the existence problem for $\partial$-fold maps and to the non-singular extension problem.

We study the asymptotic null spectrum of Gaussian-weighted tests of univariate normality with estimated location and scale. Closed Hermite coefficients give equations both away from and at the unperturbed poles. Every positive eigenvalue of the fully standardized covariance is simple, and the odd and even eigenvalues alternate strictly. For each unperturbed even pole except the largest, there is exactly one weight parameter at which it belongs to the perturbed even spectrum. These parameters are strictly ordered and converge to the boundary of the parameter interval. A signed total-positivity argument proves simplicity more generally for consecutive Gaussian covariance corrections and positive even integrable weights. We also derive the limiting quadratic forms directly from degenerate kernels at standardized observations. A derivative summability condition controls the complete spectral tail, including signed kernels, and is verified for characteristic-function kernels with a finite fourth weight moment. A real Hilbert-space factorization identifies the observation and Fourier spectra. Numerical calculations assess asymptotic calibration.

An old conjecture in Hermitian geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature is Kähler when the constant is nonzero and Chern flat when the constant is zero. Yang–Zheng introduced the real bisectional curvature as a generalization of the holomorphic sectional curvature and conjectured that, a compact Hermitian manifold with constant real bisectional curvature has zero constant and is Chern flat. They proved that the constant cannot be positive. In this paper, we first prove their conjecture. For a negative constant, we show that the metric is pluriclosed and use comparison with a negative Kähler–Einstein metric to obtain a contradiction. For the zero constant, we combine a torsion identity of Lin–Ren with the Bochner formula of Zhou–Zheng to prove Chern flatness. We also prove that a pluriclosed metric with constant negative holomorphic sectional curvature on a compact Kähler manifold is Kähler.

We introduce high-order Winfree models for oscillatory associative memory. We characterize memory performance through the local-stability dichotomy between stored and unstored patterns. Using this criterion, we show that crosstalk noise drives a cascade of ferromagnetic, mixed, and spin-glass transitions as the memory load increases. For both polynomial and exponential Winfree models, we derive the phase boundaries and develop auxiliary-field modulation schemes adapted to the different phases. We further present corresponding circuit designs and numerical simulations that support our theoretical predictions.

Let $Q$ be a finite acyclic valued quiver. We establish an alternative cluster multiplication formula in the quantum cluster algebra associated with $Q$, which exhibits certain symmetries similar to 2-Calabi–Yau properties of cluster categories. Motivated by this formula, we introduce a Hall algebra $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ via a certain quotient of a derived Hall subalgebra associated with $Q$, serving as an algebra counterpart to cluster categories of hereditary algebras. As our main result, we prove that the quantum cluster algebra associated with $Q$ is isomorphic to the exceptional subalgebra of $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$, namely, the subalgebra generated by exceptional objects. Furthermore, when $Q$ is of Dynkin type, the quantum cluster algebra is naturally isomorphic to $\mathcal{D}\mathcal{H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ itself. This establishes a Hall algebra realization for all acyclic quantum cluster algebras. As an application, we introduce the Hall subalgebra of $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ generated by projectives, yielding a Hall algebra analogue of the lower bound quantum cluster algebras. Then we show that the standard monomials of quantum projective cluster variables form a basis of the quantum cluster algebras via Hall algebra approach. Moreover, we show that the Auslander–Reiten translation of cluster categories induces an algebra automorphism of $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$. Finally, we apply the BGP-reflections to $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$, and prove that the Hall algebra $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ is invariant up to isomorphism under the mutations at sink vertices of $Q$.

We prove quantitative instability results for the light ray transform in the Minkowski setting, and locally in the smooth and Gevrey categories. In the smooth Sobolev setting, we show that no modulus of continuity can be better than $t^\alpha$ for any $\alpha \in(0,1)$, while in the Gevrey setting we obtain an explicit logarithmic lower bound. On the other hand, using the relation between stationary geometries and magnetic(-potential) systems, we obtain a stability result for "time moments" of the light ray transform.

We establish Jacobi-type continued fraction expansions for (p,q)-analogues of the bi-Stirling Eulerian polynomials, which refine classical Eulerian statistics on permutations. These polynomials admit equivalent definitions in terms of descents or excedances. The parameters p and q encode refined permutation statistics, including (ldes,rasc) and (cros,nest). Our approach is based on weighted Motzkin path models arising from variants of the bijections of Françon–Viennot and Foata–Zeilberger. As applications, we obtain new results on total positivity and gamma-positivity for several families of enumerative polynomials. Our results unify and extend a number of recent works in the literature.

We introduce a new type of boundary condition for electrical networks by specifying a cohomology class on the underlying cell complex. As special and limiting cases, we recover the Dirichlet boundary condition, multiport condition, and prescribed voltage drops along nontrivial cycles on a surface. In this setting, we obtain a combinatorial formula for minors of the response matrix, generalizing the matrix-tree theorem, the Kenyon-Wilson formula, and a recently discovered formula for networks on surfaces due to Lam et al. To prove this formula, we develop a statistical-physics toolbox, including a new parafermionic observable for the uniform spanning tree model. This is the first parafermionic observable that is intrinsic, in the sense that it does not require an embedding of the graph in the plane or on a surface. Other ingredients are an intrinsic Temperley correspondence and source-synchronized networks, the latter providing a completely elementary reformulation of our results.

We give a counterexample to completeness descent from ideal cotorsion pairs to their objects. A radical-square-zero algebra on the two-cycle gives a finite-dimensional example. The construction is intrinsically non-weakly-idempotent-complete.

We establish a constructive and quasi-Banach framework for approximation spaces $A_\mu^\rho$ of compact $H$-operators between Banach and quasi-Banach spaces. By leveraging delayed Riesz means $V_{2^{bn}}$ generated by self-adjoint differential operators $P(D)$, we replace abstract best approximants with explicit linear operator decompositions. Furthermore, we extend the theory of $H$-operator approximation spaces to quasi-Banach settings ($0 < p < 1$), bypassing the collapse of Peetre's $K$-functional by employing localized moduli of smoothness $\omega_\varphi^r(f,t)_p$. Integrating seminal spectral bounds due to Markus \cite{M1966} and factorization techniques for operator ideals, we prove that operators belonging to $A_\mu^\rho$ possess complete systems of root vectors whose expansions are Abel summable.

A chemical transition state is a phase-space bottleneck: trajectories approach along stable directions and leave along unstable directions. Lagrangian betweenness (LB), introduced in finite-time transport theory for fluid flows, combines backward and forward deformation in a way suggestive of this gather-and-disperse geometry. We ask what LB measures when the transition-state geometry is known. For a linear rank-one saddle, LB is spatially constant even though the stable and unstable manifolds are present. Relative nonlinear corrections remain small on shrinking initial neighborhoods, including observation times on the logarithmic local escape scale. At fixed spatial resolution, however, increasing observation time can concentrate the spatial variation of LB near a stable or unstable manifold. A separable quartic Hamiltonian makes these two limits explicit. A nonseparable Hamiltonian shows local flattening near a hyperbolic periodic orbit. We then use HCN/CNH isomerization, where dividing surfaces and gap times are established by phase-space transition-state theory. Before any exit, larger LB generally accompanies longer eventual gap times. The incoming reactive action provides a dynamical interpretation: initial conditions closer to the normally hyperbolic invariant manifold have longer local passages and greater accumulated stretching during passage. Among trajectories still inside at 0.5 ps that exit before 5 ps, early LB does not usefully rank exit times. LB therefore characterizes transition-state-organized transport in this example; a large value is neither an intrinsic manifold marker nor a general measure of molecular residence time.

In this paper, we determine the nonpositive spectrum of the Yang-Mills Jacobi operator in Coulomb gauge at the standard $G_2$-instanton on $S^7$. We prove that the only negative eigenvalue is $-3$, with multiplicity $8$, and the kernel has dimension $20$. Thus the connection attains the sharp lower bound for the Morse index of nonflat Yang-Mills connections on $S^7$. We also give an explicit description of the kernal of the Jacobi operator, which contains Waldron's $15$-dimensional deformation space for the fixed nearly parallel $G_2$-structure. The 5 additional zero modes are isometry-induced variations that do not satisfy the corresponding linearized instanton equation.

A digraph $F$ is avoidable if, for every positive integer $k$, there exists an integer $d$ such that every digraph of minimum out-degree at least $d$ contains an $F$-free subdigraph of minimum out-degree at least $k$. We prove that no digraph admitting a height function is avoidable, where a height function increases by one along every arc. This answers a question of Christoph, Janzer, Petrova, and Steiner and, together with an additional avoidance argument, determines which orientations of $C_4$ are avoidable. Motivated by a further question of Christoph, Janzer, Petrova, and Steiner, we also study Eulerian avoidability, in which the host digraph is required to have equal in-degree and out-degree at every vertex. We show that no one-directed complete bipartite digraph with nonempty parts is Eulerian-avoidable. In contrast, the orientation of $C_4$ consisting of two directed paths of length two with common endpoints is Eulerian-avoidable. Consequently, the one-directed complete bipartite orientation is the only orientation of $C_4$ that is not Eulerian-avoidable, completing the classification of $C_4$-orientations.

We study spectral and nonlinear instability for some steady states of a two-dimensional inertial magnetohydrodynamic system on the torus. For equilibria with aligned velocity and magnetic fields, the linearized problem reduces to a matrix recurrence of Fourier coefficients, and a stable matrix continued fraction produces a smooth eigenmode. For a purely magnetic family, the square of the linearized operator admits a compact self-adjoint reduction on each transverse Fourier blocks that classifies the entire unstable spectrum. In an invariant symmetry class, we identify the unstable branches and a spectral gap below the leading eigenspace. The spectral gap then permits an approximate-trajectory construction that proves nonlinear instability on the logarithmic time scale. Our study is motivated by the physics literature, where the nonlinear instability of purely magnetic tearing modes is invoked to explain magnetic reconnection.

In an earlier work, Yau and Zheng proposed a Hermitian analogue of a weak form of the Frankel conjecture, which states that any compact Hermitian manifold with positive Strominger-Bismut sectional curvature must be biholomorphic to the complex projective space. In this article, we confirm the conjecture in complex dimension two. This is achieved by a vanishing theorem for anti-self-dual harmonic 2-forms on such surfaces, utilizing a curvature decomposition formula by Ferreira and a refined Kato inequality.