Papers

Let $\mathbf{G}=\{G_1,\dots,G_{n}\}$ be a collection of not necessarily distinct $n$-vertex graphs with a common vertex set $V$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq n$ is called rainbow in $\mathbf{G}$, if there exists an injection $\phi\colon E(C)\to [n]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(C)$. The graph collection $\mathbf{G}$ is said to be rainbow pancyclic if it contains a rainbow cycle of every length from 3 to $n$. In this paper, we show that if $e(G_i)\ge \binom{n-1}{2}+1$ for each $i\in[n]$ with $n\ge 3$, then $\mathbf{G}$ is rainbow pancyclic, apart from three explicitly described exceptional graph collections. This answers Problem $1$ of [Discrete Math., 348(2025), 114600] and strengthens the result from rainbow Hamiltonicity to rainbow pancyclicity. As a consequence, we obtain that if $\rho(G_i)>n-2$ for each $i\in[n]$, then $\mathbf{G}$ is rainbow pancyclic unless $G_1=G_2=\dots=G_n\cong K_1\vee(K_{n-2}\cup K_1)$, which improves Theorem $5$ of [Discrete Math., 348(2025), 114600]. We also characterize all graph collections that are not rainbow pancyclic under the condition $\rho(G_i)\ge n-2$ for each $i\in[n]$.

We study the following curvature equation with two singular sources on a flat torus $$ \Delta u+e^u=8\pi(\delta_p+\delta_{-p})\quad\text{on } E_\tau:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau), $$ where $\delta_p$ denotes the Dirac measure at $p$. The case $\wp'(2p)=0$ was solved by Kuo (J. Differential Geom. 2026). In this paper, we study the general case $\wp'(2p)\neq 0$. A novel phenomenon for $\wp'(2p)\neq 0$ is that the Hill discriminant of the associated generalized Lamé equation has an essential singularity. We develop new ideas to determine the intersection of two conditional stability sets completely, which is quite surprising since it seems impossible to determine the exact structure of these two sets due to the essential singularity. This leads to a sharp criteria for the existence and non-existence of solutions when $E_\tau$ is a rectangular torus and $\wp(p)\in \mathbb{R}$.

We study weighted pluricomplex extremal functions with a fixed weight and a variable logarithmic growth parameter. We show that the regular parameters at a point form the empty set, the smallest contact parameter alone, or the whole contact ray. For continuous weights, we represent the right parameter derivative by probability measures satisfying quasi-everywhere inequalities. This gives a criterion for equality with the unweighted Green function. We construct two counterexamples to preservation of weighted regularity, using smooth plurisubharmonic weights and polynomially convex compact sets that are non-pluripolar and non-plurithin at every point. For the product example, we compute the derivative on the entire exceptional polydisc through planar Green functions and determine exactly where equality holds. The second example is globally pluriregular but has a positive right derivative. We also characterize the points where local pluriregularity fails without plurithinness, using negligible subsets and the plurifine topology, and prove preservation of regularity under local strict plurisubharmonicity of the extending weight.

In this paper, we study the homological critical points for persistence hypergraphs and persistence simplicial complexes. With the help of the relative homology groups, we define the homological critical points for persistence morphisms between persistence hypergraphs and persistence simplicial maps between persistence simplicial complexes. We prove some commutative diagrams of subset relations for the homological critical points of persistence hypergraphs as well as persistence morphisms between them. We also prove some commutative diagrams of subset relations for the homological critical points of persistence simplicial complexes as well as persistence simplicial maps between them. As examples, we use the parametric configuration spaces to construct the parametric independence complexes as the space of parametric packings and construct the parametric dominating hypergraphs as the space of parametric coverings.

We construct a transferred $L_\infty$-structure on the shifted Chouhy–Solotar cochain complex of a finite-dimensional skew-gentle algebra and give explicit basis-level formulas for its binary operation $l_2$ in all shifted degrees. We further prove that the Hochschild dg Lie algebra of an $A_n$-type skew-gentle algebra is homotopy abelian, and hence $L_\infty$-formal, and contrast this with Wong's non-formality result for the dual numbers $\mathbb{K}[\theta]/\langle \theta^2 \rangle$, leading to a general formality classification problem.

We provide direct constructions of nontrivial root elations of thick generalized quadrangles embedded in thick polar spaces of higher finite rank and show that all of these can be expressed as self-projectivities of length $4$. This reveals a connection between the little projective groups and the special projectivity groups of thick polar spaces of finite rank explicitly. We also prove that the constructed root elations extend to the ambient polar space. Furthermore, we introduce point elations and line elations and use them to give an equivalent definition of the Moufang property for thick polar spaces, without using the notion of apartments. This, in turn, allows us to define the Moufang property for thick polar spaces of infinite rank. Consequently, we show that all thick polar spaces of finite and infinite rank are Moufang.

In this paper, we construct nonunital universal enveloping Hom-associative algebras for multiplicative Hom-Lie algebras equipped with $\alpha$-derivations. We prove that the prescribed derivation extends to the enveloping algebra and that the resulting universal property yields a left adjoint to the commutator functor between the corresponding categories. When the twisting map is bijective, we establish a canonical isomorphism between this enveloping algebra and the twist of the nonunital universal enveloping algebra of the associated untwisted Lie algebra. This isomorphism preserves the twisting maps and the derivations. Consequently, the classical Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra for the untwisted product with the positive-degree symmetric algebra. When the Hom-Lie algebra is finite-dimensional, the resulting PBW basis consists of ordered monomials of positive length. The twisting map, the extended $\alpha$-derivation, and its untwisted counterpart preserve the PBW filtration.

Let $m,n\ge 2$. For $m\times n$ complex matrices $A$, $C$ and an $n\times m$ matrix $B$, Nobori (Linear Algebra Appl. 725 (2025) 135–144) proposed the following conjecture on the generalized Böttcher-Wenzel inequality: $$ \|ABC-CBA\|_F^2 \le 2\|B\|_2^2\|A\|_{(2),2}^2\|C\|_F^2, $$ where $\|\cdot\|_F$ is the Frobenius norm, $\|\cdot\|_2$ is the spectral norm, and $\|\cdot\|_{(2),2}$ is the $(2,2)$-norm defined by $\|X\|_{(2),2}=\sqrt{\sigma_1(X)^2+\sigma_2(X)^2}$, in which $\sigma_1(X)$ and $\sigma_2(X)$ are the largest and second largest singular values of $X$, respectively. This paper proves the conjecture and, as a consequence, establishes the Kronecker product inequality that Nobori derived from it. In addition, we show that for each fixed $m,~n\ge 2$, the constant $2$ is sharp. We also give necessary and sufficient conditions for equality, namely, the equality condition in the Böttcher–Wenzel inequality and some matrix identities derived from a fixed singular value decomposition.

The proximal point algorithm for finding a zero of a maximal monotone mapping is the engine that drives augmented Lagrangian methods and various splitting schemes in convex optimization, where proximal terms that are merely positive semidefinite can be helpful in preconditioning. Here a semidefinite extension of that algorithm is developed with new features which guarantee convergence, typically even linear convergence, in broader territory than previously foreseen. Moreover, inexact minimization is allowed in the subproblems in a variable-metric implementation which might open up quasi-Newton-like approaches. The key to the extension is applying an advanced form of the positive definite proximal point algorithm to a projected monotone mapping derived from the given mapping, and combining that with a completion step. For this to work, the maximality of the projected mapping is essential, but maximality is not automatic. The circumstances that guarantee maximality are definitively analyzed and convenient criteria for maximality are thereby identified. Consequences of the extended algorithm are worked out for application to convex inf-projection, augmented Lagrangian steps, and bracketed progressive decoupling based on Spingarn's partial inverse as a scheme for problem decomposition.

We prove weak existence and uniqueness in law of $[0,1]$-valued solutions for a class of scalar stochastic heat equations with measurable drifts $b : [0,1] \to \mathbb{R}$, driven by Wright–Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities $-Cu\leq b(u)\leq C(1-u)$ for some $C<\infty$, compatible with the necessary condition $b(1)\leq 0\leq b(0)$. These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.

Let $R$ be a continuous von Neumann regular ring with no nonzero abelian idempotents. We prove that every element of $R$ is a single additive commutator. In particular, every operator affiliated with a type $\mathrm{II}_1$ von Neumann algebra is a commutator of operators affiliated with the same algebra. Our commutator theorem, together with a theorem of Jaikin-Zapirain and López-Álvarez, implies that, if $R$ is a nonzero unital algebra over a field $\mathbb{K}$ of characteristic zero, the quotient division ring of the first Weyl algebra over $\mathbb{K}$ embeds unitally into $R$. We also give a direct proof of this consequence.

Let $M$ be a complete, non-compact Riemannian manifold. We prove that its Riesz transform is of weak type $(1,1)$, with constant $2$ for real-valued functions. Consequently, it is bounded on $L^p(M)$ for $1<p\leq2$, with constants depending only on $p$, which proves the Coulhon–Duong conjecture. The proof uses an obstacle decomposition for positive self-adjoint operators with sub-Markovian semigroups. Applying this decomposition to the shifted square-root Laplacian and using locality of the Sobolev differential yields the endpoint estimate without geometric or heat kernel assumptions. We also obtain the corresponding result for Dirichlet spaces admitting a local Hilbertian differential calculus.

In this paper, we study a finite-data inverse nodal optimization problem for radial Schrödinger operators in an arbitrary fixed angular-momentum sector. The analysis is built directly at the Friedrichs endpoint and in the physical weighted space $L_d^p$, so that the singular radial geometry is retained rather than replaced by a regular one-dimensional model. The main purpose of this paper is to provide a singular Friedrichs finite-data variational framework valid in every angular-momentum sector, thereby extending the existing finite-data variational theories concerning either regular one-dimensional operators or the radial sector $\ell=0$. By means of a Volterra representation of the Friedrichs branch, we prove weak continuity and continuous Fréchet differentiability of nodal radii, exact realization of compatible same-mode nodal data, existence of optimal potentials, and finite-codimensional constraint geometry. The same framework also incorporates mixed angular-momentum and spectral–nodal observations through finite-dimensional transversality. Remarkably, a global uniqueness theorem is established for inward displacements of the unique interior node of the second mode in the radial sector \(\ell=0\). For a constant reference potential and \(p>(d+2)/2\), every such displacement admits a unique global optimizer. Unlike local inverse-mapping or one-dimensional integrability arguments, the proof first selects the admissible critical sign globally and then reduces every minimizer to a scalar mass-balance equation between a focusing ball branch and a logistic annulus branch. The strict opposite monotonicity of the two weighted masses makes the balance parameter unique, providing a global rigidity mechanism over the entire inward-displacement regime.

We construct closed embedded ${\lambda}$-hypersurfaces in ${\mathbb{R}^{n+1}}$ of topological type ${S^1\times M}$ for any ${\lambda \ge 0}$, where ${M\subset\mathbb{S}^n}$ is an isoparametric hypersurface with equal principal curvature multiplicities. This extends previous results of Angenent, McGrath, and Riedler (for ${\lambda = 0}$) as well as Cheng-Wei and Ross (for ${\lambda > 0}$), and in particular completes the missing case of Riedler's construction.

We introduce various classical, foundational results on the regularity theory of harmonic maps, with focuses on the energy minimising harmonic maps. Topics include inner and outer variations, the monotonicity identity, the $\varepsilon$-regularity theorem, blow-ups and tangent maps, and the dimension estimate and stratification of singular sets. We also discuss the analogous theory for stationary harmonic maps and mention some significant recent developments.

We study Borel-Tukey morphisms between almost-homogeneity relations. For finite $r,s\ge1$ and $m,n\ge2$, we give the complete one-stage classification: $\mathbf{Hom}^r_m\to_{\mathrm{BT}}\mathbf{Hom}^s_n$ holds exactly when $r>s$, or when $r=s$ and $m\ge n$. Our main result is that binary triple homogeneity requires exactly three sequential binary pair stages: $\mathbf{Hom}^2_2;\mathbf{Hom}^2_2;\mathbf{Hom}^2_2\longrightarrow_{\mathrm{BT}}\mathbf{Hom}^3_2$ and $\mathbf{Hom}^2_2;\mathbf{Hom}^2_2\not\longrightarrow_{\mathrm{BT}}\mathbf{Hom}^3_2$. We also construct Borel morphisms $\mathbf{Hom}^2_2;\mathbf{Hom}^2_3\longrightarrow_{\mathrm{BT}}\mathbf{Hom}^3_2$ and $\mathbf{Hom}^2_2;\mathbf{Hom}^2_2\longrightarrow_{\mathrm{BT}}\mathbf{Hom}^2_k$ for every finite positive $k$.

This paper establishes the existence of local martingale solutions for a three-dimensional stochastic fluid–structure interaction (FSI) problem coupling a viscous compressible isentropic fluid with an elastic shell through Navier-slip boundary conditions. The fluid and shell are perturbed by compatible Stratonovich transport noise. A stochastic-flow transformation removes the stochastic transport integrals and produces a pathwise random FSI system whose pullback coefficients are only Hölder continuous in time. We combine a multi-layer approximation scheme with localization on bounded-flow events and freezing of the random coefficients on short deterministic time subintervals. The shell transport fields are not required to generate isometries: the bending-energy martingale and Itô correction are controlled at the natural $H^2$ level by commutator identities without derivative loss. Tightness is obtained in Jakubowski spaces, pressure compactness is proved through localized effective viscous flux, and the final artificial-pressure and penalty limit uses boundary-pressure tightness and $H^2$-graph trace tools. For $\gamma>3/2$ and non-colliding initial data, we obtain a martingale solution up to an almost surely positive stopping time.

We prove that for all real numbers $\kappa,\tau$ with $\kappa>8\tau^2$ and $\tau\neq 0$, the Berger sphere $\mathbb{E}^3(\kappa,\tau)$ contains smooth closed embedded rotational surfaces of genus one with constant mean curvature which are contained in an open hemisphere. This disproves a conjecture of Fernández and Mira. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system, and then verified by the authors.

We study the nonlinear evolution equation $$ \partial_t u+(G+G^\delta)u=I_\alpha(|u|^\rho) \quad\text{on }\mathbb{R}^{N+k}, $$ where $G$ is the nonnegative self adjoint realization of Grushin operator, $0<\delta<1$, and $I_\alpha$ is the potential operator with kernel $|z|^{-\alpha}$, $0<\alpha<N+k$. We develop a well-posedness theory for initial data in Marcinkiewicz spaces, allowing singular profiles outside the corresponding Lebesgue spaces. Using spectral calculus, subordination, and interpolation, we establish Lebesgue and Lorentz smoothing estimates for the mixed semigroup $e^{-t(G+G^\delta)}$, which distinguish the second-order behaviour at short times from the fractional decay at large times. In the subcritical regime, considering $$ \frac1\beta=\frac2Q+1-\frac{\alpha}{d}, \qquad \frac1{\beta_\delta}=\frac{2\delta}{Q}+1-\frac{\alpha}{d}, $$ where $Q$ is homogenous dimension, we prove local existence, uniqueness, Lipschitz dependence on the initial data, and a blow-up alternative in $L^{p,\infty}$ when $$ \rho<1+\frac{p}{\beta}. $$ At the critical relation $\rho=1+p/\beta$, a Yamazaki-type estimate and Lorentz duality yield global mild solutions for sufficiently small data in $L^{p,\infty}$. In the supercritical regime $\rho>1+p/\beta$, well-posedness is recovered in higher-integrability Marcinkiewicz spaces $L^{q,\infty}$ satisfying $$ \beta(\rho-1)<q<\beta_\delta(\rho-1), $$ with global existence for sufficiently small initial data. In all cases, the solutions attain their initial data in the weak-$*$ sense. These results extend the Marcinkiewicz-space theory to mixed Grushin diffusion with a spatially nonlocal potential source and reveal the role of the two competing diffusion scales in determining the admissible integrability regimes.

We provide a complete algebraic characterization of homogeneous self-affine measures whose Fourier transform does not tend to $0$ at infinity. Moreover, we prove that an integral self-affine measure has power Fourier decay if and only if it is absolutely continuous. For a given integral affine IFS, we also provide an algorithm for determining those probability vectors for which the associated self-affine measure is absolutely continuous.

We identify a directional tangency mechanism responsible for anomalous far-field decay in Schrödinger equations. For constant-coefficient operators, we introduce a directional decomposition of the limiting resolvent based on the geometry of the Fermi surface and the observation direction. In two dimensions, we show that an analytic source produces a super-algebraically small tangency contribution, whereas a transverse non-analyticity located on the directional tangency set generates an algebraic contribution of order \(R^{-(2\alpha+1)}\). Shifting the singularity away from the tangency set restores super-algebraic decay, showing that geometric alignment is essential to the anomalous behavior. We further construct a bounded, decaying real-valued potential for which the same mechanism occurs in the first Born correction through the effective source \(Vu_0\). In a suitable long-range regime, the resulting tangency contribution decays more slowly than the conventional two-dimensional far-field term.

A Yau-type gradient estimate is proved for positive $p$-harmonic functions on complete Riemannian manifolds with compact boundary, under a Ricci lower bound on the Riemannian manifold and a mean curvature lower bound on the boundary, assuming the Dirichlet condition and a sign condition on the outward normal derivative. The result extends the estimate for harmonic functions by Kunikawa and Sakurai to the full range of $p$-Laplace operators and yields a Liouville theorem when the curvature hypotheses are nonnegative. The cutoff method of the linear theory does not extend when the exponent differs from two, since it requires a directional Hessian of the distance to the boundary that Laplacian comparison cannot control. That cutoff is replaced here by a radial barrier at the boundary and an intrinsic maximum principle in the interior.

For connected closed Lagrangian submanifolds of the homogeneous nearly Kähler sphere $\mathbb S^6(1)$, we first establish a new integral inequality. Then, as applications we prove sharp pinching theorems for the scalar curvature $\tau$ and Ricci curvature Ric, respectively: $\tau \ge \tfrac{23}{8}$ implies either $\tau=6$ with it totally geodesic, or $\tau=\tfrac{23}{8}$ with it congruent to the embedded Berger sphere of Dillen–Verstraelen–Vrancken; whereas ${\rm Ric}\geq\frac18$ implies that the submanifold is either totally geodesic, or has constant sectional curvature $K=\tfrac{1}{16}$, or is congruent to the embedded Berger sphere of Dillen-Verstraelen-Vrancken.

Using N. Honda's construction of twistor spaces, bimeromorphic invarince of $\partial\bar\partial$-lemma for threefolds and embedded resolution, we find a smooth compact complex threefold satisfying the $\partial\bar\partial$-lemma that contains a smooth compact surface for which the lemma fails. This disproves heredity in the smallest possible ambient dimension and yields counterexamples to L. Alessandrini's modification question in every complex dimension at least four. We establish a derived blow-up formula for generalized Bott–Chern complexes with coefficients, compatible with the comparison maps to coefficient cohomology. We also obtain a Dolbeault blow-up formula with bounded derived coefficients. For compact complex manifolds, we introduce a relative $\partial\bar\partial$-property, characterize it using the Frölicher spectral sequence and Hodge filtrations, and prove a blow-up criterion. Finally, counterexamples show the limitations of the classical formulae for coherent coefficients and singular ambient spaces, and the failure of the naive Bott–Chern Künneth formula.

We study exact sign recovery for PCA connectivity analysis in mixture models, using the projection onto the first $(K-1)$ principal components. For any fixed number of components with affinely independent means, we derive the oracle signal projection explicitly. Under anisotropic sub-Gaussian noise, with a random number of observations in each component, a rowwise eigenspace bound yields a sufficient condition, governed by the weakest signal direction, for recovering all connectivity signs without requiring a bounded condition number of the nonzero signal spectrum. Under isotropic Gaussian noise, when the configuration of the component means is fixed up to scaling and orthogonal transformations, we identify an asymptotically sharp threshold for the signal strength: exact sign recovery succeeds above the threshold and fails below the threshold, with probability tending to one when the signal strength stays a fixed relative distance from the threshold. The threshold has an explicit formula when the mixing proportions are equal and the means form a regular simplex.

We establish a resolvent factorization of the hyperbolic projection associated with the finite-range dual operators, by a kernel with exponential off-diagonal decay. As an application, we prove that, for every irrational frequency and every analytic potential, each subcritical Type I energy satisfying the gap-labelling condition is an endpoint of an open spectral gap. This confirms the conjecture of Ge–Jitomirskaya–You [GJY,You] in the subcritical regime.

Given any norm $\|\cdot\|_{\mathbb R^2}$ on $\mathbb R^2$, we prove that the Lagrange spectrum for $2$-vectors contains a Hall ray. Equivalently, every sufficiently small $\rho\ge0$ occurs as $\liminf_{q\to\infty}q^{1/2}\min_{p\in\mathbb Z^2}\|qx-p\|_{\mathbb R^2} $ for some $x\in\mathbb R^2\setminus\mathbb Q^2$. Combined with a recent result of Kleinbock \cite{Kleinbock}, this implies that the $2$-dimensional Lagrange spectrum is either a ray, mirroring the behavior for Dirichlet spectra, or a ray preceded by a dense part.

Two-dimensional irregular bin packing combines combinatorial bin-assignment decisions with difficult geometric feasibility constraints, making exact optimization challenging. This paper develops an exact combinatorial Benders decomposition framework for the two-dimensional irregular bin packing problem with convex polygons, coupling a pattern-based master problem with an exact single-bin geometric feasibility oracle. The dynamically strengthened Benders master is solved by a tailored exact branch-and-price procedure that incorporates objective-layered search and adaptive exact pricing. Geometric information obtained from the oracle is further fed back to the master and pricing processes through dynamically generated Benders feasibility cuts, which progressively restrict the subsequent pricing problems. Computational experiments are conducted on 540 benchmark instances from 18 classes. The proposed method obtains the optimal solution for 318 instances across 12 classes within a 3600-second time limit. On these classes, the proposed method solves more instances than a direct mixed-integer programming formulation and a baseline combinatorial Benders decomposition approach.

Quantum teleportation and Bell measurements are considered in a noisy relay setting. We introduce two quantum relay-channel models with closed-form capacity formulas, motivated by the canonical quantum communication protocols of teleportation and superdense coding. In both models, the sender transmits a qubit, while the relay observes side information about the Pauli errors that occur in the channel and communicates with the receiver through an orthogonal rate-limited link. We develop a compress-forward coding scheme in which the relay compresses the Pauli-error sequence into bins and sends the bin index to the receiver. The receiver uses this partial error information to reduce the effective channel noise before quantum decoding. We show that this strategy is optimal by establishing matching converse bounds for both models.

We study the nonlinear stability of a composite wave consisting of a traveling wave and a rarefaction wave for a parabolic-hyperbolic system arising from chemotaxis. We prove that if the initial value is a small $H^1$-type perturbation of composite wave, then the system admits a global solution that converges toward the composite wave with an absolutely continuous shift. The proof combines the weighted relative-entropy mechanism for viscous shocks with the energy structure of rarefaction waves. A key ingredient is the inclusion of a rarefaction modulation factor in the weighted relative entropy; its derivatives combine with the terms generated by the rarefaction profile to produce a rarefaction dissipation. Moreover, the spatial separation of the two waves yields time-integrable interaction errors caused by the non-exact superposition. Since diffusion acts only on the density component, the full $H^1$ estimate is closed by exploiting the coupling structure of the system to recover the missing dissipation of the hyperbolic component.