Papers

In previous work by several authors, the behavior of the condition numbers of banded Toeplitz matrices was studied as the matrix size tends to infinity. In the present contribution, two main directions are pursued. As a first step, we extend this study to block Toeplitz matrices with blocks of fixed size $N$. As in the scalar case, we show that even when the symbol generates a Fredholm infinite Toeplitz operator, the condition numbers of the finite matrices may grow at least exponentially. Upper and lower bounds for the condition numbers are obtained, and examples showing that they may grow arbitrarily fast are presented. Then, as a second step, we apply the developed theory to the stability analysis of space-time Galerkin methods, where in time an Isogeometric approach is used with regularity $r$, $1\le r\le p-1$, $p$ being the employed polynomial degree. These stability issues are related exactly to the conditioning of block Toeplitz-like matrices with blocks of size $N=p-r$. Specific examples are treated in detail and related numerical experiments are presented and critically discussed. We finally present a short list of relevant open problems.

We exhibit an explicit closed seven-dimensional Riemannian manifold \[ (M,g)=\mathbb S^2(1)\times \mathbb S^5\left(\frac1{100}\right), \] where the displayed parameters denote sectional curvatures, for which \(\Ric_g>0\), and hence \(Q_g^{(2)}>0\). Moreover, \[ Q^{(4)}_g>0,\qquad Q^{(6)}_g>0, \] and the sixth-order GJMS operator \(P_{6,g}\) is strictly positive as a self-adjoint operator, but nevertheless \(P_{6,g}\) fails the strong maximum principle. The failure is caused by a nonconstant positive eigenvalue of \(P_{6,g}\) lying strictly below the eigenvalue of the constant mode. The example also has \(Y_2(M,[g])>0\) and \(Y_4(M,[g])>0\), while \(P_{6,g}\) does not have a positive Green function. It disproves Conjecture~1 of Andrade, Piccione, and Wei and its general-order formulation by Case and Gover.

Motivated by both the research on Lipschitz-free spaces and on Lipschitz harmonic functions on graphs, we study invariant pointwise closed subspaces of spaces of Lipschitz functions over graphs, with special emphasis on finitely generated groups as graphs. Such spaces form a natural class of $\text{weak}^*$-closed subspaces, that can be fully described in some cases, and hence have canonical quotient preduals of the corresponding Lipschitz-free spaces. We show that these spaces are described by finite local constraints; in the group case as Lipschitz solutions of systems of convolution equations. We describe and characterize their preduals via a universal property and show that whenever they contain a non-zero element with a $c_0$-gradient, then they contain $\ell_\infty$, and consequently their preduals contain a complemented copy of $\ell_1$. A guiding question is whether this class of Lipschitz spaces and their preduals contains an infinite-dimensional reflexive Banach space. In this regard, the main result of the paper is the following dichotomy proved using abstract harmonic analysis. Every translation-invariant pointwise closed subspace of the Lipschitz space over $\mathbb{Z}^d$ is either finite-dimensional or it is non-separable –in particular, the corresponding predual is either finite-dimensional or non-reflexive.

We study how irreducible factors associated with main eigenvalues constrain the connected components of generalized cospectral mates. Let $M_G(x)$ be the main polynomial of a graph $G$, and let $\kappa_{\mathrm m}(G)$ denote the sum of the multiplicities in $\phi_G(x)$ of the irreducible factors dividing $M_G(x)$. We prove that every graph $H$ with the same characteristic polynomial and main polynomial as $G$ satisfies $c(H)\leq \kappa_{\mathrm m}(G)$. Consequently, if $T$ is a tree and $H$ is generalized cospectral with $T$, then $\beta(H)=c(H)-1\leq \kappa_{\mathrm m}(T)-1$. We develop the case $\kappa_{\mathrm m}(T)=2$ in detail. Any disconnected generalized cospectral mate is the union of a tree and a connected bipartite unicyclic graph, and the coronals of its two components are uniquely prescribed by the canonical decomposition of the coronal of $T$ with respect to the two irreducible main factors. This turns the existence of a disconnected mate into a component-realizability problem. Factor moments yield realizability and tree-forcing obstructions, while matching data provide complementary characteristic-polynomial information. In particular, the unique cycle of any disconnected mate has length at least $6$, and finitely many matching identities, together with component-coronal realizability, certify generalized cospectrality under an explicit degree bound. As an application, we show that the double star $D(2m,m+1)$ is DGS but not DS whenever $m\geq2$ and neither $m$ nor $2m+2$ is a perfect square. In particular, $D(8t+4,4t+3)$, $t\geq0$, gives an explicit infinite family of such graphs.

Let $d \ge 2$ and $1 \le k \le d-1$. We show that if $f:\mathbb{R}^d\to\mathbb{R}$ is nonnegative and measurable and $0<m\le M <\infty$ then it is impossible that on almost all affine $k$-planes $P$ in $\mathbb{R}^d$ the integral of $f$ on $P$ lies between $m$ and $M$. Let $G = \mathbb{Z}^k \times \{0\}^{d-k}$. Using $m=M=1$ and $f$ being the indicator function of a measurable set in $\mathbb{R}^d$ this then implies that there is no measurable Steinhaus set for the group $G$ in $\mathbb{R}^d$. In other words there is no measurable set $S \subseteq \mathbb{R}^d$ such that $S$ tiles $\mathbb{R}^d$ with $T(G)$, for all $T \in O(d)$. We also generalize our impossibility results concerning the size of line-integrals of functions to integrals on strips in the plane.

Phase-field modeling of brittle fracture removes the need to track cracks explicitly by recasting their evolution as the minimization of an energy functional. In return it requires a discretization dense enough to resolve a localization band whose width is set by a regularization length and whose path is not known in advance. We propose a mesh-free discretization in which a single neural network represents the displacement and phase fields and is trained by minimizing the incremental energy directly. The coordinates enter the network through a multiresolution feature encoding built from $C^1$ quadratic B-spline grids, so the finest scale the representation can express is set by choice rather than reached through slow training, and the energy is estimated by stratified Monte Carlo integration on points redrawn at every optimizer iteration. This pairing proves critical, since the crack fails to advance both when the integration points are held fixed and when the encoding is too coarse to represent the band, while each ingredient tolerates a wide range of settings once the other is in place. Because the representation is globally $C^1$, the second- and the fourth-order fracture energy densities run on the identical discretization. Across six problems, from single-edge-notched tension and shear to a thick-walled ring on a single spline patch, the computed load-displacement curves follow staggered finite element references at matched regularization length, with peak loads within about 1% on the single-edge-notched tests and within 8% where the crack pattern changes topology. On a public benchmark dataset of random multi-crack configurations the method classifies the active or dormant state of 90% of the seeded cracks in twenty zero-shot runs, where the deep Ritz baseline of the dataset authors fails.

Results on holomorphic mappings of finitely connected planar domains are often proved using Koebe's famous circle mapping theorem. A prominent example is Julia's bound on the size of the automorphism group of finitely connected planar domains of connectivity $k\geq3$ and the sharp bound obtained later by Heins. The purpose of this article is to explore whether it is possible to give a proof of these results, and several other related results, without using Koebe's theorem. The main result is a rigidity theorem for proper holomorphic maps between finitely connected domains of connectivity higher than $2$ that immediately yields a bound on the number of proper holomorphic mappings between two such domains in terms of their connectivities. Using the same ideas, we also provide a new elementary proof of the planar Riemann–Hurwitz formula that does not use the Euler characteristic. Most of our proofs rely only on the Riemann mapping theorem, the Schwarz reflection principle, and the argument principle.

Let $C/K$ be a curve of genus 2 over an algebraically closed field $K$. Every such curve comes equipped with a canonical quadratic form $q_C$ called its refined Humbert invariant. In the case that the Jacobian $J_C$ of $C$ is isogenous to the self-product $E \times E$ for an elliptic curve $E/K$ with complex multiplication (CM), we provide an explicit formula for the finite number $N_C$ of isomorphism classes of genus $2$ curves $C'/K$ whose refined Humbert invariant is equivalent to $q_C$. This formula implies that $N_C$ is unbounded for such curves, and that there are only finitely many isomorphism classes of such curves $C/K$ with a given value of $N_C$. A key step in our approach is a characterization of when $J_C$ is isogenous to a self product of a CM elliptic curve, formulated purely in terms of properties of the refined Humbert invariant $q_C$. We establish that an analogous characterization also holds for superspecial curves of genus 2. The paper concludes with explicit examples illustrating cases where a genus 2 curve is uniquely determined by the invariant $q_C$.

Splines, which are piecewise polynomial functions with given smoothness, are used in numerous applications such as computer aided geometric design, algebraic geometry and for implementation of the finite element method. In many applications, it is important to know the dimension of the space of splines over a domain, but this continues to be an open problem. Strang's conjecture predicts the dimension of the space of bivariate $C^1$ splines of degree at most $k$ over a polygonal domain with no holes. For the cases, $k=4$ and $k \geq 5$ the conjecture has been proven true. For the case $k=2$, John Morgan and L. Ridgway Scott provided a counterexample. In this project, we begin by relating the space of bivariate $C^1$ splines of degree at most $k$ to various other function spaces. With the aid of some intrinsic operators on these spaces, we obtain a complex that yields an equivalence statement to Strang's conjecture. The equivalent statement admits a computational approach to the conjecture and thus we present an NGSolve code to check Strang's conjecture for a given mesh. In addition, we prove that if the simplicial complex associated to a triangulation is strongly collapsible (with a non-collinear condition), then Strang's conjecture holds for $k=2$.

Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincaré inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincaré spaces with the Euclidean results of Brezis et al. [Anal. PDE 17 (2024), 943-979]. The main novelty is that the finiteness of a weak-type norm, which only refers to differences or mean oscillations of $f$ without assuming any smoothness a priori, already guarantees the membership of $f$ in the relevant Sobolev or BV space. This distinguishes our contribution from the recent work of F. Dai et al. [Adv. Math. 502 (2026), Paper No. 111153], where the related norm-equivalence was obtained under the a priori Lipschitz assumption on $f$. A key intermediate step in our approach is a new localized Bourgain-Brezis-Mironescu type characterization. More precisely, we prove that, if $p\in(1,\infty)$ and $\gamma\in\mathbb R\setminus\{0\}$, then, for any $f\in L^1_{\mathrm{loc}}(\mu)$, \begin{equation*}\tag{$*$} \|f\|_{\dot W^{1,p}(\mu)} \sim \|\rho^{-1}\phi^{-\gamma}F\|_{L^{p,\infty}(\phi^{\gamma p}V^{-1})}, \qquad F\in\{\Delta f,m_f\},\quad \phi\in\{\rho,V\}, \end{equation*} where the homogeneous Sobolev space $\dot{W}^{1,p}(\mu)$ is defined by the minimal $p$-weak upper gradient and, for any $x,y\in X$, we denote $V(x,y):=\mu(B(x,\rho(x,y)))$ and $\Delta f(x,y):=|f(x) - f(y)|$, and $m_f(x,y)$ is the mean oscillation of $f$ on the ball $B(x,\rho(x,y))$. For $p=1$, the equivalence $(*)$ holds after replacing $\|f\|_{\dot W^{1,1}(\mu)}$ by a bounded variation norm and restricting the parameters to the optimal ranges $\gamma\in(-\infty,-1)\cup(0,\infty)$ for $\phi=\rho$ or $\gamma\in (-\infty,-\frac1d)\cup(0,\infty)$ for $\phi=V$, where $d\in(0,\infty)$ is the lower dimension of $X$.

In this study, we present a thermodynamically consistent framework for modelling polyatomic gases by integrating the general equation for non-equilibrium reversible-irreversible coupling (GENERIC) with an 11-moment description. Our formulation unifies rational irreversible thermodynamics (RIT) and extended irreversible thermodynamics (EIT) into a single GENERIC-compatible structure that guarantees compliance with the second law of thermodynamics through the explicit construction of Poisson and friction matrices. By including translational and internal temperatures, as well as a temperature tensor, the model offers a well-structured and physically meaningful representation of non-equilibrium behavior in polyatomic gases. The entropy production is inherently non-negative, and all closure relations emerge naturally from thermodynamic principles. As a test of applicability, we compute the shock wave structure in nitrogen gas, validating the framework against Direct Simulation Monte Carlo (DSMC) simulations and experiments up to moderate Mach numbers.

Let $G$ be a simple graph of order $n$ satisfying the following Ore-type condition: For any two nonadjacent vertices $x$ and $y$ of $G$, $d_G(x)+d_G(y)\geq n+k-2$, where $1\leq k\leq n-1$, $kn$ is even and $d_G(x)$ is the degree of $x$ in $G$. It is well known that $G$ has a $k$-factor for $k=1$ or $2$. Lu and Ning (J. Graph Theory, 94(2020), 307-319) proved that if $k\geq n/2$, then $G$ has a $k$-factor. In this paper, we show that $G$ has a $k$-factor for any $1\leq k\leq n-1$.

Fix an integer $d\ge1$. We prove that, for all sufficiently large $m$, every digraph $D$ with $m$ arcs and minimum semidegree at least $d$ admits a bisection $V(D)=V_1\cup V_2$ with $\bigl||V_1|-|V_2|\bigr|\le1$ such that \[ \min\{e(V_1,V_2),e(V_2,V_1)\} \ge \frac{d(m+d+1)}{2(2d+1)}. \] Here, for disjoint vertex sets $A,B\subseteq V(D)$, $e(A,B)$ denotes the number of arcs directed from $A$ to $B$. For each $d$, equality holds for infinitely many values of $m$, showing that the additive term $d+1$ is best possible. This answers a question of Liu, Ma and Zu by removing the asymptotic error in the minimum-semidegree bound, strengthens the result to the bisection setting, and determines the optimal additive correction. We also prove that every $n$-vertex digraph with $m$ arcs admits a bisection in which both directed cuts have size at least $(m-n+1)/4$, and that this universal bound is sharp for out-stars.

We investigate an inverse coefficient problem for a multidimensional free-boundary model of ductal carcinoma in situ (DCIS), in which the tumor interface is governed by the nonlinear coupling of nutrient concentration, tissue pressure and curvature, and the unknown nutrient consumption function is recovered from a temporal trace of the nutrient concentration obtained by needle aspiration biopsy. For the forward problem, we establish uniform local well-posedness over an admissible class of consumption functions. The inverse problem is recast as a fixed-point problem: approximating the admissible set by finite-dimensional spaces yields discrete iteration operators, for which we prove the existence of fixed points, and the strong convergence of a subsequence of discrete fixed points to a fixed point of the continuous operator, which solves the inverse problem under a consistency condition. To approximate these fixed points, we develop a homotopy-continuation method combining a linearly convergent Picard iteration with a cubical Sperner search, without differentiating an objective functional or computing an adjoint state. Several numerical experiments on radially symmetric and non-symmetric DCIS models corroborate the theoretical findings.

Motivated by dissipative approximations of the three-dimensional focusing energy-critical nonlinear Schrödinger equation, we study the inviscid and zero-dispersion limits for general energy-critical complex Ginzburg-Landau equations in dimensions $3\le d\le6$. We develop a new framework for these limits that does not require a global theory for the limiting equation. For limiting data in $H^1$, we prove strong convergence in the critical space, in particular in $L_t^\infty H_x^1$, on every compact subinterval of the maximal lifespan of the limiting solution. This includes strong $H^1$ inviscid convergence to the 3D focusing energy-critical NLS without higher regularity of the limiting data. The diffusion and nonlinear coefficients may vary independently, and both focusing and defocusing cases are treated. Both limits use coefficient-uniform critical stability. The inviscid argument uses low-frequency truncation to overcome the derivative loss at $H^1$, whereas uniform parabolic smoothing yields a quantitative $H^1$ estimate in the zero-dispersion limit. Moreover, for the inviscid limit, we prove the required homogeneous Strichartz estimates and obtain the retarded estimates by establishing a uniform retarded double-endpoint bound for the critical forcing space.

We study modular transformation formulas from the viewpoint of Mellin space. The basic observation is that the functional relation between the Hurwitz zeta function and the polylogarithm can be used to organize modular transformations as reflection identities of Mellin kernels, while the accompanying polynomial anomalies arise from contour deformations. We first illustrate this mechanism for the $q$-$\theta$ function and then extend it to the elliptic Gamma function. In the latter case, independently Mellin transforming the two elliptic directions leads to a trilinear reflection identity relating the three elliptic Gamma functions appearing in the SL$(3,\mathbb{Z})$ modular formula, while the associated contour deformation reproduces the cubic Bernoulli polynomial. Further reflection formulas lead to a weighted $q$-Pochhammer type function with a modular transformation analogous to that of the $q$-$\theta$ function, as well as a transposed trilinear reflection identity analogous to the Mellin space structure underlying the SL$(3,\mathbb{Z})$ transformation. Our results suggest that Mellin space reflection identities provide a useful organizing principle for constructing and studying modular special functions beyond the standard multiple elliptic Gamma hierarchy.

We study inequalities for the volume of orthogonal projections and their relation to Firey $L_p$-sum, together with their determinant-power analogues, motivated by the Dembo–Cover–Thomas conjecture. For $L_p$-zonoids $K,L\subset\mathbb{R}^n$ and $u\in S^{n-1}$, we consider the inequality \[ \left( \frac{|K\oplus_p L|} {|P_{u^\perp}(K\oplus_p L)|} \right)^p \geq \left( \frac{|K|}{|P_{u^\perp}K|} \right)^p + \left( \frac{|L|}{|P_{u^\perp}L|} \right)^p . \] For every $1<p<2$, we prove that this inequality fails in every dimension $n\geq2$. In contrast, the weak one-term inequality, obtained by omitting the second term on the right-hand side, holds in dimension two throughout the full range $1\leq p\leq2$. The proof of this planar result uses a sharp estimate for the normalized duality map. We also classify the corresponding determinant-power inequalities in the range $0<p<2$. The strong two-term inequality holds in dimension two and fails in every dimension $n\geq3$. The weak one-term inequality holds for $0<p\leq1$ in dimensions $n\leq3$ and fails for $n\geq4$; for $1<p<2$, it holds only in dimension two.

In this paper, we consider generalized Nash equilibrium (GNE) seeking in non-cooperative games with coupled constraint sets. Specifically, we aim to enforce safety for distributed GNE seeking, whereby the safety specifications are encoded in the coupled constraint set. To achieve this, we introduce the control barrier function (CBF) in the design of the GNE seeking dynamics. We design the dynamics for both full- and partial-information setting, where each player has knowledge of the decision information of all other players or only neighboring players, respectively. We justify the proposed dynamics by showing that the coupled constraint set is forward invariant, the equilibrium of the dynamics coincides with the exact GNE of the game, and the dynamics is asymptotically stable. Furthermore, we extend the approach to games where the agents are multi-integrators. Numerical simulations are provided to verify our results.

We study the vanishing-regularization limit of the regularized subjective-surface model introduced for touching and dividing cell nuclei and previously analyzed for fixed positive regularization parameters. On a smooth bounded domain $\Omega\subset\mathbb R^d$, the model is \[ \begin{aligned} \partial_tu^{\eps,\nu} ={}&\nu\Delta u^{\eps,\nu} +A_\eps(\nabla u^{\eps,\nu}) \operatorname{div}\!\left( G(x)\frac{\nabla u^{\eps,\nu}} {A_\eps(\nabla u^{\eps,\nu})} \right) -\mu R_\eta(x,u^{\eps,\nu}),\\ &A_\eps(p)=\sqrt{\eps^2+|p|^2}, \qquad R_\eta(x,r)=\Lambda(x)H_\eta(r-q), \end{aligned} \] subject to homogeneous Dirichlet data and a prescribed initial profile. We formulate the limiting weighted level-set mean-curvature equation as a viscosity initial-boundary value problem and establish global existence, uniqueness, preservation of $[0,1]$, and nonexpansive dependence on the initial data. We determine the direct upper and lower limiting values of the regularized principal operators at zero gradient and identify the additional argument needed to recover the geometric directional values. For every $T>0$, the regularized solutions converge uniformly to the unique viscosity solution on $[0,T]\times\overline\Omega$ as $\eps,\nu\to0$, independently of their relative rates of decay. This determines the geometric limit of the regularized model used in 3D and 3D+time microscopy segmentation.

For a non-smooth toric flop of the simplest type, we provide the $\textit{quantum product formulas}$ and construct the $\textit{degree-preserving}$ quantum correspondence via $\textit{generalized}$ analytic continuation which is achieved through a $\textit{regularization}$ map. Moreover, the generating functions of Gromov–Witten invariants with ancestors are invariant for $\textit{all genera}$.

Let $(M^3, g_{\mathrm{hyp}})$ be a closed oriented hyperbolic three-manifold normalized so that $\operatorname{sec}_{g_{\mathrm{hyp}}} \equiv -1$. We prove a sharp lower bound for the volume of hypersurfaces in $M^3 \times \mathbb{S}^1$ representing the slice class $[M^3 \times \{ \mathrm{pt} \}] \in H_3(M^3 \times \mathbb{S}^1; \mathbb{Z})$, and we classify the equality case. If $g$ is a smooth Riemannian metric on $M^3 \times \mathbb{S}^1$ with the scalar curvature $\operatorname{Sc}_g \geq -6$, then every closed embedded hypersurface $\Sigma$ representing the slice class $[M^3\times\{\mathrm{pt}\}]$ satisfies $\operatorname{vol}_g(\Sigma) \geq \operatorname{vol}_{g_{\mathrm{hyp}}}(M^3)$. The bound is attained by the product metric $g_{\mathrm{hyp}}+h$, with $h$ any metric on $\mathbb{S}^1$. Conversely, if equality holds for some $\Sigma$, then up to a diffeomorphism preserving the slice class, $g=g_{\mathrm{hyp}}+h$ and $\Sigma = M^3 \times \{\mathrm{pt}\}$.

Immersed finite element (IFE) methods provide an effective framework for solving interface problems on unfitted meshes. The basic idea underlying IFE methods is to modify standard finite element spaces by enforcing interface conditions at certain points. It is known that, for nodal degrees of freedom, unisolvence of linear IFE basis functions on triangular elements is subject to a non-obtuse-angle condition for scalar diffusion coefficients. This limitation is more severe for tensor diffusion coefficients, for which the non-obtuse-angle condition is no longer sufficient for unisolvence. Even on rectangular meshes, conventional bilinear IFE basis functions may not be unisolvent for tensor diffusion coefficients. In this paper, we develop and analyze a nodal isoparametric IFE method on quadrilateral and hexahedral meshes and show that the unisolvence issue can be overcome by appropriately selecting the enforcement point of the discrete flux condition. The key observation is that, unlike triangular elements, the discrete flux in quadrilateral and hexahedral elements is not constant, which provides the flexibility to select such an enforcement point to ensure unisolvence. We provide a systematic procedure for this selection that not only ensures unisolvence of the IFE basis functions on general quadrilateral and hexahedral elements with either scalar or tensor-valued diffusion coefficients but also preserves the optimal approximation properties of the resulting IFE space. The proposed IFE method offers several advantages: flexibility for complex geometries, the absence of angle restrictions, and applicability to tensor diffusion coefficients, thereby overcoming the limitations of existing nodal IFE methods on rectangular and triangular meshes. Optimal error estimates are established and confirmed by numerical experiments.

In this paper, after reviewing recent progress on the cohomology of $\overline{{\cal M}}_{0,n}$, we further our investigation on the cohomology of moduli spaces of pointed curves in continuation of [2,4,5,6,7,8]. In particular, we prove that the Betti number distribution of the Fulton-MacPherson compactification $C[n]$ of the space of $n$ ordered distinct points on any smooth projective curve $C$ is asymptotically Gaussian as $n$ goes to infinity.

A graph is called 10-tough and $(2P_2 \cup P_1)$-free if every vertex set whose deletion leaves at least two components has cardinality at least ten times the number of those components and if the graph has no induced subgraph consisting of two disjoint edges and an isolated vertex. We prove that every finite simple 10-tough $(2P_2 \cup P_1)$-free graph on at least three vertices is Hamiltonian. The proof splits according to whether some edge has joint neighbourhood of order at most $4n/11$. In the small-neighbourhood case, a matched path-cover is compressed to a prescribed matching. In the large-neighbourhood case, an asymmetric analysis of the two components left by a putative small cut yields the required connectivity bound. A Hamilton cycle through the prescribed edges is then expanded, and a cycle-extension lemma inserts the remaining vertices.

Immanants are a class of generalized matrix functions associated with the irreducible characters of the symmetric group. Bürgisser [SIAM J. Comput., 30 (2000), pp. 1023–1040] proved that the computation of hook immanants and immanants corresponding to rectangular Young diagrams of polynomially growing width is VNP-complete under $p$-projections. And he posed an open problem: whether the family of immanants corresponding to rectangular Young diagrams of width $2$ is VNP-complete under $p$-projections. This paper gives a solution to this problem. We prove that, over any field of characteristic zero, the immanant families associated with rectangular Young diagrams of width $2$ and of length $2$ are both VNP-complete under $p$-projections.

For primes $n\equiv1\pmod{24}$, we study the depth of explicit, factorization-free searches for a decomposition of $4/n$ into three positive unit fractions. Let $E_2(N;J)$ be the set of such primes $n\le N$ for which no coprime pair $1\le u,a\le J$ admits a divisor $s\mid an+u$ with $s\equiv-1\pmod{4ua}$. For every fixed $J$, we prove $|E_2(N;J)|\ll_J N/(\log N)^{1+A(J)/2}$, where $A(J)$ counts the coprime pairs in $[1,J]^2$. The proof uses a fixed-point-free involution and identifies the exact dimension of the covering. We also establish a dual Type I divisor criterion and an unconditional obstruction: at the fixed shift $c=7$, both branches of the cofactor criterion fail for $\asymp N/(\log N)^{3/2}$ primes. At growing depth, an exact reciprocal-totient asymptotic with leading constant $9/(2\pi^2)$ improves the explicit large-sieve bounds to $O(N\exp(-0.0942(\log N)^{2/3}))$ at cutoff $p>J^5$, and to $O_\varepsilon(N\exp(-(0.1093-\varepsilon)(\log N)^{2/3}))$ at cutoff $p>J^{4+\varepsilon}$ for fixed sufficiently small $\varepsilon>0$. Finally, combining a truncated Type I pass with a sparse residual search yields a complete, factorization-free batch decision procedure for all primes in $[N,2N]$, using $O(N^2(\log N)^2\exp(-c_0(\log N)^{2/3}))=o(N^2)$ arithmetic operations for some $c_0>0$. The conjecture itself remains open.

For two digraph properties $P_1$ and $P_2$, a $(P_1,P_2)$-arc-decomposition of a digraph $D$ is a partition $A(D)=A_1\mathbin{\dot\cup}A_2$ such that the spanning subdigraphs $D[A_1]$ and $D[A_2]$ have properties $P_1$ and $P_2$, respectively. We prove that it is NP-complete to decide whether a digraph admits an arc-decomposition with properties $(P_1,P_2)$ where $(P_1,P_2)\in \{$(perfect matching, having no odd cycle), (perfect matching, strong), (perfect matching, having an out-branching), (cycle factor, having no odd cycle)$\}$. All four problems remain NP-complete even when restricted to strongly connected digraphs. We also prove that every strongly connected digraph with minimum semi-degree at least three contains two vertex-disjoint cycles whose simultaneous arc deletion leaves the digraph strongly connected. These results settle some open problems posed by Bang-Jensen, Bessy, Gonçalves, and Picasarri-Arrieta [Theoret. Comput. Sci. 928 (2022), 167–182].

We propose a hybrid sharp-diffuse interface representation for modeling intermittent and isolated topological transitions during Cahn-Hilliard phase coarsening. Away from topological events, the evolution is approximated by the Mullins-Sekerka sharp-interface limit system and computed using a boundary integral formulation. When diffuse transition layers overlap, a novel interface surgery algorithm resolves topology changes through a localized Cahn-Hilliard pseudo-time evolution, after which the sharp-interface calculation is resumed. We develop the mathematical formulation underlying this decomposition, present a simple two-dimensional numerical implementation, and simulate a mass-exchange problem with interface coalescence. By localizing the diffuse evolution to topological events, the method achieves a speedup of two to three orders of magnitude over conventional diffuse-interface simulations.

Controllability of nonlinear systems has been extensively studied by using the Lie algebra methods. Although local controllability can be proved, global controllability is in general hard to obtain even for bilinear systems. Recently, a new approach is developed to study controllability of bilinear systems by checking whether the reduced rank points corresponding to the bilinear terms form hyperplanes or hypersurfaces, so that near-controllability, in the global sense, can be algebraically verified once no hypersurface exists. In this paper, we show that there is a flaw in the algorithm for checking the reduced rank hypersurface germ condition (RRHGC) that some special cases are overlooked. We thus propose a new algorithm for testing the RRHGC based on a Darboux-polynomial characterization, which can be used to deal with all cases. An example is provided to demonstrate the proposed new algorithm.

We show that various partitions of the canonical basis of quantum groups constructed by Lusztig and by Kashiwara coincide. Using this partition, we show that the subset corresponding to open Richardson varieties equals to the intersection of the subsets corresponding to the Schubert cells. We also show that in type $A$, the weights from open Richardson varieties are saturated in the corresponding Bruhat interval polytope.