Papers

We study the problem of realizing line configurations as collections of 2-spheres smoothly embedded in the complex projective plane. Building upon prior work by Ruberman and Starkston on topological realizations, we establish a stronger obstruction in the smooth category. Our proof relies on lattice-theoretic arguments based on Donaldson's diagonalization theorem.

Maximizing a symmetric quadratic form under simultaneous L1 norm inequality and L2 norm equality constraints is a standard and widely used continuous relaxation for Sparse Principal Component Analysis (SPCA). This paper settles the computational complexity of this continuous formulation by proving it is NP-hard. Furthermore, the variant with both L1 and L2 norm inequalities is also shown to be NP-hard.

In this paper, we study the problem of optimizing Markov processes that interpolate between two prescribed probability distributions while minimizing a given cost. The main computational challenge is the curse of dimensionality: in high-dimensional state spaces, representing the full distribution is intractable. To address this, we reformulate the problem in terms of sequential couplings and develop convex relaxations based on local marginals and cluster moments. These relaxations exploit locality and sparse interaction structure, provide computable lower bounds, and recover low-order statistics of the intermediate laws. We identify dynamic optimal transport as a special case of our Markov process optimization problem and develop a procedure for recovering the underlying Benamou–Brenier dynamics from the relaxed solution. We also show that the procedure extends to more general Markov processes and illustrate it with a constrained process between Ising models.

In this paper, we study several incomplete entropy measures, namely the Incomplete Weighted Cumulative Residual Entropy, the Incomplete Cumulative Residual Tsallis Entropy and its weighted version, and introduce the associated partial orderings. Their connections with certain well-known stochastic orderings are also investigated. Based on these characterizations, a class of nonparametric tests for stochastic equality against ordered alternatives is developed. The asymptotic properties of the proposed tests are derived, while their finite-sample performances are assessed through extensive Monte Carlo simulations under various alternative models and sample sizes. These tests are further compared with the test based on incomplete cumulative residual entropy proposed by Zardasht (2015).

Let $\IB$ denote the intersection body operator on star bodies in $\R^n$. A recent theorem of Milman, Shabelman and Yehudayoff establishes that for $n\ge 3$ the equation $\IB^2 K = cK$ holds if and only if $K$ is a centered ellipsoid, thereby resolving the fixed–point problem for $\IB^2$ and, as a consequence, the long–standing conjecture $\IB K = cK \Leftrightarrow K$ is a ball. We complement this qualitative rigidity with a quantitative analysis in a neighbourhood of the ball. Linearizing the associated shape dynamics on $L^2(\Sph)$, we compute the full spectrum of the operator $\IB^2$ at the ball in closed form for every dimension: the degree–two (ellipsoidal) harmonics are neutral with multiplier exactly $1$, while all higher harmonics are contracted, with a sharp spectral gap \[ \mathrm{gap}(n)\;=\;\frac{(n-2)(n+4)}{(n+1)^2}. \] This yields an explicit linear stability constant $C(n)=(n+1)^2/\big((n-2)(n+4)\big)$, and, via a center–manifold reduction, a local quantitative stability statement for $\IB^2$ near the ball valid in each fixed dimension $n\ge 3$. The gap degenerates precisely as $n\to 2^+$, giving a transparent dynamical explanation of the well–known exceptional status of the plane, where $\IB K = 2K$ for every origin–symmetric star body. We also record the reduced normal form of $\IB$ on the ellipsoidal directions and observe that the centered ellipsoids constitute a normally attracting invariant manifold for the shape under iterated intersection bodies. The methods are perturbative and do not address the global periodic problem $\IB^m K = cK$ for $m\ge 3$, which we discuss.

We develop a rigid $\mathcal{N}=(1,1)$ superspace formulation for multifield scalar models coupled to localized impurities through spurion superfields in two-dimensional spacetime. The spurionic completion gives a manifestly supersymmetric action and provides a systematic framework for describing interacting scalar fields in the presence of impurity backgrounds. In this setting, supersymmetry acts as an organizing principle for a controlled bosonic half-BPS sector, with the preserved projector selecting the impurity-compatible first-order equations. We derive the corresponding coupled BPS equations, energy density, boundary conditions, and Bogomol'nyi bound, showing that localized impurities deform the BPS profiles and redistribute the local energy density while leaving the total BPS energy fixed by the topological boundary term. We illustrate the formalism through representative one-, two-, and three-field models, analyzing the resulting BPS configurations and their local energy density profiles.

We investigate constraints on the existence of topological and smooth realisations of combinatorial line arrangements and $(n_k)$-configurations in the complex projective plane. By replacing complex lines with locally-flatly or smoothly embedded 2-spheres, we explore the extent to which classical geometric results, such as Hirzebruch's inequality, persist in the topological or smooth category. We introduce two classes of special line arrangements that we call odd and even. We provide constraints for any smoothly realised, non-trivial, odd arrangement via Furuta's 10/8-Theorem. By looking at branched double covers and using the G-signature theorem, we study topologically realised, non-trivial, even arrangement. Finally, we establish a new lower bound for $(n_k)$-configurations, showing that for any topologically realised configuration we have $n \geq k^2-5$, which implies the non-existence of topological realisations for finite projective planes.

In this work, we investigate the mechanisms that trigger phase transitions in both discrete-time and continuous-time dynamical systems. We prove a simple topological condition that implies the existence of Hölder continuous potentials displaying phase transitions. As a consequence, we show that phase transitions are typical in several scenarios. Specifically, for $C^1$ diffeomorphisms and $C^1$ vector fields, we show that phase transitions are generic among non-transitive systems. In low dimensions, we conclude that they are generic for non-Anosov systems. Finally, in the $C^0$ setting, we prove that on any compact topological manifold of dimension different from 4, there exists a dense set of homeomorphisms with finite entropy that display phase transitions.

Let $G = (V,E)$ be a graph of order $n$. A bijection $f : V \rightarrow \{1,2,\cdots,n\}$ is a distance magic labeling of $G$ if there exists a positive integer $k$ such that $\sum_{u \in N(v)}f(u) = k$ for all $v \in V$, where $N(v)$ is the neighborhood of $v$. Any graph which admits a distance magic labeling is called a distance magic graph. In this article, we give a partial solution to the problem by Rao et al.[10] to predict all distance magic labelings of cartesian product of two cycles, $C_m \Box C_m$, where $m\equiv 2 \mod 4$. Further, we prove that the number of distance magic labelings of a distance magic graph is a multiple $| Aut(G)|$ where $Aut(G)$ is the automorphism group of the distance magic graph $G$.

We consider the inverse problem for the quantitative reconstruction of physical properties in the context of passive imaging, where ambient wavefields are used to infer a medium. The data are modeled as a superposition of waves generated by stochastic sources. In this work, we focus on time-harmonic acoustic wave propagation and assume that the stochastic sources exciting the medium are zero-mean and spatially uncorrelated. Under these assumptions, the expected value of the cross-correlation between signals recorded at two locations can be related to the deterministic Green's function and the covariance of the source terms. We follow a first-order formulation of the wave equation, which enables the treatment of correlations between different types of wavefields. A numerical framework is developed for the resulting nonlinear inverse problem. The quantitative reconstruction is carried out using an iterative minimization scheme, in which the gradient of the misfit functional is computed via the adjoint-state method. Numerical experiments in two and three dimensions are performed using synthetic data, and inversions based on the expected value of cross-correlations are compared with those relying on direct wavefield measurements from active-source acquisitions.

Let $G$ be a simple graph of maximum degree $d$, and let $\mu(G)$ denote the largest eigenvalue of its Laplacian matrix. For a fixed integer $k\geq 2$, Aharoni, Alon, and Berger (2016) asked whether every graph containing no induced copy of $K_{1,k}$ satisfies $\mu(G)\leq (2 - \frac{2}{k} + o(1)) d$. We answer this question by proving the stronger sharp bound \[ \mu(G)\leq \left(2-\frac{2}{k}\right)(d+1). \] The proof combines a sign decomposition of a Laplacian Rayleigh vector with a weighted local Caro-Wei type inequality for independent sets.

We provide an elementary proof of a symmetric property of the top dimensional components of the quiver subvariety $A_n\dots A_1=0$.

We give a direct BLM-coordinate description of the centers of generic $q$-Schur algebras for type $A$ and type $B$. For the $q$-Schur algebras ${\boldsymbol{\mathcal{S}}_q(n,r)}$ of type $A$, we show that any central element is supported on some matrices satisfying certain balance properties, and that its coefficients are precisely the solutions of an explicit finite linear system derived from the left and right multiplication formulas for the Chevalley generators of ${\boldsymbol{\mathcal{S}}_q(n,r)}$. This yields an effective reduced row echelon form construction of a $\mathbb{Q}(\upsilon)$-basis of the center, a basis that we then compare with the central bases obtained by Fu via quantum Schur-Weyl duality. For the $q$-Schur algebras ${\boldsymbol{\mathcal{S}}^\jmath(n,r)}$ of type $B$, we first apply a method similar to that for type $A$ to provide a basis for the center. In addition, we provide another structural description of the center by decomposing ${\boldsymbol{\mathcal{S}}^\jmath(n,r)}$ into the $q$-Schur algebras of type $A$, and these two descriptions are compatible. The center of a variant ${\boldsymbol{\mathcal{S}}^\imath(n,r)}$ of ${\boldsymbol{\mathcal{S}}^\jmath(n,r)}$ is also discussed.

Given a topologically stratified space $X$, we develop a categorical framework for extensions of perverse sheaves over a closed stratum $S$. We introduce the notion of $S$-small extensions and extension pairs relative to a fixed perverse sheaf on $X\smallsetminus S$. By replacing the homotopical assumptions $\pi_1(S)=\pi_2(S)=0$ in the work of MacPherson and Vilonen (Invent. Math., 84(2):403-435, 1986) with the categorical condition that the category of local systems on $S$ is semisimple, we construct an equivalence of additive categories between $S$-small extensions and extension pairs. This extends the MacPherson–Vilonen description to a more general setting and yields a maximal extension functor, generalising Beilinson's construction. As a consequence, we obtain a structural classification of indecomposable $p$-perverse sheaves on $X$: every such object is either an extension by zero of an indecomposable local system on $S$, or arises from an extension pair whose canonical morphism does not decompose as a direct sum.

Let $s_\mathbf{w}$ be the weighted binary sum-of-digits function associated with an arbitrary sequence of complex weights $\mathbf{w}=(w_j)_{j\geq 0}$. We investigate Hankel determinants $\mathcal{H}_\mathbf{w}(n) = \det [s_{\mathbf{w}}(i+j)]_{0\leq i,j<n}$ and derive a general recursion that allows us to effectively compute $\mathcal{H}_\mathbf{w}(n)$ for all $n$. Applying it to the ordinary binary sum-of-digits, that is, $w_j=1$, we express $\mathcal{H}_\mathbf{w}(n)$ in a closed form for several sequences of indices, including the remarkably simple $$ \mathcal{H}_\mathbf{w}(\lceil 2^{k+2}/3\rceil)= (-1)^{\frac{(k+2)(k+3)}{2}}(k+1). $$ This yields an infinite family of explicit evaluations, giving a partial solution to a problem posed by Allouche and Shallit. Moreover, we closely study the specialization $w_j=t^j$, where the determinants become polynomials in $t$, and investigate their vanishing. For $t=2\zeta$, where $\zeta$ is a root of unity, we show that the determinants vanish on a large structured set of indices, while the complementary is sparse but infinite. In addition to $\mathcal{H}_\mathbf{w}(n)$, we consider Hankel determinants associated with the first difference of $s_{\mathbf{w}}$, obtaining an explicit product formula. This generalizes the results by Fokkink, Kraaikamp, and Shallit concerning Hankel determinants for the period-doubling sequence.

Nielsen and Zhou (Mean size of wavelet packets, ACHA 13 (2002), 22–34) conjectured that for every Daubechies filter of length at least four the associated wavelet packets fail to be a Schauder basis of $L^p(\mathbb{R})$ for every $p\neq2$; their own $\ell^1/\ell^\infty$ estimate only reaches extreme exponents. We prove the Schauder-basis-failure half of the conjecture in full, for every $1<p<\infty$ with $p\neq2$, in the length-four case. The proof combines convexity of the wavelet packet pressure function with the uniqueness of equilibrium states for irreducible matrix families due to Feng and Käenmäki (2011) and an exact algebraic separation of two periodic spectral growth rates of the high-pass transition matrices.

Let $p_1,...,p_k$ be a rationally independent sequence of integer valued nonlinear polynomials. We show that for all $E\subseteq \mathbb{N}$, every Folner sequence $\Phi$, and every $\varepsilon>0$, the set $$\left\{n\in \mathbb{N} : d_{\Phi}\left(E\cap(E+p_1(n))\cap\cdots\cap (E+p_k(n))\right) > d_{\Phi}(E)^{k+1}-\varepsilon\right\}$$ intersects every IP generated by a sequence with rational spectrum. Our methods involve the study of the characteristic factors for multiple ergodic polynomial averages along IPs. In particular, we also prove a pointwise convergence theorem for polynomial averages along IPs with rational spectrum, generalizing a well known result of Leibman.

We prove a Miniowitz–Zalcman rescaling principle for quasiregular curves into calibrated manifolds. We have two main applications. First, we introduce Brody hyperbolicity adapted to our setting and prove its equivalence to the normality of the family of quasiregular curves from the Euclidean unit ball into the target. When normality holds, we quantify the local modulus of continuity for quasiregular curves using an injectivity radius lower bound and a sectional curvature upper bound of the target. Second, in the special case of conformal curves into closed calibrated manifolds, we prove the equivalence of Kobayashi and Brody hyperbolicity. This answers a question posed by Broder–Iliashenko–Madnick. As an intermediate result, we prove an analogue of Marty's theorem from complex analysis in this setting. Additionally, we construct new examples of non-constant entire quasiregular curves factoring through a Special Lagrangian submanifold of a closed Calabi–Yau manifold, an associative submanifold of a closed $G_2$ manifold, and four-dimensional analogues thereof, providing obstructions to Brody hyperbolicity.

We study scalar-on-function linear regression when each covariate curve is observed only through finitely many noisy point evaluations. Our goal is to characterize the minimax estimation and prediction risks as joint functions of the number of trajectories $n$ and the within-trajectory resolution $m$. Working in a fixed trigonometric eigenbasis, with covariance eigenvalues decaying at rate $\alpha$ and slope function of Sobolev smoothness $s$, we derive matching minimax upper and lower bounds under two canonical sampling schemes. Under an independent random design, the minimax prediction rate is $n^{-\frac{2\alpha+2s}{2\alpha+2s+1}} + (nm)^{-\frac{2\alpha+2s}{4\alpha+2s+1}}$. The first term is the fully observed functional linear regression benchmark, while the second term captures the cost of noisy point evaluations after amplification by the inverse covariance operator. Under a common design on an equally spaced grid, the shared sampling geometry introduces additional obstructions, and the minimax prediction rate becomes $n^{-\frac{2\alpha+2s}{2\alpha+2s+1}} + (nm)^{-\frac{2\alpha+2s}{4\alpha+2s+1}} + m^{-(2\alpha+2s)} + m^{-4\alpha}$. Here the third term represents discretization error induced by the fixed grid, whereas the fourth reflects the cost of identifying unknown eigenvalues from observations on a common grid. We further construct data-driven adaptive estimators that screen the covariance scale and threshold blockwise prediction energy, attaining these rates without prior knowledge of the eigenvalue sequence or the smoothness indices. The results reveal a sharp phase transition that depends on the sampling resolution under independent design and a richer phase diagram under common design. Numerical simulations and a real data example illustrate the theoretical findings.

The Irrelevant Vertex Technique is one of the cornerstones of algorithmic graph theory, underlying Robertson and Seymour's algorithm for Disjoint Paths and much of the algorithmic Graph Minors theory. We show that, in the setting of spanning routing, this technique exhibits an exact combinatorial limitation. Unlike classical routing problems, spanning routing is not governed by the number of distinguished vertices but by the way they are distributed throughout the graph. The input is a triple $(G,R,\mathcal{T})$ where $(G,R)$ is an annotated graph and $\mathcal{T}$ is a set of terminal pairs. The goal is to determine if $G$ contains a family of internally disjoint paths connecting the pairs in $\mathcal{T}$ such that the union of the paths spans the set $R$. We identify a new structural parameter of annotated graphs, called $\mathsf{depth}_2$, that measures precisely this phenomenon. Our main result is a complete combinatorial dichotomy: for every red-minor-closed class of annotated graphs, the Irrelevant Vertex Technique applies to Spanning Disjoint Paths if and only if the class has bounded $\mathsf{depth}_2$. Thus $\mathsf{depth}_2$ forms the exact structural boundary between classes where the Robertson-Seymour paradigm survives and those where it breaks down. Our proof combines a new local structure theorem for annotated graphs of bounded $\mathsf{depth}_2$ with a spanning analogue of the celebrated Vital Linkage Theorem. The resulting algorithm solves Spanning Disjoint Paths in time $2^{2^{\mathbf{poly}(k+d)}}\cdot n^2$ where $d$ is the $\mathsf{depth}_2$ of the input instance. We provide matching lower bounds showing that beyond bounded $\mathsf{depth}_2$ no irrelevant-vertex rule can exist, even on planar graphs. In particular, $\mathsf{depth}_2$ is the exact combinatorial barrier for the Irrelevant Vertex Technique under spanning constraints.

Combining the Jung–van der Kulk theorem with the conjugacy invariance of the Pascal finite class, we show that every polynomial automorphism $F$ of the plane over an arbitrary field $K$, satisfying $F(0) = 0$, decomposes into the form $F = \diag(\det J_F, 1) \circ P_1 \circ \dots \circ P_s$, where all $P_i$ are Pascal finite automorphisms. Since every Pascal finite automorphism has Jacobian determinant equal to 1, the diagonal factor is the only obstacle: $F$ is a composition of Pascal finite maps if and only if $\det J_F = 1$. In particular, Question~3.1 from \cite{ABCH2} has a positive answer in dimension 2 in any characteristic, which constitutes an analogue of the Exponential Generators Conjecture in positive characteristic. In characteristic $p$, the factors can be chosen to have an order dividing $p^2$.

Here we present PCPOP, a Julia package for polynomial optimization that supports non-commutative optimization, tracial polynomial optimization, trace polynomial optimization and state polynomial optimization. PCPOP fully supports exact arithmetic computations and incorporates convenient functionalities such as algebraic reductions based on Gröbner basis methods, automatized symmetrization via Wedderburn decompositions, and Jordan algebra reductions. As a distinguished feature, PCPOP implements a specialized framework for polynomial computations in partially commutative variables that provides significant computational advantages for problems appearing in quantum information.

Multistable dynamical systems are ever-prevalent, used to model for example ecosystems, power grids, climate elements, neurons, and more. When perturbed, such systems may “tip” from one state of operation to another, often with abrupt, irreversible, and high-impact consequences in each context. Traditionally, these systems are analysed via bifurcation diagrams, the result of a process we refer to as local continuation, as it only captures the linear (local) system response to infinitesimal perturbations. Local continuation requires substantial expertise, constant interventions, and may yield inaccurate assessment of the system's response to large perturbations that is crucial for tipping analysis. To address some inherent challenges of local continuation and to provide fundamentally new information during a continuation, this paper introduces global continuation as a complement suitable for the study of multistability, critical transitions and real-world-oriented applications. Global continuation finds and continues in parallel (practically) all system attractors and their response to finite perturbations by synthesising information from the whole state space, while placing a focus on the qualities or observables of a dynamical system that the practitioner cares about in context. Global continuation does not require deep expertise and is effortless to use and troubleshoot, making it attractive to applied scientists from different disciplines. We highlight several unique advantages that allow global continuation to complement the status quo and exemplify them through a plethora of representative examples. Global continuation is also implemented as open source software in DynamicalSystems.jl, enhancing its accessibility.

The Einstein-de Sitter universe is the prevailing model used in cosmology to describe the cold dark matter-dominated epoch of the universe. This model is a spatially homogeneous and isotropic spacetime undergoing decelerated expansion, and is linearly unstable under the Einstein-Euler equations with a pressureless fluid equation of state. We show that every initial data set for the Einstein-Euler equations on $\mathbb{T}^3$ with a near-flat metric and positive fluid energy density converges to a flat metric under the Einstein-Euler flow with a polytropic equation of state. This means the metric asymptotes to an Einstein-de Sitter spacetime. In particular, this settles the question of whether the Einstein-de Sitter model can be nonlinearly stable for an appropriate matter model.

In this note we investigate a notion of stability for spacelike cross sections of a null cone under area preserving variations that has been introduced in previous work by Kröncke and the author. Here, we consider null cones with spherical cross sections in a $4$-dimensional spacetime and show that the Hawking energy of a stable cross section admits a non-negative lower bound provided the dominant energy condition holds. Similar to a recent work by Peñuela Diaz, we show that under an additional assumption the Hakwing energy is zero if and only if the stable cross section embeds isometrically into the Minkowski lightcone. As a main result, we show that the only stable cross sections of the standard Minkowski lightcone are round spheres.

This paper introduces Bernoulli-type and Euler-type Mersenne-Lerch interpolation families associated with Apostol-Mersenne polynomial families. Their construction is based on the Mersenne translation polynomials \(P_{n,M}(x;m)\), defined by \[ e_M^{xt}(e_M^t)^m = \sum_{n=0}^{\infty}P_{n,M}(x;m)\frac{t^n}{M_n!}. \] Explicit formulas for these polynomials are derived, including an expansion in terms of M-Stirling numbers of the second kind. At nonpositive integers, the resulting interpolation values recover the Apostol-Mersenne-Bernoulli and Apostol-Mersenne-Euler polynomials of order \(r\). Derivative, integral, addition, and difference formulas are also obtained for these values. A comparison with \(q\)-calculus shows that the M-factorial, M-binomial coefficient, M-derivative, and M-exponential are obtained by setting \(q=2\) in the corresponding \(q\)-calculus expressions.

We define a model of multi-type Galton-Watson processes in dynamical environments where the environment evolves according to a dynamical system (X,T). Three behaviours are possible: uniformly subcritical, critical, and uniformly supercritical. We provide a criterion to determine the regime of a multi-type Galton-Watson process in dynamical environments. We study the continuity of the probability of extinction q in the uniformly supercritical case.

In this paper, we address control design of Networks of nondeterministic and metric finite state Systems (NoS) where some systems are plants, others act as controllers, and some are uncontrolled processes. The control architecture of NoS extends decentralized control architectures. The problem addressed consists in designing local controllers for enforcing on the corresponding plants, local specifications given by regular languages over plants' state alphabet, up to desired accuracies. The approach used to solve the problem is termed incremental and consists in finding solutions on an increasing sequence of NoS extracted from the original NoS.

We adopt a modern approach to extend a comparison inequality of Chevet for Gaussian processes to a broader class of functions. Furthermore, we consider centered Gaussian random vectors associated with a family of $n+1$ vectors on the unit sphere $S^{n-1}$ and investigate configurations of the vertices for which these new functionals could be maximum for the regular simplex.

We present a polynomial-time algorithm for optimally coloring perfect graphs that is based entirely on graph-theoretic operations. At its core, the algorithm decides whether a perfect graph contains a clique of a given size by iteratively counting walks in the graph with certain weights assigned to its edges and nonedges. These weights are initialized according to a uniform scheme and then updated in each iteration based on the walk counts from the previous iteration.