Papers

In this note, we show that, for all $n\ge 2$, the number of distinct rooted binary phylogenetic $X$-trees displayed by a binary tree-child network $\mathcal{N}$ on $X$ with $n$ leaves is at most $2^{n-1}-1$ and that this upper bound is sharp. Furthermore, if $\mathcal{N}$ displays exactly $2^{n-1}-1$ such trees, then exactly one rooted binary phylogenetic $X$-tree is displayed twice, and this tree can be canonically found by iteratively replacing a reticulated cherry with a cherry.

The main result of this paper is a generalization of a theorem of Nakajima-Landweber-Stong to the modular invariant rings of transvection groups over Dedekind domains. More precisely, let $A$ be a Dedekind domain and $K$ be its field of fractions. Assume that $A$ contains a finite field $\mathbb{F}_q$ with $q=p^r$ elements for a prime $p$. Let $n\geq 2$ and consider a finite subgroup $G$ of $\mathrm{GL}(A^n)$ such that every non-identity element of $G$ inside $\mathrm{GL}(K^n)$ is a transvection. Consider the ring $A[X_1,X_2,\dots, X_n]$ and let $G$ act linearly on the ring (fixing $A$). Then $(A[X_1,X_2,\dots, X_n])^{G}$ is regular.

We study an inverse problem for the time-dependent nonlinear fourth-order Schrödinger equation on both compact Euclidean domains and compact Riemannian manifolds, (say) $M$. This model arises in nonlinear fiber optics and the theory of optical solitons in gyrotropic media. Our main objective is the identification of unknown coefficients from the associated source-to-solution map, which assigns to each source term $f$, supported in $(0, T)\times \Gamma$, the corresponding solution $u$ restricted to the same set, where $\Gamma \subset M$ is a neighborhood of $\partial M$. We prove that the zeroth-order term, the second-order coefficient, and the nonlinear coefficient are uniquely determined by this map. Moreover, the recovery of the symmetric second-order tensor reduces to the inversion of a divergent beam transform.

In this paper, we establish Riesz–Fejér type inequalities for the $\alpha$-harmonic functions $f=P_{\alpha}[f^*]$ in $\mathbb B^n$, where $f^*\in L^{p}(\mathbb{S}^{n-1})$ and $1<p<\infty$. More precisely, for $n\geq2$ and $\alpha>-1$, we prove the existence of a constant $\mathcal{C}_{n,p,\alpha}$ such that $\int_{-1}^{1} |f(r\eta)|^p(1-r^2)^{n-2}\,dr \leq \mathcal{C}_{n,p,\alpha} \int_{\mathbb S^{n-1}}|f^*(\xi)|^p\,d\sigma(\xi)$. Moreover, in the range $\alpha>\max\left\{ -\frac{n-1}{p},\,n-2-\frac{2(n-1)}{p} \right\}$, we determine the sharp constant explicitly. The result generalize and extend the corresponding results of Ahmed et al. (J. Math. Anal. Appl., 563:13, 2026), Hu et al. (Anal. Math. 51:15, 2025) and Long (arXiv: 2410.12137).

Let \((w_{ij})_{i,j\ge1}\) be a single infinite array of independent identically distributed real- or complex-valued entries of mean zero, variance \(\sigma^2\), and finite fourth moment. Set \(W_n=(w_{ij})_{1\le i,j\le n}\) and \(X_n=n^{-1/2}W_n\). For every fixed \(k\ge1\), we identify the almost sure limiting operator norm of several fixed products built from this family. Define the \(k\)-th freeness coefficient by \[ \gamma_k:=\sqrt{\frac{(k+1)^{k+1}}{k^k}}. \] Then we prove \[ \|X_n^k\|\to\sigma^k\gamma_k \qquad \text{almost surely}. \] The same limit holds for products sampled with replacement from any fixed finite pool of independent copies of \(X_n\); in particular, it holds for the product of \(k\) independent copies. Thus, the freeness coefficient captures the non-commuting characteristic between large random matrices %powers and independent or fixed-pool sampled products under the finite fourth moment assumption. The improvement of the classical Bai–Yin-type power estimate from the scale \(\sigma^k(k{+}1)\) to \(\sigma^k \sqrt{k{+}1}\) is a direct corollary of our result. The main technical challenge is to prove the upper bound using a high-moment expansion of %the upper bound is proved by a high-moment expansion of \(\E\Tr((X_n^kX_n^{*k})^m)\). The leading zero-defect trace words are tree-like and are counted by the Fuss–Catalan number \[ F_{k,m}= \frac1{km+1}\binom{(k+1)m}{m}. \] The combinatorial tool helps to devise a defect-sensitive global enumeration: if \(L=km\) and \[ r=(L+1-v)+(L-q), \] then the number of admissible word classes with defect \(r\) is at most \(F_{k,m}(Cm)^{Dr}\). This polynomial-in-\(m\) loss, with degree proportional to the defect, is summable in the logarithmic moment range.

Cooperative massive multiple-input multiple-output (MIMO) promises large gains over cellular deployments, but existing comparisons of different architectures often mix antenna distribution, inter-site coordination, and processing assumptions. This paper introduces a graph-based framework for fair comparison of cellular, coordinated, and cell-free massive-MIMO systems. We differentiate between two key properties, namely antenna distribution and inter-site cooperation, which yields seven representative system types. We derive compatible uplink and downlink spectral efficiency (SE) expressions, including an uplink bound for detectors with mixed instantaneous and statistical effective channel state information (CSI), and adapt scalable user association and processing rules to all considered architectures. We evaluate these systems using extensive numerical simulations and show that for a fair comparison much larger simulation areas (at least 2.5 $\times$ 2.5 km2) than commonly used are required. We introduce the relative capacity, which measures how closely each architecture approaches centralized cell-free processing. The results show that coordinated, phase-aligned beamforming across spatially distributed antennas is the main source of cooperation gains. In dense deployments with few antennas per access point (AP), coordinated Distributed Antenna System (DAS) and hybrid cell-free architectures achieve much of the centralized cell-free performance while requiring substantially weaker midhaul assumptions.

We consider a hydrodynamic model of flocking-type with pressure on the torus, with integrable interaction kernel and density bounded away from zero. We prove that, if an entropy weak solution exists, then its $L^2$ norm decays exponentially fast in time towards the mean values on the period. The proof relies on the study of a suitable energy functional that combines a strictly convex entropy for the system and a potential term, and this allows us to treat the nonlocal source term for a class of strictly positive convolution kernels in $L^1$.

We study the logarithmic Schrödinger equation \[ -\Delta u+V(x)u=u\log u^2,\qquad x\in\mathbb R^N,\ N\ge3. \] Since the logarithmic energy is not \(C^1\) on the natural space \(H_V^1(\mathbb R^N)\), direct invariant-set minimax arguments for sign-changing solutions are not available. We introduce an \(L^p\)-regularization perturbation, which restores a \(C^1\) variational structure while preserving the logarithmic nonlinearity, and prove via a limiting argument that the original equation admits infinitely many sign-changing weak solutions.

We study the semiclassical limit of the Polyakov-Liouville measure $\boldsymbol{\nu}_\gamma$, which is a non-Gaussian measure on $H^{-\eps}(M)$ that has recently been extended from Riemann surfaces to general Riemannian manifolds $(M,g)$ of even dimension. We show that under an appropriate rescaling in the semiclassical limit as $\gamma\to0$, the normalized Polyakov-Liouville measure $\Q_\gamma$ concentrates on the unique smooth weight $u$ for which the conformal metric $e^{2u}g$ on $M$ has constant $Q$-curvature.

Inclusion minimal convex bodies $K$ with the property that the integer translates of $K$ cover the space are studied. Such bodies are referred to as minimal covering bodies and it is shown that, while they are not necessarily tiles, they are polytopes with at least $2d$ facets, if $d$ is the dimension of $K$. Moreover, minimal covering bodies are related to covering properties of Minkowski combinations of convex bodies. Two sharp Brunn Minkowski type inequalities are established for the covering radius of the Minkowski sum of planar convex bodies.

We introduce a new convex optimization framework for logistic scalar-on-matrix regression which incorporates nuclear and $\ell_1$ norm penalties to enforce simultaneously low-rank and sparse structures in the estimated coefficient matrix. The proposed method enables interpretable modeling of high-dimensional matrix-valued predictors in the presence of binary responses. We derive a custom algorithm based on the Alternating Direction Method of Multipliers (ADMM) to efficiently solve the resulting convex optimization problem and establish the theoretical properties of the obtained solution. Numerical experiments clearly demonstrate the effectiveness of our method in recovering meaningful predictive patterns. Finally, we apply our method to the brain imaging data to identify structures in functional brain connectivity matrices that are characteristic of subjects with a family history of alcohol use disorders (AUDs).

Shifts of finite type (SFTs) play a central role in the theory of shadowing. Good and Meddaugh showed that SFTs serve as basic building blocks in the structural theory of shadowing systems; in particular, every compact metric dynamical system with shadowing is a factor of the inverse limit of an inverse sequence consisting of SFTs. We first show that, for this factor representation alone, neither shadowing nor metrizability is needed: every compact Hausdorff dynamical system is a factor of the inverse limit of an inverse system consisting of SFTs. Thus, being a factor of an SFT inverse limit is not the structural feature genuinely forced by shadowing. What shadowing provides is stronger stability: in the metric case, every compact shadowing system is a factor of the inverse limit of an inverse sequence of SFTs with surjective bonding maps. Hence the associated zero-dimensional extension still has shadowing. We also prove that every compact Hausdorff shadowing system is conjugate to an inverse limit of metrizable shadowing systems with factor bonding maps.

Based on a variational characterization of the local slope of a functional, we extend the celebrated Brezis-Ekeland-Nayroles null-minimization principle to curves of maximal slope in metric spaces. In the time continuous setting, we check that the ensuing global-in-time functionals admit minimizers.

Partial differential equations (PDEs) regulate the behaviour of countless spatiotemporal systems in the physical and life sciences. In many cases, they encode the coupling between the system's degrees of freedom, leading to nonlinear equations whose solution space is challenging to explore exhaustively. Systematic approaches to PDE model exploration are a holy grail of computational science. In this article, we formulate a criterion for increasing the diversity of a search campaign, based on the PDE residual behaviour under solution deformation. We develop a practical formalism to compute this property and illustrate its role in a few cases of interest.

Let $(M, [g])$ be a $3$-dimensional conformal manifold. The space of knots $\operatorname{Kn}(M)$ in $M$ is an infinite-dimensional manifold that is known to carry an almost complex structure. This structure is formally integrable by a result of Brylinski. We study finite dimensional holomorphic submanifolds in $\operatorname{Kn}(M)$. We define an holomorphic family of knots in $(M, [g])$ parametrised by a finite-dimensional complex manifold $(X, I_X)$, and construct several series of examples. We show that the base $(X, I_X)$ is Kähler, and if $X$ is compact, it is a projective variety of complex dimension at most $2$. In this case the conformal structure $[g]$ uniquely determines the complex structure $I_X$ and vice versa. We prove that if an holomorphic family of knots in $(M, [g])$ over a compact base $(X, I_X)$ defines a foliation on $\mathbf{S}(TM)$, then $(X, I_X) \simeq \mathbb{C}\mathbf{P}^1 \times \mathbb{C}\mathbf{P}^1$, the manifold $(M, [g])$ is conformally equivalent to either $S^3$ or $\mathbb{R}\mathbf{P}^3$ with round metric, and all knots are geodesic in some round metric in the class.

A locally polyhedral manifold $M$ is a manifold which is locally diffeomorph to open subsets of convex polytopes. Manifolds with corners are a special case of locally polyhedral manifolds. We turn the group $\text{Diff}^{\infty}(M)$ of all smooth diffeomorphisms $f\colon M\to M$ for compact smooth $M$ into a regular Lie group.

Vizing's conjecture from 1963, considered by many the most important open problem in the field of graph domination, states that all graphs $G$ and $H$ satisfy $$\gamma(G\square H)\ge \gamma(G)\gamma(H),$$ where $\gamma$ denotes the domination number and $\square$ the Cartesian product. In a seminal result, Clark and Suen (2000) proved an approximate form of the conjecture, namely that $\gamma(G\square H)\ge \frac{1}{2}\gamma(G)\gamma(H)$ for all graphs $G$ and $H$. Despite several lower-order improvements of this bound and improvements for special classes of graphs $G$ and $H$, no absolute constant $c>\frac{1}{2}$ such that $\gamma(G\square H)\ge c\gamma(G)\gamma(H)$ for all graphs $G$ and $H$, has been known thus far. In this paper, we obtain the first constant-factor improvement of the Clark-Suen bound by proving that for all graphs $G$ and $H$, we have $$\gamma(G\square H)\ge c\gamma(G)\gamma(H),$$ where $$c=\frac{5+\sqrt{73}}{24}\approx 0.5643.$$ Along the way, we prove another lower bound on $\gamma(G\square H)$ which outperforms the above bound for many graphs.

We show that number of bistellar moves and sparse degree-two edge collapses for 3-sphere is NP-hard. It follows that a similar problem for an arbitrary 3-manifold is NP-hard as well. This is the first NP-hardness result concerning moves between two triangulations of a 3-manifold.

Reciprocity-based downlink beamforming is imperative for a scalable time-division duplex massive multiple-input multiple-output~(MIMO) deployment. Specifically, for a dual-antenna repeater-assisted massive MIMO system, a mismatch between forward and reverse path gains at the repeater can exacerbate the overall calibration error between the user equipments (UEs) and the base station (BS), which potentially also contains calibration errors of their individual radio-frequency chains. This paper models the effects of such calibration errors, underpins the relations between the uplink and downlink channels for repeater-assisted systems with calibration errors clubbed with the over-the-air channel estimation errors, and derives analytical expressions of the downlink spectral efficiency. The presented results can then be simplified to several special cases, underscoring situations wherein such errors can become pronounced.

Recent results have shown that domain-uniform stabilizability and detectability imply spatial exponential decay of perturbations in optimally controlled hyperbolic PDEs on one-dimensional domains. This domain-uniform stabilizability and detectability can be achieved only if the control domain is distributed over the whole spatial domain such that the distance between neighboring control intervals is bounded from above. Motivated by these insights we investigate whether analogous effects can be observed for optimal boundary control in higher dimensions. For this purpose we conduct a simulation study on a two-dimensional wave equation on expanding square domains which is driven by a localized perturbation of the initial displacement. We compare two control geometries: a control acting only on the outer boundary and a control acting on a regular grid of interior line interfaces combined with the boundary. The problem is discretized by conforming P1 finite elements in space and the implicit midpoint rule in time. Instead of assembling the full space-time KKT system, we use a condensed formulation and solve the reduced optimality system by preconditioned conjugate gradients. We show that domain-uniform spatial decay of the perturbation can only be observed in the scenario with a regular grid of line controls. This is because finite propagation velocity requires a uniform bound on the distance that a wave can travel without reaching the control domain.

We prove that, for a classically free vertex algebra, the $C_2$-algebra is naturally isomorphic to the associated graded algebra of Zhu's algebra. As an application, we show that non-trivial holomorphic VOAs of CFT type are not classically free. This includes the affine vertex operator algebra of $E_8$ at level one, as well as the Moonshine VOA.

This paper investigates the fundamental limits and optimal signal distribution design for Integrated Sensing and Communication (ISAC) systems operating under strictly noncoherent conditions. Unlike conventional coherent frameworks that rely on perfect channel state information, we consider a block-fading MIMO channel where the channel realizations are unknown to both the transmitter and the receiver. We adopt a realization-wise perspective to characterize the noncoherent performance tradeoff across different signal-to-noise ratio (SNR) regimes. In the high-SNR regime, we derive a lower bound for the noncoherent mutual information and define a metric, termed sensing-induced rate loss, to quantify the communication penalty incurred by sensing-oriented beamforming. We then employ a projected gradient algorithm to optimize the spatial power allocation, balancing the conflict between the unitary space-time modulation-based structure for communication and the task-oriented spatial power allocation for sensing. Conversely, in the low-SNR regime, we perform a first-order asymptotic analysis of the ergodic minimum mean squared error (EMMSE). Our theoretical derivation reveals a fundamental synergy: the sensing-optimal strategy collapses to a rank-one transmission along the dominant eigenvector of the target response, which incurs no first-order communication loss in the low-SNR regime. This result demonstrates that the conflicting tradeoff observed at high SNR vanishes asymptotically at low SNR, enabling perfect alignment between sensing and communication objectives.

This paper continues our previous work on duality for Anderson $t$-modules. We study two dimensional triangular t-modules in a specific reduce form. Computer - assisted symbolic computations with matrices over a skew field revealed a reduction pattern for t-modules satisfying the ALD condition. Using this pattern, we prove that such t-modules are isomorphic to their double duals and extend the validity of the Cartier-Nishi theorem and the Weil-Barsotti formula to a substantially broader class of t-modules.

We prove polynomial bounds on the growth of high Sobolev norms of solutions to the defocusing fractional nonlinear Schrödinger equation with cubic and Hartree-type nonlinearities in two and three dimensions. Our result is based on the upside-down I-method and the method of higher modified energies. The analysis of the fractional case in higher dimensions is possible due to a higher-dimensional analogue of a resonance inequality, which allows us to control the nonresonant frequency contributions.

We discuss probabilistic measures of concordance between two probability distributions based on the expected misclassification rate (EMR). The focus is on comparing a given reference distribution with other distributions in a parametrised class, and optimising concordance by identifying parameter values maximising EMR or a regularised variant. EMR is a practical and decision-theoretically meaningful measure, and its optimisation has direct interpretation as a Bayesian decision analysis with a bounded utility function. We explore theoretical properties of EMR, discuss relationships with other measures including Küllback-Leibler divergence, and recognise that its optimisation has a synthetic Bayesian emulation interpretation that aids understanding and specification of regularisation penalties. A main area of methodology is in mixture synthesis where the parametrised family is a discrete mixture of given distributions. A detailed example comes from scenario forecasting in macroeconomic policy settings, a key applied area motivating the new methodology. Theoretical developments underlie efficient numerical optimisation and analysis is easily implemented using direct Monte Carlo simulation.

Recently, the Turán problem for graphs with bounded clique number has attracted considerable attention. Since a graph can be regarded as a signed graph without negative edges, it is natural to extend the study of such Turán-type problems to signed graphs. With this motivation, we investigate the Turán-type problem for unbalanced $C_{2k+1}^{-}$-free signed graphs with bounded clique number. In fact, we establish a general result via the classical stability theorem. Specifically, for sufficiently large $n$ and a color-critical graph $F$, we determine the maximum number of edges among all $n$-vertex $C_{2k+1}^{-}$-free unbalanced signed graphs whose underlying graphs are $F$-free.

Building upon the semi-implicit multilevel spectral deferred correction (SI-MLSDC) method introduced by Pfister and Stiller [38], this work presents a 1D space-time adaptive high-order method combining discontinuous Galerkin spectral element discretizations with multilevel spectral deferred corrections. The proposed approach enables genuine arbitrary-order accuracy in space and time while dynamically balancing spatial and temporal discretization errors to reduce computational cost. A key contribution is the development of a novel temporal error estimator that in combination with a spectral error estimator in space provides a reliable basis for adaptive refinement decisions. The error estimator is compared with two alternative refinement criteria to assess their impact on accuracy, computational efficiency and their suitability for complex problems. The performance of the adaptive method is demonstrated for nonlinear conservation laws ranging from Burgers' equation to the Euler equations. Numerical results show substantial runtime reductions while maintaining the desired accuracy. In particular, significant computational savings are achieved for Burgers' equation, and challenging benchmark problems such as the Shu-Osher shock-fluctuation benchmark. These results demonstrate the potential of adaptive SI-MLSDC methods for efficient high-order space-time adaptive simulations of complex flow problems.

The classification and enumeration of short mesh patterns have emerged as two central directions in the area. We make substantial progress on both fronts. We construct an involution and a bijection that establish distributional equivalences for two classes of length-$2$ mesh patterns, thereby resolving a conjecture from 2019 and a recent conjecture. As a consequence, the best known upper bounds for the numbers of distribution-equivalence and Wilf-equivalence classes drop to $106$ and $47$, respectively. Combined with the known lower bounds of 105 and 46, conjectured to be exact, these results leave both classifications hinging on a single distribution-equivalence question conjectured in 2019, whose resolution would at once settle the remaining Wilf-equivalence case. We further conjecture that this unresolved equidistribution also holds for involutions, a subclass of all permutations. We also determine the distributions of three additional classes of length-$2$ mesh patterns through a detailed structural analysis. Our work combines bijective techniques with generating-function methods, yielding new insights into the structure and enumeration of short mesh patterns.

Consider the annulus $\mathbb{A}_r=\{z\in\mathbb{C}:r^{-1}<|z|<r\}$ for $r>1$ and the quantum annulus \[ Q\mathbb{A}_r=\{T:\ T \text{ is an invertible operator and} \ \|T\|, \|T^{-1}\|\leq r\}. \] McCullough and Pascoe proved that $T\in Q\mathbb{A}_r$ if and only if $\beta(T^*,T)=(r^2+r^{-2})-T^*T-(T^*T)^{-1}\ge0$. We call an invertible operator $T$ a quantum annulus unitary if $\beta(T^*,T)=0$. In this article, we construct an explicit doubly commuting $d$-tuple of quantum annulus unitaries that simultaneously extends a given doubly commuting $d$-tuple of operators in $Q\mathbb{A}_r$. We introduce the notion of a regular quantum annulus unitary dilation and show that the dilation arising from our construction is regular. As an application of the dilation theorem, we show that $\overline{\mathbb A}_r$ is a complete $K_t$-spectral set for operators in $Q\mathbb{A}_r$ and $\overline{\mathbb{A}}_r^d$ is a complete $K_{dc}^{(d)}$-spectral set for doubly commuting $d$-tuples of operators in $Q\mathbb{A}_r$, where \[ K_t=2\left(1+\frac{2r^2}{(r^2+1)\sqrt{r^4-1}}\right) \quad \text{and} \quad K_{dc}^{(d)}=\left[2\left(1+\frac{2r^2}{(r^2+1)\sqrt{r^4-1}}\right)\right]^d. \] We further prove that every doubly commuting tuple of operators in $Q\mathbb{A}_r$ is similar to a commuting tuple having $\overline{\mathbb{A}}_r^{d}$ as a complete spectral set. In addition, we establish bounds for the optimal spectral constants and show that they converge to $2^d$ as $r\to\infty$. We also obtain an alternative characterization of operators in $Q\mathbb{A}_r$ and quantum annulus unitaries, and prove that $\overline{\mathbb{A}}_r^d$ is a $K$-spectral set for a subclass of commuting $d$-tuples in $Q\mathbb{A}_r$.

In this paper, we prove lower bounds for the Gaussian-weighted \(L^2\)-curvature integral of embedded self-shrinkers. The proof combines normal coordinate functions with weighted Poincaré inequalities arising from first-eigenvalue estimates of Ding–Xin and Brendle–Tsiamis. For closed self-shrinkers, the estimate gives an explicit lower bound in terms of entropy and, together with the entropy gap theorem of Colding–Ilmanen–Minicozzi–White, yields a strict curvature gap for nonspherical closed shrinkers. In dimension two, we combine this estimate with the classification theorem and entropy gap theorem of Bernstein–Wang to obtain the corresponding gap statement for complete embedded self-shrinkers with polynomial volume growth.