Papers

We prove that hereditary diameter lower bounds are stable under convex combinations in real $L_1$-spaces over arbitrary measure spaces. More precisely, if every weakly open piece of each set has diameter at least $\delta>0$, then every finite or countable convex combination has the corresponding hereditary lower bound $\delta/4$. In the diameter-two case, the bound is preserved exactly. Consequently, for every topology containing the relative weak topology, the diameter two and strong diameter two properties are equivalent on bounded convex subsets of the unit ball, as are the convex point of continuity property and strong regularity.

We use equivariant singular instanton Floer theory with the Chern–Simons filtration to obstruct same-sign unknotting operations. We show that, for a large class of slice knots obtained through ribbon concordance, any unknotting sequence of null-homologous twists must contain both signs. The same method gives a $3$–manifold analogue, obstructing certain homology $3$–spheres from surgery on a knot and providing evidence for the monotonicity of the Dehn surgery number under ribbon homology cobordism.

Let \(A\) be a cyclic operator on an \(r\)-dimensional vector space over a field \(k\), and let \(z\) be a cyclic vector. Their Krylov code has parity-check matrix \((z,Az,\ldots,A^{n-1}z)\). For \(r\ge 3\) and \(n\ge r+3\), we prove that an MDS orbit segment lies on a rational normal curve precisely when the projective pair \((A,[z])\) is conjugate to one arising from the \((r-1)\)-st symmetric-power action of \(\mathrm{PGL}_2\). Over finite fields, for companion operators, this gives a complete classification of the generalized Reed–Solomon locus into split semisimple, two nonsplit semisimple, and unipotent families. Over an algebraically closed field \(k\), the Zariski closure \(\GRSsurf_{r,k}\) of the semisimple GRS coefficient locus is an irreducible rational surface, generically parameterized two-to-one by a two-dimensional torus of geometric-progression root sets; reversal is the generic ambiguity. The affine quotient of the parameter torus by reversal is the normalization of \(\GRSsurf_{r,k}\cap D(a_0)\), its nonzero-constant-term open part. The codimension in the space of monic degree-\(r\) polynomials is \(r-2\). Frobenius descent gives an exact formula for the number of GRS polynomials over \(\mathbb F_q\). A canonical remainder parity-check matrix defines the MDS locus by a principal open condition. For fixed \(r\ge3\) and \(n\ge r+3\), the proportion of all monic degree-\(r\) polynomials over \(\mathbb F_q\) whose companion codes are MDS and non-GRS tends to one as \(q\to\infty\) through prime powers.

A charge simulation method, a variant of the method of fundamental solutions, is applied to approximate the multi-phase Mullins–Sekerka flow in $\mathbb R^{2}$ and in a half-plane bounded by a Neumann wall. In the underlying mathematical model, interfaces driven by their curvature are coupled through a harmonic chemical-potential field. Each chemical potential is represented by fundamental solutions centered at charge points off the curve, so no bulk mesh is required. The scheme treats curve networks separating several phases at triple junctions, including phases that occupy more than one region; on the half-plane boundary, the homogeneous Neumann condition is imposed exactly by image charges, and mobile contacts stay orthogonal to the wall. The discretization is structure-preserving in the sense that every bounded phase area is conserved to machine precision at the velocity level by a null-space projection of the discrete area constraints. The proposed scheme is assessed through a convergence test against an exact three-concentric-circle solution.

We study an elliptic nonlocal problem driven by a source term which is a Radon measure. We prove the existence of a weak solution by a weak convergence method. In the process, we define a new function space which we name the Walker space. The question about the existence of infinitely many nontrivial solutions leads to an interesting conjecture.

For positive integers $n$ and $M$, Sundaram defined $ F_{n,M}=\sum_{\substack{d\mid n\\d\le M}} Lie_{n/d}[p_d] $ and conjectured that every coefficient in its Schur expansion is a nonnegative integer. We prove the conjecture. The power-sum expansion isolates the contribution of the identity class, and the proof divides Young diagrams at the intrinsic majority threshold. If the first row or first column contains more than half of the boxes, the Bernstein creation formula separates each rectangular-cycle character value into a hook constant and a remainder supported only on short cycles. The restrictions of the trivial and sign characters to the cyclic subgroup evaluate the constants exactly, whereas a uniform binomial contraction controls every distance from the boundary at once. If neither a row nor a column has a majority, Swanson's opposite-hook estimate gives a Catalan-scale lower bound for the dimension. The Fomin–Lulov rectangular-character estimate, together with uniform bounds for the truncated Möbius weights, then shows that the identity class dominates. Sundaram's plethystic identity consequently implies that $\prod_{r=1}^M(1-p_r)^{-1}$ is Schur-positive for every fixed $M$. We also record a degree-$12$ counterexample to a separate congruence-class product conjecture attributed to Richard Stanley.

We prove the following theorem: let $\mathcal{M}_d$ denote the set of $d \times d$ matrices over an infinite field $K$, and let ${(K^d)^*}$ be the set of row vectors. Define an action of $\mathrm{SL}_d(K)$ on $X:= \mathcal{M}_d \oplus (K^d)^*$ by \[ g \cdot (A,\phi) = (gAg^{-1}, \phi g^{-1}).\] Then $K[X]^{\mathrm{SL}_d}$ is a polynomial ring, generated by the coefficients of the characteristic polynomial of $A$ and one further invariant, namely $\Delta(A,\phi):= \det(\phi,\phi A,\phi A^2,\ldots, \phi A^{d-1})^t.$ Our proof is entirely classical in nature, but we give an interpretation of the result and its proof in terms of quiver representation theory.

This paper investigates the homogenization of the 3D Navier–Stokes–Cahn–Hilliard (NSCH) system in domains containing a large number of solid obstacles (named holes). Each hole has diameter of order $\varepsilon^{\alpha}(\alpha>3)$, where $\varepsilon > 0$ denotes the small length scale for inter-hole separation. Both viscosity and mobility depend on the phase-field variable. We establish two distinct asymptotic regimes: if the capillary strength $\lambda_\varepsilon\to \lambda>0$ as $\varepsilon\to 0$, the limit system coincides with the original NSCH system; if $\lambda_\varepsilon\to 0$ as $\varepsilon\to 0$, the scaled velocity, phase field and chemical potential converge to a weak solution to a Stokes–Cahn–Hilliard (SCH) system. To the best of our knowledge, this work constitutes the first rigorous homogenization analysis for evolutionary NSCH flows with phase-dependent viscosity and mobility under the subcritical hole scaling.

We prove that any finite collection of at least three isotopy classes of knots in a 3-manifold $M$ is realizable as the components of a genus-zero link in $M$, provided that an obvious requirement on their conjugacy classes in $\pi_1(M)$ is met. This condition is vacuously satisfied for $M = \mathbb S^3$, and in this case we also control the pairwise linking numbers of the components. Replacing the 3-genus with the 4-genus, we obtain an analogous result where only two knot isotopy classes are prescribed.

Let $\Phi$ be an irreducible crystallographic root system of rank $r$, with Coxeter number $h$, Weyl group $W$, Cartan matrix $C_\Phi$, and invariant degrees $2=d_1\leq\cdots\leq d_r=h$. We prove that Au's normalized Witten zeta function has a simple pole at $2/h$ and evaluate its residue as $\frac{2(2\pi)^{r/2}\sqrt{\det C_\Phi}}{h|W|}\cdot\frac{\prod_{i<r}\Gamma(1-d_i/h)}{\Gamma(1-1/h)^r}$. The central step evaluates the critical chamber integral in gamma values from the boundary pole of the Macdonald-Mehta-Opdam identity; two preparatory sections pass from the dominant-weight lattice to that convergent integral. This proves Au's conjecture on algebraic multiples of products of gamma values at rational arguments, including his $A_4$ prediction.

Let $\mathcal B(\mathcal X)$ be the algebra of all bounded linear operators on a real or complex Banach space $\mathcal{X}$ with $\dim\mathcal X \ge 3$. In this paper, we first explore the ascent (descent) of upper triangular block operator matrices and certain special algebraic operators, and then establish characterizations for the ascent (descent) of rank-one and rank-two operators. Based on these results, we characterize features for some special operators by the ascent (descent) of Jordan products. As an application, we give the structure of all maps with range containing all bounded operators of rank at most three preserving the ascent (descent) of operator Jordan product on $\mathcal B(\mathcal X)$.

To reveal the effect of the inner product choice, we present a unified formulation of saddle dynamics for the functional F with a mass constraint under different inner products. We establish the equivalence between the index-k saddle points and the linearly stable steady states of the corresponding dynamics. Further, we present the dynamics with discrete H^{-1} and L^2 inner products and numerically verify the convergence orders of both dynamics. Finally, we apply the method to a phase field model with driving force under Neumann and periodic boundary conditions. The results uncover previously unreported saddle points and their connectivity, highlighting how the choice of inner product enriches the solution landscape in conservative systems.

This paper proposes a superimposed transmission strategy for cooperative cellular and cell-free massive MIMO systems. By classifying users into near and far, the base station transmits an additional data symbol for each near user, superimposed on the signals from distributed access points. Successive interference cancellation is employed at near-user receivers to decode both symbols. The proposed strategy achieves the highest peak spectral efficiency while maintaining fairness at the cell edge, thereby outperforming all the existing network configurations in system capacity.

We show that the bicategory of operads and bimodules can be embedded into the bicategory of symmetric 2-rigs, a categorification of commutative rings. In order to do this, we introduce the notion of an operadic 2-rig and show that the full sub-bicategory of symmetric 2-rigs spanned by operadic 2-rigs has the universal property of being a completion under Eilenberg-Moore-Kleisli objects.

Pinching antenna systems (PASS) employing dielectric waveguides have recently emerged as a promising flexible antenna architecture for high-frequency wireless communications. While prior work has focused primarily on millimeter-wave regimes, extending PASS to the terahertz (THz) band introduces distinct electromagnetic phenomena that invalidate conventional modeling assumptions. This paper develops the first analytical framework for THz-PASS that integrates in-waveguide propagation attenuation, evanescent coupling via coupled-mode theory, and THz-specific free-space effects including molecular absorption and its re-radiation noise. Using this model, we benchmark THz-PASS against conventional phased arrays under identical propagation scenarios. Our comparative evaluation reveals that THz-PASS achieves effective gains in spectral efficiency through proximity exploitation, making it particularly well-suited for confined and linear deployment topologies.

The information geometric regularization (IGR) is an inviscid regularization of the compressible Euler equations that alters the geometry of Lagrangian characteristics to prevent trajectories from crossing in finite time. Previous work on IGR established global strong solutions in one dimension, explored thermodynamic effects of the model, and enabled large-scale simulations of compressible flow. However, a fundamental question that remains is how this regularization alters the structure and regularity of a shock-like solution. We prove existence, uniqueness modulo translation, and regularity of transonic compressive IGR shock profiles in one spatial dimension. The analysis applies to the full thermodynamic compressible Euler–IGR model with a general equation of state, subject to mild convexity hypotheses. A traveling-wave ansatz reduces the Euler–IGR equations to a degenerate second-order scalar equation for the density profile. At the sonic crossing, the elliptic coefficient degenerates: the density profile remains continuous, but its derivative diverges. The profile is a classical solution away from this single point, while at the degeneracy it retains quantified Hölder and Sobolev regularity. We also analyze the vanishing-regularization limit, showing that the shock width scales like $\sqrt{{\alpha}}$ and that the IGR profiles converge to the entropy-admissible Euler shock.

We obtain the optimal convergence rates in the subsonic limit of the three-dimensional Zakharov system for initial data belonging to the low-regularity Sobolev space $\HH^s=H^s\times H^{s-1}\times H^{s-1}$. For the Schrödinger component, we prove first-order convergence in $L^2$ for initial data in $\HH^3$, and second-order convergence under the compatibility condition for data in $\HH^4$. For the wave component, we obtain first-order convergence in $L^2$ for data in $\HH^3$ and second-order convergence for data in $\HH^4$. The obtained rates are optimal and coincide with those predicted by the formal asymptotic expansion. No localization assumptions, smallness or high-order regularity hypotheses are required. This improves all previous results on the subsonic limit of the Zakharov system and resolves the optimality issue at the Sobolev regularity level. The proof relies on a uniform local well-posedness theory that remains valid in the subsonic limit. A key ingredient is a refined normal form analysis combined with bilinear Strichartz estimates in atomic function spaces, which allows us to fully exploit the dispersive structure of the Zakharov system at low regularity and to overcome the derivative losses arising from the singular coupling.

We study coexponentiability in the context of the cocartesian 2-category RIG of symmetric 2-rigs, symmetric strong monoidal cocontinuous functors, and symmetric monoidal natural transformations. Our results characterize the coexponentiable symmetric 2-rigs as those that are deformation retracts of presheaf categories over small categories. As an application, we give an account of the cartesian closure of two full sub-2-categories of the dual of RIG arising from the theory of combinatorial species and the theory of symmetric operads.

Let $p\geq7$ be a prime number. We build a non-trivial Euler system for the symmetric square of a $p$-adic Hida family of modular forms interpolating the Euler system constructed by Loeffler-Zerbes for the symmetric square of a $p$-ordinary newform. As a second contribution, we prove an algebraic functional equation for dual Selmer groups in this setting. Finally, building on recent work by Büyükboduk-Ganguly on functional equations of algebraic (Rankin-Selberg) $p$-adic $L$-functions, we prove a divisibility result towards the Iwasawa main conjecture for the symmetric square of a Hida family.

We study the negative $K$-groups of complex varieties from a mixed Hodge-theoretic perspective, making use of the theory of higher singularities, Chow groups, and the Minimal Model Program.

For the two-dimensional Aharonov-Bohm potential $A_\beta$ with flux $\beta\notin\mathbb{Z}$ and $1<p<2$, Cazacu, Krejčiřík, Lam and Laptev proved by a compactness argument that their constant $\lambda_\beta(p)$ in the $L^p$-Hardy inequality strictly exceeds the free constant $\big(\tfrac{2-p}{p}\big)^p$, and asked for a constructive proof with explicit estimates and for comparability of $\lambda_\beta(p)$ with a quantity depending on $\text{dist}(\beta,\mathbb{Z})$. We answer both questions by using a compactness-free two-sided bound for the twisted angular constant. Our explicit Hardy constant is $$\big[\big(\tfrac{2-p}{p}\big)^{2}+\big(\tfrac{\sin(\pi\text{dist}(\beta,\mathbb{Z}))}{\pi}\big)^{2}\big]^{p/2},\quad 1<p<2.$$ As a byproduct we observe that when $p\ge 2$ the Aharonov–Bohm field produces an $L^p$-Hardy inequality with the usual homogeneous weight $|x|^{-p}$. Our approach also provides new $L^p$-Hardy inequalities with explicit constants for the complex AB potentials.

We give a lower bound for the canonical height associated to Hénon maps $\phi(X,Y)=(Y,X+Y^D+B)$ of non-periodic points when $D>2$ in the spirit of conjectures of Lang and Silverman. This is followed by an application and extends previous work for $D=2$ by Ingram.

The spiked mixture model (SMM) has been introduced as a probabilistic model that generalizes the single-spike (Wishart) model to a mixture model form. With applications ranging from imaging mass spectrometry in the life sciences to hyperspectral imaging in computer vision, it is crucial to understand under which circumstances its signals can be recovered from noisy measurements. The highly multiplexed nature of these measurement types furthermore necessitates such analysis to hold in highdimensional settings. In this paper, we prove that the extreme eigenvalues of the covariance matrix from the SMM exhibit a phase transition in high-dimensional regimes. We show that this phase transition, and thus signal recovery by extreme eigenvalues, depends on several interacting factors: the correlation between spikes (i.e., how similar in content underlying signals are), the energy parameters (i.e., the absolute strength of each underlying signal), and the mixture probabilities (i.e., how likely it is to encounter each underlying signal). This work provides sharp information-theoretic bounds on the parameters needed to detect one or more spikes from extreme eigenvalues of the SMM covariance matrix, and these guarantees could potentially impact any application of the SMM. Understanding this interplay could serve as a tool for driving experimental design in analytical chemistry and life sciences.

We study a scheduled-arrival process in which deterministic arrival times are perturbed by i.i.d. unpunctuality random variables. For the empirical mean of independent copies of the resulting counting process, we prove a sample-path large deviations principle on $D([0,T])$, equipped with the Skorokhod topology. As an application, we derive large deviation principles for the corresponding workload process under deterministic and random service setting. In the deterministic service case, we also identify the exponentially tilted arrival path associated with a rare workload overflow event.

In this paper we introduce Hermite spectral approximation for functions with endpoint singularities using exponential transforms, including single exponential (SE), double exponential (DE) and error function (EF) transforms, and present a comprehensive convergence analysis for these approximations without and with scaling. In the case without scaling, we show that these methods converge at some root-exponential rate. In the case with scaling, we derive optimal scaling factors for each of exponential transforms and show that the convergence rate of Hermite spectral approximation can be significantly improved. Numerical comparisons with sinc method are present and it is shown that Hermite method has comparable or superior accuracy performance when using the same number of terms. Extensions to quadrature and rootfinding algorithm are also discussed.

The pinching-antenna system (PASS) uses dielectric particles along a low-loss waveguide as reconfigurable passive radiators. Existing analyses conclude that the in-waveguide attenuation is negligible at low frequencies and millimeter wave bands; we show this fails at terahertz (THz), where realizable waveguide losses are dramatically larger. We develop a unified wideband THz-PASS propagation model integrating in-waveguide attenuation, atmospheric absorption, molecular re-radiation noise, and beam squint. Closed-form results follow: a band-averaged coherence factor; a cluster-center placement satisfying a band-edge SINR equalization condition; an associated placement-inversion threshold; and a proposed Switched-Feed PASS (SF-PASS) architecture in which a centrally located radio-frequency switch routes the signal among multiple waveguide segments, with a closed-form insertion-loss payoff threshold. Numerical evaluation at the best PASS-compatible THz operating point shows that SF-PASS substantially outperforms single-feed PASS in spectral efficiency and is competitive with a large-scale antenna array at much lower hardware costs.

We study multi Danielewski varieties, which are the generalization of Danielewski varieties and Double Danielewski varieties. The paper contains description of the Makar-Limanov invariant of these varieties and of their isomorphism classes.

We generalize the contour integral methods (CIM) framework to the time integration of linear dynamical systems that are subject to algebraic constraints at all times during their evolution. The proposed approach relies on applying the Laplace transform to the Cauchy problem associated with a linear system of differential-algebraic equations (DAE), and subsequently reconstructing the time-domain solution by approximating the inverse Laplace transform via a suitable quadrature rule. This procedure yields an efficient and accurate alternative to classical Runge-Kutta schemes, which are well known to exhibit order reduction in accuracy when applied to DAE. In the second part of the paper, we address linear parametric DAE and propose an efficient strategy for tuning the integration contour in the CIM framework using suitable structured-unstructured pseudospectral computations. This allows the identification of a single integration profile capable of approximating an entire family of parametric solutions, thereby facilitating the efficient application of model order reduction techniques. Finally, numerical experiments are presented to validate the proposed methodology and support the theoretical findings.

We propose an architecture for learning the dynamics of mechanical systems based on discrete forced Euler-Lagrange equations on Lie groups using only position data. By formulating the dynamics directly on manifold-valued configuration spaces, the method naturally respects the geometric structure of the systems and preserves geometric invariants and conservation laws. The reliance on position measurements alone makes the framework applicable in settings where velocity data are unavailable or noisy. The approach extends naturally to multibody systems, accommodates external control inputs, and demonstrates strong performance on both synthetic and real-world datasets.

Let $\mathcal{A}$ and $\mathcal{B}$ be unital $C^*$-algebras, and let $V_0(a)=\{f(a): f\in\mathcal S(\mathcal A), f(a^*a)=\|a\|^2\}$ be the algebraic maximal numerical range of $a\in\mathcal{A}$, where $\mathcal S(\mathcal A)$ is the set of all states of $\mathcal A$. We study the properties of $V_0(a)$ and characterize surjective maps preserving $V_0$ of triple products. We show that if $\Phi\colon\mathcal{A}\to\mathcal{B}$ satisfies \(V_0(\Phi(a)\Phi(b)\Phi(c))=V_0(abc) \text{~for all~} a,b,c\in\mathcal{A},\) then the map $a\mapsto \Phi(1_{\mathcal{A}})^{-1}\Phi(a)$ is a multiplicative bijection. Furthermore, for von Neumann algebras without central summands of type $I_1$ or prime $C^*$-algebras of real rank zero, such preservers are precisely $*$-isomorphisms multiplied by a central element $u\in Z(\mathcal{B})$ with $u^3=1$.