Papers

Let $K$ be a hyperbolic knot and let $\rho_n$ be the $n$-dimensional irreducible representation induced from a lift of its holonomy representation. Motivated by Goda's asymptotic volume formula and the complexified Volume Conjecture, we study whether the higher-dimensional twisted Alexander invariants associated with $\rho_n$ detect the complex volume of the knot complement. We compute $$ \frac{\pi}{2} \log \left( \frac{A_{K,n-2}(1)A_{K,n+2}(1)} {A_{K,n}(1)^2} \right) $$ for all hyperbolic knots with at most six crossings. Our numerical experiments indicate that these values approach $$ \operatorname{Vol}(S^3\setminus K) +i\,2\pi^2\operatorname{CS}(S^3\setminus K) $$ modulo $i\pi^2\mathbb{Z}$. Based on these computations, we propose a complexified analogue of Goda's asymptotic volume formula.

For every $n\ge2$ and $1<p<\infty$, we show an existence of a nonconstant homogeneous entire $p$-harmonic function whose only critical point is the origin. The angular parts of these functions are axially symmetric and even across the equator. To establish the solution we use contraction argument at the pole in conjuction with a shooting argument. It follows that in every dimension $n\ge3$ and for $p<2$ sufficiently close to $2$, the natural gradient has pointwise Hölder exponent strictly below one at the origin. Such examples settle the remaining scalar case of the conjecture of Balci, Diening, and Weimar by disproving both the $C^1$ assertion and the linear $L^2$ mean oscillation estimate.

The constant-Q transform (CQT) represents audio on a logarithmic frequency axis. Its nonstationary Gabor formulation is exactly invertible, but the unequal numbers of time coefficients in its bands complicate GPU computation. An exact factorisation combines spectral selection, conjugation, windowing, and reordering into a fixed map between one packed Fourier transform and the shorter band inverse transforms. The factors give waveform reconstruction, real adjoints for backpropagation, and bounds on arithmetic depth and block width; overlapping slices permit streaming with bounded memory. Tests on two GPU models show that Flash-CQT reduces analysis-synthesis round-trip time by factors of two to eight relative to a baseline computing the same CQT. The proposed implementation also uses over 30% less peak temporary workspace and reaches a negligible reconstruction error, with a signal-to-noise ratio of about 130 dB, in single-precision floating-point arithmetic. These advances make Flash-CQT a practical, computationally efficient front end for spectral analysis and modern audio machine-learning systems.

We study the existence and nonexistence of meromorphic solutions of nonlinear differential equations involving differential polynomials and finite sums of exponential-type functions with entire coefficients of controlled growth. Employ ing Nevanlinna theory, we obtain structural results showing that any admissible meromorphic solution must reduce to an exponential-type function governed by the dominant exponential term. The obtained lemma and main result strengthen several existing results in nonlinear complex differential equations with exponential terms.

Let $\nu$ be a Krull valuation of arbitrary rank on a field with valuation ring $R_\nu$, and let $\theta$ be a root of the irreducible polynomial $F(x)=(x^k+c)^m-ax^n\in R_\nu[x]$ and $1\leq n<km$. We establish necessary and sufficient conditions for the integral closedness of $R_\nu[\theta]$, expressed explicitly in terms of the coefficients $a$, $b$, and the integers $m$, $n$, and $k$. In particular, when $\nu$ is the $p$-adic valuation on $\mathbb{Q}$, our results yield criteria for determining the primes dividing the index $[\mathbb{Z}_K:\mathbb{Z}[\theta]],$ where $K=\mathbb{Q}(\theta)$ and $\mathbb{Z}_K$ denotes the ring of integers of $K$.

Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{G}_m$. We prove that $C$ is $f$-periodic in the latter case, as expected by the dynamical Mordell-Lang conjecture.

In this paper, we study the existence, nonexistence, and growth be haviour of solutions of certain nonlinear differential and differential-difference equa tions involving the term fnf'. Using tools from Nevanlinna theory, we establish several nonexistence results for finite-order entire solutions of equations involving ff'. In particular, we describe all the possible entire solutions of the type f2f' in which right hand side involves exponential polynomials. The results obtained by us improve several known results in the literature and we have them demonstrated through illustrative examples.

The winding number is a simple topological invariant. In the case of chiral symmetry it characterises gapped phases of Fermions. We study statistical properties of this topological index or invariant in a chiral symplectic setting using Random Matrix Theory. We consider ensembles of Hamilton matrices in the symmetry class CII (quaternionic matrices) according to the classification in the tenfold way. We set up a parametric random matrix model and derive expressions for parametric correlations of the winding number density as well as for the discrete winding number distribution. We found a super-universality in the limit of large matrix dimensions for the one- and two-point correlators for the bulk of parameters, meaning that the results agree up to rescaling with those of the class AIII (complex matrices). In this context we employ a new method of unfolding and discover the Gaussian behaviour of the winding number distribution.

The main objective of this paper is to investigate orthonormal Strichartz estimates for Schrödinger equation on the Wiener amalgam space $\mathcal{W}(\mathcal{F} L^p, L^q)$. More precisely, we first examine improvements in the time-integrability exponent of the existing Strichartz estimates established by Cordero and Nicola in \cite{NFC} for Schrödinger equations associated with $\mathcal{H}^{+}=-\frac{1}{4\pi}\Delta+|x|^2$. We then extend these improved Strichartz estimates from a single initial datum to systems of orthonormal families, providing, to the best of our knowledge, the first results in this direction in the setting of Wiener amalgam spaces. We further extend the classical Strichartz estimates in Wiener amalgam spaces from a single initial datum to systems of orthonormal families of initial data for the Schrödinger equation associated with the operator $\mathcal{H}^{-}=-\frac{1}{4\pi}\Delta-|x|^2$ and Hamiltonian operator of the form $\mathcal{H}_{\mathcal{A}} = -\frac{1}{4\pi} B \nabla \cdot \nabla$, where $\mathcal{A} = \begin{pmatrix} 0 & B \\ 0 & 0 \end{pmatrix} \in \mathrm{Sp}(d,\mathbb{R})$ with $B = B^*$ and $\det B \neq0$. A key ingredient of our approach is Stein's complex interpolation theory for Wiener amalgam spaces, combined with a duality argument inspired by the work of Frank and Sabin. As an application of these orthonormal estimates associated with $\mathcal{H}^{+}, \mathcal{H}^{-}$, and $H_{\mathcal{A}},$ we establish local and small-data global well-posedness for the Hartree equation with infinitely many particles, for non-trace-class initial data.

This paper develops a compact fourth-order positivity-preserving active flux (AF) method for the one- and two-dimensional compressible Navier–Stokes equations on Cartesian meshes. The method retains the cell averages and shared point values of the standard third-order AF method as its degrees of freedom. To avoid the order reduction that can arise when diffusion is discretized using operators from the standard third-order AF method, while maintaining compactness, the divergence of the viscous flux is discretized directly using compact fourth-order operators. For the inviscid part, incorporating a downwind point value into the biased stencil yields fourth-order accuracy. A monolithic flux limiting blends high-order total numerical fluxes with low-order positivity-preserving counterparts, treating the inviscid and viscous fluxes jointly while maintaining local conservation. Together with a scaling limiter for point values, this procedure preserves density and pressure positivity for both cell averages and point values. Numerical experiments demonstrate fourth-order convergence, positivity preservation, and accurate resolution of shocks and viscous flow structures. For the two-dimensional viscous shock tube, a comparison with a discontinuous Galerkin method shows improved computational efficiency in resolving complex interaction of shock waves and boundary layers.

The first two authors derived the Weitzenböck-type Formula for a CR Dirac operator. In this paper, we apply the above-mentioned formula and follow the process as Riemannian case to derive a Reilly-type inequality. We give two applications for the CR Reilly inequality. One is a lower bound estimate of the p-mass in terms of the first eigenvalue of the modified hypersurface CR Dirac operator. Another one is to solve the CR Dirac equation on a bounded domain with an $S^{1}$-action and an APS-type boundary condition.

In this paper, we investigate the existence of pairs of orthogonal quantum Latin squares, each with maximal cardinality. Using characters and full-rank factorizations of finite abelian groups, together with a product construction, we establish two infinite families of orders for which such pairs exist.

This paper studies the time optimal control problem for the heat equation with singular inverse-square potentials under the Hardy critical condition. We first establish an observability inequality for the singular parabolic equation from general space-time measurable sets of positive Lebesgue measure. Rather than depending on the still-unknown Lebeau-Robbiano spectral inequality for the underlying singular operator, our proof combines Carleman-based observability results over open cylinders, real-analyticity estimates of solutions away from the singular origin, propagation of smallness estimates for real-analytic functions, and a telescoping-series technique. Using this observability result, we derive the null-controllability with controls supported on measurable subsets. Finally, we prove that the corresponding time optimal control is unique and obeys the bang-bang property almost everywhere over the control domain.

Links in projective space $\mathbb{R}P^3$ can be represented by diagrams in $\mathbb{R}P^2$ where the projective plane $\mathbb{R}P^2$ is represented by a disk with antipodal identifications on the boundary. In this context if we place an $n-$strand braid $\beta$ on this disk keeping its end points on the boundary then due to the identification of antipodal points it naturally represents a link diagram in $\mathbb{R}P^3$. We call this the projective closure of the braid $\beta$. In this paper we show that not all links in $\mathbb{R}P^3$ possess a diagram represented by projective closure of some braid.

Spatial correlations can suppress scattering in disordered chains and produce perfectly transmitting resonances. The practical value of this protection, however, depends not on the resonance peak itself but on the width of the surrounding transmission window and its sensitivity to local ordering errors. We show that a resonance can remain exactly transparent at its center while arbitrarily rare adjacent-swap errors restore an inverse localization length proportional to the square of the energy detuning. For lossless single-channel chains assembled from independent blocks of fixed length and composition, this positive quadratic term is guaranteed by a local scattering invariant and holds for every arrangement and every fixed swap probability between zero and one. With exact tuning and matched contacts, the central transmission remains unity. A microscopic quantum chain exhibits both higher-order suppression of scattering in the ideal recursive arrangement and the predicted response to local exchanges. These results reveal a limitation of spatial ordering that is invisible to a measurement at the resonance alone: spatial ordering protects the resonance peak, not the transport around it. The effect can therefore be tested experimentally by measuring transmission spectra before and after exchanges, without identifying microscopic defects.

We disprove almost everywhere convergence of the Schrödinger evolution on the standard torus \(\mathbb{T}^d\) for initial data \(f\in H^s(\mathbb{T}^d)\) when \(s<d/(d+2)\), in all dimensions \(d\ge2\). We construct normalized data with frequencies of size $N$ whose evolution attains size $N^{d/(d+2)}$ on a set of uniformly positive measure. The key ingredient in the construction is that the frequencies lie in an affine congruence class, so that at suitable rational times their phases coincide on a family of well-separated spatial points. The result matches the known estimate for $s>d/(d+2)$ with $d\geq2$. By integer dilation and uniform boundedness, we also obtain a single datum in $H^s$ whose evolution is unbounded along a sequence of times tending to zero whenever $s<d/(d+2)$. We also record a logarithmic upper bound in $2D$ at the critical frequency power and a lower bound on shrinking time intervals.

Pilot-aided channel estimation is a decisive block in orthogonal frequency-division multiplexing (OFDM) receivers for both 5G New Radio (5G-NR) and Long-Term Evolution (LTE). A large body of estimators exists, from simple least-squares (LS) interpolation to statistically optimal linear minimum-mean-square-error (LMMSE) variants and, more recently, deep convolutional denoisers, yet no single estimator is uniformly best: the winner depends on the propagation scenario, the numerology, the operating signal-to-noise ratio (SNR), the mobility (Doppler), and the antenna configuration. In this paper, we quantify this fact through a unified study of eight literature estimators evaluated over the 3GPP TR~38.901 Urban-Macro (UMa), Urban-Micro (UMi), and Rural-Macro (RMa) channels generated with NVIDIA Sionna, for both 5G-NR and LTE numerologies, in single-input single-output (SISO) and $8\times2$ multiple-input multiple-output (MIMO) settings. We then propose a condition-adaptive multi-agent orchestrator that treats each estimator as an independent agent and dispatches, per operating condition, to the agent that is best on a validation split without any genie knowledge. The orchestrator tracks the per-realization oracle to within $1.07$~dB and improves the normalized mean-square error (NMSE) over the best fixed strategy by up to $3.6$~dB at high SNR, where the low-SNR champion is no longer optimal. Because the agents are independent, running them concurrently delivers this best-of-eight accuracy at essentially single-estimator latency: a data-parallel partition scales the wall-clock nearly as $1/K$ with $K$ workers (up to $6.9\times$), whereas naive by-algorithm partitioning is Amdahl-limited by the heaviest agent. The results substantiate multi-agent orchestration as a practical route to robust channel estimation across heterogeneous 5G-NR/LTE deployments.

We develop a novel pathwise approach to study the convergence of the law of killed diffusion processes conditioned on non-absorption, towards a quasi-stationary distribution (QSD) as time goes to infinity. We start from the general observation that the dynamics of an absorbed Markov process conditioned upon survival up to time $T>0$ is the minimizer of the pathwise relative entropy with respect to its unconditioned dynamics, under a simple distributional constraint at that time; in other words, a Föllmer process. We then show how this result applies to a Brownian diffusion process softly-killed at a state-dependent regular rate, and characterize the associated drift change. In the case when the diffusion process is moreover reversible, we leverage this idea and recent results on the propagation of weak log-concavity of HJB semigroups to prove that, under strict asymptotic convexity of the potential, the conditioned dynamics satisfy a contractivity property in $1$-Wasserstein distance, uniformly in $T>0$. Under a general ergodicity condition on the associated Feynman-Kac semigroup, we then establish the existence of a QSD with a large domain of attraction, and the exponentially fast convergence to it of the conditioned semigroup in the $1$-Wasserstein distance as $T$ goes to infinity. Finally, we deduce the exponentially fast convergence, also in $1$-Wasserstein distance, of the law of the corresponding Q-process towards its equilibrium.

Wetting transitions on textured substrates play a central role in the design of functional surfaces, but resolving their transition mechanisms requires efficient exploration of complex energy landscapes. In particular, saddle points are essential for revealing the connectivity among metastable states and transition pathways. In this work, we develop a diffuse-domain accelerated saddle dynamics (DD-ASD) framework for mass-constrained wetting transitions on complex textured substrates. The diffuse-domain formulation embeds complex solid geometries into a Cartesian grid, avoiding body-fitted mesh construction in repeated saddle-point searches. An orthogonal projection is applied consistently to the phase-field state and unstable-direction dynamics, preserving the prescribed droplet mass at the discrete level. Momentum acceleration is incorporated into the saddle dynamics to improve the efficiency of high-index saddle searches. Under the stated local spectral and exact-eigenspace assumptions, we establish a local convergence rate of $1-\mathcal{O}(1/\sqrt{\kappa})$ on the effective mass-conserving subspace. Numerical experiments demonstrate the effectiveness of the proposed method in resolving wetting solution landscapes, transition pathways, and energy barriers on textured substrates. We further extend the framework to fully three-dimensional wetting-landscape computations.

The eigenvalue problem for a probability measure $\mu$ with compact support in $\R$ is whether there exist a countable set $\Lambda$ and a nonzero real $t\ne 1$ such that both $\Lambda$ and $t\Lambda$ are spectra of $\mu$, that is, the family $$E_{a\Lambda}=\{e^{-2\pi i a\lambda x}:\lambda\in\Lambda\}$$ is an orthonormal base for $L^2(\mu)$ for $a=1, t$. The eigenvalue problem was discovered independently by Strichartz (Str00), Laba and Wang (LW02) for the Cantor measures $\mu_{4,\{0,1\}}$ and $\mu_{6,\{0,1,2\}}$, respectively. In this paper, we investigate the spectral eigenvalue problem for the general spectral Cantor measure $\mu_{b,\mathcal{D}}$. This topic is naturally related to elementary number theory. Unexpectedly, however, our main results depend on the theory of integers, especially Artin's primitive root conjecture. To some extent, our results suggest that Artin's primitive root conjecture may hold and confirms some viewpoints implied by Minkowski in (Min57).

We prove that a lattice $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb C)$ has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense. Our method also gives a similar result for lattices in $\mathrm{SL}_d(\mathbb R)$, for every $d\ge3$: a lattice $\Gamma<\mathrm{SL}_d(\mathbb R)$ has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree $d$ over $\mathbb Q$ that splits over $\mathbb R$. We also discuss spectral consequences and other applications of the results and techniques.

In this article, we consider the stochastic nonlinear heat equation driven by Lévy space-time white noise in dimension one. For the spatial average of the solution, we prove quantitative and functional central limit theorems under $m_1+m_{2p}<\infty$ for some $p\in(1,\frac{3}{2})$. These results extend the Gaussian fluctuation theory for the parabolic Anderson model to the nonlinear setting. The main new feature is a minimum-type term in the second Malliavin derivative estimate caused by the nonlinear coefficient.

We prove that the language of palindromic reduced words in a Coxeter system is regular if and only if the Coxeter group is a direct product of a finite Coxeter group with finitely many copies of the infinite dihedral group.

We obtain a large number of exact hyperbolic, trigonometric, and periodic solutions in terms of Jacobi elliptic functions as well as algebraic solutions with a power law tail of the integrable local Fokas-Lennels equation and integrable nonlocal Fokas-Lennels equation. Further, we consider a one-parameter family of generalized Fokas-Lenells equations and obtain a few of their exact solutions.

A multispecies, collisionless plasma is modeled by the relativistic Vlasov-Maxwell system. Assuming the plasma is globally neutral, we show that solutions can be constructed with arbitrarily fast, polynomial rates of decay, depending upon the cancellation properties of the limiting spatial averages, as well as the masses and charges of the particles. In doing so, we establish a countably infinite number of asymptotic profiles for the charge and current density, electric and magnetic fields, and their derivatives, each of which is necessarily realized by a sufficiently smooth solution. Our approach relies upon a refined field representation that features self-similar behavior away from the boundary of the light cone. In each case we establish a linear scattering result in $L^\infty$ for every particle distribution function, namely we show that they converge as $t \to \infty$ along the transported spatial characteristics at increasingly faster rates. In the case that charge cancellation does not occur, our methods can be further extended to demonstrate polyhomogeneous expansions with logarithmic corrections.

We prove a Torelli theorem for full moduli stacks of vector bundles of rank at least three with fixed determinant on a smooth, proper, geometrically connected, tame Deligne–Mumford curve of coarse genus at least two over an arbitrary field $k$. We also prove a Torelli theorem for the full moduli stack of vector bundles of rank at least three of fixed-degree under an explicit compatibility hypothesis. The method is intrinsic to the stack: faithful special-linear linearizations recover Picard stacks, and determinants of tangent complexes recover the canonical principal polarization. This is the first part in a series of two papers.

We study weak solutions to degenerate parabolic double phase equations with growth exponents $2\le p<q$ and a modulating coefficient that is Hölder continuous with exponent $\alpha$. If the solution itself is Hölder continuous with exponent $\gamma$, we prove higher integrability of its gradient under the gap condition $q\le p+\alpha/(1-\gamma)$ together with $q<p+1$. This is the first gap bound of Hölder interpolative type for parabolic double phase problems, and it is the parabolic counterpart of the corresponding bound for elliptic problems. For $\alpha<1$ it allows exponents beyond the range $q\le p+\alpha$ known for bounded solutions.

This paper investigates the null controllability of the complex cubic Ginzburg–Landau equation with controls supported on general space-time measurable sets. Starting from a positive-measure control set, we use a slicing argument to obtain time slices of uniformly positive spatial measure. This yields uniform spectral inequalities on the slices, which are the key ingredient in constructing quantitative stabilizing feedbacks. A density-point argument provides a sequence of shrinking intervals accumulating at the density point, on which the active times have a uniform positive density. We then apply a time-iteration scheme with piecewise feedback laws and increasing decay rates. Iterating the resulting stabilization estimates shows that the closed-loop solution reaches the zero state at the selected density point. This provides a constructive approach to the local null controllability on measurable control sets for nonlinear parabolic equations.

Motivated by the work of Druţu–Sapir, we introduce a definition of relatively hyperbolic TDLC groups that depends only on their Cayley-Abels graphs and show that this new definition is equivalent to one introduced by Arora–Martínez-Pedroja. Then, we prove that relative hyperbolicity for TDLC groups is invariant under quasiisometry.

We prove sharp metric entropy estimates for spaces of Kähler potentials. In complex dimension $n$, normalized potentials have Kolmogorov entropy of order $\e^{-n}$ in the background $L^1$ metric. On a polarized manifold, a relative-entropy sublevel has the same order in the Mabuchi–Darvas $d_1$ metric, including its full finite-energy closure. The upper bound is $C_X(1+B)^{n+1}\e^{-n}$ for entropy budget $B$. For toric potentials, the sharp exponent is $n/2$.