Papers

Let $D$ be a finite simple minimally strongly $k$-connected digraph, where $k\ge 2$, and let $n=|V(D)|$. Mader proved in 1985 that $|A(D)|\le 2k(n-k)$ for all sufficiently large $n$, with equality if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$ (JCTB 1985). We prove that this bound holds whenever $n\ge 3k-1$ and that the order threshold $3k-1$ is best possible. Moreover, if $n\ge 3k$, then equality holds if and only if $D\cong \overleftrightarrow{K}_{k,n-k}$. The threshold $3k$ is also best possible. These results resolve the conjecture of Mader in (JCTB 1985).

This paper is concerned with the inverse scattering problem of determining a real-valued potential for the three-dimensional Schrödinger equation from multi-wavenumber data. Under suitable a priori bounds, we establish local increasing stability from near-field Dirichlet data on a fixed sphere for polynomially decaying potentials, and global increasing stability from full far-field amplitudes for exponentially decaying potentials. The stability estimates consist of a noise term and a remainder term, with the latter decreasing as the upper measured wavenumber increases. We also obtain global uniqueness from exact near-field data under weaker regularity assumptions. The analysis combines analytic continuation, weighted resolvent estimates, and half-ray identities whose differences at chord endpoints cancel the unknown exterior contribution.

We provide a unified algebraic classification of approximate controllability, approximate null controllability, exact null controllability, and exact controllability for finite-dimensional linear stochastic control systems with constant coefficients and multidimensional Brownian noise. The controls may enter both the drift and the diffusion terms, and the joint diffusion-control matrix may have arbitrary rank. A finite decreasing sequence of subspaces defines a common geometric obstruction. Its vanishing characterizes the two approximate properties and exact null controllability. Exact controllability to arbitrary random targets additionally requires the joint diffusion-control matrix to be surjective. The codimension of the obstruction is given by a rank difference of explicit polynomial block matrices, yielding finite tests in the original coefficients. When the joint diffusion-control matrix is surjective, we obtain a finite Kalman-type full-rank condition for exact controllability. If the obstruction vanishes, a linear state feedback with a deterministic time-dependent gain drives every initial state to zero at any prescribed positive time with finite energy and polynomial small-time cost. An equivalent orthogonal elimination algorithm computes the rank differences with polynomial storage and cost, avoiding exponentially large block matrices. A two-noise example relates the input classification to terminal errors and control energy in penalized stochastic linear-quadratic control problems.

Let $(A,\mathfrak{m})$ be a Noetherian local ring and let $I\subseteq \mathfrak{m}$ be an ideal in $A$. Let $\ell(I)$ be the analytic spread of $I$. Let $G_I(A) = \bigoplus_{n \geq 0}I^n/I^{n+1}$ be the associated graded ring of $I$ and let its depth be $t$. We prove $\text{depth} \ A/I^n \geq t - \ell(I)$ for all $n\geq 1$.

We study uniqueness for parabolic motions in the Newtonian $N$-body problem with a prescribed minimal normalized central configuration at infinity. A McGehee compactification identifies their tails with the strong stable manifold of the corresponding outgoing zero-energy equilibrium. The configuration projection of this manifold is locally a smooth graph, yielding a smooth initial-velocity field and uniqueness among all parabolic motions confined to a sufficiently narrow truncated cone, without any minimizing assumption. For parabolic geodesic rays, a renormalized action excess and a quantitative no-exit argument remove the confinement hypothesis when the initial configuration lies sufficiently far inside a smaller cone. We then prove that the set of collision-free initial configurations admitting more than one such ray has Lebesgue measure zero. The global argument propagates the local stable manifold backward, discards critical values of the configuration projections by Sard's theorem, and separates distinct regular branches through an explicit differential formula for their renormalized excess. No nondegeneracy assumption on the prescribed minimal central configuration is required. The proofs combine invariant-manifold theory with direct action comparisons and do not use weak KAM theory, Busemann functions, or the differentiability theory of Hamilton–Jacobi solutions.

We formulate offline tuning of Model Predictive Path Integral (MPPI) controllers as a chance-constrained black-box optimization problem and develop an update-aware framework based on Matrix Adaptation Evolution Strategy (MA-ES). We use statistical information from noisy controller evaluations to determine when the evolutionary update is sufficiently reliable. When the recombination direction is certified, a derivative-free nonmonotone line search (NMLS) is used to refine the candidate solution, and a separate validation stage assesses the final frozen controller. We establish a worst-case arithmetic-complexity bound for both online MPPI control and offline tuning. Numerical experiments on TurtleBot 4, Jackal-like, and Ackermann models show that the proposed NMLS refinement is competitive with standard MA-ES, transfers across the three vehicle models without model-specific retuning, and improves several cost and validation measures. Both MA-ES variants also compare favorably with the tested randomized derivative-free line search (VRBBO) configuration. The results support the practical applicability and transferability of the proposed framework, while not claiming universal superiority or hardware-level safety.

We study a non-convex, nonlinear integer mathematical program with a linear constraint. This problem originates in finance and consists of selling a large predetermined quantity of a financial asset over a fixed time horizon, while maximizing the total proceeds. The selling price is modeled using a deterministic penalty function capturing the market's adverse reaction to the selling flow. From a complexity perspective, the problem is in NP. We first solve the problem exactly using dynamic programming and propose several heuristics that yield tight numerical lower bounds. We then reformulate the problem into an equivalent factorable form and derive a tight convex relaxation using McCormick inequalities. This relaxation is embedded within an adapted Branch & Bound algorithm to find the global optimum in finite time. Numerical experiments show that our methods outperform several widely used generic solvers (LocalSolver, Scip, Baron, Couenne, and Bonmin) for selected instances, but do not yield tight upper bounds, beyond small instances.

Discretization can change a sharp stability threshold even when the continuum threshold is recovered at infinity. We study this phenomenon for the Hardy inequality on the integer lattice in dimensions at least three: its best coefficient agrees with the continuum value through dimension eight and is smaller from dimension nine onwards. We characterize this transition by the total critical defect, the summable error obtained by testing the lattice operator against the critical continuum power. A theta-function representation proves that the defect is negative and strictly decreasing from dimension nine; explicit positive Hardy remainders establish its positivity in dimensions three through eight. An exact energy identity relates this defect to the energy saved by optimizing the profile outside the neighbours of the origin. This exterior remains stable in dimensions eight and nine, locating the first loss of positivity in its coupling to those neighbours. We give self-contained proofs of both dimensional ranges. We then describe how local attraction creates a negative eigenvalue in a general class of lattice operators with critical inverse-square tails. Its magnitude is exponentially small near the threshold, with both the rate and the multiplicative prefactor determined by zero-energy solutions. The result allows connected finite modifications of the lattice and nonradial corrections to the tail. The same analysis controls the spread of the emerging state and gives a two-dimensional return scale for the associated Markov chain in every ambient dimension.

It is a classical fact in spectral geometry that the eigenvalues of the Dirichlet Laplacian depend continuously on the geometry of the underlying domain, under suitable geometric restrictions. We show that this theoretical property is reflected in a concrete numerical situation: the convergence rate of iterative solvers for convection-diffusion problems remains stable under deformations of the domain. The connection is a superlinear estimate for preconditioned GMRES/GCR in which the relative residual errors are bounded by an explicit expression of the eigenvalues of the Laplacian, or of a symmetric positive elliptic operator in divergence form. We prove that this bound depends continuously on the domain, uniformly in the iteration index, under either regularity hypotheses on the boundary or a topological constraint in dimension two. We also establish corresponding statements on the discrete level. Two examples delimit the theory: domains bounded by a fractal curve, where continuity still holds, and perforated domains, where it fails. The theoretical results, including both examples with rough domains, are confirmed by numerical experiments.

We present a mass-weighted meshless formulation for the shifted incompressible lattice Boltzmann method (LBM) whose streaming step is exactly unitary, combining with the Carleman embedding. In traditional meshless LBM, on a scattered node set the semi-Lagrangian streaming step is an interpolation, which is not unitary; a generic block encoding of this matrix therefore requires a normalization proportional to its sparsity and logarithmically many ancillas. To overcome the non-unitarity, we instead express the interpolation as the exponential exp(a Dx + b Dy) of mimetic derivative operators. Endowing each node with its Voronoi mass and applying a mass-weighted similarity transform renders these derivative operators skew-Hermitian by enforcing moment constraints together with low-wave number trigonometric differentiation conditions, so that the interpolation becomes exactly unitary. Such unitarization of interpolation leave the local collision step unchanged up to a rescaling of its quadratic term, where the same weights define the amplitude encoding of the distribution functions. For quasi-uniform node sets, the mass-weighting diagonal remains within a unit-order normalization that degrades only mildly as the Carleman truncation order grows, thereby mitigating the cost of block encoding. Numerical experiments on the Taylor Green vortex confirm that the method reproduces the full meshless LBM, with the velocity error decreasing as the Carleman order increases.

We study how local coherenAAce of abelian categories behaves under gluing. For AB5 locally finitely presentable abelian categories, we give conditions on the gluing functors under which local coherence is transferred along a gluing $(\mathcal{A}, \mathcal{B}, \mathcal{C})$: from $\mathcal{A}$ and $\mathcal{C}$ to $\mathcal{B}$, and conversely from $\mathcal{B}$ to $\mathcal{A}$ or $\mathcal{C}$. The coherence theorem for augmented homotopes of Bondal and Zhdanovskiy is recovered as a special case, the coherence theorem for trivial extensions of Fossum, Griffith, and Reiten is a companion result in the same direction. We also introduce quasiprojective pairs of adjoint functors and prove a coherence criterion that generalises the criterion of Gomez Pardo and Martinez Hernandez to arbitrary Grothendieck categories. Finally, we extend Stenström's characterisation of coherence via FP-injective modules to compactly assembled Grothendieck categories: such a category is coherently assembled if and only if the class of CM-injective objects, defined through compact morphisms, is closed under filtered colimits. Along the way we revisit the various equivalent conditions for the coherence of a ring, collecting them in a single place. The proofs make use of the compactly assembled categories of Efimov.

We present a unified treatment of the data-fitting problems with quadratic (Hilbertian) regularization that arise in approximation theory, machine learning, engineering, and statistics. The generic problem is an optimization in the continuum: given $M$ linear measurements of an unknown function, one minimizes a convex data-fidelity term that is penalized by $\|\mathrm{L} f\|^2_{L_2}$, where $\mathrm{L} $ is a differential-like (spline-admissible) operator. The natural domain of the minimization is the native space $\mathcal{H}_\mathrm{L}$ of $\mathrm{L} $ - the largest Hilbert space over which the regularizer remains meaningful - for which we provide a concise and directly usable characterization. Central to our formulation is a measurement-adapted (conditioned) basis that decouples the null-space and spline components of the solution. This basis serves in three places. First, it tightens the formulation of our representer theorem, which confines the unique solution to an $M$-dimensional space without any orthogonality constraint on the coefficients. Second, it streamlines the implementation, as the exact discretization of the problem becomes an unconstrained convex program with closed-form solutions in the least-squares case, which covers ridge regression, Tikhonov regularization, and smoothing splines. Third, it makes the bridge with the minimum-mean-square-error estimation of generalized Gaussian processes, which is the topic of a companion paper.

We prove that the realization functors obtained by alternating forward and backward HRS tilts of any torsion pair over a finite-dimensional algebra are equivalences from some step onward. The number of steps is bounded by the lengths of the indecomposable projective modules. The even hearts define an increasing filtration of ideals contained in a square-zero ideal. For a torsion pair induced by a two-term silting complex, we identify the successive layers with Buan-Zhou kernel ideals and compute them by reduction modulo the successive ideals. We also describe the largest quotient over which the complex becomes tilting and determine the exact stabilization index from the two parity heights. We construct algebras for which the HRS height is arbitrarily large, and compute their ideal filtrations.

In this paper, new existence and multiplicity results for positive periodic solutions of singular second-order problems under sublinear and superlinear growth assumptions at infinity are established. Two aspects of our results should be highlighted in comparison with the existing literature. On the one hand, they allow for the consideration of systems exhibiting mixed behavior: sublinear in one component and superlinear in another. On the other hand, in the superlinear case, the existence of up to four positive solutions is established, whereas only two were obtained in previous works. Our approach is based on a recently established vectorial version of the Krasnoselskii-Guo fixed point theorem in product spaces.

The variational theory of splines requires a native space: a Hilbert space over which the regularization energy $\|\mathrm{L} f\|_{L_2}^2$ is well defined and the minimum-energy interpolation problem admits a spline solution. In this paper, we characterize the operators $\mathrm{L}$ that enable the construction of such spaces. Our framework is based on the completion of a space $\mathcal{S}$ of test functions in a controlled Gelfand-triple configuration $\mathcal{S}\subset\mathcal{H}\subset\mathcal{S}'$, which we call a Hilbert-Gelfand space. This departs from the classical reproducing-kernel techniques. Our construction is substantially more general, as it supports arbitrary linear operators and singular behaviors, and it recovers the reproducing-kernel setting as a particular case. It is also complete and canonical: the admissibility conditions on $\mathrm{L}$ are necessary and sufficient, and $\mathrm{L}$ determines its native space as the unique Hilbert-Gelfand space on which it extends to a bounded surjection onto $L_2$. We first show that Hilbert-Gelfand spaces are in one-to-one correspondence with compatible Riesz pairs $(\mathrm{A},\mathrm{A}^{-1})$ of positive-definite operators $\mathcal{S}\to\mathcal{S}'$. We then extend this correspondence to conditional Riesz pairs, where $\mathrm{A}$ is only conditionally positive-definite, which is the scenario that is relevant to splines. The theory identifies the attainable null space of the energy as the unique admissible conditioning, realizes the abstract completions as concrete spaces of distributions, and provides a verifiable admissibility condition in terms of a dual energy functional. The framework is illustrated with the fractional Laplacian, which covers polyharmonic and thin-plate splines, and with ordinary differential operators, which cover exponential splines.

To solve a family of large linear systems where the system matrix depends on the family parameter quadratically, we propose a new solution approach. Our method employs the derivatives of solutions with respect to the parameter. It turns out that the accumulated solution derivatives of increasing order, evaluated at a single specially chosen parameter value, provide successively better approximations to solution subspaces for the whole linear system family. We show that the derivatives can be computed by a recurrence, involving, at each step, solving a linear system with the same matrix. Constructing an orthonormal basis for the subspace spanned by the derivatives then leads to an efficient solver where just a single (sparse) LU factorization (or a single preconditioner setup in an iterative scheme) suffices to solve the whole linear system family. We call the new method ODB (orthonormal derivative basis method). As shown in numerical tests, compared to conventional solution methods, the ODB method allows to drastically reduce computational costs.

We study the Gibbs measure of $H(h) = \Re\sum_{p^{j} \leq X}U_{p}^{j}/(j p^{j(\sigma + i h)})$, a random field modeling large values of the log-modulus of the Riemann zeta function on and near the critical line, where the $U_{p}$ are independent and uniformly distributed on the complex unit circle. Writing $t = \log \log T$, the cutoff $X$ and real part $\sigma$ may vary with $T$ subject to $T^{1/t^{\Phi}} \leq X \leq T^{1/t^{\phi}}$ and $0 \leq \sigma - 1/2 \leq C_{\sigma}t^{\Phi}/\log T$, for fixed $\Phi \geq \phi > 1/2$ and $C_{\sigma} > 0$. On $I_{\theta} = [ - (\log T)^{\theta},(\log T)^{\theta}]$, define the Gibbs measure $G_{\beta,\sigma,T}^{\mathrm{H}}(\mathrm{d}h) := e^{\beta H(h)}\mathrm{d}h/\int_{I_{\theta}}e^{\beta H(h')}\mathrm{d}h'$. For every fixed $\theta > - 1$ and inverse temperature $\beta > 0$, the limiting joint distributions of correlations (overlaps) among points sampled from this measure are determined after averaging over the Haar phases. At and below $\beta_{c}(\theta) = 2\sqrt{1 + (\theta \vee 0)}$, the limiting two-overlap distribution is concentrated at $q_{0} = |\theta \wedge 0|$. Above this threshold, it assigns masses $\beta_{c}(\theta)/\beta$ and $1 - \beta_{c}(\theta)/\beta$ to $q_{0}$ and $1$, respectively, and the limiting joint overlap distribution is described by a Poisson-Dirichlet process of parameter $\beta_{c}(\theta)/\beta$. Approximate Ghirlanda-Guerra identities are shown to hold uniformly for bounded deterministic functions of the sampled positions. This paper is the first in a series devoted to extending these results to $\log|\zeta|$ near the critical line.

Let $\Gamma \le \mathrm{SL}(d+1,\mathbb R)$ be a torsion-free hyperbolic group dividing a properly and strictly convex domain $\Omega \subset \mathbb R\mathrm P^d$, and let $\rho: \Gamma \to \mathrm{SL}(d+1,\mathbb R)$ be the inclusion. We study representations of $\Gamma$ into the coaffine group that are deformations of $\rho\oplus 1 \in \mathrm{Hom}(\Gamma, \mathrm{GL}(d+2,\mathbb R))$ and that lie on the boundary of the locus of projective Anosov representations. In this setting, we identify a canonical limit set $\Lambda$ for the action and study its fine geometric features which are, in a suitable sense, stable. We prove the existence of an ending lamination that encodes the failure of the Anosov condition and the non-existence of a Cannon–Thurston map. Under certain conditions, we compute the Hausdorff dimension and show that the corresponding measure is positive.

We study how many bits are needed to write down an elementary analytic expression that uniformly approximates a Lipschitz function. Expressions are finite trees over the basis $\{+,-,\times,\div,\exp\}$ with the variables and rational constants, and their size is $\dc(f)$, the length of the shortest description of $f$ in a fixed binary description language. For functions of one variable we prove that every $L$-Lipschitz $F:[0,1]\to[0,1]$ has an everywhere-defined approximant of accuracy $\Delta$ with $\dc(f)\le C\,(L/\Delta+\lg(1/\Delta+2))$ for all $L,\Delta>0$, by a linear spline smoothed with logistic ramps in a description language with one repetition construct. We prove a matching lower bound valid for every description language: for $\Delta\le 1/36$ and every $L>0$ some $L$-Lipschitz function needs at least $\tfrac1{16}(L/\Delta+\lg(1/\Delta+2))$ bits. Hence the worst-case description complexity is $\Theta(L/\Delta+\lg(1/\Delta+2))$. For functions on $[0,1]^d$ the same method gives $\Theta_d\bigl((L/\Delta)^d+\lg(1/\Delta+2)\bigr)$, using a tensor-product spline whose coefficients are mixed second differences and cost $O(d)$ bits each. Without the repetition construct the upper bounds carry an additional logarithmic factor, whose necessity for plain expression trees is left open. A reference encoder, decoder and evaluator confirm the constructions numerically, and figures and tables report the checks.

Let $G$ be a finite bipartite graph and let $\widetilde G$ be an $M$-fold cover. We compare independent sets of $\widetilde G$ with those of the trivial cover, consisting of $M$ disjoint copies of $G$, after prescribing how many vertices are selected above each vertex of $G$. A conjecture from 2011, motivated by the Bethe approximation, asserts that the trivial cover maximizes every such count. We prove this for every feasible occupation profile whose active graph has treewidth at most two or is chordal bipartite. Equality holds exactly when the restricted cover is flat, meaning that fibre relabelings make it trivial. For $M\ge2$ and positive feasible occupations, a nonflat cover of a connected bipartite graph of treewidth at most two has a deficit of at least twice the coefficient of $p_G^{M-2}$ with every occupation reduced by one. A holonomy-dependent bound improves the factor to three for odd-order holonomy; both uniform constants are sharp. Comparison and rigidity survive finite compositions along an edge or a vertex, and quantitative bounds from one piece propagate to the whole graph, giving results beyond the two original classes. On any bipartite base, the first nonzero homogeneous defect is determined by the shortest cycles with nontrivial holonomy. At singleton occupations, our counts are numbers of DP-colorings of full covers. The dual DP color function on these classes and their finite seam compositions equals the chromatic polynomial, with equality attained precisely by canonical labelings. Dong's chromatic-polynomial inequality then yields the Dual-DP Shameful inequality on the same graph class. The main theorems have been formalized in Lean 4.

This paper presents wideband indoor channel measurements and measurement-based statistical modeling of channel parameters in the upper 6 GHz (U6G) band using directional and quasi-omnidirectional antenna pairs. Measurements span 6.425-7.125 GHz at 32 matched positions per configuration in an open hall and an L-shaped corridor. System calibration and frequency-dependent peak-gain normalization retain antenna radiation-pattern weighting. Close-in and floating-intercept models are fitted to line-of-sight (LOS) and non-LOS (NLOS) path loss, respectively. Shadow-fading residuals, RMS delay spread (DS), and equivalent K factors derived from power delay profiles are characterized. LOS exponents range from 1.98 to 2.14. Despite similar attenuation slopes, directional links exhibit smaller LOS residuals and DS and higher equivalent K factors. Hall DS medians are 1.78 and 36.52 ns for directional and quasi-omnidirectional links. Corridor NLOS slopes are -1.12 and 2.54, respectively, while DS ordering varies by position. These results characterize antenna-dependent parameters for U6G indoor channels.

We establish the holomorphic dependence of the Helmholtz Calderón projector on closed $C^{1,1}$ surfaces and of the operator associated with the Maxwell electric field integral equation (EFIE) on closed and open surfaces of the same regularity. More precisely, after transporting the boundary integral operators (BIOs) to fixed reference spaces, we prove that the resulting maps admit bounded holomorphic extensions to complex neighborhoods of compact sets of admissible boundary transformations, a property referred to as shape holomorphy. A key step in our proof consists in extending the results in [Lanza de Cristoforis and Rossi, Analytic Methods of Analysis and Differential Equations: AMADE 2006 (2008), pp. 193–220] and [Dalla Riva, Luzzini, and Musolino, Inverse Problems 38 (2022), 055004] from Schauder to fractional Sobolev spaces. For the EFIE, the single layer representation and the surface Piola transformation give the result in Maxwell trace spaces, whereas the result for open surfaces or screens is obtained by introducing suitable restriction and extension by zero operators. As opposed to existing results, we do not rely on an affine-parametric representation of the boundary deformations, thus allowing for more general parametric shape variations. Finally, we establish shape holomorphy of the shape-to-solution map for the EFIE, subject to interior electric Maxwell nonresonance on closed surfaces but with no such condition on screens. Under suitable summability and stability assumptions, these shape holomorphy results provide an analytic basis for the dimension-robust convergence of interpolation and quadrature methods used in computational uncertainty quantification (UQ) of scattering problems with high-dimensional, shape-parametric uncertain geometry.

Deep learning provides a flexible mesh-free framework for solving elliptic interface problems. However, domain-specific neural approaches often rely on separate networks for individual subdomains, causing the model complexity to grow rapidly as the number of subdomains increases. To address this limitation, we propose a Domain-Specific Adaptation Network (DSAN), which combines a shared backbone with lightweight subdomain-specific adapters to provide local represen- representation flexibility without replicating a complete network for each subdomain. We further establish a stability and error analysis showing that the continuous physics-informed residual controls the solution error, and decompose the resulting error into approximation, optimization, and sampling components. Numerical experiments on two- and three-dimensional problems with multiple regular and irregular interfaces demonstrate that DSAN achieves accuracy comparable to or better than fully decomposed multi-network methods while requiring substantially fewer trainable parameters. More- over, the parameter savings become increasingly pronounced as the number of subdomains grows, demonstrating the scalability of the proposed domain-specific adaptation framework.

Implicit time integration is the standard way to overcome the tight CFL time step restriction when solving nonstationary PDEs numerically. However, within an adaptive time accurate integration, the time step size selection is governed not only by stability but also by accuracy requirements. Therefore, an eligible question to ask is: is (unconditionally) stable implicit time stepping really needed if the time step is restricted by accuracy considerations anyway? And how much do we then lose (in terms of the time step size) when switching from the expensive implicit time stepping to stabilized explicit schemes? An answer to this question is also important because such a switch is often done when solving large scale nonstationary PDEs on massively parallel computers. This paper attempts to answer these questions by numerical tests for two particular adaptive time accurate settings: (i)~nonstationary convection-diffusion equations and (ii)~nonlinear heat equations.

This paper develops adaptive movable-antenna (MA) array structures to enhance wireless sensing performance under arbitrary SNR conditions, using the AoA estimation MSE as the sensing metric. Due to the presence of strong sidelobes in the ambiguity function of sparse MA arrays, the classical CRB of AoA estimation MSE fails to characterize the true MSE in the low and medium SNR regimes, rendering CRB-optimized antenna position vector designs suboptimal for MA-enabled sensing. We derive a deterministic MSE approximation that captures the local estimation error within the mainlobe of the ambiguity function and the outlier error induced by sidelobes, providing a tractable and accurate optimization objective. We then propose three adaptive MA structures that are tailored to different hardware and computational requirements. For systems with fully independent motors, a movable free array is optimized by a two-stage greedy algorithm, which approaches the performance of the globally optimal exhaustive search but with lower computational complexity. Halving the required motors, a movable nested array fixes the dense inner subarray and adjusts only the outer-subarray spacing, reducing the optimization to a one-dimensional search. As the most hardware-efficient design, a movable twin array is proposed to adjust the inter-subarray spacing only with a single motor, while still achieving significant MSE reduction over fixed array geometries. Numerical results show that the derived approximation closely matches the true MSE compared with existing performance bounds, and that the proposed structures outperform the CRB-optimized designs by orders of magnitude in MSE across the tested low-to-medium SNR range. Moreover, it reveals that in the low-SNR regime, APV reconfiguration is more effective for improving estimation accuracy than simply increasing the number of fixed antennas.

We establish dimension-free, pointwise optimal lower-deviation bounds for convex functions under spherically symmetric Cauchy-type measures. These bounds use either mean-variance normalization or moments about the median, the latter covering the Cauchy case where the mean and variance are unavailable. By approximation with a parametrized family the Cauchy-type measures, our results recover the Gaussian setting of Paouris and Valettas \cite{PaourisValettas18} and strengthen their variance-sensitive estimate to a sharp bound, with a complete equality characterization. This goes beyond the partial improvements obtained in subsequent work. We also prove an Ehrhard-type inequality for arbitrary convex sets under these Cauchy-type measures in every dimension and characterize all equality cases. This resolves a conjecture posed by Byczkowski, Małecki and Żak \cite{ByczkowskiMaleckiZak22}. The proof follows from a variational argument combined with the joint concavity of fiber rearrangement. The geometric inequality reduces distributions of convex observables to one-dimensional convex transforms, yielding the sharp lower-deviation bounds.

Within the framework of multivalued dynamical systems, we compute the Conley index for a class of limit cycles in planar Filippov systems. We classify these cycles according to their interaction with the switching manifold, including crossing, sliding, and mixed regimes. Using the Conley index for multivalued admissible maps developed by Mrozek, we construct the appropriate index pairs and determine the corresponding indices for each class. These computations enlarge the class of Filippov invariant sets for which the Conley index is known and extend previous results on low-codimension singularities to periodic dynamics.

Let $A=\mathbb{Q}[t^{\frac{1}{2^n}}:n\geq 0]$ and consider $A[u,u^{-1};\sigma]$, a skew Laurent polynomial ring, where $\sigma(t^q)=t^{2q}$. We study the kernel of derived extension of scalars along $ A[u,u^{-1};\sigma]\longrightarrow\mathbb{Q}[u,u^{-1}], $ the quotient by $\mathfrak{m}A[u,u^{-1};\sigma]$, with $\mathfrak{m}=(t^{\frac{1}{2^n}}:n\geq 0)$. A finite free resolution of $\mathfrak{m}A[u,u^{-1};\sigma]$ over $ A[u,u^{-1};\sigma]\otimes_{\mathbb{Q}} \bigl(A[u,u^{-1};\sigma]\bigr)^{\mathrm{op}} $ proves smoothness. A specialization inequality for matrix ranks supplies a nonzero exact colimit-preserving functor that vanishes on every compact object, proving failure of compact generation.

The iHowe duality for iquantum groups of type AIII/IV was established in \cite{LX22} via two approaches, based on partial flag varieties and quantum coordinate coalgebras respectively. In this paper, we provide a new construction of this iHowe duality via quantum differential operators. Moreover, by deriving explicit action formulas, we prove that our construction coincides with the two constructions given in \cite{LX22}.

In this paper, we prove a global $C^{2,\alpha}$ estimate for the complex Monge–Ampère equation with a positive $C^\alpha$ right-hand side on a compact Kähler manifold.