Papers

The notion of $d$-stable dg categories axiomatizes $d$-cluster tilting subcategories of stable dg categories. We establish an Auslander correspondence for $d$-stable dg categories: we characterize the $d$-stability of an additive connective dg category in terms of coherence, weak global dimension, and a duality on finitely presented modules. This gives a homological characterization of $d$-stability and reveals it as a twisted form of $(d+1)$-Calabi–Yau duality. For locally finite connective dg algebras, this interpretation becomes particularly transparent under Koszul duality, where $d$-stability corresponds to a shifted self-injectivity condition on the Koszul dual. Following the constructions of Amiot, Guo and Keller, for a $d$-stable dg category $M$, we introduce its $d$-cluster dg category $\mathcal C_{d,{\rm dg}}(M):=\operatorname{per}_{\rm dg}M/^\mathbb{L}\mathcal D^b_{\rm fp, dg}(M)$. Using our Auslander correspondence, we show that $\mathcal C_{d,{\rm dg}}(M)$ contains $M$ as a $d$-cluster tilting subcategory. In particular, every $d$-stable dg category can be realized as a $d$-cluster tilting subcategory of a stable dg category. We then develop cluster Morita theory: a pretriangulated dg category equipped with a $d$-cluster tilting subcategory $M$ is quasi-equivalent to $\mathcal C_{d,{\rm dg}}(M)$. Thus the connective dg structure of a cluster tilting subcategory determines its ambient dg category up to quasi-equivalence. As an application of cluster Morita theory, we prove a Morita-theoretic variant of Amiot's conjecture. More precisely, we establish a Calabi–Yau correspondence: for a locally finite $d$-stable dg category $M$ over a field, right $(d+1)$-Calabi–Yau structures on $\mathcal D^b_{\rm fp, dg}(M)$ are in bijection with right $d$-Calabi–Yau structures on $\mathcal C_{d,{\rm dg}}(M)$.

We investigate stability properties of distribution-dependent stochastic Volterra equations with respect to changes in the coefficients, the Volterra kernels, and the initial condition. Under Lipschitz continuity assumptions on the coefficients, we first derive quantitative stability estimates for strong solutions with explicit error bounds. We then prove a general convergence theorem for strong solutions under substantially weaker assumptions, replacing Lipschitz continuity by a continuity assumption together with uniform linear growth of the approximating sequence. Finally, we study the associated distribution-dependent Volterra local martingale problem and prove the stability of its solutions under convergence of the coefficients, kernels, and initial distributions.

The Lagrange spectrum is a classical object in number theory, defined as the set of values of $\liminf_{q\to\infty} q\, \mathrm{dist}(q\alpha,\mathbb{Z})$ where $\alpha$ runs through irrational numbers. It has a complicated structure, with the discrete part, Hall's ray, and a transitional part in between. One can similarly define Lagrange spectrum in the multidimensional set-up, and until now not much has been understood about it. In this paper we prove that, unlike in the one-dimensional case, the closure of the multidimensional Lagrange spectrum is equal to the interval between $0$ and its supremum. The proof relies on a correspondence between Diophantine approximation and dynamics on the space of unimodular lattices and proceeds by studying a dynamical counterpart of the Lagrange spectrum that we call dynamical Lagrange spectrum. The latter is shown to be equal to the interval between $0$ and its maximum by means of an argument utilizing the higher rank nature of the set-up. A passage from full dynamical spectrum to the density of the Diophantine spectrum is achieved by applying equidistribution of expanding translates of horospheres in the space of lattices.

We introduce and study the so-termed {\it $\sqrt{\Delta}$-fine rings}, a new class of rings that generalizes the classical fine rings introduced by Călugăreanu-Lam in J. Algebra & Appl. (2016) by requiring that every nonzero element $r \in R$ can be written as $r = u + a$, where $u$ is a unit and $a \in \sqrt{\Delta(R)}$. We establish that every such ring is simple, every abelian $\sqrt{\Delta}$-fine ring is indecomposable, and most notably, the matrix ring $M_n(R)$ over a $\sqrt{\Delta}$-fine ring $R$ is again $\sqrt{\Delta}$-fine for every $n \ge 1$. As a consequence, we characterize all semi-local $\sqrt{\Delta}$-fine rings as those rings which are precisely the simple Artinian rings. We also examine group rings, providing conditions under which they are either $\sqrt{\Delta}$-fine or generalized fine, where the latter class was introduced by Zhou in J. Algebra & Appl. (2022), and conclude our work with the difficult open question asking of whether each $\sqrt{\Delta}$-fine ring is necessarily fine.

For two Gaussian random elements on a separable Hilbert space with common covariance operator, the fourth-order moment tensor structure of their mixture admits an operator-valued representation obtained from the purely quadratic component of the class-conditional second moment. We identify the tensor mechanism generating the completely diagonal coefficients of this fourth-order representation after covariance standardization, and show that these coefficients are governed entirely by the coordinatewise Cameron-Martin energy of the mean shift. This yields a necessary and sufficient condition, together with an explicit eigenvalue, for the Fisher discriminant to be an eigenfunction of the induced coordinatewise operator. We further introduce aggregated fourth-order spectral functionals for two complementary constructions, one built from fourth-order moments and the other from a product-of-expectations counterpart, and prove that they are asymptotically equivalent precisely when the underlying Gaussian measures are mutually singular. These results provide a spectral realization of the classical Cameron-Martin criterion through higher-order moment operators, explaining the probabilistic origin of fourth-order spectral quantities previously proposed for Gaussian discrimination and functional data classification and relating them to the "near-perfect classification" regime.

Let $A=(\xi_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb E\xi_{ij}=0,\qquad \mathbb E\xi_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|\xi_{ij}-z|<a)\le b \] for fixed $a>0$ and $b\in(0,1)$. We prove that, for every $\delta\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,\delta$, such that \[ \mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\!\left(-c\min\{tl,n\}\right) \] for every $t\ge1$ and every $1\le l\le(1-\delta)n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l/\sqrt n$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l/\sqrt n$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\sqrt{N+1}-\sqrt{n-l+1}$ for $N\times n$ matrices whenever $N-n+l\le(1-\delta)N$.

We investigate the eigenfrequency asymptotics of a bounded acoustic cavity containing a shrinking high-contrast inclusion. This configuration is motivated by contrast-enhanced ultrasound imaging, in which microbubbles are employed as acoustic contrast agents. We identify dimension-dependent material scalings that keep the inclusion-induced resonance in a fixed order-one frequency regime as the inclusion shrinks. Our main result gives a complete asymptotic description of the spectrum in any prescribed bounded frequency window. Two distinct spectral mechanisms arise: the background Dirichlet eigenfrequencies persist as perturbed eigenvalue clusters, while the singular material contrast creates an additional Minnaert eigenvalue branch. The limiting Minnaert frequency is explicit in both dimensions, with a capacitance-based expression in three dimensions and an area-based expression in two dimensions. Two-dimensional numerical experiments confirm both the perturbation of the background Dirichlet eigenfrequencies and the emergence of the additional Minnaert branch.

In doubly selective channels, the joint delay and Doppler dispersion generally induces coupling among transmitted symbols, thereby increasing receiver equalization complexity. Nevertheless, by using appropriately designed waveforms, channels with certain delay-Doppler (DD) supports can be diagonalized for one-tap equalization. The whole picture of such DD supports and their corresponding waveforms is still unclear, except for several examples identified in literature. In this paper, under cyclic-prefix (CP)-based block transmission and assuming that the modulation waveforms form an orthonormal basis, we identify all such channel supports by an elementary expression, and derive the corresponding waveforms in closed form.

We prove that a point atom in an endpoint kinetic-energy measure of a three-dimensional incompressible Navier-Stokes flow on the flat torus forces quantitative concentration of the actual pressure. At each restart time tau, a same-state constraint-response comparison evolves u(tau) by the pressure-free componentwise advection-diffusion equation driven by u, and subtracts the resulting passive field z_tau. Writing r_tau = u - z_tau and g_tau = Q z_tau, we obtain an exact relative pressure-work identity. If the atom has mass m, Nash smoothing makes z_tau terminally non-atomic while r_tau retains terminal atomic mass at least m; hence, after every fixed restart, the terminal limsup of the accumulated pressure work is at least m/2. Mesoscopic localization yields scale-explicit lower bounds for gauge-invariant L2 pressure oscillation and for the local L2 mass of the pressure gradient in shrinking terminal cylinders. Consequently, on every terminal neighborhood of the atomic point, the pressure modulo functions of time and its gradient fail to be square integrable. The relative pressure-work principle extends to constrained solenoidal Oseen evolutions: as tau approaches T*, the zero-initial responses vanish in weak-tail topologies, although every fixed restart retains the same lower bound. This identifies a necessary pressure-work mechanism, generated by the incompressibility constraint, that links endpoint atoms to local pressure concentration, without requiring a full regularity hypothesis.

Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.

We solve the oriented asymptotic Plateau problem for integral currents of arbitrary codimension in conformally compact asymptotically locally hyperbolic manifolds without global curvature assumptions. This extends Anderson's theorem in hyperbolic space. A new ingredient is the construction of a barrier hull near conformal infinity, which serves as a barrier for area minimizing currents and stationary varifolds. In the conformally compact setting, the associated convex hull identity extends Anderson's result beyond globally pinched negative curvature.

The Poisson–Nernst–Planck (PNP) equations strongly couple ionic transport and electrostatic interactions through the Poisson equation, posing substantial numerical challenges under small permittivity and complex potential boundary conditions. Underlying these equations is a natural Wasserstein gradient-flow structure, in which the Poisson equation serves as a local realization of the nonlocal electrostatic interaction energy. Exploiting this structure, we formulate each time step as a constrained convex minimization problem where the ionic continuity equations and the Poisson equation are incorporated as linear constraints, allowing the concentrations, fluxes, and electrostatic potential to be updated simultaneously. The variational structure of the scheme intrinsically guarantees the dissipation of the original free energy, mass conservation, and nonnegativity of ionic concentrations under general electrostatic boundary conditions. Moreover, the framework is structurally modular: extending from classical to modified PNP models with steric interactions and concentration-gradient corrections requires only modifying the energy functional, while all structure-preserving properties are automatically retained. To efficiently solve the resulting large-scale constrained problems, we develop preconditioned and transformed primal–dual algorithms equipped with tailored fast dual solvers, namely DCT-based direct and Schur-complement iterative methods, that exploit the coupled block structure of the PDE constraints. Numerical experiments on classical and modified PNP systems demonstrate the accuracy and structure-preserving properties of the scheme, and show that the proposed algorithms converge reliably in strongly coupled small-permittivity regimes without significant growth in computational cost.

We show that if $(X,d,\mathscr H^n)$ is an $RCD(K,N)$ space for some $K\in\mathbb{R}$ and $N\in [1,\infty)$, then it is also a (non-collapsed) $RCD(K,n)$ space. In other words, unweighted finite-dimensional $RCD$ spaces enjoy an automatic improvement of the dimension bound to the Hausdorff dimension of the space. More generally, we prove that this holds also when $N=\infty$, provided that we assume in addition the $n$-rectifiability of the space.

We provide a general description of the module $T^1_X$ of first-order infinitesimal deformations of a not necessarily normal affine toric variety $X$ with a special emphasis on seminormal varieties satisfying Serre's $(S_2)$ condition. In the surface case we are able to give a very detailed and concrete picture which includes the dimension of each homogeneous component.

We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $\mu$ is critical and attracted to an $\alpha$-stable law. This is done via the infinite invariant BRW introduced by Le Gall & Lin and a new criterion which relates this growth rate of the range to a notion of dimension of the underlying tree in a general way. Then, in the transient case (that is, when the range does grow linearly), we extend the notion of branching capacity to this $\alpha$-stable case. We show that it is still related to the asymptotic probability that a BRW (or its infinite version) reaches a distant set in $\mathbb Z^d$, and we estimate the $\alpha$-stable branching capacity of balls.

We study higher regularity and its relation to nonresonant behavior for time-periodic solutions of boundary value problems for one-dimensional linear and nonlinear nonautonomous first-order integro-differential strictly hyperbolic systems. The boundary conditions include integral operators and various types of boundary reflections. We prove that continuous and classical solutions have $C^k$-regularity, provided the coefficients are sufficiently smooth and a suitable number of nonresonance conditions is satisfied. In the linear case, these conditions involve the principal coefficients, the diagonal lower-order coefficients, and the boundary reflection coefficients. In the nonlinear case, they also depend on the nonlinearities and on the solution itself. For nonautonomous hyperbolic systems, higher regularity generally requires additional nonresonance conditions, whose number depends on the desired order of differentiability. These conditions are not only sufficient but, in general, also necessary, revealing a distinctive feature of nonautonomous hyperbolic PDEs. By contrast, in the autonomous case, a single nonresonance condition (if one is needed at all) suffices to obtain arbitrarily high regularity. We also identify a class of nonautonomous hyperbolic problems for which no nonresonance conditions are required. In this case, the higher regularity of solutions is determined solely by the regularity of the data. The main technical tool underlying the proofs is an abstract regularity principle formulated in the setting of vector spaces.

By analyzing a stochastic Lotka-Volterra (LV) equation, we show that the underlying logic behind the diversity-stability debate in ecology can be framed as a dimensionality-induced critical phase transition, that is, a transition separating the regime dominated by statistical averaging and a systemic tipping point. This phase transition framework unifies the opposing ecological predictions in the diversity-stability debate and sets an intrinsic diversity ceiling for stochastic ecological communities.

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $\delta$-shell interaction of strength $\tau\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $\Sigma\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $\Sigma$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $\Sigma$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $\tau$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.

We complete the classification of non-Kähler steady pluriclosed soliton on compact complex surfaces, as initiated by Streets. The underlying complex structure must be a Hopf surface, with any finite primary cover being of class 1. Moreover, the soliton metric is unique up to biholomorphism.

For an integer $t\geq-1$, let $\theta_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and set $R_t=\mathbb{Z}[\theta_t]\subseteq\mathcal{O}_t=\mathcal{O}_{\mathbb{Q}(\theta_t)}$, $N_t=[\mathcal{O}_t:R_t]$, and $\varepsilon_t=[\mathcal{O}_t^\times:R_t^\times]$. We determine $\varepsilon_t$ when $N_t$ is squarefree, $27$, or $343$: the only nontrivial indices in these cases are $\varepsilon_3=3$, $\varepsilon_5=7$, $\varepsilon_{12}=13$, and $\varepsilon_{54}=19$. Local conductor calculations give $\varepsilon_t\mid N_t$ for squarefree $N_t$. For arbitrary $N_t$, a regulator comparison shows that $t\geq2N_t$ implies $\varepsilon_t=1$. When $N_t=p^3$ with $p\neq3$ a rational prime, this bound and the index criterion leave at most four possible parameters with $\varepsilon_t>1$ for each fixed $p$. For arbitrary additive index, we also prove that $13\mid\varepsilon_t$ if and only if $t=12$ or $66$, with unit index $13$ in both cases. The proof determines the rational solutions of a plane quartic equation by an explicit genus-two descent and a local Chabauty argument, proving Louboutin's Conjecture 19 on its integral solutions. When $N_t$ is a rational prime, we determine the Picard kernel, the cardinality and fibers of the ideal class monoid over $\operatorname{Pic}(\mathcal{O}_t)$, and the corresponding integral matrix-conjugacy classes.

Hyperanalytic functions, in the sense established by the mathematician Avron Douglis, are Douglis algebra-valued functions defined via a hypercomplex structure rather than the standard Cauchy-Riemann equations characteristic of traditional complex analysis. The classes of polyhyperanalytic and meta-hyperanalytic functions represent advanced generalizations of Douglis's analysis. They are employed in the study of partial differential equations and elasticity, extending the concept of the classical holomorphic function through higher-order iterations and non-homogeneous terms. The aim of this work is to find solvability conditions for a fundamental Riemann-type boundary value problem for spaces of poly-hyperanalytic and meta-hyperanalytic functions defined on an open, bounded, simply connected subset of the complex plane, where the boundary need only be a closed d-summable curve. In fractal geometry, d-summability is a geometric property used to define the boundaries of fractal domains, enabling advanced mathematical integration and calculus on complex structures defined by Jenny Harrison and Alec Norton.

In a remarkable work published in Analysis and PDE 8 (2015) 1541-1563, M. Fazly, J. Wei, and X. Xu establish the following pointwise estimate \[-\Delta u \geq \frac 2{n-4} \frac{|\nabla u|^2}u + \sqrt{\frac 2{p+1- \frac 8{n(n-4)}}} |x|^{\frac \sigma 2} u^\frac{p+1}2 \quad \text{in } \mathbf R^n\] for any bounded, positive, $C^4$-solution $u$ to the the fourth-order Hénon equation \[\Delta^2 u = |x|^\sigma u^p \quad \text{in } \mathbf R^n\] with $n \geq 5$, $\sigma \geq 0 $ but implicitly near $0$, and $p>(n+4+2\sigma)/(n-4)$. Their argument relies on a sophisticated iteration argument in the fashion of the standard Moser proof. In this paper, we provide an alternative approach which relies on the Bernstein technique via the maximum principle. The heart of our argument is a suitable choice of an auxiliary function allowing us not only achieving the same pointwise estimate, which still holds even for $p = (n+4+2\sigma)/(n-4)$, but also relaxing the boundedness of solutions. In addition, the pointwise inequality actually holds for any $\sigma \geq 0$. This estimate leads to an interesting consequence of the existence of conformal metric of $\mathbf R^n$ having positive scalar and $Q$ curvatures. We also show that the coefficient $2/(n-4)$ is sharp, hence providing the first answer to the question in the op. cit. paper. Our proof appears to be simpler and constructive.

We solve the Nielsen realisation problem for high-dimensional aspherical manifolds in a new class of special cases. Mainly, we focus on actions of cyclic groups of prime order. A novelty is that it gives topological solutions to the problem with certain high-dimensional fixed point sets, whereas previous solutions of this type were restricted to discrete fixed point sets. This is enabled by recent joint work with Kirstein on isovariant Poincaré duality spaces, and Farrell-Lück-Steimle's obstruction theory to finding approximate fibrations within a homotopy class.

The problem of global minimization of nonconvex quadratic functions subject to box constraints is studied. Applying Gershgorin theorem we reduce the main matrix A to its diagonal form which is used to obtain difference of convex representation of the quadratic function. Then {\epsilon}-subdifferentials of DC components are studied and used to design a method for minimizing the quadratic function globally.

We prove an effective joint equidistribution theorem for $d$-dimensional $K$-subspaces, $1\le d<n$, of a fixed $n$-dimensional rigid adelic space over a number field $K$. The parameters are the height, the image in the archimedean Grassmannian, and the isometry classes of the subspace and quotient after each is rescaled to height $1$. For bounded Lipschitz test functions, the error term is power-saving with an explicit exponent. As an application, our theorem refines Thunder's asymptotic formula for subspaces of bounded height. We also obtain a power-saving asymptotic for subspaces containing $d$ linearly independent projective points of bounded height, with a leading constant expressed as a moment of the last successive minimum.

Let $\zeta(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erdős and Gimbel asked whether, for $G_n\sim G(n,1/2)$, the difference $\chi(G_n)-\zeta(G_n)$ tends to infinity with high probability. We resolve this problem along the full sequence $n\to\infty$ and prove that $\mathbb P(\chi(G_n)-\zeta(G_n)\ge ((\log 2)^2/4)\log(200/153)\,n/(\log n)^3)\to1$. This gives a lower bound at the conjectured scale $n/(\log n)^3$. We also obtain a phase-resolved refinement: if $\delta_n$ is the fractional part of the standard independence-number center, then the coefficient may be replaced by $(\log 2)^2A_4(\delta_n)/4-o(1)$, where $A_4$ is explicit, continuous, nonconstant, and satisfies $A_4(\delta)>\log(200/153)$ for every $\delta\in[0,1]$. The proof uses signed cocoloring profiles supported on four consecutive class sizes and remains uniform across jumps of the natural class-size cutoff. An exact signed-overlap identity separates local cell rewards from a binary cycle-space factor. A canonical decomposition into high cells and a capped residual matching, together with an endpoint-table comparison and an injective restriction of residual even edge sets, yields the required second-moment bound. A bounded-differences argument then amplifies the resulting rare signed witness to a high-probability cocoloring.

We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-\Delta)$ of two-dimensional Schrödinger operators $H=-\Delta+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón–Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind—an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

The independence complex and perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining their homotopy types is generally difficult, with explicit descriptions known mainly for highly structured graph families. In this article, we determine the homotopy types of these complexes for the Möbius ladder graphs $M_{2n}$ and circular ladder graphs $\mathcal{C}_{2n}$. The Möbius ladder graphs $M_{2n}$ are highly symmetric cubic graphs obtained from a $2n$-cycle by joining opposite vertices, while the circular ladder graphs $\mathcal{C}_{2n}$ are the Cartesian products of an $n$-cycle and a path of length one. We show that $\operatorname{Ind}(M_{2n})$ and $\operatorname{Ind}(\mathcal{C}_{2n})$ have the homotopy type of wedges of spheres, with the numbers and dimensions of the spheres exhibiting periodic behavior according to $n$ modulo $4$. For the perfect matching complex $\mathcal{M}_p(M_{2n})$, its homotopy type is a wedge of two copies of $\mathbb{S}^{(n-2)/2}$ when $n$ is even, while for odd $n$ it has the homotopy type of a wedge of spheres whose numbers and dimensions depend periodically on $n$ modulo $6$. The perfect matching complex $\mathcal{M}_p(\mathcal{C}_{2n})$ is contractible for odd $n$, whereas for even $n$ its homotopy type is a wedge of spheres, with the numbers and dimensions determined periodically by $n$ modulo $6$. Thus, we obtain explicit homotopy types for the independence and perfect matching complexes of two highly symmetric families of cubic graphs, which are also relevant in crystallization theory and the combinatorial representation of PL manifolds.

Let $\mathcal{A}$ be the normalized class of analytic functions $f$ in the unit disc $\mathbb{D} := \{z \in \mathbb{C} : \vert{}z\vert{} < 1\}$. For $\beta > 1$, let $\mathcal{N}(\beta)$ denote the subclass of $\mathcal{A}$ satisfying $\text{Re}\{1 + z f''(z)/f'(z)\} < \beta$ for $z \in \mathbb{D}$. The main purpose of this paper is to establish sharp bounds for the pre-Schwarzian norm $\Vert{}P_f\Vert{}$ and Schwarzian norm $\Vert{}S_f\Vert{}$ for functions $f \in \mathcal{N}(\beta)$, parametrized by $f''(0)$, with special emphasis on the case $f''(0) = 0$. In addition, sharp growth, distortion, and radius results (convexity and concavity) for $\mathcal{N}(\beta)$ are obtained. As an application, we determine the sharp pre-Schwarzian norm estimate for harmonic mappings $f = h + \bar{g}$ whose analytic part $h$ belongs to $\mathcal{N}(\beta)$.

The concept of an $E$-frame, recently introduced in frame theory, is obtained by applying an infinite invertible complex matrix to a sequence of elements of a Hilbert space $\mathcal{H}$. Here, $E$ is considered as a matrix mapping on the sequence space $\bigoplus_{n=1}^{\infty}\mathcal{H}$. A natural question is to determine the conditions on a sequence $\Psi=\{\psi_k\}_{k=1}^{\infty}$ and a matrix $E$ under which $E\Psi$ becomes a frame. In this paper, we show that if $E$ acts surjectively on $\bigoplus_{n=1}^{\infty}\mathcal{H}$, then every ordinary frame is an $E$-frame.