Papers

We prove that if a forward or backward complete Berwaldian or reversible Finslerian manifold $(M,F)$ admits a non-trivial (non-homothety) conformal concircular transformation (circle-preserving transformation or {\cpt} for short), $e^{\sigma} F$, where $\sigma$ has at least one critical point, then, $(M,F)$ is Riemannian. Consequently, $(M,g)$ is conformally diffeomorphic to either 1) the standard sphere, 2) the Euclidean space, or 3) the hyperbolic space. In particular, a compact Berwaldian or reversible Finslerian manifold does not admit any non-trivial conformal {\cpt}s, unless it is conformally diffeomorphic to the standard sphere.

In many modern experiments, latent interactions drive multicomponent stochastic systems, yet the data are available only as pooled measurements that obscure these dependencies. Whether such interactions can be identified and inferred from aggregate signals remains largely unexplored. Motivated by multi-channel electrophysiological recordings, we address this problem by introducing sum-dependent Markov chains, a class of finite-state continuous-time multivariate Markov processes whose transition rates encode interactions through the aggregate state. Under natural structural conditions, we establish identifiability of the latent dynamic parameters from the aggregate process. We define a cooperativity index that distinguishes positive cooperativity, negative cooperativity and independence, and construct its consistent estimators. For discretely and noisily observed pooled data, we develop likelihood-based inference through a hidden Markov model, address the associated embedding problem, and prove consistency and asymptotic normality. We further propose a stepdown test for cooperativity with asymptotic size control and power guarantees. Simulations and real-data analyses, demonstrate the scope and effectiveness of the methodology.

The Beyond Endoscopy strategy, proposed by Langlands, aims to establish the principle of functoriality by analyzing the trace formula. Recently, Deng and Espinosa advanced this program for $\mathrm{GL}_3(\mathbb{Q})$ by isolating the contribution of the trivial representation from the elliptic regular part. Their work relies on a conjectural factorization formula for the $L$-function associated with a cubic order, which yields the functional equation for the completed $L$-function. In this paper, we provide an unconditional proof of this functional equation for every Gorenstein order in a cubic number field. As a consequence, their isolation of the trivial representation for $\mathrm{GL}_3(\mathbb{Q})$ becomes fully unconditional.

We introduce and study the Baer transform for finite skew braces whose additive group has odd order and nilpotency class at most $2$. Starting from Baer's classical construction for nilpotent groups of odd order and nilpotency class $2$, we replace the additive group $(X,+)$ by an abelian group $(X,\oplus)$ on the same underlying set and prove that \[ Br(X)=(X,\oplus,\cdot) \] is a finite skew brace of abelian type. We show that the Baer transform preserves automorphisms, strong left ideals, ideals and central ideals, and we establish its compatibility with quotients. We then compare structural properties of $X$ and $Br(X)$. As applications, we show that direct product decompositions into ideals are preserved by the Baer transform, giving a criterion for indecomposability of skew braces. Finally, for finite $p$-skew braces with $p$ odd, we apply Thompson critical subgroups together with the Baer transform to embed suitable automorphism groups into automorphism groups of skew braces of abelian type.

We study spectral invariance and Gevrey regularity for convolution operators with kernels in suitable weighted function spaces on locally compact groups equipped with a locally bounded length function $\ell$. The main analytic scale is given by the subexponential weights. For groups whose volume growth is bounded above by $e^{R^\gamma}$ for some $0<\gamma<1$, we establish spectral comparison result for compactly supported functions. For compactly supported Hermitian functions, we prove spectral radius invariance across the symmetric $q$-pseudofunction $*$-algebra, the weighted and unweighted group algebras, and the full and reduced group $C^*$-algebras. For unimodular groups satisfying strong subexponential growth of exponent at most $\beta$, we construct a Gevrey-Beurling operator algebra inside the unitized $q$-pseudofunction algebra. We prove that this algebra is inverse-closed and that its inclusion induces an isomorphism in topological $K$-theory. The inverse-closedness theorem may be viewed as a quantitative Gevrey-type noncommutative Wiener lemma. As an application, we show that whenever a convolution operators with kernels in the corresponding weighted Gevrey-Beurling space is invertible in the unitized $q$-pseudofunction algebra, then its inverse belongs to the same Gevrey-Beurling operator algebra and satisfies explicit Gevrey seminorm estimates. We also develop a relative theory for pairs of finitely generated groups using Schreier graph lengths and quasi-regular representations. This provides a subexponential analogue of rapid decay for group pairs, when subgroup is normal, it reduces to the usual theory on the quotient. The framework can apply to intermediate-growth examples, including the Grigorchuk group, and is stable under products with polynomial growth groups and under compact extensions.

Robust optical design demands quantitative knowledge of how manufacturing and alignment tolerances degrade system performance. We show that analysing the perturbation eigenmodes of the Hessian matrix gives qualitative insight about the mechanisms of performance degradation (such as couplings) that is invisible to classical sensitivity-matrix analysis based on the Jacobian alone. Via the envelope theorem, we prove that the first-order sensitivity of the fully compensated system is identical to that of the uncompensated one; refocusing only acts at second order through the Schur complement of the Hessian. We propose the trace of the tolerance-scaled Hessian %,$\Tr(\mathbf{S}\mathbf{H}\mathbf{S})$, as a single scalar robustness metric. Demonstrated on an off-axis three-mirror anastigmat and scaled to a twenty-three-parameter surface-figure model, eigenmode decomposition reveals the dominant sensitivity directions and yields deterministic tolerance budgets that replace costly Monte Carlo sampling.

This study investigates the propagating direction of bistable traveling waves in the two-species Lotka-Volterra competition-diffusion model under strong competition. From an ecological perspective, the sign of the wave speed is critical, as it dictates which species eventually prevails. We focus on a near-symmetric scenario where intrinsic growth rates and inter-specific competition coefficients are identical, leaving diffusion rates as the sole source of asymmetry. This framework is motivated by the conjecture "Unity is not strength" as described by Alzahrani et al., Girardin and Nadin, and Girardin, which proposes that the species with the higher diffusion rate gains a competitive advantage, directly dictating the wave speed's sign. While extensive literature, including the significant recent progress by Nakamura and Ogiwara, has validated this conjecture under specific assumptions, a comprehensive proof remains elusive. In this paper, we explore the subtle regime where inter-specific competition weakens, approaching the strong-competition borderline. Leveraging our previous finding, a minimax formulation for the zero-wave-speed condition, we successfully analyze the asymptotic behavior of the wave and construct a sharp test function to determine the sign of the wave speed. Consequently, we verify "Unity is not strength" conjecture within this new parameter regime and derive explicit bounds that characterize how the zero-wave-speed condition is influenced by the interplay between competition strength and diffusion rates.

We study the spectral asymptotics of elliptic operators with quasiperiodic coefficients by exploiting projections from higher-dimensional periodic functions. Using the framework of two-scale convergence adapted to cut-and-project quasiperiodic structures we establish that, in the low-frequency (homogenization) regime, the spectrum converges to that of a homogenized operator with effective coefficients determined by a cell problem on the higher-dimensional torus. In the high-frequency regime, we introduce a rescaling approach that transforms the problem to an expanding domain with asymptotically frozen coefficients. In the critical scaling, the rescaled spectrum converges to the union of the Bloch spectra arising from the quasiperiodic bulk and a boundary layer spectrum consisting of eigenfunctions concentrated near the boundary of the macroscopic domain. This boundary spectrum can be characterised as a subset of the spectrum of a family of half-space operators with frozen macroscopic coefficients. For any non-critical scaling, the rescaled spectrum fills the positive real line.

We consider the aggregation-diffusion equation in the whole space with a mildly singular interaction kernel K = K(x) which behaves like |x|^k near the origin for some k $\in$ (0, 2). This equation, supplemented with nonnegative, bounded, and integrable initial data, possesses a global-in-time solution. We prove that the family of nonnegative, radially symmetric solutions of this equation, all sharing the same initial datum, focuses around the origin over a common finite time interval as $\epsilon$ ___ 0.

We study weighted total variation (TV) regularization for inverse problems in which the forward operator has a large null space, a setting in which standard (unweighted) TV is known to produce systematic reconstruction artifacts such as spatial bias. To address this issue, we consider spatially varying weights derived from a sensitivity analysis of the forward operator expression via an associated Green's function. These weights rebalance the TV penalty and thereby compensate for the inhomogeneous sensitivity of the forward operator. We introduce and analyze two related models: an isotropic weighted TV functional, where the weight reflects worst-case directional sensitivity, and a directionally weighted refinement in which the weighting depends on the actual jump direction of the solution. In addition to the standard Tikhonov formulation, we also study corresponding basis-pursuit problems. We derive optimality conditions and prove exact recovery results under suitable structural assumptions. Furthermore, we investigate how recoverability depends on the geometry of the solution and its distance to the observation boundary. Numerical experiments for inverse source problems and related applications demonstrate that the proposed weighted formulations significantly reduce artifacts present in unweighted TV reconstructions, yielding improved localization and size recovery. These results highlight the crucial role of spatial weighting in TV regularization for inverse problems with large null spaces.

Let $k$ be an algebraically closed field of characteristic different from $2$ and $3$, and let $f(t)=t^3+at+b\in k[t]$ be square-free. For $X\in M_m(k)$ set $A=f(X)$. This paper studies the full matrix equation $Y^2=A$, without imposing $[X,Y]=0$. Existence is the classical square-root problem for the fixed matrix $A$. The only obstruction is the nilpotent square-root pairing criterion on the zero-primary component of $A$. The main result is a fiberwise commutativity classification: if the fiber $\{Y\in M_m(k):Y^2=A\}$ is nonempty, then every such $Y$ commutes with $X$ if and only if $X\in k[A]$. The forward direction restricts the classical centralizer criterion of Thompson (1969) to the square root fiber. The new content is the converse: when $X\notin k[A]$, an explicit square root $Y$ of $A$ with $[X,Y]\ne 0$ is exhibited. The witnesses arise from nonzero spectral collision under $f$, critical nilpotent collapse, or a multicolor zero-primary fiber.

Product structure theory aims to understand complex graphs by embedding them into products of simpler graphs. In this direction, Campbell, Distel, Gollin, Harvey, Hendrey, Hickingbotham, Mohar and Wood (2022) put forth the conjecture that all graphs of degree-$d$ polynomial growth (i.e., where balls of radius $r$ have $\mathcal{O}(r^d)$ vertices) can be embedded into the strong product of $d$ trees, each with linear growth, and a constant-size clique. In this paper, we disprove this conjecture for $d = 4$. The counterexamples are finite subgraphs of a Cayley graph of the discrete $3$-dimensional Heisenberg group $\mathbb{H}(\mathbb{Z})$. These graphs were first proposed by Huang and McCarty as potential counterexamples to the conjecture. A key technical tool of our proof is the ”quantitative central collapse” theorem due to Cheeger, Kleiner and Naor (2011), guaranteeing that every Lipschitz map from the continuous Heisenberg group $\mathbb{H}$ to the function space $L_1$ collapses along a central line.

Exceptional points (EPs) have long promised enhanced sensing of physical signals, but have practically been limited by simultaneous enhancement of noise. Aligning an EP's non-analytic response with target perturbations while suppressing noise has, however, remained challenging. Here we show that fully passive, phase-tuned multi-port scattering networks enable scattering EPs with tailored anisotropic response to perturbations. We leverage projection-induced non-unitarity to realize effective non-Hermitian behavior when measuring only a subset of system ports. By formulating EP design in terms of the discriminant of the projected scattering sub-block and its directional derivatives, we give control-counting rules relating the number of programmable link phases to achievable Riemann surface topologies. We demonstrate our framework in a four-port photonic network by designing both an anisotropic EP and a Dirac-type EP with linear splitting along two parametric directions. We further suppress the global thermal drift response of a network-based sensor to a 3/2 power-law scaling while retaining square-root sensitivity to localized signals. Since the effective non-Hermiticity arises purely from port projection, our approach transfers to integrated photonic and microwave meshes, acoustic circuits, and projected metasurfaces, offering a phase-only route to reconfigurable non-Hermitian response and noise-robust EP sensing.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded domain, and let $\rho:\Omega\to[-1,0)$ be a smooth strictly plurisubharmonic exhaustion function. We consider the logarithmic potential $g=-\log(-\rho)$ and the associated complete Kähler metric $\omega=dd^{c} g$. We prove that if $\rho$ satisfies the finite weighted Monge-Ampère mass condition with weight $(-\rho)^{\varepsilon}$ for every $\varepsilon>0$, then the spectrum and the essential spectrum of the Laplace-Beltrami operator of $(\Omega,\omega)$ both coincide with $[n^{2},\infty)$. In particular, the bottom of the spectrum satisfies $\lambda_{0}(\Delta_{\omega},\Omega)=n^{2}$, and $n^{2}$ is not an $L^{2}$-eigenvalue. For each $\alpha > \frac{n}{2}$, we set $f=(-\rho)^{\alpha}$ and prove that $f$ belongs to $W^{1,2}(\Omega,\omega)$ if and only if the Monge-Ampère mass of $\rho$ weighted by $(-\rho)^{2\alpha-n}$ is finite. The integral identities underlying this equivalence, applied to the complex powers $(-\rho)^{\alpha+it}$, give, after normalization, a singular Weyl sequence at every $n^{2}+4t^{2}$, $t\in\mathbb{R}$, as $\alpha\downarrow\frac{n}{2}$. Combining our theorem with Cegrell's finite-mass exhaustion theorem, we obtain a complete Kähler metric with spectrum and essential spectrum equal to $[n^{2},\infty)$ on every bounded hyperconvex domain.

The chromatic threshold of Erdős and Simonovits asks when a minimum-degree condition forces every \(H\)-free graph to have bounded chromatic number. Thomassen's homomorphism threshold strengthens this by requiring a bounded \(H\)-free homomorphic image. The recently introduced blowup threshold \(\delta_{\mathrm B}(H)\) asks for a still more rigid conclusion: when must every sufficiently dense maximal \(H\)-free graph be an actual blowup of a bounded graph? Thus the blowup threshold measures when quotient-level structure can be upgraded to exact bounded-template structure. We show that, although chromatic and homomorphism thresholds are often hard to separate, the stronger blowup threshold diverges from the chromatic threshold in several fundamental ways. First, we prove that \(\delta_{\mathrm B}(H)>0\) for every non-bipartite graph \(H\). Hence, unlike the chromatic threshold, the blowup threshold never vanishes outside the bipartite world. Second, we prove that $\delta_{\mathrm B}$ is not monotone under taking induced subgraphs. This shows that the blowup threshold is sensitive to global features of the forbidden graph and cannot be classified by a direct analogue of the monotonicity-based strategy used for chromatic thresholds. Third, we prove that $\delta_{\mathrm B}(H)=\frac{1}{4}$ for a natural family of \(3\)-chromatic constrained blowups of odd cycles. This gives a new exact blowup-threshold value beyond the chromatic-threshold spectrum.

We investigate quantitative rapid stabilization for the one-dimensional Allen–Cahn equation and develop a quantitative modal decomposition approach that makes explicit the dependence of the feedback laws and stabilization costs on the prescribed decay rate. We construct an explicit feedback law on the finite-dimensional unstable modes via Ackermann's formula. The explicit structure of the feedback allows us to derive quantitative low-frequency estimates, which, combined with the frequency Lyapunov method, yield quantitative stabilization estimates. Together with the stabilization framework of [37], the resulting estimates can be adapted to a broader class of one-dimensional parabolic models. We further construct piecewise feedback laws that yield the null controllability with control costs and finite-time stabilization.

We propose a subspace curvature-scaling high-index saddle dynamics (SCS-HiSD) method to accelerate high-index saddle dynamics (HiSD) for locating ill-conditioned saddle points. The key observation is that HiSD already computes approximations of the unstable Hessian eigenvectors during iteration, which can be used at negligible additional cost to construct an inverse-Hessian approximation on the unstable subspace. This subspace curvature information is incorporated to adaptively scale the dynamics along each unstable direction, eliminating the dependence of the convergence rate on the smallest-magnitude negative eigenvalues and thereby substantially accelerating the convergence for ill-conditioned saddle points. We establish the linear stability of the continuous SCS-HiSD system and provide a local convergence analysis for the discrete iterative scheme. This method extends naturally to address slow convergence caused by small positive eigenvalues. Numerical experiments on benchmark problems and a liquid-crystal model demonstrate that SCS-HiSD substantially accelerates the computation of ill-conditioned saddle points, particularly in severely ill-conditioned cases.

We give two general constructions for $2$-designs, that can be used recursively, and interchangeably, to produce new infinite families of $2$-designs admitting block-transitive groups of automorphisms which preserve arbitrarily large posets of partitions of the point-set. The only arbitrarily large posets for which constructions were previously known are chains of arbitrary length. Using the constructions we exhibit new infinite families of poset-imprimitive block-transitive $2$-designs corresponding to several different arbitrarily large posets, as well as constructions for most posets with four nodes.

A set system $\mathcal{F}$ is called $t$-intersecting if $|A\cap B|\ge t$ for every pair of sets $A,B\in \mathcal{F}.$ A set system $\mathcal{F}$ is $k$-Sperner if it does not contain a chain of length $k+1$. Balogh, Linz and Patkós (Combinatorial Theory, 2023) conjectured an extremal result for $t$-intersecting $k$-Sperner families when $n+t$ is odd. In this note we give an explicit construction that is $t$-intersecting and $k$-Sperner, and whose size exceeds that of the conjectured fixed-star construction for infinitely many values of $n$. Consequently, we disprove the Balogh-Linz-Patkós conjecture for all $t\ge 2$ and $k\ge 2$ satisfying $k(t-1)\ge t+1$.

Let $G$ be a group and $K$ be a normal subgroup of $G$. A sequence over $G$ is a finite collection of terms from $G$, where repetition is allowed, and the order is disregarded. A product-$K$ sequence is a sequence whose terms can be ordered such that their product in $G$ belongs to $K$. The set $\mathcal B_K (G)$ of all product-$K$ sequences over $G$ forms a monoid, called the monoid of product-$K$ sequences, under the operation of sequence concatenation. In this paper, we investigate the algebraic and arithmetic properties of the monoid $\mathcal B_K (G)$. Among our main results, we provide precise characterizations of when $\mathcal B_K (G)$ satisfies key properties, namely being (transfer) Krull, a (seminormal) C-monoid, or (half-)factorial. After that, we study the Isomorphism Problem for monoids of product-$K$ sequences. Our results generalize existing frameworks, making them applicable to both the classical abelian and the more recently developed non-abelian settings.

This survey article discusses the emerging interaction between conformally symplectic topology and dynamics, with a focus on recent developments in convex hypersurface theory.

Stability of a high-speed gas flow past a compliant solid is impacted by two distinct features: high solid-to-fluid density ratio ($\rho_r$), and flow compressibility when flow speeds are comparable to acoustic speed. This study investigates the linear stability of a shear-driven compressible gas flow past a compliant substrate modelled as a continuum Neo-Hookean solid. Numerical solutions of the eigenvalue problem reveal that at high density ratios, the dominant instabilities are the elastic shear-waves of the solid. Our study shows that flow compressibility exerts a non-monotonic effect on the growth rate of the elastic modes; the growth rate increases with increase in Mach number up to $\mbox{Ma} \approx 2$ before subsequently decreasing. Furthermore, for compressible flows, strong thermal-coupling renders the base state highly sensitive to both the solid-to-fluid thermal conductivity ratio and the substrate's bottom-surface temperature. Numerical results demonstrate that increasing the conductivity ratio destabilizes the system, whereas increasing the bottom wall temperature is stabilizing. The stability equations, analyzed in the asymptotic limit of $\rho_r \mbox{Re} \gg 1$, reveals that the fluid-solid system de-couples at the leading order with the elastic modes emerging as a solution of the linear elasticity equations under free-shear condition at the interface. We derive a closed-form expression for the leading-order growth rate of the instability, which shows an excellent agreement with the numerical solution. This expression explicitly quantifies the influence of fluid stresses at the interface, which can in-turn be expressed as integrals of the flow solution isolating the distinct physical mechanisms driving the instability.

The Ramsey community number $r_\kappa$ is the smallest network size at which a graph is better described by a partition into communities than by no partition, under a prescribed detection rule. On a scale-free graph this question is confounded: a block model can split the network merely to absorb its degree distribution. I compute $r_\kappa$ analytically for the deterministic pseudofractal scale-free web of Dorogovtsev, Goltsev, and Mendes, separating genuine community structure from degree heterogeneity with two closed-form detection rules. Under a plain Bernoulli stochastic block model, the web's natural recursive bipartition is unpreferred while small and breaks at $r_\kappa=1095$ nodes, with a log-evidence growing as $(\ln 3-\tfrac{2}{3}\ln 2)n$. Under a degree-corrected model tested against the configuration-model null, the same partition survives, breaking far earlier at $r_\kappa=42$, with a log-evidence growing as $(2\ln 3-\tfrac{4}{3}\ln 2)n$ – exactly twice the plain slope, and independent of the prior. Degree correction reverses the ordering of the candidate cuts, demoting the hub-leaf split and elevating the recursive one. Because the web is self-similar, the best description is not two communities but a nested hierarchy: the degree-corrected evidence keeps rising as the partition is refined, and is maximised at of order $\sqrt{n}$ communities of $\sim\sqrt{n}$ nodes. A purely local recursive rule thus builds true hierarchical community structure, over and above the scale-free degree sequence it also produces, in an exactly solvable setting.

This work concerns a quantitative form of Landau's eigenvalue theorem for spatio-spectral limiting operators. We isolate a simple mechanism that converts the problem of estimating the distribution of eigenvalues of a limiting operator into the problem of bounding the trace of the difference between the operator and its square. This mechanism allows us to analyze limiting operators for domains with fractal boundaries. When the boundaries have finite perimeter, we recover the expected optimal dependence on the scaling parameter.

We study the long-time asymptotics of the three-dimensional Hartree equation with Yukawa potential \[ V_\mu(x)=\frac{e^{-\mu|x|}}{|x|}, \qquad 0\leq\mu\leq1. \] The Coulomb case corresponds to $\mu=0$, while $\mu>0$ introduces the screening length $\mu^{-1}$. In the limit $\mu\to0$ and $t\to\infty$, the asymptotic behavior depends on the comparison between the observation scale $t$ and the screening length $\mu^{-1}$, equivalently on the parameter $\mu t$. This leads to three distinct asymptotic regimes, according as $\mu t\to0$, $\mu t\to L\in(0,\infty)$, or $\mu t\to\infty$, with different modified scattering phases in each case.

We determine the cellular $\mathbb{A}^1$-chain complex of full and partial flag varieties of split semisimple simply connected groups over a perfect field of characteristic different from 2. With compatible integral normal orientations and reindexing by opposite Bruhat cells, the differential in each degree at least two is represented by the motivic Hopf map $\eta$ times one half of the signed cellular boundary matrix of the corresponding real flag manifold. Consequently, the Smith data of the resulting integer complex give all positive-degree cellular homology sheaves in terms of Milnor–Witt $K$-theory and the kernels and cokernels of integer multiples of $\eta$. We obtain a uniform computation of the second cellular $\mathbb{A}^1$-homology of $SL_n/B$ for $n\geq3$, together with complete calculations for $SL_3/B$ and the full flags of types $F_4$ and $G_2$.

We develop a two-dimensional structural local-defect theory for scalar non-divergence advection–diffusion operators \(Lu=-a:D^2u+b\cdot\nabla u\), where \(a=a^{\mathrm{per}}+a^e\) and \(b=b^{\mathrm{per}}+b^e\). The periodic coefficients and the bounded local defects are uniformly Hölder continuous, the interpolating matrices \(a_t=a^{\mathrm{per}}+t a^e\) are symmetric and uniformly elliptic, and \(a^e\in L^r(\mathbb{R}^2)\), \(b^e\in L^s(\mathbb{R}^2)\), where \(1<r,s<2\). Under the periodic centering condition \(\langle m^{\mathrm{per}}b^{\mathrm{per}}\rangle=0\), global harmonic coordinates remove the periodic drift. After blow-down, the transformed defect drift is small in the critical local \(L^2\) space. Critical-drift compactness then yields a finite-energy Liouville theorem and closes the continuation argument, giving a whole-space estimate for \(1<q<2\) and \(1/q^*=1/q-1/2\). This estimate provides defect correctors and, by duality, a positive invariant density \(m=m^{\mathrm{per}}+m^e\). A planar Hodge construction in harmonic coordinates, followed by a Piola pull-back, produces a skew-symmetric field \(B=B^{\mathrm{per}}+B^e\) such that \(mLu=-\operatorname{div}((ma-B)\nabla u)\). If \(M=\max\{r,s\}\) and \(M^*=2M/(2-M)\), then, for some \(\beta>0\), the defects \(B^e\) and \(A^e:=ma-B-(m^{\mathrm{per}}a^{\mathrm{per}}-B^{\mathrm{per}})\) belong to \(L^{M^*}\cap L^\infty\cap C_{\mathrm{unif}}^{0,\beta}\) and vanish uniformly at infinity. This supplies the missing two-dimensional structural reduction in the scalar regular non-endpoint regime.

We prove an analogue of Mackey's irreducibility criterion for compactly induced locally analytic representations of $p$-adic groups, where we induce from an open subgroup which is compact mod centre. We also discuss several examples.

For a graph $\Gamma$, the splitting field of $\Gamma$ is defined as the splitting field of the characteristic polynomial of $\Gamma$ over rationals. The algebraic degree of $\Gamma$ is defined by the extension degree of its splitting field over rationals. Let $k$ be a positive integer. We call a finite group $G$ \textit{Cayley $k$-integral} if, for every inverse-closed subset $S$ of $G$, the algebraic degree of the Cayley graph $\Cay(G,S)$ does not exceed $k$. We give a complete classification of all finite Cayley $2$-integral groups. It is shown that a finite abelian group is Cayley $2$-integral if and only if it is isomorphic to one of the following forms: $G \cong \mathbb{Z}_2^r \times \mathbb{Z}_5^s$, $\mathbb{Z}_2^r \times \mathbb{Z}_4^s \times \mathbb{Z}_8^t$, or $\mathbb{Z}_2^r \times \mathbb{Z}_3^s \times \mathbb{Z}_{12}^t$, where $r, s, t \geq 0$. Furthermore, we prove that the set of finite non-abelian Cayley $2$-integral groups consists of the infinite family $Q_8 \times \mathbb{Z}_2^n$, with $n \geq 0$, and $22$ specific groups.

Problems complete for the existential theory of the reals ($\exists \mathbb{R}$) arise throughout discrete geometry. We introduce satisfiability modulo realizability, a SAT-based approach for solving satisfiable instances of $\exists \mathbb{R}$ whose solutions correspond to realizable geometric configurations. Our method encodes an underapproximation of a geometric problem as a SAT instance over abstract order types. Since almost all abstract order types are unrealizable, naive search is infeasible. We guide the search toward realizable order types using diversity-driven sampling, partial realizability feedback, and a novel flippability heuristic that passes only limited information between components. We apply our method to discrete geometry problems and resolve an open problem by showing that the largest set of points avoiding empty convex hexagons and convex heptagons is of size 23.