Papers

We give new constructions of cycles of pairing-friendly elliptic curves with a view towards unbounded recursive pairing-based proof systems. Unlike the only known prior cycle of elliptic curves - the ordinary MNT cycle - our construction uses elliptic curves that are supersingular. A trade-off of our approach is that the supersingular cycles are defined over extension fields (quadratic extensions in the optimal case), which makes elements and computations in $\mathbb{G}_1$ less compact and efficient than those in the MNT cycle. On the other hand, the supersingular cycles in this paper offer a key advantage over their MNT counterpart: every instance of our infinite family of supersingular curves can be efficiently constructed via Bröker's algorithm, whereas it is only feasible to construct a relatively small, bounded number of MNT instances via the CM method. In other words, while both constructions give infinite families of pairing-friendly curves in theory, only the supersingular construction gives rise to infinite numbers of cycles that can be realised in practice. Supersingular cycles offer benefits that are relevant in the context of recursive pairing-based proof systems. They afford flexibility in the choices of underlying finite fields; one can choose primes $p$ for which the underlying field arithmetic is efficient and for which $p-1$ is divisible by a large power of 2. Or, as we study in detail, using supersingular cycles unlocks the possibility of connecting the cycle with other pairing-friendly elliptic curves that are defined over much smaller finite fields, where proof system arithmetic is much more efficient. Indeed, constructing these so-called lollipops of pairing-friendly curves was the motivating problem (posed by researchers back in 2019) that inspired the present work.

In this paper, we prove the continued fraction conjecture posed by Han, Mao and Zeng for the generating function of permutations with respect to the number of descents of type $2$ and the number of cycles, thereby settling their reformulation of a conjecture originally due to Baril and Kirgizov. We further establish the symmetry of the bistatistic $(\des_2, \ear)$ over $\mathfrak{S}_n$ as conjectured by Han, Mao and Zeng and strengthen this result by exhibiting five equidistributed companions for $(\des_2, \ear)$. Here the statistic $\des_2$ denotes the number of descents of type $2$, and the statistic $\ear$ denotes the number of exclusive antirecord cycle peaks originally introduced by Sokal and Zeng.

Let $G$ be the Boolean hypercube which carries uniform measure $\lambda$, and let $T_\mu$ denote convolution by a finite positive measure $\mu$ on $G$. For $\psi_\mu(u)=\sup\{u\lambda(\{T_\mu f\geq u\}):f\geq 0,\|f\|_1=1\},$ we prove Talagrand's convolution conjecture (Talagrand, 1989): if $\mu_a=((1+a)\delta_1/2+(1-a)\delta_{-1}/2)^{\otimes n}$ and $0<a<1$, then $\psi_{\mu_a}(u)\leq C_a/\sqrt{\log u}$ for every $u > 1$ and $n\geq1$, where $C_a$ depends only on $a$. The proof utilizes the reverse-heat and Boolean-bridge framework of Chen (2025) and the localized terminal-discrepancy method of Xiang and Zhang (2026). We introduce a new power coupling: each reverse edge ratio is split into two geometric powers. This choice produces a switched exponential weight which restores the exact reverse jump rate of the perturbed coordinate. The resulting endpoint comparison yields an anti-concentration profile estimate without the iterated-logarithmic factor. The proof was discovered by the Odin Automatic AI Research Agent.

We establish optimal Casorati inequalities for pointwise slant Riemannian submersions. As such a submersion carries two distinct distributions, the vertical, tangent to the fibers, and the horizontal, governed by O'Neill's tensors T and A, we treat them separately. For submersions from generalized complex and generalized Sasakian space forms we bound the normalized scalar curvature of each distribution by its normalized Casorati curvatures. Special cases recover sharp inequalities for real, complex, Kähler, Sasakian, Kenmotsu, cosymplectic, and almost and almost space forms. Equality is characterized geometrically: invariantly quasi-umbilical fibers in the vertical case and integrability in the horizontal case. The invariant and anti-invariant limits reproduce known results, and examples illustrate sharpness.

Let $F$ be a graph. A graph $G$ is called $F$-free if $G$ does not contain $F$ as a subgraph. Let ${\rm EX}(n,F)$ denote the set of $F$-free graphs of order $n$ with the maximum edges. In this paper, we characterize the graphs in ${\rm EX}(n,F)$ for large $n$, where $F$ is the disjoint union of paths and stars. This generalizes a result in \cite{LLP}.

The goal of this paper is to characterize the boundedness property of the uncentered Hardy–Littlewood maximal operator on ball Banach function spaces and their Köthe associate spaces in terms of the vector-valued maximal inequality. The main result of this paper supplements the recent results by Nieraeth.

We determine the Gromov–Hausdorff compactification of the moduli space of unit-diameter Ricci-flat Kähler metrics on Enriques surfaces. We show that all metric degenerations are encoded by the adjoint Satake compactification of the Enriques period domain. There are 32 boundary components added in this Satake compactification, and the GH limit at each boundary is determined. As a corollary, all the GH limits of Kähler Ricci-flat metrics on Enriques surfaces are classified, including those for a fixed complex structure or a fixed polarization.

We construct complete, one-ended Riemannian manifolds in dimensions four and five having strictly positive Ricci curvature, with fundamental group isomorphic to the Prüfer $2$-group $C_{2^\infty}$ and any subgroup $\Gamma\leq \mathbb Q/\mathbb Z$, respectively. This gives counterexamples to Milnor's conjecture in the two remaining dimensions.

In this article we consider the Drinfeld-Sokolov hierarchy of type $A$ and its similarity reduction. Then the similarity reduction is expressed as $2$ types of Hamiltonian systems, one is with a Poisson bracket and another a Poisson bracket of canonical coordinates. We also investigate a connection between the similarity reduction and the isomonodromy deformation system.

We combine results of Rickard and Shaul with methods of intrinsic homological algebra and the theory of purity to prove that the finiteness of the big finitistic dimension of an Artin algebra is an intrinsic property of the singularity category of its opposite algebra, via its category of modules. This `object-free' approach complements work of Dey–Šťovíček and arrives at the same conclusion: the finiteness of the finitistic dimension of Artin algebras is a singular invariant (of the opposite algebras). In fact, our characterisation can be used to prove that finite finitistic dimension descends along certain fully faithful functors between singularity categories. We include an appendix where we explain how our methods can be used to prove non-existence of bounded t-structures on singularity categories.

We study the algebra of formal finite multiple zeta values by imposing the stuffle and linear shuffle relations of Kaneko and Zagier. Our first results are a surjective homomorphism to the algebra of formal symmetric multiple zeta values and a parity reduction in depth at most four. In even weight we construct a quotient of the formal double zeta space which is isomorphic to the space of formal finite multiple zeta values of depth at most four. The proof combines an explicit depth-three reduction with a motivic lower bound. We also attach to every even period polynomial an explicit relation among the values $\zeta_{A}(2a,1,2b,1)$, and these give all relations among the corresponding formal values.

Randomized sketching reduces the computational cost of large ridge-regression problems, but the coefficient error depends on the realized sketch. We extend the paired-row bootstrap from randomized least squares to ridge regression, enabling coefficient-error estimation using only compressed data. Under a fixed coefficient dimension and increasing data and sketch sizes, we derive asymptotic linear representations and Gaussian limits for the sketched estimator and the conditional bootstrap distribution. These results establish uniform consistency of the bootstrap error distribution and asymptotically exact coverage when the estimator and error bound are computed from the same sketch. For sketches with independent, mean-zero, variance-one entries, an explicit covariance formula separates the effects of the residual, regularization, and fourth moment of the sketch entries, and shows that Rademacher entries minimize the leading covariance matrix in the Loewner order. We also develop a fast linearized bootstrap, an order-statistic correction for finitely many bootstrap replicates, and a Bonferroni rule for selecting from a fixed set of sketch sizes. Experiments on two real and two synthetic data sets support the proposed methods. With a sketch size 15 times the number of coefficients, 199 bootstrap replicates, and nominal coverage of 95%, the bootstrap with refitting attains coverage between 92.0% and 95.3%; after the order-statistic correction, coverage ranges from 94.7% to 97.7%.

In this paper, we investigate algebraic properties of two classes of monomial ideals associated with graphs, namely path ideals and closed neighborhood ideals. Our primary focus is on the strong persistence property and several homological invariants, including depth and the $v$-number. We first prove that the path ideal of length $n-1$ associated with a path graph on $n\geq4$ vertices attached to an arbitrary graph satisfies the strong persistence property. We then study closed neighborhood ideals and establish the strong persistence property for graphs containing a leaf vertex. Furthermore, we identify a broader class of graphs satisfying this property, namely those whose vertex set admits a decomposition $V(G)=C\sqcup\{v\}\sqcup X,$ where $C$ is a nonempty clique, every vertex of $C$ is adjacent to each vertex of $X\cup\{v\}$, and the vertex $v$ has no neighbors in $X$. In addition, we investigate algebraic invariants of ideals associated with Helm graphs. Specifically, we compute the depth of the edge ideal and determine the Krull dimension, the depth, and the $v$-number of the closed neighborhood ideal of the Helm graph. These results contribute to a deeper understanding of the interplay between graph-theoretic structures and the algebraic properties of their associated monomial ideals.

We show that there exists an $F$-finite four-dimensional local domain $(R,\mathfrak{m})$ of characteristic two which is not $F$-injective but which admits a nonzerodivisor $f\in \mathfrak{m}$ such that $R/fR$ is $F$-injective.

This paper addresses the challenging problem of output feedback stabilization for nonlinear nonminimum phase (NMP) systems in the presence of parametric uncertainties and external disturbances. The proposed framework integrates three distinct observers: a reduced-order observer for reconstructing the unmeasured states of the internal (zero) dynamics, a high-gain observer for estimating output derivatives, and an observer for estimating the aggregated effect of parametric uncertainties and disturbances. Leveraging these estimates, a sliding mode control law is synthesized to ensure global asymptotic stability of the entire system using only output measurements. The control design requires only partial model knowledge, significantly relaxing the restrictive assumptions common in existing literature. Numerical simulations illustrate the effectiveness of the proposed output-feedback strategy and corroborate the theoretical developments.

Lesniewski and Ruskai conjectured that the contraction coefficient of every monotone Riemannian metric under a unital stochastic map equals the Hilbert–Schmidt contraction on the traceless subspace. We disprove the conjecture with an explicit entanglement-breaking qutrit channel induced by a doubly stochastic $3\times3$ matrix. A faithful diagonal state and a commuting traceless tangent give, simultaneously for every normalized monotone metric, the exact lower bound $\eta^{\mathrm{Riem}}_{\kappa}(\Phi_K)\ge 8896/20007>(62+2\sqrt{61})/225=\Lambda_2(\Phi_K^{\dagger}\Phi_K)$. The counterexample is entirely classical on a maximal abelian subalgebra. A theorem of Hiai and Ruskai establishes the conjectured identity for all unital qubit maps, so dimension three is minimal among full matrix algebras.

Let $R$ be a field and $A$ the endomorphism algebra of a finite direct sum of cyclic modules over a finite-dimensional commutative local principal ideal $R$-algebra. We construct a central quotient showing that the support $\tau$-tilting poset of $A$ is isomorphic to the poset of the symmetric group with the weak order. We show that the center of $A$ and degree-zero Hochschild homology, viewed as a module over the center, determine the truncated local algebra and the multiset of successive length gaps. For centralizer matrix algebras, the support $\tau$-tilting poset determines the multiset of distinct-exponent counts of the primary blocks. The corresponding algebra–module pair also recovers their local algebras and gap multisets. Combining this reconstruction with the known derived equivalence classification, we characterize derived equivalence by isomorphism of these algebra-module pairs. We apply the results to Morita reconstruction in the string and gentle classes.

We obtain asymptotic formulas for additive congruences \[ \sum_{i=1}^r m_i x_i^{-s}\equiv \lambda \pmod p, \] where the \(m_i\) range over arbitrary subsets of \(\mathbb F_p^\ast\) and the \(x_i\) over shifted intervals. For five terms, in the balanced case of common cardinality \(N\), the asymptotic holds uniformly in \(\lambda\) whenever \[ N>p^{14/29+\varepsilon}, \] giving a genuine sub-square-root range. The main input is a centered fourth-moment estimate for the associated double exponential sums. The same method yields sub-square-root thresholds for every fixed \(r\ge 5\), including \(N>p^{8/17+\varepsilon}\) for six terms, with \[ \alpha_r=\frac13+\frac{4}{9\sqrt r}+O(r^{-1}) \] as \(r\to\infty\).

It is well known the local integral inequality which is called Poincaré–Sobolev inequality on each domain cube (or ball). After that many authors investigated the generalization for this inequality with Muckenhoupt-type weights or global weights which are independent of the domain cube. In this paper, we investigated similar weighted generalization for this inequality with the local weights which are depending on the domain cube without assuming the Muckenhoupt condition. Moreover, we considered the weighted generalization for the homogeneous and inhomogeneous-type Sobolev embedding theorems as some applications.

Let $K=\{x\in\R^d:Ax\ge b\}$ be a bounded full-dimensional polytope with $m$ inequalities. For targets on $K$ with density proportional to $\exp(-\vartheta^\transpose x)$, we prove a $\widetilde O(d^2)$ bound, measured in Markov chain iterations, for the chi-square mixing of the $1/2$-lazy Metropolis Dikin walk based on a polylogarithmically rescaled Lee–Sidford metric. The estimate holds for every initial law with finite chi-square divergence. The proof controls the reverse Gaussian proposal through a quantitative average self-concordance estimate at the required $O(1/d)$ scale. We establish this estimate from uniform analytic derivative bounds for the Lewis-weight subspace and an all-order projection-weighted contraction inequality for Gaussian chaoses. For order $j$, the latter bounds each base-point polynomial at scale $d^{j/2}$, up to $j$-dependent and logarithmic factors. Truncating at logarithmic order then absorbs the remaining pathwise loss. This resolves, in the affirmative, the open problem of whether the Gaussian Lee–Sidford Dikin walk mixes in $\widetilde O(d^2)$ iterations: the previously stated quadratic bounds \cite[Theorem~1.4]{LLV20}, \cite[Theorem~1.2]{GKM24} rest on reverse-proposal steps identified as unresolved in \cite[Section~1]{Kook26}, whose $\widetilde O(d^{9/4})$ bound was the best rigorously established rate prior to this work. The proof is found by ChatGPT 5.6 Pro.

We study lower tails of support functions of isotropic convex bodies. An endpoint negative moment estimate, combined with a spherical cap argument, gives polynomial deviation estimates for the inradius of random projections in every dimension; in the unconditional class this yields the optimal order of the median inradius and shows that the cube is extremal. The same argument gives estimates for convex hulls of independent rotations and, after an additional projection, a mixed rotation-projection theorem. We also prove weighted lower and upper estimates for Minkowski averages of rotated polars and their random projections, as well as an inradius dependent estimate for the geometric distance of global averages from the Euclidean ball. A family of isotropic product cylinders shows that the latter estimate is sharp, up to absolute constants, throughout the possible range of the inradius. For the cube, an explicit negative moment computation gives a quantitative projected $(m,k,n)$-profile and, in full dimension, recovers the classical order $n/\ln(en)$ for bounded geometric distance. Finally, we determine the sharp volume profile of intersections of independent rotations in the unconditional isotropic class and show that the cube is extremal.

For a real Banach space $X$. The Dehghan–Rooin magnitude compares the angular distance $\alpha[x,y]=\|x/\|x\|-y/\|y\|\|$ together with the skew angular distance $\beta[x,y]=\|x/\|y\|-y/\|x\|\|$. Motivated by the recent use of Birkhoff orthogonality in the Massera–Schaffer inequality, we introduce and study the Birkhoff-restricted Dehghan–Rooin constant \[ \operatorname{DR}_{B}(X)=\sup\left\{\frac{\alpha[x,y]}{\beta[x,y]}:x,y\in X\setminus\{0\},\ x\perp_{B} y,\ \beta[x,y]\ne0\right\}. \] The restriction $x\perp_{B} y$ turns the global comparison of $\alpha$ and $\beta$ into a directional invariant which detects the interaction between supporting functionals and the radial normalization map. We obtain a scale-profile formula, sharp elementary bounds, stability under subspaces and ultrapowers, a variational characterization of the extremal case $\operatorname{DR}_{B}(X)=1$, estimates in terms of the classical Dehghan–Rooin constant, and consequences for uniformly convex and uniformly smooth spaces. In smooth two-dimensional spaces we derive an explicit formula through the normalized duality mapping, and in Radon planes we reduce the constant to a one-dimensional optimization on an arc of the unit circle. Several model computations are included, including Hilbert spaces, polyhedral planes, and two-dimensional $p$-norm spaces. The final section gives normal-structure type consequences through a companion Birkhoff angular modulus and lists several problems suggested by the new constant.

In a high-dimensional Funk ball, the endpoint potential makes the forward and reverse distances behave differently, while symmetrization erases the distinction that survives in pyramid limits. For Euclidean beta-type radial measures, we determine the natural-scale limits in all three parameter phases. The divergent phase yields directed Gaussian pyramids indexed by the balance between dimension and radial concentration. A positive limiting parameter yields a Gaussian–chi Funk horocone pyramid, built from Gaussian bases, chi-distributed heights, and a directed endpoint-potential increment, while a vanishing parameter yields the universal directed star pyramid. In the last phase, the dimension-dependent logarithmic contributions cancel exactly, so the conclusion requires no further rate condition.

Firstly, for branched covering maps $f\colon Y\to X$ between smooth, separated schemes of finite type over the integral ring $O_K$ of a non-archimedean local field $K$, the ramification divisor is found to coincide with the divisor of the Radon-Nikodym derivative of the Radon measure associated with an algebraic differential form on $Y$ against the pullback measure of one on $X$, both taking Borel sets of the space $Y(O_K)$ of $O_K$-rational points of $Y$ as input values. Secondly, the ultracontractivity of semigroups generated by Laplacian integral operators having a geodetic-distance-power kernel function is proven, using heat kernel estimates for ultrametric operators on $p$-adic balls. Thirdly, a series of $p$-adic Neumann Boundary Value Problems, depending on algebraic and pluricanonical differential forms with poles on spaces $X(O_K)$ coming from schemes is formulated and solved, extending previous work of the author. These results are then used to solve reconstruction problems on schemes: the divisor of a pluricanonical form with poles with at worst log-terminal singularities, as well as Weierstrass points of projective algebraic curves can be reconstructed via repeatedly finding weak solutions of $p$-adic Neumann Boundary Value Problems.

We compare isotypic (n-SiC/n-Si) and anisotypic (n-SiC/p-Si) heterojunctions using Kelvin probe force microscopy, scanning tunnelling spectroscopy, precision electrical measurements with a Keithley 2450 SourceMeter, and a 0.1 farad supercapacitor charge-storage test. Two-wire probing shows an enhanced low-bias current in the isotypic structures; four-wire Kelvin sensing removes the voltage-drop contribution of external leads and contacts and collapses this current to a much lower bulk and interface limited level. The high density of interface states detected by KPFM and STS, up to five times ten to the twelfth per square centimetre per electronvolt in dislocation-rich regions, provides a plausible origin for the enhanced local response. Anisotypic junctions exhibit a genuine diode-like current-voltage characteristic and an open-circuit voltage of about 1.2 volts under beta irradiation. In the capacitor test with the patented charge-pump converter, the anisotypic structure reaches 0.25 volts versus 0.05 volts for the isotypic one after 30 minutes, giving an energy ratio of about 25. We formulate a measurement-interpretation framework based on four-wire Kelvin sensing, generator polarity with cathode to LO, and high-impedance output-off state. Our results show that anisotypic SiC/Si heterojunctions are the preferred platform for betavoltaic conversion when measured correctly.

We compare the specific dissipation $Q^{-1}\left( \omega \right) $ for the Becker, Lomnitz, and Lambert models in linear viscoelasticity. Graphically, the specific dissipation of these models behaves similarly as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$. Furthermore, we obtain analytic formulas for the asymptotic behavior of $Q^{-1}\left( \omega \right) $ in the Becker and Lomnitz models as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$, where the latter limit coincides with the specific dissipation of the Maxwell model. We also derive a novel closed-form expression for the specific dissipation in the generalized Becker model, $Q_{\nu }^{-1}\left( \omega \right) $, when the parameter $\nu \in \left( 0,1\right] $ is rational. When $\nu \rightarrow 0$, we recover the specific dissipation of the Maxwell model, whereas when $\nu =1$, we recover that of the original Becker model. In addition, we derive two new closed-form expressions for the specific dissipation for the extended Jeffreys–Lomnitz model $Q_{\alpha }^{-1}\left( \omega \right) $, the first being valid for $\alpha \in \left( 0,1\right] $, and the second for $\alpha \in \left( -\infty ,1\right] $. When $\alpha \rightarrow 0$, we recover the specific dissipation of the Lomnitz model, whereas when $\alpha =1$, we recover that of the Maxwell model. Finally, we conclude that the asymptotic behavior of $Q_{\alpha }^{-1}\left( \omega \right) $ as $\omega \rightarrow +\infty $ coincides with the specific dissipation of the Maxwell model. Keywords:

Morphing flexible multibody aircraft require structural strain, aerodynamic geometry, surface velocity, and generalized loading to remain compatible as joints and flexible components change configuration. A compositional formulation is developed around an assumed material attachment between each lifting surface and a geometrically exact beam. Separating the component root pose from the section field shows that the body strain and elastic potential of a component depend on its own elastic coordinates, while upstream motion enters kinetic terms and external-load pullbacks. At element level, an exact relative logarithm $d$ supplies strain and potential energy, whereas a reference-anchored section coordinate $\sigma$ supplies deformed section geometry. Finite-order expansions retain the finite reference geometry exactly and truncate only endpoint perturbations. The attachment map then generates surface points, tangents, normals, velocities, and force Jacobians from common section kinematics. Euler–Poincare beam balance, moving-surface potential-flow relations, graph cotangent assembly, and the associated semidiscrete power identity are stated in a common twist–wrench convention. Collocation, pressure, equivalent-load, and structural-station sites are distinguished to expose their approximation errors. Verification gives the expected $N_d+1$ convergence order for degree-$N_d$ relative-log expansions. In a geometrically nonlinear cantilever comparison, a cubic static-manifold correction reduces mean full-record displacement error from $0.479$ to $0.255$ over four completed load cases. These results provide structural and interface-level evidence rather than validation of a complete aircraft aeroelastic prediction.

A $k$-irregular graph is a graph with maximum degree $k$ such that vertices of degree $k$ are not adjacent. A $2$-distance $k$-coloring of a graph is a coloring of the vertices using $k$ colors in which any two vertices at distance at most $2$ receive distinct colors. The $2$-distance chromatic number of $G$, denoted by $\chi_{2}(G)$, is the minimum integer $k$ such that $G$ admits a $2$-distance $k$-coloring. Zhu \cite{zhu} proved that $\chi_2(G)\leq 13$ for planar graphs with maximum degree at most $4$. We prove that for a 4-irregular planar graph $G$, we have $\chi_2(G) \leq 10$.

We construct smooth toric Fano $n$-folds of Picard number $n+2$ whose tangent bundles are slope-stable with respect to the anticanonical polarization, for every $n\ge4$. The construction gives toric fibrations with fibre the del Pezzo surface of degree six over products of projective lines, parametrized by multisets of roots of $A_2$. Every nonempty multiset of nonzero roots with vanishing sum yields a stable tangent bundle. Among these multisets, the resulting variety is Kähler-Einstein if and only if the multiset is invariant under negation or under the order-three rotation of the root hexagon. We also construct stable, non-Kähler-Einstein examples with nonzero twist sum in every even dimension at least four. The proofs reduce slope inequalities and barycenter computations to integrals over the moment hexagon. For a smooth toric Fano variety with strictly semistable tangent bundle, we prove that polystability is equivalent to decomposition as a nontrivial product of smooth toric Fano varieties with anticanonically stable tangent bundles. Exact evaluation of Klyachko's criterion extends the stability classification to dimensions five and six, recovering the classifications of Steffens and Reynolds in dimensions three and four. Together with the product criterion, this determines polystability for all $8{,}630$ varieties in dimensions three through six, and hence the existence of Hermitian-Einstein metrics on their tangent bundles with respect to an anticanonical Kähler form.

Let $X_p$ be the quotient of the closed disk obtained by identifying boundary points under rotation through angle $2\pi/p$. For every $2\leq p\leq8$, we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to $X_p$ that admits a discrete, faithful, convex cocompact reflection representation into Isom$(\mathbf H^5)$, whose limit set is homeomorphic to the index-$p$ Pontryagin surface $\Pi_p$. Dimension five is optimal, since $\Pi_p$ does not embed in $\mathbb S^3$. For $p=2,3$, the constructions are analytic and yield cyclically symmetric infinite families.