Wanda Szmielew showed in 1959 that A−−, plane absolute geometry with the elementary continuity axiom schema, has precisely two models: Euclidean planes and hyperbolic planes over real-closed fields. In this note we determine the reason why this surprising result holds. We find that two axioms are responsible for it, the circle axiom and Aristotle’s axiom; both hold in A−−.
There are only two known strongly invertible L-space knots whose non-trivial Dehn surgery never yields the double branched cover of a knot or link with thin Khovanov homology. In this paper, we give another example of such an L-space knot.
We determine all possible exact noncommutative symplectic structures for certain path algebras with relations. These algebras are the path algebras of a cyclic quiver with r+1 arrows quotiented by the ideal generated by the paths of length l. The main result is that there are exact noncommutative symplectic structures only when l=s(r+1)+1 with s≥1. In this case a description of the open subset of one-forms z such that dz is a non-degenerate symplectic form is given, by reducing it to the case s=1.
The numbers n for which both n−2 and n+2 are sums of two integral squares are characterized in two ways. These characterizations are applied to draw some corollaries on three consecutive sums of two squares. The divergence of the series ∑a,b,c,d∈Z,ad−bc=11a2+b2+c2+d2 is also proved.
A symmetric quandle is a quandle equipped with a good involution. In this paper, we study the structure of the associated groups of symmetric quandles. We examine the relationship between the associated group of a symmetric quandle and that of its underlying quandle. As a consequence, we obtain three distinct structural properties of the associated group of the underlying quandle in terms of that of the symmetric quandle. We also provide a group-theoretic characterization of the associated group of a symmetric quandle and compute its abelianization, which coincides with the first symmetric quandle homology. Furthermore, we express the second quandle homology of a quandle in terms of the associated group of the corresponding symmetric quandle. Finally, we show that a symmetric quandle is embeddable in its associated group if and only if its underlying quandle is.
We construct a degree 12 homogeneous invariant of the complex reflection group G29 (in Shephard–Todd’s notation) whose associated surface has 320 singularities of type A2, thereby improving previous records for dodecic surfaces.
In [J. Class. Anal. 26 (2025), 63–76], we proved that the discrete Riesz potential Iα is a bounded operator Hp(Zn)→Hq(Zn) for n−1n
We prove that there exists a closed universal set E⊆X×Y for the nowhere dense sets iff the Polish space X is not σ-compact. We construct closed universal sets for the collection of finite sets of size ≤n and of size precisely n. We develop a technique of construction of universal sets similar to the one of Cieślak and Michalski (2018) but using the properties of the Vietoris topology.
We find the first weak Ziegler pair in the class of conic-line arrangements with ordinary quasi-homogeneous singularities defined over the rational numbers.
In the first part of the paper we study smooth solvability properties of linear equations. We prove an extension of Mather’s theorem (1973) to skew-symmetric smooth function matrices. For the proof of the skew-symmetric case as well as of Mather’s original results for the cases where the coefficient matrices are general matrices or symmetric matrices, the algebraic methods of Bochnak (1973) are applied. In the second part using criteria for solutions of linear equations we obtain sufficient conditions for smooth solvability of generalized Hamiltonian systems on smooth constraints.
We show how the Huygens, Cusa, Wilker and Mitrinović–Adamović inequalities and their counterparts involving hyperbolic, inverse trigonometric and inverse hyperbolic functions can be deduced from elementary properties of convex functions and an elementary integral inequality.
Grassmann cactus variety is a common generalisation of Grassmann secant variety and cactus variety. In their definitions one considers the vector spaces of fixed dimension that are contained in the linear span of some finite schemes. We prove that to characterise Grassmann cactus varieties it is enough to use finite schemes that locally have low socle dimension. This motivates the study of parameter spaces of such schemes and simplifies calculations of examples of Grassmann cactus varieties.
We study regularity of the time-delayed coordinate maps \[\phi_{h,k}(x) = (h(x), h(Tx), \ldots, h(T^{k-1}x))\] for a diffeomorphism $T$ of a compact manifold $M$ and smooth observables $h$ on $M$. Takens' embedding theorem shows that if $k > 2\dim M$, then $\phi_{h,k}$ is an embedding for typical $h$. We consider the probabilistic case, where for a given probability measure $\mu$ on $M$ one allows self-intersections in the time-delayed embedding to occur along a zero-measure set. We show that if $k \geq \dim M$ and $k > \dim_H(\text{supp} \mu)$, then for a typical observable, $\phi_{h,k}$ is injective on a full-measure set with a pointwise Lipschitz inverse. If moreover $k > \dim M$, then $\phi_{h,k}$ is a local diffeomorphism at almost every point. As an application, we show that if $k > \dim M$, then the Lyapunov exponents of the original system can be approximated with arbitrary precision by almost every orbit in the time-delayed model of the system. We also give almost sure pointwise bounds on the prediction error and provide a non-dynamical analogue of the main result, which can be seen as a probabilistic version of Whitney's embedding theorem.
We introduce the notion of a Lie superheaps as a generalisation of Lie supergroups. We show that the well-known `groupification' and `heapification' functors generalise to the ambience of supergeometry. In particular, we show that there is an isomorphism between the categories of pointed Lie superheaps and Lie supergroups. To do this we make extensive use of the functor of points.
In this note, we consider the stochastic heat and Schrödinger equation, and show that, at time t, the onset of the chaos occurs on the scale of 1/t, and the Fourier spectrum of the solution is asymptotically Gaussian after centering and rescaling.
Using the addition technique, we present polynomial identities for the Betti and Poincaré polynomials of reduced plane curves.
Up to now there has been no proof in the literature of the often quoted fact that the Jewett-Krieger theorem is valid for all countable amenable groups. In this brief note I will close this gap by applying a recent result of B. Frej and D. Huczek [FH].
In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or equal to the inductive dimension of its boundary, and consequently may be strictly bigger than the covering dimension of the boundary. We also give a condition, called complete traceability, on the boundary of the coarse proximity space under which the asymptotic inductive dimension of a coarse proximity space and the inductive dimension of its boundary coincide. Finally, we show that spaces whose boundaries are Z-sets and spaces admitting metrizable compactifications have completely traceable boundaries.
Any collection of non -blocking squares with total area not greater than 5/9 can be packed into the unit square.
We view all smooth metrics g on a closed surface Σ through their Nash isometric embeddings f_g: (Σ,g) → (𝕊^ñ, g̃) into a standard sphere of large, but fixed, dimension ñ. We define the Willmore functional 𝒲_f_g over this space of metrics on Σ in terms of the extrinsic quantities of f_g. Its infimum over metrics in a conformal class is an invariant of the class varying differentiably with it. If Σ is oriented of genus k, when k=0, we use the gap theorem of Simons to show that there is a unique conformal class of metrics on Σ, whose invariant 16π is the value for the standard totally geodesic embedding of 𝕊^2 ↪𝕊^ñ, and we have that 𝒲_f_g(Σ) ≥ 16π, with the lower bound achieved if, and only if, f_g is conformally equivalent to this standard geodesic embedding of 𝕊^2, and area_g(Σ) ≤ 4π, while when k≥ 1, the Lawson minimal surface (ξ_k,1,g_ξ_k,1) fixes the scale, and we show that 𝒲_f_g(Σ) ≥ 4 area_g_ξ_k,1 (ξ_k,1), with the lower bound achieved by f_g if, and only if, f_g is conformally equivalent to f_g_ξ_k,1 : (ξ_k,1, g_ξ_k,1) → (𝕊^3,g̃) ↪ (𝕊^ñ,g̃), and area_g(Σ)≤ area_g_ξ_k,1 (ξ_k,1). For a nonoriented Σ, we prove a likewise estimate from below for 𝒲_f_g(Σ), and characterize conformally the surface that realizes the optimal lower bound.
