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We show that a function (f: X rightarrow {mathbb {R}}) defined on a closed uniformly polynomially cuspidal set X in ({mathbb {R}}^n) is real analytic if and only if f is smooth and all its composites with germs of polynomial curves in X are real analytic. The degree of the polynomial curves needed for this is effectively related to the regularity of the boundary of X. For instance, if the boundary of X is locally Lipschitz, then polynomial curves of degree 2 suffice. In this Lipschitz case, we also prove that a function (f: X rightarrow {mathbb {R}}) is real analytic if and only if all its composites with germs of quadratic polynomial maps in two variables with images in X are real analytic; here it is not necessary to assume that f is smooth.

We show that, if (-A) generates a bounded holomorphic semigroup in a Banach space X, (alpha in [0,1)), and (f:D(A)rightarrow X) satisfies (Vert f(x)-f(y)Vert le LVert A^alpha (x-y)Vert ), then a non-constant T-periodic solution of the equation ({dot{u}}+Au=f(u)) satisfies (LT^{1-alpha }ge K_alpha ) where (K_alpha >0) is a constant depending on (alpha ) and the semigroup. This extends results by Robinson and Vidal-Lopez, which have been shown for self-adjoint operators (Age 0) in a Hilbert space. For the latter case, we obtain - with a conceptually new proof - the optimal constant (K_alpha ), which only depends on (alpha ), and we also include the case (alpha =1). In Hilbert spaces H and for (alpha =0), we present a similar result with optimal constant where Au in the equation is replaced by a possibly unbounded gradient term (nabla _H{mathscr {E}}(u)). This is inspired by applications with bounded gradient terms in a paper by Mawhin and Walter.

Herzog, Hibi, and Zheng classified the Cohen-Macaulay edge ideals of chordal graphs. In this paper, we classify Cohen-Macaulay edge ideals of (vertex) weighted oriented chordal and simplicial graphs, a more general class of monomial ideals. In particular, we show that the Cohen-Macaulay property of these ideals is equivalent to the unmixed one and hence, independent of the underlying field.

We show that in the theory of Daniell integration iterated integrals may always be formed, and the order of integration may always be interchanged. By this means, we discuss product integrals and show that the related Fubini theorem holds in full generality. The results build on a density theorem on Riesz tensor products due to Fremlin, and on the Fubini–Stone theorem.

For every (nge 6), we give an example of a finite subset of (mathbb {P}^{2}) of degree n which does not descend to any Brauer–Severi surface over the field of moduli. Conversely, for every (nle 5), we prove that a finite subset of degree n always descends to a 0-cycle on (mathbb {P}^{2}) over the field of moduli.

As a continuation of the study of the Steinberg algebra of a Hausdorff ample groupoid ({mathcal {G}}) over commutative semirings by Nam et al. (J. Pure Appl. Algebra 225, 2021), we consider here the Steinberg algebra (A_S({mathcal {G}})) with coefficients in a Clifford semifield S. We obtain a complete characterization of the full k-ideal simplicity of (A_S({mathcal {G}})). Using this result for the Steinberg algebra (A_S({mathcal {G}}_Gamma )) of the graph groupoid ({mathcal {G}}_Gamma ), where (Gamma ) is a row-finite digraph and S is a Clifford semifield, we characterize the full k-simplicity of the Leavitt path algebra (L_S(Gamma )).

If a toric foliation on a projective ({mathbb {Q}})-factorial toric variety has an extremal ray whose length is longer than the rank of the foliation, then the associated extremal contraction is a projective space bundle and the foliation is the relative tangent sheaf of the extremal contraction.

In this paper, we focus on covering and illumination properties of a specific class of convex polytopes denoted by (mathcal {P}). These polytopes are obtained as the convex hull of the Minkowski sum of a finite subset of (mathbb {Z}^n) and ((1/2)[-1,1]^n). Our investigation includes the verification of Hadwiger’s covering conjecture for (mathcal {P}), as well as the estimation of the covering functional for convex polytopes in (mathcal {P}). Furthermore, we demonstrate that when an integer M is sufficiently large, the elements belonging to (mathcal {P}) that are contained in (M[-1,1]^n) serve as an (varepsilon )-net for the space of convex bodies in (mathbb {R}^n), equipped with the Banach–Mazur metric.

For a given closed nonempty subset E of a Hilbert space H, the singular set (Sigma _E) consists of the points in (Hsetminus E) where the distance function (d_E) is not Fréchet differentiable. It is known that (Sigma _E) is a weak deformation retract of the open set (mathcal {G}_E={xin H: d_{overline{{text {co}}},E}(x)< d_E(x)}). This short paper sheds light on the relationship between the connected components of the three sets (Sigma _Esubset mathcal {G}_Esubseteq H{setminus } E).

Rédei and Megyesi proved that the number of directions determined by a p-element subset of ({mathbb F}_p^2) is either 1 or at least (frac{p+3}{2}). The same result was independently obtained by Dress, Klin, and Muzychuk. We give a new and short proof of this result using a lemma proved by Kiss and the author. The new proof relies on a result on polynomials over finite fields.