Nontrivial p p -polygonal equalities impose certain conditions on the geometry of a metric space ( X , d ) (X,d) and so it is of interest to be able to identify the values of p ∈ [ 0 , ∞ ) p \in [0,\infty ) for which such equalities exist. Following work of Li and Weston [Positivity 14 (2010), no. 3, 529–545], Kelleher, Miller, Osborn, and Weston [J. Math. Anal. Appl. 415 (2014), no. 1, 247–268] established that if a metric space ( X , d ) (X,d) is of p p -negative type, then ( X , d ) (X,d) admits no nontrivial p p -polygonal equalities if and only if it is of strict p p -negative type. In this note we remove the underlying premise of p p -negative type from this theorem. As an application we show that the set of all p p for which a finite metric space ( X , d ) (X,d) admits a nontrivial p p -polygonal equality is always a closed interval of the form [ ℘ , ∞ ) [\wp , \infty ) , where ℘ > 0 \wp > 0 or the empty set. It follows that for each q ≠ 2 q \not = 2 , the Schatten q q -class C q \mathcal {C}_{q} admits a nontrivial p p -polygonal equality for each p > 0 p > 0 . Other spaces with this same property include C [ 0 , 1 ] C[0, 1] and ℓ q ( 3 ) \ell _{q}^{(3)} for all q > 2 q > 2 .

We use the slice filtration to study the M U MU -homology of the fixed points of connective models of Lubin–Tate theory studied by Hill, Hopkins, and Ravenel and Beaudry, Hill, Shi, and Zeng. We show that, unlike their periodic counterparts E O n EO_n , the M U MU homology of B P ( ( G ) ) ⟨ m ⟩ G BP^{((G))}\langle m\rangle ^G usually fails to be even and torsion free. This can only happen when the height n = m | G | / 2 n=m|G|/2 is less than 3 3 , and in the edge case n = 2 n=2 , we show that this holds for t m f 0 ( 3 ) tmf_0(3) but not for t m f 0 ( 5 ) tmf_0(5) , and we give a complete computation of the M U ∗ M U MU_*MU -comodule algebra M U ∗ t m f 0 ( 3 ) MU_*tmf_0(3) .

The purpose of this short note is to prove a convenient version of Itô’s formula for the Rearranged Stochastic Heat Equation (RSHE) introduced by the two authors in a previous contribution. This equation is a penalized version of the standard Stochastic Heat Equation (SHE) on the circle subject to a colored noise, whose solution is constrained to stay within the set of symmetric quantile functions by means of a reflection term. Here, we identity the generator of the solution when it is acting on functions defined on the space P 2 ( R ) {\mathcal P}_2({\mathbb R}) (of one-dimensional probability measures with a finite second moment) that are assumed to be smooth in Lions’s sense. In particular, we prove that the reflection term in the RSHE is orthogonal to the Lions (or Wasserstein) derivative of smooth functions defined on P 2 ( R ) {\mathcal P}_2({\mathbb R}) . The proof relies on nontrivial bounds for the gradient of the solution to the RSHE.

We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the works of Bishop, Carleson, Garnett, and Jones and Browder and Wermer relating tangent points of Jordan curves, harmonic measure and Dirichlet algebras. Our approach also provides a new proof of a result of Ransford, Younsi, and Ai on the variation of analytic capacity under holomorphic motions. In addition, we show that extremal functions for continuous analytic capacity may not exist.

We extend the class of ultrafilters U U over countable sets for which U ⋅ U ≡ T U U\cdot U\equiv _T U , extending several results from \cite{Dobrinen/Todorcevic11}. In particular, we prove that for each countable ordinal α ≥ 2 \alpha \ge 2 , the generic ultrafilter G α G_\alpha forced by P ( ω α ) / f i n ⊗ α P(\omega ^\alpha )/\mathrm {fin}^{\otimes \alpha } satisfy G α ⋅ G α ≡ T G α G_\alpha \cdot G_\alpha \equiv _T G_\alpha . This answers a question posed in \cite[Question 43]Dobrinen/Todorcevic11. Additionally, we establish that Milliken–Taylor ultrafilters possess the property that U ⋅ U ≡ T U U\cdot U\equiv _T U .

We provide in this paper a constructive proof of optimal L ∞ L_{\infty } star discrepancy values in dimension 2 for up to 21 points and up to 8 points in dimension 3. This extends work by White (Numer. Math. 27 (1976/77), no. 2, 157–164) for up to six points in dimension 2 and of Larcher and Pillichshammer (J. Comput. Appl. Math. 206 (2007), no. 2, 977–985) for two points in arbitrary dimensions. We show that these optimal sets have a far lower discrepancy than the previous references and, perhaps more importantly, present a very different structure.

A monotone surjective map F F is continuous and defines a spectral family and unitary group on L 2 L^{2} . We use the H 0 \mathcal {H}_{0} splitting theorem (Int. J. Dyn. Syst. Differ. Equ. 10(2020), no. 4, 358–372) to extend the Lebesgue decomposition F = F s + F a F \!=\! F_{s} \!+\! F_{a} to the singular component of F F , F s = F m + F w s F_{s} \!=\! F_{m} + F_{w_{s}} . The components (recurrent: F m F_{m} , and stable: F w s F_{w_{s}} ) are unique up to additive constants and are characterized by their Fourier–Stieltjes transforms. Several examples are included. Section 7.2 shows that each Riesz product has a nontrivial recurrent component.

In 2015, Borodzik, Némethi, and Ranicki proved that an interior critical point can be pushed to the boundary, where it splits into two boundary critical points. In this paper, we show that two critical points at the boundary can be, under specific assumptions, merged into a single critical point in the interior. That is, we reverse the original construction.

Let k k be an algebraically closed field of positive characteristic and G G a simple algebraic group over k k . Under the assumption that the characteristic is a good prime for G G , we determine which unipotent elements u ∈ G u \in G , with u u of order p p , satisfy the property that any two A 1 A_1 -subgroups of G G containing u u are G G -conjugate. This result establishes to what degree an analog of the theorems of Jacobson–Morozov and Kostant for Lie algebras are valid for simple algebraic groups defined over fields of (good) positive characteristic.

We prove that every orientation-preserving homeomorphism of Euclidean space can be expressed as a commutator of two orientation-preserving homeomorphisms. We give an analogous result for annuli. In the annulus case, we also extend the result to the smooth category in the dimensions for which the associated sphere has a unique smooth structure. As a corollary, we establish that every orientation-preserving diffeomorphism of the real line is the commutator of two orientation-preserving diffeomorphisms.