Integrable nonlocal nonlinear Schrödinger equations associated with 𝑠𝑜(3,ℝ)

We construct integrable PT-symmetric nonlocal reductions for an integrable hierarchy associated with the special orthogonal Lie algebra so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) . The resulting typical nonlocal integrable equations are integrable PT-symmetric nonlocal reverse-space, reverse-time and reverse-spacetime nonlinear Schrödinger equations associated with so ⁡ ( 3 , R ) \operatorname {so}(3,\mathbb {R}) .

Some new multi-cell Ramsey theoretic results

We extend an old Ramsey Theoretic result which guarantees sums of terms from all partition regular linear systems in one cell of a partition of the set N \mathbb {N} of positive integers. We were motivated by a quite recent result which guarantees a sequence in one set with all of its sums two or more at a time in the complement of that set. A simple instance of our new results is the following. Let P f ( N ) \mathcal {P}_{f}(\mathbb {N}) be the set of finite nonempty subsets of N \mathbb {N} . Given any finite partition R {\mathcal R} of N \mathbb {N} , there exist B 1 B_1 , B 2 B_2 , A 1 , 2 A_{1,2} , and A 2 , 1 A_{2,1} in R {\mathcal R} and sequences ⟨ x 1 , n ⟩ n = 1 ∞ \langle x_{1,n}\rangle _{n=1}^\infty and ⟨ x 2 , n ⟩ n = 1 ∞ \langle x_{2,n}\rangle _{n=1}^\infty in N \mathbb {N} such that (1) for each F ∈ P f ( N ) F\in \mathcal {P}_{f}(\mathbb {N}) , ∑ t ∈ F x 1 , t ∈ B 1 \sum _{t\in F}x_{1,t}\in B_1 and ∑ t ∈ F x 2 , t ∈ B 2 \sum _{t\in F}x_{2,t}\in B_2 and (2) whenever F , G ∈ P f ( N ) F,G\in \mathcal {P}_{f}(\mathbb {N}) and max F > min G \max F > \min G , one has ∑ t ∈ F x 1 , t + ∑ t ∈ G x 2 , t ∈ A 1 , 2 \sum _{t\in F}x_{1,t}+\sum _{t\in G}x_{2,t}\in A_{1,2} and ∑ t ∈ F x 2 , t + ∑ t ∈ G x 1 , t ∈ A 2 , 1 \sum _{t\in F}x_{2,t}+\sum _{t\in G}x_{1,t}\in A_{2,1} . The partition R {\mathcal R} can be refined so that the cells B 1 B_1 , B 2 B_2 , A 1 , 2 A_{1,2} , and A 2 , 1 A_{2,1} must be pairwise disjoint.

Bounded complexes of permutation modules

Let k k be a field of characteristic p > 0 p > 0 . For G G an elementary abelian p p -group, there exist collections of permutation modules such that if C ∗ C^* is any exact bounded complex whose terms are sums of copies of modules from the collection, then C ∗ C^* is contractible. A consequence is that if G G is any finite group whose Sylow p p -subgroups are not cyclic or quaternion, and if C ∗ C^* is a bounded exact complex such that each C i C^i is a direct sum of one dimensional modules and projective modules, then C ∗ C^* is contractible.

Strichartz estimates for the Schrödinger equation with a measure-valued potential

We prove Strichartz estimates for the Schrödinger equation in R n \mathbb {R}^n , n ≥ 3 n\geq 3 , with a Hamiltonian H = − Δ + μ H = -\Delta + \mu . The perturbation μ \mu is a compactly supported measure in R n \mathbb {R}^n with dimension α > n − ( 1 + 1 n − 1 ) \alpha > n-(1+\frac {1}{n-1}) . The main intermediate step is a local decay estimate in L 2 ( μ ) L^2(\mu ) for both the free and perturbed Schrödinger evolution.

Radial symmetry for an elliptic PDE with a free boundary

In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u ) in B 1 , 0 ≤ u > M , in B 1 , u = M , on ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.

A construction principle for proper scoring rules

Proper scoring rules enable decision-theoretically principled comparisons of probabilistic forecasts. New scoring rules can be constructed by identifying the predictive distribution with an element of a parametric family and then applying a known scoring rule. We introduce a condition which ensures propriety in this construction and thereby obtain novel proper scoring rules.

A note on the asymptotic behavior of radial solutions to quasilinear elliptic equations with a Hardy potential

The quasilinear elliptic equation with a Hardy potential d i v ( | x | α | ∇ u | p − 2 ∇ u ) + μ | x | p − α | u | p − 2 u = 0 in R N − { 0 } \begin{equation*} {\mathrm {div}}(|x|^\alpha |\nabla u|^{p-2}\nabla u) + \frac {\mu }{|x|^{p-\alpha }}|u|^{p-2}u = 0 \quad \text {in} \ {\mathbf {R}}^N-\{0\} \end{equation*} is considered, where N ∈ N N\in {\mathbf {N}} , p > 1 p>1 and α ∈ R \alpha \in {\mathbf {R}} , μ ∈ R − { 0 } \mu \in {\mathbf {R}}-\{0\} . In this note, the asymptotic behaviors of radial solutions are obtained divided into three case μ > | ( N − p + α ) / p | p \mu >|(N-p+\alpha )/p|^p , μ = | ( N − p + α ) / p | p \mu =|(N-p+\alpha )/p|^p and μ > | ( N − p + α ) / p | p \mu >|(N-p+\alpha )/p|^p . This equation also appears as the Euler-Lagrange equation related to the weighted Hardy inequality ∫ Ω | ∇ u ( x ) | p | x | α d x ≥ | N − p + α p | p ∫ Ω | u ( x ) | p | x | α − p d x \begin{equation*} \int _\Omega |\nabla u(x)|^p |x|^\alpha dx \ge \Biggl | \frac {N-p+\alpha }{p} \Biggr |^p \int _\Omega |u(x)|^p |x|^{\alpha -p} dx \end{equation*} for u ∈ C c ∞ ( R N ) u \in C_c^\infty ({\mathbf {R}}^N) and N − p + α ≠ 0 N-p+\alpha \ne 0 , where Ω \Omega is a domain of R N {\mathbf {R}}^N . The rectifiability of oscillatory solutions to the ordinary differential equation with one-dimensional p p -Laplacian is also studied, and an answer to an open problem is given.

Slice monogenic functions of a Clifford variable via the 𝑆-functional calculus

In this paper we define a new function theory of slice monogenic functions of a Clifford variable using the S S -functional calculus for Clifford numbers. Previous attempts of such a function theory were obstructed by the fact that Clifford algebras, of sufficiently high order, have zero divisors. The fact that Clifford algebras have zero divisors does not pose any difficulty whatsoever with respect to our approach. The new class of functions introduced in this paper will be called the class of slice monogenic Clifford functions to stress the fact that they are defined on open sets of the Clifford algebra R n \mathbb {R}_n . The methodology can be generalized, for example, to handle the case of noncommuting matrix variables.

Basic properties of 𝑋 for which the space 𝐶_{𝑝}(𝑋) is distinguished

In our paper [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99] we showed that a Tychonoff space X X is a Δ \Delta -space (in the sense of R. W. Knight [Trans. Amer. Math. Soc. 339 (1993), pp. 45–60], G. M. Reed [Fund. Math. 110 (1980), pp. 145–152]) if and only if the locally convex space C p ( X ) C_{p}(X) is distinguished. Continuing this research, we investigate whether the class Δ \Delta of Δ \Delta -spaces is invariant under the basic topological operations. We prove that if X ∈ Δ X \in \Delta and φ : X → Y \varphi :X \to Y is a continuous surjection such that φ ( F ) \varphi (F) is an F σ F_{\sigma } -set in Y Y for every closed set F ⊂ X F \subset X , then also Y ∈ Δ Y\in \Delta . As a consequence, if X X is a countable union of closed subspaces X i X_i such that each X i ∈ Δ X_i\in \Delta , then also X ∈ Δ X\in \Delta . In particular, σ \sigma -product of any family of scattered Eberlein compact spaces is a Δ \Delta -space and the product of a Δ \Delta -space with a countable space is a Δ \Delta -space. Our results give answers to several open problems posed by us [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99]. Let T : C p ( X ) ⟶ C p ( Y ) T:C_p(X) \longrightarrow C_p(Y) be a continuous linear surjection. We observe that T T admits an extension to a linear continuous operator T ^ \widehat {T} from R X \mathbb {R}^X onto R Y \mathbb {R}^Y and deduce that Y Y is a Δ \Delta -space whenever X X is. Similarly, assuming that X X and Y Y are metrizable spaces, we show that Y Y is a Q Q -set whenever X X is. Making use of obtained results, we provide a very short proof for the claim that every compact Δ \Delta -space has countable tightness. As a consequence, under Proper Forcing Axiom every compact Δ \Delta -space is sequential. In the article we pose a dozen open questions.

Connecting a direct and a Galerkin approach to slow manifolds in infinite dimensions

In this paper, we study slow manifolds for infinite-dimensional evolution equations. We compare two approaches: an abstract evolution equation framework and a finite-dimensional spectral Galerkin approximation. We prove that the slow manifolds constructed within each approach are asymptotically close under suitable conditions. The proof is based upon Lyapunov-Perron methods and a comparison of the local graphs for the slow manifolds in scales of Banach spaces. In summary, our main result allows us to change between different characterizations of slow invariant manifolds, depending upon the technical challenges posed by particular fast-slow systems.