On a Characterization of Idempotent Distributions on Discrete Fields and on the Field of <Emphasis Type="Italic">p</Emphasis>-Adic Numbers

We prove the following theorem. Let X be a discrete field, and \(\xi \) and \(\eta \) be independent identically distributed random variables with values in X and distribution \(\mu \). The random variables \(S=\xi +\eta \) and \(D=(\xi -\eta )^2\) are independent if and only if \(\mu \) is an idempotent distribution. A similar result is also proved in the case when \(\xi \) and \(\eta \) are independent identically distributed random variables with values in the field of p-adic numbers \({\mathbf {Q}}_p\), where \(p>2\), assuming that the distribution \(\mu \) has a continuous density.     

Monotonicity and non-monotonicity results for sequential fractional delta differences of mixed order

We consider the discrete fractional sequential difference \(\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)\), where \(t\in \mathbb {N}_{3-\mu -\nu +a}\), in two separate cases, where in each case we require that \(\mu +\nu \in (1,2)\). In the first case, we show that when \(\mu \in (0,1)\) and \(\nu \in (1,2)\) it follows that the condition \(\Delta _{1+a-\mu }^{\nu }\Delta _a^{\mu }f(t)\ge 0\) implies that f is an increasing map when we impose that \(f(a)\ge 0\), \(\Delta f(a)\ge 0\), and \(\Delta f(a+1)\ge 0\). On the other hand, when \(\mu \in (1,2)\) and \(\nu \in (0,1)\) we demonstrate that the situation is very different and that this type of monotonicity result only holds when restricted to a proper subregion of the \((\mu ,\nu )\)-parameter space coupled with some additional auxiliary conditions.     

Chains,antichains, and complements in infinite partition lattices

We consider the partition lattice \(\Pi (\lambda )\) on any set of transfinite cardinality \(\lambda \) and properties of \(\Pi (\lambda )\) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly \(\lambda \); (II) there are maximal chains in \(\Pi (\lambda )\) of cardinality \(> \lambda \); (III) a regular cardinal \(\lambda \) is strongly inaccessible if and only if every maximal chain in \(\Pi (\lambda )\) has size at least \(\lambda \); if \(\lambda \) is a singular cardinal and \(\mu ^{< \kappa } < \lambda \le \mu ^\kappa \) for some cardinals \(\kappa \) and (possibly finite) \(\mu \), then there is a maximal chain of size \(< \lambda \) in \(\Pi (\lambda )\); (IV) every non-trivial maximal antichain in \(\Pi (\lambda )\) has cardinality between \(\lambda \) and \(2^{\lambda }\), and these bounds are realised. Moreover, there are maximal antichains of cardinality \(\max (\lambda , 2^{\kappa })\) for any \(\kappa \le \lambda \); (V) all cardinals of the form \(\lambda ^\kappa \) with \(0 \le \kappa \le \lambda \) occur as the cardinalities of sets of complements to some partition \(\mathcal {P} \in \Pi (\lambda )\), and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.     

Associate space with respect to a locally $$\sigma $$-finite measure on a $$\delta $$δ-ring and applications to spaces of integrable functions defined by a vector measure

We show that for a locally \(\sigma \)-finite measure \(\mu \) defined on a \(\delta \)-ring, the associate space theory can be developed as in the \(\sigma \)-finite case, and corresponding properties are obtained. Given a saturated \(\sigma \)-order continuous \(\mu \)-Banach function space E, we prove that its dual space can be identified with the associate space \(E ^\times \) if, and only if, \(E^\times \) has the Fatou property. Applying the theory to the spaces \(L^p (\nu )\) and \(L_w^p (\nu )\), where \(\nu \) is a vector measure defined on a \(\delta \)-ring \(\mathcal {R}\) and \(1 \le p < \infty \), we establish results corresponding to those of the case when the vector measure is defined on a \(\sigma \)-algebra.     

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).     

A note on Wilson’s functional equation

Let S be a semigroup, and \(\mathbb {F}\) a field of characteristic \(\ne 2\). If the pair \(f,g:S \rightarrow \mathbb {F}\) is a solution of Wilson’s \(\mu \)-functional equation such that \(f \ne 0\), then g satisfies d’Alembert’s \(\mu \)-functional equation.     

Directed graphs of inner translations of semigroups

A mapping \(\alpha :S\rightarrow S\) is called a Cayley function if there exist an associative operation \(\mu :S\times S\rightarrow S\) and an element \(a\in S\) such that \(\alpha (x)=\mu (a,x)\) for every \(x\in S\). The aim of the paper is to give a characterization of Cayley functions in terms of their directed graphs. This characterization is used to determine which elements of the centralizer of a permutation on a finite set are Cayley functions. The paper ends with a number of problems.     

On the regularity of weak solutions to Burgers’ equation with finite entropy production

Bounded weak solutions of Burgers’ equation \(\partial _tu+\partial _x(u^2/2)=0\) that are not entropy solutions need in general not be BV. Nevertheless it is known that solutions with finite entropy productions have a BV-like structure: a rectifiable jump set of dimension one can be identified, outside which u has vanishing mean oscillation at all points. But it is not known whether all points outside this jump set are Lebesgue points, as they would be for BV solutions. In the present article we show that the set of non-Lebesgue points of u has Hausdorff dimension at most one. In contrast with the aforementioned structure result, we need only one particular entropy production to be a finite Radon measure, namely \(\mu =\partial _t (u^2/2)+\partial _x(u^3/3)\). We prove Hölder regularity at points where \(\mu \) has finite \((1+\alpha )\)-dimensional upper density for some \(\alpha >0\). The proof is inspired by a result of De Lellis, Westdickenberg and the second author : if \(\mu _+\) has vanishing 1-dimensional upper density, then u is an entropy solution. We obtain a quantitative version of this statement: if \(\mu _+\) is small then u is close in \(L^1\) to an entropy solution.     

A characterization of cut locus for $$C^1$$ hypersurfaces

Let \(\Omega \) be an open set in \(\mathbb {R}^n\) with \(C^1\)-boundary and \(\Sigma \) be the skeleton of \(\Omega \), which consists of points where the distance function to \(\partial \Omega \) is not differentiable. This paper characterizes the cut locus (ridge) \(\overline{\Sigma }\), which is the closure of the skeleton, by introducing a generalized radius of curvature and its lower semicontinuous envelope. As an application we give a sufficient condition for vanishing of the Lebesgue measure of \(\overline{\Sigma }\).     

Small Littlewood–Richardson coefficients

We develop structural insights into the Littlewood–Richardson graph, whose number of vertices equals the Littlewood–Richardson coefficient \(c_{\lambda ,\mu }^{\nu }\) for given partitions \(\lambda \), \(\mu \), and \(\nu \). This graph was first introduced in Bürgisser and Ikenmeyer (SIAM J Discrete Math 27(4):1639–1681, 2013), where its connectedness was proved. Our insights are useful for the design of algorithms for computing the Littlewood–Richardson coefficient: We design an algorithm for the exact computation of \(c_{\lambda ,\mu }^{\nu }\) with running time \(\mathcal {O}\big ((c_{\lambda ,\mu }^{\nu })^2 \cdot {\textsf {poly}}(n)\big )\), where \(\lambda \), \(\mu \), and \(\nu \) are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge t\) whose running time is \(\mathcal {O}\big (t^2 \cdot {\textsf {poly}}(n)\big )\). Even the existence of a polynomial-time algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge 2\) is a nontrivial new result on its own. Our insights also lead to the proof of a conjecture by King et al. (Symmetry in physics. American Mathematical Society, Providence, 2004), stating that \(c_{\lambda ,\mu }^{\nu }=2\) implies \(c_{M\lambda ,M\mu }^{M\nu } = M+1\) for all \(M \in \mathbb {N}\). Here, the stretching of partitions is defined componentwise.     

The deformation of symplectic critical surfaces in a Kähler surface-II—compactness

In this paper we consider the compactness of \(\beta \)-symplectic critical surfaces in a Kähler surface. Let M be a compact Kähler surface and \(\Sigma _i\subset M\) be a sequence of closed \(\beta _i\)-symplectic critical surfaces with \(\beta _i\rightarrow \beta _0\in (0,\infty )\). Suppose the quantity \(\int _{\Sigma _i}\frac{1}{\cos ^q\alpha _i}d\mu _i\) (for some \(q>4\)) and the genus of \(\Sigma _{i}\) are bounded, then there exists a finite set of points \({{\mathcal {S}}}\subset M\) and a subsequence \(\Sigma _{i'}\) which converges uniformly in the \(C^l\) topology (for any \(l<\infty \)) on compact subsets of \(M\backslash {{\mathcal {S}}}\) to a \(\beta _0\)-symplectic critical surface \(\Sigma \subset M\), each connected component of \(\Sigma \setminus {{\mathcal {S}}}\) can be extended smoothly across \({{\mathcal {S}}}\).     

New nonbinary code bounds based on divisibility arguments

For \(q,n,d \in \mathbb {N}\), let \(A_q(n,d)\) be the maximum size of a code \(C \subseteq [q]^n\) with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds \(A_5(8,6) \le 65\), \(A_4(11,8)\le 60\) and \(A_3(16,11) \le 29\). These in turn imply the new upper bounds \(A_5(9,6) \le 325\), \(A_5(10,6) \le 1625\), \(A_5(11,6) \le 8125\) and \(A_4(12,8) \le 240\). Furthermore, we prove that for \(\mu ,q \in \mathbb {N}\), there is a 1–1-correspondence between symmetric \((\mu ,q)\)-nets (which are certain designs) and codes \(C \subseteq [q]^{\mu q}\) of size \(\mu q^2\) with minimum distance at least \(\mu q - \mu \). We derive the new upper bounds \(A_4(9,6) \le 120\) and \(A_4(10,6) \le 480\) from these ‘symmetric net’ codes.     

Some new expansions for certain truncated <Emphasis Type="Italic">q</Emphasis>-series

We introduce the notion of \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials, where \(\mathcal {R}_{\mu }\) is the degree raising shift operator for the sequence of Laguerre polynomials of parameter \(\mu \). Then we show that the Laguerre polynomials \(L^{(\mu )}_n(x), \ \mu \ne -m, \ m\ge 0\), are the only \(\mathcal {R}_{\mu }\)-classical orthogonal polynomials.     

On <Emphasis Type="Italic">f</Emphasis>-non-parabolic ends for Ricci-harmonic metrics

In this paper, we study gradient Ricci-harmonic soliton metrics and quasi Ricci-harmonic metrics (both metrics are called Ricci-harmonic). First, we prove that all ends of \(\tau \)-quasi Ricci-harmonic metrics with \(\tau >1\) should be f-non-parabolic if \(\lambda =0,\mu >0\), or \(\lambda <0, \mu \ge 0\). For the case that \(\lambda<0, \mu < 0\), we can also arrive at the f-non-parabolic property if we add a condition about the scalar curvature. Furthermore, we discuss the connectivity at infinity for quasi Ricci-harmonic metrics. We also conclude that all ends of steady or expanding gradient Ricci-harmonic solitons should be f-non-parabolic, based on which we establish structure theorems for these two solitons.     

Removability of the logarithmic singularity for the elliptic PDEs with measurable coefficients and its consequences

This paper introduces the notion of log-regularity (or log-irregularity) of the boundary point \(\zeta \) (possibly \(\zeta =\infty \)) of the arbitrary open subset \(\Omega \) of the Greenian deleted neigborhood of \(\zeta \) in \({\mathbb {R}}^2\) concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the log-harmonic measure of \(\zeta \) is null (or positive). A necessary and sufficient condition for the removability of the logarithmic singularity, that is to say for the existence of a unique solution to the Dirichlet problem in \(\Omega \) in a class \(O(\log |\cdot - \zeta |)\) is established in terms of the Wiener test for the log-regularity of \(\zeta \). From a topological point of view, the Wiener test at \(\zeta \) presents the minimal thinness criteria of sets near \(\zeta \) in minimal fine topology. Precisely, the open set \(\Omega \) is a deleted neigborhood of \(\zeta \) in minimal fine topology if and only if \(\zeta \) is log-irregular. From the probabilistic point of view, the Wiener test presents asymptotic law for the log-Brownian motion near \(\zeta \) conditioned on the logarithmic kernel with pole at \(\zeta \).     

Typical Behavior of the Harmonic Measure in Critical Galton–Watson Trees with Infinite Variance Offspring Distribution

We study the typical behavior of the harmonic measure in large critical Galton–Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index \(\alpha \in (1,2]\). Let \(\mu _n\) denote the hitting distribution of height n by simple random walk on the critical Galton–Watson tree conditioned on non-extinction at generation n. We extend the results of Lin (Typical behavior of the harmonic measure in critical Galton–Watson trees, arXiv:1502.05584, 2015) to prove that, with high probability, the mass of the harmonic measure \(\mu _n\) carried by a random vertex uniformly chosen from height n is approximately equal to \(n^{-\lambda _\alpha }\), where the constant \(\lambda _\alpha >\frac{1}{\alpha -1}\) depends only on the index \(\alpha \). In the analogous continuous model, this constant \(\lambda _\alpha \) turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for \(\lambda _\alpha \), we are able to show that \(\lambda _\alpha \) decreases with respect to \(\alpha \in (1,2]\), and it goes to infinity at the same speed as \((\alpha -1)^{-2}\) when \(\alpha \) approaches 1.     

Fractional Schrödinger Equation in Bounded Domains and Applications

We establish the existence and uniqueness of a positive solution to the Schrödinger equation involving the fractional Laplacian \(\Delta ^{\frac{\alpha }{2}}u=\mu \,u\) in smooth bounded domains of \(\mathbb {R}^d\) for a large class of nonnegative perturbations \(\mu \). We then use this result to give some new facts about the fractional semilinear equation \(\Delta ^{\frac{\alpha }{2}}u= u^\gamma \), \(\gamma >0\).     

Preservation of semistability under Fourier–Mukai transforms

For a trivial elliptic fibration \(X=C \times S\) with C an elliptic curve and S a projective K3 surface of Picard rank 1, we study how various notions of stability behave under the Fourier–Mukai autoequivalence \(\Phi \) on \(D^b(X)\), where \(\Phi \) is induced by the classical Fourier–Mukai autoequivalence on \(D^b(C)\). We show that, under some restrictions on Chern classes, Gieseker semistability on coherent sheaves is preserved under \(\Phi \) when the polarisation is ‘fiber-like’. Moreover, for more general choices of Chern classes, Gieseker semistability under a ‘fiber-like’ polarisation corresponds to a notion of \(\mu _*\)-semistability defined by a ‘slope-like’ function \(\mu _*\).     

On Hecke groups,Schwarzian triangle functions and a class of hyper-elliptic functions

Let m be a positive integer \(\ge \)3 and \(\lambda =2\cos \frac{\pi }{m}\). The Hecke group \(\mathfrak {G}(\lambda )\) is generated by the fractional linear transformations \(\tau + \lambda \) and \(-\frac{1}{\tau }\) for \(\tau \) in the upper half plane \(\mathbb H\) of the complex plane \(\mathbb C\). We consider a set of functions \(\mathfrak {f}_0, \mathfrak {f}_i\) and \(\mathfrak {f}_{\infty }\) automorphic with respect to \(\mathfrak {G}(\lambda )\), constructed from the conformal mapping of the fundamental domain of \(\mathfrak {G}(\lambda )\) to the upper half plane \(\mathbb H\), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of \(\lambda =1\) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of \(\pi \) in terms of the hypergeometric series.     

Compact $$\lambda $$-translating Solitons with Boundary

A \(\lambda \)-translating soliton with density vector \(\mathbf {v}\) is a surface \(\varSigma \) in Euclidean space \(\mathbb {R}^3\) whose mean curvature H satisfies \(2H=2\lambda +\langle N,\mathbf {v}\rangle \), where N is the Gauss map of \(\varSigma \). In this article, we study the shape of a compact \(\lambda \)-translating soliton in terms of its boundary. If \(\varGamma \) is a given closed curve, we deduce under what conditions on \(\lambda \) there exists a compact \(\lambda \)-translating soliton \(\varSigma \) with boundary \(\varGamma \) and we provide estimates of the surface area depending on the height of \(\varSigma \). Finally, we study the shape of \(\varSigma \) related with the geometry of \(\varGamma \), in particular, we give conditions that assert that \(\varSigma \) inherits the symmetries of its boundary \(\varGamma \).