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 \(\varGamma \).     

Appell polynomial sequences with respect to some differential operators

We present a study of a specific kind of lowering operator, herein called \(\Lambda \), which is defined as a finite sum of lowering operators and might be presented by various configurations. We characterize the polynomial sequences fulfilling an Appell relation with respect to \(\Lambda \), and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are \(\Lambda \)-Appell, with \(\Lambda \) defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also \(\Lambda \)-Appell, when \(\Lambda \) is the lowering operator \(\Lambda =a_{0}D+a_{1}DxD+a_{2}\left( Dx\right) ^2D\), where \(a_{0}\), \(a_{1}\) and \(a_{2}\) are constants and \(a_{2} \ne 0\). The case where \(a_{2}=0\) and \(a_{1} \ne 0\) is also naturally recaptured.     

Pairs of Semisimple Lie Algebras and their Maximal Reductive Subalgebras

Let \(\mathfrak g\) be a semisimple Lie algebra over a field \(\mathbb K\), \(\text{char}\left( \mathbb{K} \right)=0\), and \(\mathfrak g_1\) a subalgebra reductive in \(\mathfrak g\). Suppose that the restriction of the Killing form B of \(\mathfrak g\) to \(\mathfrak g_1 \times \mathfrak g_1\) is nondegenerate. Consider the following statements: ( 1) For any Cartan subalgebra \(\mathfrak h_1\) of \(\mathfrak g_1\) there is a unique Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) containing \(\mathfrak h_1\); ( 2) \(\mathfrak g_1\) is self-normalizing in \(\mathfrak g\); ( 3) The B-orthogonal \(\mathfrak p\) of \(\mathfrak g_1\) in \(\mathfrak g\) is simple as a \(\mathfrak g_1\)-module for the adjoint representation. We give some answers to this natural question: For which pairs \((\mathfrak g,\mathfrak g_1)\) do ( 1), ( 2) or ( 3) hold? We also study how \(\mathfrak p\) in general decomposes as a \(\mathfrak g_1\)-module, and when \(\mathfrak g_1\) is a maximal subalgebra of \(\mathfrak g\). In particular suppose \((\mathfrak g,\sigma )\) is a pair with \(\mathfrak g\) as above and σ its automorphism of order m. Assume that \(\mathbb K\) contains a primitive m-th root of unity. Define \(\mathfrak g_1:=\mathfrak g^{\sigma}\), the fixed point algebra for σ. We prove the following generalization of a well known result for symmetric Lie algebras, i.e., for m=2: (a) \((\mathfrak g,\mathfrak g_1)\) satisfies ( 1); (b) For m prime, \((\mathfrak g,\mathfrak g_1)\) satisfies ( 2).     

Traces of the Nevanlinna Class on Discrete Sequences

We show that a discrete sequence \(\Lambda \) of the unit disk is the union of n interpolating sequences for the Nevanlinna class \(\mathcal {N}\) if and only if the trace of \(\mathcal {N}\) on \(\Lambda \) coincides with the space of functions on \(\Lambda \) for which the divided differences of order \(n-1\) are uniformly controlled by a positive harmonic function.     

Knaster and friends I: closed colorings and precalibers

The productivity of the \(\kappa \)-chain condition, where \(\kappa \) is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970s, consistent examples of \(\kappa \)-cc posets whose squares are not \(\kappa \)-cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, \(\textsf {ZFC}\) examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which \(\kappa = \aleph _2\), was resolved by Shelah in 1997. In this work, we obtain analogous results regarding the infinite productivity of strong chain conditions, such as the Knaster property. Among other results, for any successor cardinal \(\kappa \), we produce a \(\textsf {ZFC}\) example of a poset with precaliber \(\kappa \) whose \(\omega ^{\mathrm {th}}\) power is not \(\kappa \)-cc. To do so, we carry out a systematic study of colorings satisfying a strong unboundedness condition. We prove a number of results indicating circumstances under which such colorings exist, in particular focusing on cases in which these colorings are moreover closed.     

Reducibility of operator semigroups and values of vector states

Let \(\mathcal S\) be a multiplicative semigroup of bounded linear operators on a complex Hilbert space \(\mathcal H\), and let \(\Omega \) be the range of a vector state on \(\mathcal S\) so that \(\Omega = \{ \langle S \xi , \xi \rangle \,{:}\,S \in \mathcal S\}\) for some fixed unit vector \(\xi \in \mathcal H\). We study the structure of sets \(\Omega \) of cardinality two coming from irreducible semigroups \(\mathcal S\). This leads us to sufficient conditions for reducibility and, in some cases, for the existence of common fixed points for \(\mathcal S\). This is made possible by a thorough investigation of the structure of maximal families \(\mathcal F\) of unit vectors in \(\mathcal H\) with the property that there exists a fixed constant \(\rho \in \mathbb C\) for which \(\langle x, y \rangle = \rho \) for all distinct pairs x and y in \(\mathcal F\).     

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.     

Closure properties of parametric subcompleteness

For an ordinal \(\varepsilon \), I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \(\varepsilon \)-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \(\varepsilon \)-subcomplete forcings are \(\varepsilon \)-subcomplete, and if \(\mathbb {P}\) and \(\mathbb {Q}\) are forcing-equivalent notions, then \(\mathbb {P}\) is \(\varepsilon \)-subcomplete iff \(\mathbb {Q}\) is. I formulate a Two Step Theorem for \(\varepsilon \)-subcompleteness and prove an RCS iteration theorem for \(\varepsilon \)-subcompleteness which is slightly less restrictive than the original one, in that its formulation is more careful about the amount of collapsing necessary. Finally, I show that an adequate degree of \(\varepsilon \)-subcompleteness follows from the \(\kappa \)-distributivity of a forcing, for \(\kappa >\omega _1\).     

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.     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

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.     

Directed strongly walk-regular graphs

We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly \(\ell \)-walk-regular with \(\ell > 1\) if the number of walks of length \(\ell \) from a vertex to another vertex depends only on whether the first vertex is the same as, adjacent to, or not adjacent to the second vertex. This generalizes also the well-studied strongly regular digraphs and a problem posed by Hoffman. Our main tools are eigenvalue methods. The case that the adjacency matrix is diagonalizable with only real eigenvalues resembles the undirected case. We show that a digraph \(\varGamma \) with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of \(\ell \) for which \(\varGamma \) can be strongly \(\ell \)-walk-regular, and we also construct examples of such strongly walk-regular digraphs. We also consider digraphs with non-real eigenvalues. We give such examples and characterize those digraphs \(\varGamma \) for which there are infinitely many \(\ell \) for which \(\varGamma \) is strongly \(\ell \)-walk-regular.     

Epireflective subcategories of TOP,T$$_$$UNIF,UNIF, closed under epimorphic images,or being algebraic

The epireflective subcategories of \(\mathbf{Top}\), that are closed under epimorphic (or bimorphic) images, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \) and \(\mathbf{Top}\). The epireflective subcategories of \(\mathbf{T_2Unif}\), closed under epimorphic images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is compact \(T_2 \} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{T_2Unif}\). The epireflective subcategories of \(\mathbf{Unif}\), closed under epimorphic (or bimorphic) images, are: \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid \) covering character of X is \( \le \lambda _0 \} \) (where \(\lambda _0\) is an infinite cardinal), and \(\mathbf{Unif}\). The epireflective subcategories of \(\mathbf{Top}\), that are algebraic categories, are \(\{ X \mid |X| \le 1 \} \), and \(\{ X \mid X\) is indiscrete\(\} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being varietal, are \(\{ X \mid |X| \le 1 \} \), \(\{ X \mid X\) is indiscrete\(\} \), \(\{ X \mid X\) is compact \(T_2 \} \). The subcategories of \(\mathbf{Unif}\), closed under products and closed subspaces and being algebraic, are \(\{ X \mid X\) is indiscrete\( \} \), and all epireflective subcategories of \(\{ X \mid X\) is compact \(T_2 \} \). Also we give a sharpened form of a theorem of Kannan-Soundararajan about classes of \(T_3\) spaces, closed for products, closed subspaces and surjective images.     

Quasi-classical generalized CRF structures

In an earlier paper, we studied manifolds M endowed with a generalized F structure \(\Phi \in \mathrm{End}(TM\oplus T^*M)\), skew-symmetric with respect to the pairing metric, such that \(\Phi ^3+\Phi =0\). Furthermore, if \(\Phi \) is integrable (in some well-defined sense), \(\Phi \) is a generalized CRF structure. In the present paper, we study quasi-classical generalized F and CRF structures, which may be seen as a generalization of the holomorphic Poisson structures (it is well known that the latter may also be defined via generalized geometry). The structures that we study are equivalent to a pair of tensor fields \((A\in \mathrm{End}(TM),\pi \in \wedge ^2TM)\), where \(A^3+A=0\) and some relations between A and \(\pi \) hold. We establish the integrability conditions in terms of \((A,\pi )\). They include the facts that A is a classical CRF structure, \(\pi \) is a Poisson bivector field and \(\mathrm{im}\,A\) is a (non)holonomic Poisson submanifold of \((M,\pi )\). We discuss the case where either \(\mathrm{ker}\,A\) or \(\mathrm{im}\,A\) is tangent to a foliation and, in particular, the case of almost contact manifolds. Finally, we show that the dual bundle of \(\mathrm{im}\,A\) inherits a Lie algebroid structure and we briefly discuss the Poisson cohomology of \(\pi \), including an associated spectral sequence and a Dolbeault type grading.     

Two Novel Characterizations of Self-Decomposability on the Half-Line

Two novel characterizations of self-decomposable Bernstein functions are provided. The first one is purely analytic, stating that a function \(\varPsi \) is the Bernstein function of a self-decomposable probability law \(\pi \) on the positive half-axis if and only if alternating sums of \(\varPsi \) satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that \(\varPsi \) is a self-decomposable Bernstein function if and only if a related d-variate function \(C_{\psi ,d}\), \(\psi :=\exp (-\varPsi )\), is a d-variate copula for each \(d \ge 2\). A canonical stochastic construction is presented, in which \(\pi \) (respectively \(\varPsi \)) determines the probability law of an exchangeable sequence of random variables \(\{U_k\}_{k\in {\mathbb {N}}}\) such that \((U_1,\ldots ,U_d) \sim C_{\psi ,d}\) for each \(d \ge 2\). The random variables \(\{U_k\}_{k\in {\mathbb {N}}},\) are i.i.d. conditioned on an increasing Sato process whose law is characterized by \(\varPsi \). The probability law of \(\{U_k\}_{k \in {\mathbb {N}}}\) is studied in quite some detail.     

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 \(\zeta \).     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

$$I_0$$ and combinatorics at $$\lambda ^+$$λ+

We investigate the compatibility of \(I_0\) with various combinatorial principles at \(\lambda ^+\), which include the existence of \(\lambda ^+\)-Aronszajn trees, square principles at \(\lambda \), the existence of good scales at \(\lambda \), stationary reflections for subsets of \(\lambda ^{+}\), diamond principles at \(\lambda \) and the singular cardinal hypothesis at \(\lambda \). We also discuss whether these principles can hold in \(L(V_{\lambda +1})\).     

Partial regularity and smooth topology-preserving approximations of rough domains

For a bounded domain \(\Omega \subset {\mathbb R}^m, m\ge 2,\) of class \(C^0\), the properties are studied of fields of ‘good directions’, that is the directions with respect to which \(\partial \Omega \) can be locally represented as the graph of a continuous function. For any such domain there is a canonical smooth field of good directions defined in a suitable neighbourhood of \(\partial \Omega \), in terms of which a corresponding flow can be defined. Using this flow it is shown that \(\Omega \) can be approximated from the inside and the outside by diffeomorphic domains of class \(C^\infty \). Whether or not the image of a general continuous field of good directions (pseudonormals) defined on \(\partial \Omega \) is the whole of \(S^{m-1}\) is shown to depend on the topology of \(\Omega \). These considerations are used to prove that if \(m=2,3\), or if \(\Omega \) has nonzero Euler characteristic, there is a point \(P\in \partial \Omega \) in the neighbourhood of which \(\partial \Omega \) is Lipschitz. The results provide new information even for more regular domains, with Lipschitz or smooth boundaries.