We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems, and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic 0 are nondegenerate.

Higher dimensional mixed norm type inequalities involving certain integral operators are characterized in terms of the corresponding lower dimensional inequalities.

Given a complex semisimple Lie algebra $\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{t}$ is a compact real form of $\mathfrak{g}$), let $\text{ }\pi \text{ }\text{:}\mathfrak{g}\to \mathfrak{h}$ be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra $\mathfrak{h}:=\mathfrak{t}\text{+}i\mathfrak{t}$, where $\mathfrak{t}$ is a maximal abelian subalgebra of $\mathfrak{k}$. Given $x\,\in \,\mathfrak{g}$, we consider $\text{ }\!\!\pi\!\!\text{ (Ad(}K\text{)}x)$, where $K$ is the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range $f(\text{Ad(}K\text{)}x)$, where $f$ is a linear functional on $\mathfrak{g}$. We establish the star-shapedness of $f(\text{Ad(}K\text{)}x)$ for simple Lie algebras of type $B$.

A continuumis said to be Suslinian if it does not contain uncountably many mutually exclusive non-degenerate subcontinua. Fitzpatrick and Lelek have shown that a metric Suslinian continuum $X$ has the property that the set of points at which $X$ is connected im kleinen is dense in $X$. We extend their result to Hausdorff Suslinian continua and obtain a number of corollaries. In particular, we prove that a homogeneous, non-degenerate, Suslinian continuum is a simple closed curve and that each separable, non-degenerate, homogenous, Suslinian continuum is metrizable.

Let $p$ be a prime, and let ${{\zeta }_{p}}$ be a primitive $p$-th root of unity. The lattices in Craig's family are $(p\,-\,1)$-dimensional and are geometrical representations of the integral $\mathbb{Z}[{{\zeta }_{p}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}$, where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p\,-\,1$ where $149\,\le \,p\,\le \,3001$, Craig's lattices are the densest packings known. Motivated by this, we construct $(p\,-\,1)(q\,-\,1)$-dimensional lattices from the integral $\mathbb{Z}[{{\zeta }_{pq}}]$-ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}{{\left\langle 1\,-\,{{\zeta }_{q}} \right\rangle }^{j}}$, where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

Let $\mathcal{A}$ be a line arrangement in the complex projective plane ${{\mathbb{P}}^{2}}$, having the points of multiplicity $\ge \,3$ situated on two lines in $\mathcal{A}$, say ${{H}_{0}}$ and ${{H}_{\infty }}$. Then we show that the non-local irreducible components of the first resonance variety ${{\mathcal{R}}_{1}}(\mathcal{A})$ are 2-dimensional and correspond to parallelograms $P$ in ${{\mathbb{C}}^{2}}={{\mathbb{P}}^{2}}\text{ }\backslash \text{ }{{H}_{\infty }}$ whose sides are in $\mathcal{A}$ and for which ${{H}_{0}}$ is a diagonal.

This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic.

We comment on domain conditions that regulate when the adjoint of the sum or product of two unbounded operators is the sum or product of their adjoints, and related closure issues. The quantum mechanical problem $\text{PHP}$ essentially selfadjoint for unbounded Hamiltonians is addressed, with new results.

In this paper, a fixed point equation of the compound-exponential type distributions is derived, and under some regular conditions, both the existence and uniqueness of this fixed point equation are investigated. A question posed by Pitman and Yor can be partially answered by using our approach.

A. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's ideal-related $KK$-theory in the fundamental case of a ${{C}^{*}}$-algebra with one specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain conditions. Employing certain $K$-theoretical information derivable from the given operator algebras using a method introduced here, we shall demonstrate that Bonkat's $\text{UCT}$ does not split in general. Related methods lead to information on the complexity of the $K$-theory which must be used to classify $*$-isomorphisms for purely infinite ${{C}^{*}}$-algebras with one non-trivial ideal.

A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener–Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump–Diaconis–Tracy–Widom identity is a differentiated version of the classical Jacobi–Trudi identity.

For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let ${{M}_{cb}}A(G)$ denote the completely bounded multipliers of $A(G)$, and let ${{A}_{Mcb}}\,(G)$ stand for the closure of $A(G)$ in ${{M}_{cb}}A(G)$. We characterize the norm one idempotents in ${{M}_{cb}}A(G)$: the indicator function of a set $E\,\subset \,G$ is a norm one idempotent in ${{M}_{cb}}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of ${{A}_{Mcb}}\,(G)$ with an approximate identity bounded by 1, and we characterize those $G$ for which ${{A}_{Mcb}}\,(G)$ is 1-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.)

It is shown that it follows from PFA that there is no compact scattered space of height greater than $\omega $ in which the sequential order and the scattering heights coincide.

We study infinitesimal deformations of holomorphic maps of compact, complex, Kähler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain.

Let $W$ be a Weyl group, $\sum $ a set of simple reflections in $W$ related to a basis $\Delta $ for the root system $\Phi $ associated with $W$ and $\theta $ an involution such that $\theta (\Delta )\,=\,\Delta $. We show that the set of $\theta $- twisted involutions in $W$, ${{\mathcal{J}}_{\theta }}\,=\,\{w\,\in \,W\,|\,\theta (w)\,=\,{{w}^{-1}}\}$ is in one to one correspondence with the set of regular involutions ${{\mathcal{J}}_{\text{ID}}}$. The elements of ${{\mathcal{J}}_{\theta }}$ are characterized by sequences in $\sum $ which induce an ordering called the Richardson–Springer Poset. In particular, for $\Phi $ irreducible, the ascending Richardson–Springer Poset of ${{\mathcal{J}}_{\theta }}$, for nontrivial $\theta $ is identical to the descending Richardson–Springer Poset of ${{\mathcal{J}}_{\text{ID}}}$.

Using Poincaré duality, we obtain a formula of Lefschetz type that computes the Lefschetz number of an endomorphism of a separable nuclear ${{C}^{*}}$-algebra satisfying Poincaré duality and the Kunneth theorem. (The Lefschetz number of an endomorphism is the graded trace of the induced map on $\text{K}$-theory tensored with $\mathbb{C}$, as in the classical case.) We then examine endomorphisms of Cuntz–Krieger algebras ${{O}_{A}}$. An endomorphism has an invariant, which is a permutation of an infinite set, and the contracting and expanding behavior of this permutation describes the Lefschetz number of the endomorphism. Using this description, we derive a closed polynomial formula for the Lefschetz number depending on the matrix A and the presentation of the endomorphism.

Let $L$ be a finite distributive lattice. Let $\text{Su}{{\text{b}}_{0}}(L)$ be the lattice

$$\{S\,|\,S\,\text{is}\,\text{a sublattice of }L\}\cup \{\phi \}$$

and let ${{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(L)]$ be the length of the shortest maximal chain in $\text{Su}{{\text{b}}_{0}}(L)$. It is proved that if $K$ and $L$ are non-trivial finite distributive lattices, then

$${{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(K\times L)]={{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(K)]+{{\ell }_{*}}[\text{Su}{{\text{b}}_{0}}(L)]$$

A conjecture from the 1984 Banff Conference on Graphs and Order is thus proved.

A class of generalized Marcinkiewicz integral operators is introduced, and, under rather weak conditions on the integral kernels, the boundedness of such operators on ${{L}^{p}}$ and Triebel–Lizorkin spaces is established.

We show that a partially hyperbolic diffeomorphism is plaque expansive (a form of structural stability for its center foliation) if the strong stable and unstable foliations are quasi-isometric in the universal cover. In particular, all partially hyperbolic diffeomorphisms on the 3-torus are plaque expansive.

In this paper, another relationship between the quasi-ideal adequate transversals of an abundant semigroup is given. We introduce the concept of a weakly multiplicative adequate transversal and the classic result that an adequate transversal is multiplicative if and only if it is weakly multiplicative and a quasi-ideal is obtained. Also, we give two equivalent conditions for an adequate transversal to be weakly multiplicative. We then consider the case when $I$ and $\Lambda $ (defined below) are bands. This is analogous to the inverse transversal if the regularity condition is adjoined.