Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

Clusters of varieties of completely regular semigroups

The class \({\mathcal{CR}}\) of completely regular semigroups equipped with the unary operation of inversion forms a variety whose lattice of subvarieties is denoted by \({\mathcal{L(CR)}}\). The variety \({\mathcal B}\) of all bands induces two relations \({\mathbf{B}^{\land}}\) and \({\mathbf{B}^{\lor} }\) by meet and join with \({\mathcal B}\). Their classes are intervals with lower ends \({\mathcal V_{B^{\land}}}\) and \({\mathcal V_{B^{\lor}}}\), and upper ends \({\mathcal V^{B^{\land}}}\) and \({\mathcal V^{B^{\lor}}}\). These objects induce four operators on \({\mathcal{L(CR)}}\).The cluster at a variety \({\mathcal V}\) is the set of all varieties obtained from \({\mathcal V}\) by repeated application of these four operators. We identify the cluster at any variety in \({\mathcal{L(CR)}}\).     

A Generalization of Stenger’s Lemma to Maximal Dissipative Operators

It is shown that for any maximal dissipative operator A in some Hilbert space \({\mathcal H}\) , which is the orthogonal sum \({\mathcal H=\mathcal F\oplus \mathcal G}\) of two Hilbert spaces \({\mathcal F,\, \mathcal G}\) with \({{\rm dim}\,\mathcal G < \infty}\) , the compression \({\left. T:=P_\mathcal F\,A\right|_{{\rm dom}\,A\cap\mathcal F}}\) of A to \({\mathcal F}\) is again a maximal dissipative operator in \({\mathcal F}\) .     

Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are \({H(\zeta) = U + U^{-1} + V + \zeta V^{-1}}\) and \({H_{m,n} = U + V + q^{-mn}U^{-m}V^{-n}}\), where \({U}\) and \({V}\) are self-adjoint Weyl operators satisfying \({UV = q^{2}VU}\) with \({q = {\rm e}^{{\rm i}\pi b^{2}}}\), \({b > 0}\) and \({\zeta > 0}\), \({m, n \in \mathbb{N}}\). We prove that \({H(\zeta)}\) and \({H_{m,n}}\) are self-adjoint operators with purely discrete spectrum on \({L^{2}(\mathbb{R})}\). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean \({\sum_{j\ge 1}(\lambda - \lambda_{j})_{+}}\) as \({\lambda \to \infty}\) and prove the Weyl law for the eigenvalue counting function \({N(\lambda)}\) for these operators, which imply that their inverses are of trace class.     

Derivations,Local and 2-Local Derivations on Some Algebras of Operators on Hilbert C*-Modules

For a commutative C*-algebra \({\mathcal {A}}\) with unit e and a Hilbert \({\mathcal {A}}\)-module \({\mathcal {M}}\), denote by End\(_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all bounded \({\mathcal {A}}\)-linear mappings on \({\mathcal {M}}\), and by End\(^*_{{\mathcal {A}}}({\mathcal {M}})\) the algebra of all adjointable mappings on \({\mathcal {M}}\). We prove that if \({\mathcal {M}}\) is full, then each derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is \({\mathcal {A}}\)-linear, continuous, and inner, and each 2-local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) or End\(^{*}_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation. If there exist \(x_0\) in \({\mathcal {M}}\) and \(f_0\) in \({\mathcal {M}}^{'}\), such that \(f_0(x_0)=e\), where \({\mathcal {M}}^{'}\) denotes the set of all bounded \({\mathcal {A}}\)-linear mappings from \({\mathcal {M}}\) to \({\mathcal {A}}\), then each \({\mathcal {A}}\)-linear local derivation on End\(_{{\mathcal {A}}}({\mathcal {M}})\) is a derivation.     

Invariant {\varphi}-means on left introverted subspaces with application to locally compact quantum groups

Given a Banach algebra \({{\mathcal{A}}}\), for a non-zero character \({\varphi}\) on \({{\mathcal{A}}}\), we characterize the existence of \({\varphi}\)-means on a left introverted subspace of \({{\mathcal{A}^{*}}}\) containing \({\varphi}\) in terms of certain derivations from \({{\mathcal{A}}}\) into certain Banach \({{\mathcal{A}}}\)-bimodules. We also adapt and extend a result in (Crann and Neufang, Trans Amer Math Soc 368:495–513, 2016) on locally compact quantum groups to the Banach algebra setting which, in particular, answers a question of Bédos and Tuset, concerning the amenability of locally compact quantum groups.     

On a new generalization of metric spaces

In this paper, we introduce the \({\mathcal {F}}\)-metric space concept, which generalizes the metric space notion. We define a natural topology \(\tau _{{\mathcal {F}}}\) in such spaces and we study their topological properties. Moreover, we establish a new version of the Banach contraction principle in the setting of \({\mathcal {F}}\)-metric spaces. Several examples are presented to illustrate our study.     

A semigroup related to a convex combination of boundary conditions obtained as a result of averaging other semigroups

Let \({\alpha}\) be a bounded linear operator in a Banach space \({\mathbb{X}}\), and let A be a closed operator in this space. Suppose that for \({\Phi_1, \Phi_2}\) mapping D(A) to another Banach space \({\mathbb{Y}}\), \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) are generators of strongly continuous semigroups in \({\mathbb{X}}\). Assume finally that \({A_{|{\rm ker}\, \Phi_\text{a}}}\), where \({\Phi_\text{a} = \Phi_1 \alpha + \Phi_2 \beta}\) and \({\beta = I_\mathbb{X} - \alpha}\), is a generator also. In the case where \({\mathbb{X}}\) is an L ¹-type space, and \({\alpha}\) is an operator of multiplication by a function \({0 \le \alpha \le 1}\), it is tempting to think of the later semigroup as describing dynamics which, while at state x, is subject to the rules of \({A_{|{\rm ker}\, \Phi_1}}\) with probability \({\alpha (x)}\) and is subject to the rules of \({A_{|{\rm ker}\, \Phi_2}}\) with probability \({\beta (x)= 1 - \alpha (x)}\). We provide an approximation (a singular perturbation) of the semigroup generated by \({A_{|{\rm ker}\, \Phi_\text{a}}}\) by semigroups built from those generated by \({A_{|{\rm ker}\, \Phi_1}}\) and \({A_{|{\rm ker}\, \Phi_2}}\) that supports this intuition. This result is motivated by a model of dynamics of Solea solea (Arino et al. in SIAM J Appl Math 60(2):408–436, 1999–2000; Banasiak and Goswami in Discrete Continuous Dyn Syst Ser A 35(2):617–635, 2015; Banasiak et al. in J Evol Equ 11:121–154, 2011, Mediterr J Math 11(2):533–559, 2014; Banasiak and Lachowicz in Methods of small parameter in mathematical biology, Birkhäuser, 2014; Sanchez et al. in J Math Anal Appl 323:680–699, 2006) and is, in a sense, dual to those of Bobrowski (J Evol Equ 7(3):555–565, 2007), Bobrowski and Bogucki (Stud Math 189:287–300, 2008), where semigroups generated by convex combinations of Feller’s generators were studied.     

Number fields in fibers: the geometrically abelian case with rational critical values

Let X be an algebraic curve over \({\mathbb {Q}}\) and \({t\in {\mathbb {Q}}(X)}\) a non-constant rational function such that \({{\mathbb {Q}}(X)\ne {\mathbb {Q}}(t)}\). For every \({ n \in {\mathbb {Z}}}\) pick \({P_ n \in X(\bar{{\mathbb {Q}}})}\) such that \({t(P_n)=n}\). We conjecture that, for large N, among the number fields \({\mathbb {Q}}(P_1), \ldots , {\mathbb {Q}}(P_N)\) there are at least cN distinct. We prove this conjecture in the special case when \(\bar{{\mathbb {Q}}}(X)/\bar{{\mathbb {Q}}}(t)\) is an abelian field extension and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a more famous conjecture of Schinzel.     

Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Spacing properties of the zeros of orthogonal polynomials on Cantor sets via a sequence of polynomial mappings

Let \({\varepsilon}\) be a (small) positive number. A packing of unit balls in \({{\mathbb{E}^{3}}}\) is said to be an \({\varepsilon}\)-quasi-twelve-neighbour packing if no two balls of the packing touch each other but for each unit ball B of the packing there are twelve other balls in the packing with the property that the distance of the centre of each of these twelve balls from the centre of B is smaller than \({2+\varepsilon}\). We construct \({\varepsilon}\)-quasi-twelve-neighbour packings of unit balls in \({{\mathbb{E}^{3}}}\) for arbitrary small positive \({\varepsilon}\) with some surprising properties.     

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let \(\mathcal S\) be an abelian group of automorphisms of a probability space \((X, {\mathcal A}, \mu )\) with a finite system of generators \((A_1, \ldots , A_d).\) Let \(A^{{\underline{\ell }}}\) denote \(A_1^{\ell _1} \ldots A_d^{\ell _d}\), for \({{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).\) If \((Z_k)\) is a random walk on \({\mathbb {Z}}^d\), one can study the asymptotic distribution of the sums \(\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}\) and \(\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f\), for a function f on X. In particular, given a random walk on commuting matrices in \(SL(\rho , {\mathbb {Z}})\) or in \({\mathcal M}^*(\rho , {\mathbb {Z}})\) acting on the torus \({\mathbb {T}}^\rho \), \(\rho \ge 1\), what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on \({\mathbb {T}}^\rho \) after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), \(\mathcal S\) a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

The exact asymptotic behavior of boundary blow-up solutions to infinity Laplacian equations

In this paper, we study the asymptotic behavior of viscosity solutions to boundary blow-up elliptic problem \({\Delta_{\infty}u=b(x)f(u),\, x\in\Omega,\,u|_{\partial\Omega}=+\infty,}\) where \({\Omega}\) is a bounded domain with C²-boundary in \({\mathbb{R}^{N}}\), \({b\in \rm C(\bar{\Omega})}\) is positive in \({\Omega}\), which may be vanishing on the boundary, \({f\in C^{1}([0, \infty))}\) is regularly varying or is rapidly varying at infinity.     

Additive iterative roots of identity and Hamel bases

We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}}\neq\emptyset}\) for every infinite set \({{\mathcal{H}}\subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X^X. Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.     

On Absolute Continuity of the Spectrum of a 3D Periodic Magnetic Dirac Operator

Absolute continuity of the spectrum of a 3D periodic magnetic Dirac operator is proved provided that the magnetic potential A belongs to the space \({H^q_{\mathrm{loc}},q >1 }\) , and the matrix potential \({\widehat V\in L^3_{\mathrm {loc}}}\) is represented in the form \({\widehat V=\widehat V_0+\widehat V_1}\) , where \({\widehat V_0}\) commutes and \({\widehat V_1}\) anticommutes with the Dirac matrices \({\widehat \alpha _j, j = 1, 2, 3}\) .     

Continuity of the first eigenvalue for a family of degenerate eigenvalue problems

For each \({\alpha\in[0,2)}\) we consider the eigenvalue problem \({-{\rm div}(|x|^\alpha \nabla u)=\lambda u}\) in a bounded domain \({\Omega\subset \mathbb{R}^N}\) (\({N\geq 2}\)) with smooth boundary and \({0\in \Omega}\) subject to the homogeneous Dirichlet boundary condition. Denote by \({\lambda_1(\alpha)}\) the first eigenvalue of this problem. Using \({\Gamma}\)-convergence arguments we prove the continuity of the function \({\lambda_1}\) with respect to \({\alpha}\) on the interval \({[0,2)}\).     

Infra-nilmanifolds modeled on the group of uni-triangular matrices

Let \({\mathcal {N}}_m\) be the group of \(m\times m\) upper triangular real matrices with all the diagonal entries 1. Then it is an \((m-1)\)-step nilpotent Lie group, diffeomorphic to \({\mathbb {R}}^{\frac{1}{2} m(m-1)}\). It contains all the integer matrices as a lattice \(\Gamma _m\). The automorphism group of \({\mathcal {N}}_m \ (m\ge 4)\) turns out to be extremely small. In fact, \(\mathrm {Aut}({\mathcal {N}})=\mathcal {I} \rtimes \mathrm {Out}({\mathcal {N}})\), where \(\mathcal {I}\) is a connected, simply connected nilpotent Lie group, and \(\mathrm {Out}({\mathcal {N}})={{\tilde{K}}}={(\mathbb {R}^*)^{m-1}\rtimes \mathbb {Z}_2}\). With a nice left-invariant Riemannian metric on \({\mathcal {N}}\), the isometry group is \(\mathrm {Isom}({\mathcal {N}})= {\mathcal {N}} \rtimes K\), where \(K={(\mathbb {Z}_2)^{m-1}\rtimes \mathbb {Z}_2}\subset {{\tilde{K}}}\) is a maximal compact subgroup of \(\mathrm {Aut}({\mathcal {N}})\). We prove that, for odd \(m\ge 4\), there is no infra-nilmanifold which is essentially covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\). For \(m=2n\ge 4\) (even), there is a unique infra-nilmanifold which is essentially (and doubly) covered by the nilmanifold \(\Gamma _m\backslash {\mathcal {N}}_m\).