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).     

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrödinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n?≥?3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.     

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.     

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.     

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 \({[0,2)}\).     

On regular seven-membered loops in {{\mathbb{R}}^3} with arbitrary join angle

The problem of ring molecules and macromolecules arises in a number of contexts in physical chemistry. Perhaps the simplest example of a seven-membered loop is cycloheptane \({\rm C_{7}\rm H_{14}}\), which is a molecule where the carbon–carbon bonds form a regular seven-membered loop. However, it is possible to envisage much more complicated arrangements of proteins in chains comprising straight rigid sections linked in ways that enforce the same angle at all of the joins. In this paper, we present a coordinate system that reduces the problem to four free variables and three constraints. We then survey the solutions numerically and find that there are families of solutions for all join angles \({\theta}\) between \({\pi/7}\) and \({5\pi/7}\) with fixed planar solutions existing for \({\theta = \pi/7}\), \({3\pi/7}\) and \({5\pi/7}\). The available families of solutions undergo a major reorganisation at the join angle \({\theta = \pi/3}\), where 28 intersecting solutions form a single connected network of configurations.     

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 \({\mathcal{L(CR)}}\).     

W*-rigidity for the von Neumann algebras of products of hyperbolic groups

We show that if \({\Gamma = \Gamma_1\times\dotsb\times \Gamma_n}\) is a product of \({{\rm n} \geq 2}\) non-elementary ICC hyperbolic groups then any discrete group \({\Lambda}\) which is \({W^*}\)-equivalent to \({\Gamma}\) decomposes as a direct product of n ICC groups and does not decompose as a direct product of k ICC groups when \({{\rm n} \not= {\rm k}}\). This gives a group-level strengthening of Ozawa and Popa’s unique prime decomposition theorem by removing all assumptions on the group \({\Lambda}\). This result in combination with Margulis’ normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II₁ factors.     

Waist and trunk of knots

Let K be a knot in the 3-sphere S ³. We define the waist of K as

$waist (K) = \mathop{\rm max}\limits_{F\in\mathcal{F}} \mathop{\rm min}\limits_{D\in\mathcal{D}_{F}} |D \cap K|,$

where \({\mathcal{F}}\) is the set of all closed incompressible surfaces in S ³?K and \({\mathcal{D}_F}\) is the set of all compressing disks for F in S ³. We define the trunk of K as

$trunk(K) = \mathop{\rm min}\limits_{h\in\mathcal{H}} \mathop{\rm max}\limits_{t\in\mathbb{R}} |h^{-1}(t) \cap K|,$

where \({\mathcal{H}}\) is the set of all Morse function \({h : S^3 \to \mathbb{R}}\) with two critical points. We show that

$waist (K) \le \frac{trunk(K)}{3}$

.

     

Hecke algebras for {{\rm GL}_n} over local fields

We study the local Hecke algebra \({\mathcal{H}_{G}(K)}\) for \({G = {\rm GL}_{n}}\) and K a non-archimedean local field of characteristic zero. We show that for \({G = {\rm GL}_{2}}\) and any two such fields K and L, there is a Morita equivalence \({\mathcal{H}_{G}(K) \sim_{M} \mathcal{H}_{G}(L)}\), by using the Bernstein decomposition of the Hecke algebra and determining the intertwining algebras that yield the Bernstein blocks up to Morita equivalence. By contrast, we prove that for \({G = {\rm GL}_{n}}\), there is an algebra isomorphism \({\mathcal{H}_{G}(K) \cong \mathcal{H}_{G}(L)}\) which is an isometry for the induced \({L^1}\)-norm if and only if there is a field isomorphism \({K \cong L}\).     

Asymptotics for Erdős–Solovej Zero Modes in Strong Fields

We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl–Dirac operators on \({\mathbb{R}^3}\). In particular, we are interested in those operators \({\mathcal{D}_B}\) for which the associated magnetic field \({B}\) is given by pulling back a two-form \({\beta}\) from the sphere \({\mathbb{S}^2}\) to \({\mathbb{R}^3}\) using a combination of the Hopf fibration and inverse stereographic projection. If \({\int_{\mathbb{s}^2} \beta \neq 0}\), we show that

$$\sum_{0 \leq t \leq T} {\rm dim Ker} \mathcal{D}{tB}=\frac{T^2}{8\pi^2}\,\Big| \int_{\mathbb{S}^2}\beta\Big|\,\int_{\mathbb{S}^2}|{\beta}| +o(T^2)$$

as \({T\to+\infty}\). The result relies on Erd?s and Solovej’s characterisation of the spectrum of \({\mathcal{D}_{tB}}\) in terms of a family of Dirac operators on \({\mathbb{S}^2}\), together with information about the strong field localisation of the Aharonov–Casher zero modes of the latter.

     

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.     

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem \({\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}\) in \({\mathbb{R}^3}\) related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of \({H({\rm curl};\mathbb{R}^3)}\). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and \({\Gamma}\) the variational problem can be posed in a symmetric subspace of \({H({\rm curl};\mathbb{R}^3)}\). For a defocusing case \({{\rm sup} \Gamma < 0}\) with large negative values of \({\Gamma}\) at infinity we obtain ground states by the direct minimization method. For the focusing case \({{\rm inf} \Gamma > 0}\) the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator \({\nabla \times \nabla \times +V(x)}\). Examples of cylindrically symmetric functions V are provided for which this holds.     

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 \({\widehat \alpha _j, j = 1, 2, 3}\) .     

Linear Maps Preserving Numerical Radius on Nest Algebras

A linear map \({\phi}\) of operator algebras is said to preserve numerical radius (or to be a numerical radius isometry) if \({w(\phi(A))=w(A)}\) for all A in its domain algebra, where w(A) stands for the numerical radius of A. In this paper, we prove that a surjective linear map \({\phi}\) of the nest algebra \({{\rm Alg}\mathcal N}\) onto itself preserves numerical radius if and only if there exist a unitary U and a complex number ξ of modulus one such that \({\phi(A)= \xi UAU^*}\) for all \({A\in{\rm Alg}\mathcal N}\), or there exist a unitary U, a conjugation J and a complex number ξ of modulus one such that \({\phi(A)=\xi UJA^*JU^*}\) for all \({A\in{\rm Alg}\mathcal N}\).     

The trace graph of the matrix ring over a finite commutative ring

Let R be a commutative ring and let \({n >1}\) be an integer. We introduce a simple graph, denoted by \({\Gamma_t(M_n(R))}\), which we call the trace graph of the matrix ring \({M_n(R)}\), such that its vertex set is \({M_n(R)^{\ast}}\) and such that two distinct vertices A and B are joined by an edge if and only if \({{\rm Tr} (AB)=0}\) where \({ {\rm Tr} (AB)}\) denotes the trace of the matrix AB. We prove that \({\Gamma_t(M_n(R))}\) is connected with \({{\rm diam}(\Gamma_{t}(M_{n}(R)))=2}\) and \({{\rm gr} (\Gamma_t(M_n(R)))=3}\). We investigate also the interplay between the ring-theoretic properties of R and the graph-theoretic properties of \({\Gamma_t(M_n(R))}\). Hence, we use the notion of the irregularity index of a graph to characterize rings with exactly one nontrivial ideal.     

Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

Dilating Covariant Representations of a Semigroup Dynamical System Arising from Number Theory

In Cuntz et al. (Math Ann 355(4):1383–1423, 2013. doi:10.1007/s00208-012-0826-9), studied the \({C^*}\)-algebra \({\mathfrak {T}[R]}\) generated by the left-regular representation of the \({ax + b}\)-semigroup of a number ring R on \({\ell^2(R \rtimes R^\times)}\). They were able to describe it as a universal \({C^*}\)-algebra defined by generators and relations, and show that it has an interesting KMS-structure and that it is functorial for injective ring homomorphisms. In this paper we show that \({\mathfrak {T}[R]}\) can be realized as the \({C^*}\)-envelope of the isometric semicrossed product of a certain semigroup dynamical system \({(\mathcal {A}_R, \alpha, R^\times)}\). We do this by proving that a representation of \({\mathcal {A}_R \times_\alpha^{\rm is}R^\times}\) is maximal if it is also a representation of \({\mathfrak {T}[R]}\). We also show how to explicitly dilate any representation of \({\mathcal {A}_R \times_\alpha^{\rm is}R^\times}\) to a representation of \({\mathfrak {T}[R]}\).