Partial isometries and extensions of $${{\rm A}{\mathbb T}}$$-algebras

In this paper, we will define several new isomorphism invariants for C*-algebras by hyponormal partial isometries and discuss the relation between these invariants and K-theory of C*-algebras. This study was in part inspired by the work of H. Lin and H. Su in the context of \({A\mathcal{T}}\)-algebras. An \({{\rm A}\mathcal{T}}\)-algebra often becomes an extension of an \({{\rm A}\mathbb{T}}\)-algebra by an AF-algebra. We show that there is an essential extension of a simple \({{\rm A}\mathbb{T}}\)-algebra which has real rank zero by an AF-algebra such that it has real rank zero and is not an \({A\mathcal{T}}\)-algebra.     

On nonlinear integral equations in the space $${{\rm \Phi}BV (I)}$$

We present the existence and uniqueness of global and local \({{\rm \Phi}}\)-bounded variation (\({{\rm \Phi}BV}\)) solutions as well as continuous \({{\rm \Phi}BV}\)-solutions of nonlinear Hammerstein and Volterra–Hammerstein integral equations formulated in terms of the Lebesgue integral.     

On a variational problem involving critical Sobolev growth in dimension three

In this paper we consider the following nonlinear problem: \({{-\Delta u=Ku^{5}}}\), u > 0 in \({{\Omega}}\), u =  0 on \({{\partial \Omega}}\), where K > 0 in \({{\Omega}}\), K =  0 on \({{\partial \Omega}}\) and \({{\Omega}}\) is a bounded domain of \({{\mathbb{R}^{3}}}\). We prove a version of a Morse lemma at infinity for this problem, which allows us to describe the critical points at infinity of the associated variational functional. Using a topological argument, we prove an existence result.     

Magnetic Trajectories in an Almost Contact Metric Manifold $${\mathbb{R}^{2N+1}}$$

In this paper we classify magnetic trajectories γ in \({{\mathbb{R}}^{2N+1}}\) endowed with a canonical quasi-Sasakian structure, corresponding to a magnetic field proportional to the fundamental 2-form. We prove that they are helices of order 5 and we show that there exists a totally geodesic \({{\mathbb{R}}^5}\) in \({\mathbb{R}^{2N+1}}\) such that γ lies in \({{\mathbb{R}}^5}\). Moreover, the quasi-Sasakian structure of \({{\mathbb{R}}^5}\) is that induced from the ambient manifold.     

A Global Approximation Result by Bert Alan Taylor and the Strong Openness Conjecture in $${\mathbb {C}}^n$$

We improve a global approximation result by Bert Alan Taylor in \({\mathbb {C}}^n\) for holomorphic functions in weighted Hilbert spaces. The main tools are a variation of the theorem of Hörmander on weighted \(L^2\)-estimates for the \({\overline{\partial }}\)-equation together with the solution of the strong openness conjecture. A counterexample to a global strong openness conjecture in \({\mathbb {C}}^n \) is also given here.     

A partition relation for pairs on $$\omega ^{\omega ^\omega }$$

We consider colorings of the pairs of a family \(\mathcal {F}\subseteq {{\mathrm{FIN}}}\) of topological type \(\omega ^{\omega ^k}\), for \(k>1\); and we find a homogeneous family \(\mathcal {G}\subseteq \mathcal {F}\) for each coloring. As a consequence, we complete our study of the partition relation \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,m}}\) identifying \(\omega ^{\omega ^\omega }\) as the smallest ordinal space \(\alpha <\omega _1\) satisfying \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,4}}\).     

Uncountable critical points for congruence lattices

The critical point between two classes \({{\mathcal K}}\) and \({{\mathcal L}}\) of algebras is the cardinality of the smallest semilattice isomorphic to the semilattice of compact congruences of some algebra in \({{\mathcal K}}\), but not in \({{\mathcal L}}\). Our paper is devoted to the problem of determining the critical point between two finitely generated congruence-distributive varieties. For a homomorphism \({\varphi: S \rightarrow T}\) of \({(0, \vee)}\)-semilattices and an automorphism \({\tau}\) of T, we introduce the concept of a \({\tau}\)-symmetric lifting of \({\varphi}\). We use it to prove a criterion which ensures that the critical point between two finitely generated congruence-distributive varieties is less or equal to \({\aleph_{1}}\). We illustrate the criterion by constructing two new examples with the critical point exactly \({\aleph_{1}}\).     

$${\varvec{\mathcal {}}}$$-minimaxness and local cohomology modules

Let R be a commutative Noetherian ring, \({\mathfrak {a}}\) an ideal of R, M a finitely generated R-module, and \({\mathcal {S}}\) a Serre subcategory of the category of R-modules. We introduce the concept of \({\mathcal {S}}\)-minimax R-modules and the notion of the \({\mathcal {S}}\)-finiteness dimension

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M):=\inf \lbrace f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{\mathfrak {p}}) \vert \mathfrak {p}\in {\text {Supp}}_R(M/ \mathfrak {a}M) \text { and } R/\mathfrak {p}\notin {\mathcal {S}} \rbrace \end{aligned}$$

and we will prove that: (i) If \({\text {H}}_{\mathfrak {a}}^{0}(M), \cdots ,{\text {H}}_{\mathfrak {a}}^{n-1}(M)\) are \({\mathcal {S}}\)-minimax, then the set \(\lbrace \mathfrak {p}\in {\text {Ass}}_R( {\text {H}}_{\mathfrak {a}}^{n}(M)) \vert R/\mathfrak {p}\notin {\mathcal {S}}\rbrace \) is finite. This generalizes the main results of Brodmann–Lashgari (Proc Am Math Soc 128(10):2851–2853, 2000), Quy (Proc Am Math Soc 138:1965–1968, 2010), Bahmanpour–Naghipour (Proc Math Soc 136:2359–2363, 2008), Asadollahi–Naghipour (Commun Algebra 43:953–958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860–4872, 2015). (ii) If \({\mathcal {S}}\) satisfies the condition \(C_{\mathfrak {a}}\), then

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M)= \inf \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\hbox {-}minimax\rbrace . \end{aligned}$$

This is a formulation of Faltings’ Local-global principle for the \({\mathcal {S}}\)-minimax local cohomology modules. (iii) \( \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\text {-minimax} \rbrace = \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not in } {\mathcal {S}} \rbrace \).

     

Fourth-order Schrödinger type operator with singular potentials

We prove that a deformation of a hypersurface in an (n + 1)-dimensional real space form \({{\mathbb S}^{n+1}_{p,1}}\) induces a Hamiltonian variation of the normal congruence in the space \({{\mathbb L}({\mathbb S}^{n+1}_{p,1})}\) of oriented geodesics. As an application, we show that every Hamiltonian minimal submanifold in \({{\mathbb L}({\mathbb S}^{n+1})}\) (resp. \({{\mathbb L}({\mathbb H}^{n+1})}\)) with respect to the (para-)Kähler Einstein structure is locally the normal congruence of a hypersurface \({\Sigma}\) in \({{\mathbb S}^{n+1}}\) (resp. \({{\mathbb H}^{n+1}}\)) that is a critical point of the functional \({{\mathcal W}(\Sigma) = \int_\Sigma\left(\Pi_{i=1}^n|\epsilon+k_i^2|\right)^{1/2}}\), where k_i denote the principal curvatures of \({\Sigma}\) and \({\epsilon \in \{-1, 1\}}\). In addition, for \({n = 2}\), we prove that every Hamiltonian minimal surface in \({{\mathbb L}({\mathbb S}^{3})}\) (resp. \({{\mathbb L}({\mathbb H}^{3})}\)), with respect to the (para-)Kähler conformally flat structure, is the normal congruence of a surface in \({{\mathbb S}^{3}}\) (resp. \({{\mathbb H}^{3}}\)) that is a critical point of the functional \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K+1}}\) (resp. \({{\mathcal W}\prime(\Sigma) = \int_\Sigma\sqrt{H^2-K-1}}\)), where H and K denote, respectively, the mean and Gaussian curvature of \({\Sigma}\).     

Total Rainbow Connection Number and Complementary Graph

A vertex-colored graph G is rainbow vertex connected if any two distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex connection number of G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertex connected. In this paper, we prove that for a connected graph G, if \({{\rm diam}(\overline{G}) \geq 3}\), then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. Next, we obtain that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 2}\), if G is connected, then \({{\rm rvc}(G) \leq 2}\), and this bound is tight. A total-colored path is total rainbow if its edges and internal vertices have distinct colors. A total-colored graph G is total rainbow connected if any two distinct vertices are connected by some total rainbow path. The total rainbow connection number of G, denoted by trc(G), is the smallest number of colors required to color the edges and vertices of G in order to make G total rainbow connected. In this paper, we prove that for a triangle-free graph \({\overline{G}}\) with \({{\rm diam}(\overline{G}) = 3}\), if G is connected, then trc\({(G) \leq 5}\), and this bound is tight. Next, a Nordhaus–Gaddum-type result for the total rainbow connection number is provided. We show that if G and \({\overline{G}}\) are both connected, then \({6 \leq {\rm trc} (G) + {\rm trc}(\overline{G}) \leq 4n - 6.}\) Examples are given to show that the lower bound is tight for \({n \geq 7}\) and n = 5. Tight lower bounds are also given for n = 4, 6.     

On the Class of Banach Spaces with James Constant $${\sqrt{}}$$ : Part II

The class of two-dimensional normed spaces with James constant \({\sqrt{2}}\) is studied. It is shown that, if the norm is \({\pi/2}\)-rotation invariant, then its James constant is \({\sqrt{2}}\) if and only if the norm is \({\pi/4}\)-rotation invariant. We also present a characterization of \({\pi/4}\)-rotation invariant norms using some properties of certain convex functions on the unit interval, which allow us to easily construct norms the James constant of which are \({\sqrt{2}}\). Moreover, two important examples are given, which show that neither absoluteness, symmetry nor \({\pi/2}\)-rotation invariance can be a characteristic property of the norms with James constant \({\sqrt{2}}\).     

$${\sigma_2}$$-Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations

Let \({\mathbb{X} \subset \mathbb {R}^n}\) be a bounded Lipschitz domain and consider the energy functional

$${{\mathbb F}_{\sigma_2}}[u; \mathbb{X}] := \int_\mathbb{X} {\mathbf F}(\nabla u) \, dx,$$

over the space of admissible maps

$${{\mathcal {A}_\varphi}(\mathbb{X}) :=\{u \in W^{1,4}(\mathbb{X}, {\mathbb{R}^n}) : {\rm det}\, \nabla u > 0\, {\rm for}\, {\mathcal {L}^n}{\rm -a.e. in}\, \mathbb{X}, u|_{\partial \mathbb{X}} =\varphi \}},$$

where the integrand \({{\mathbf F}\colon \mathbb M_{n\times n}\to \mathbb{R}}\) is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when \({{\mathbf F}(\xi):=\frac{1}{2}\sigma_2(\xi)+\Phi(\det\xi)}\). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of \({\mathbb F_{\sigma_2}}\) and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler–Lagrange equations associated to \({{\mathbb F}_{\sigma_2}}\), we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group \({\mathbf {SO}(n)}\). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one.

     

M-$${\mathcal{}}$$-Injective (Flat) and Strongly $${\mathcal{B}}$$B-Injective (Flat) Modules

Let M be a left R-module, \({\mathcal{A}}\)be a family of some submodules of M and \({\mathcal{B}}\)be a family of some left R-modules. In this article, we introduce and characterize \({\mathcal{A}}\)-coherent, \({P\mathcal{A}}\), \({F\mathcal{A}}\), M-\({\mathcal{A}}\)-injective (flat) and strongly \({\mathcal{B}}\)-injective (flat) modules, which are generalizations of coherent, PS, FS, M-injective (flat) and strongly M-injective modules, respectively. We extend some known results to this general structure.     

The $${\square_{b}}$$ heat equation and multipliers via the wave equation

Recently, Nagel and Stein studied the \({\square_{b}}\) -heat equation, where \({\square_{b}}\) is the Kohn Laplacian on the boundary of a weakly pseudoconvex domain of finite type in \({\mathbb{C}^2}\) . They showed that the Schwartz kernel of \({e^{-t\square_{b}}}\) satisfies good “off-diagonal” estimates, while that of \({e^{-t\square_{b}}-\pi}\) satisfies good “on-diagonal” estimates, where π denotes the Szegö projection. We offer a simple proof of these results, which easily generalizes to other, similar situations. Our methods involve adapting the well-known relationship between the heat equation and the finite propagation speed of the wave equation to this situation. In addition, we apply these methods to study multipliers of the form \({m(\square_{b})}\) . In particular, we show that \({m(\square_{b})}\) is an NIS operator, where m satisfies an appropriate Mihlin–Hörmander condition.     

The Holditch-Type Theorem for the Polar Moment of Inertia of the Orbit Curve in the Generalized Complex Plane

In this study, we first calculate the polar moment of inertia of orbit curves under one-parameter planar motion in the generalized complex plane \({{\mathbb{C}_p}}\) and then give the Holditch-type theorem for \({{\mathbb{C}_p}}\): When the fixed points \({X}\) and \({Y}\) on the moving plane \({{\mathbb{K}_p} \subset {\mathbb{C}_p}}\) trace the same curve \({k}\) with the polar moment of inertia \({{T_X}}\), the different point \({Z}\) on this line segment \({XY}\) traces another curve \({{k_Z}}\) with the polar moment of inertia \({{T_Z}}\) during the one-parameter planar motion in the fixed plane \({{\mathbb{K}'_p} \subset {\mathbb{C}_p}}\). Thus, we obtain that the difference between the polar moments of inertia of these curves \({( {{T_Z} - {T_X}} )}\) depends on the only the \({p}\)-distances of this points and \({p}\)-rotation angle of the motion, \({{T_X} - {T_Z} = {\delta _p}ab.}\)     

Local Maximum Modulus Property for Polyanalytic Functions

Let \(\Omega \subset {\mathbb {C}}\) be an open subset and let \({\mathcal {F}}\) be a space of functions defined on \(\Omega \). \({\mathcal {F}}\) is said to have the local maximum modulus property if: for every \(f\in {\mathcal {F}},p_0\in \Omega ,\) and for every sufficiently small domain \(D\subset \Omega ,\) with \(p_0\in D,\) it holds true that \(\max _{z\in \overline{D}}\left| f(z)\right| = \max _{z\in \Sigma \cup \partial D}\left| f(z)\right| ,\) where \(\Sigma \subset \Omega \) denotes the set of points at which \(\left| f\right| \) attains strict local maximum. This property fails for \({\mathcal {F}}=C^{\infty }.\) We verify it however for the set of complex-valued functions whose real and imaginary parts are real analytic. We show by example that the property cannot be improved upon whenever \({\mathcal {F}}\) is the set of n-analytic functions on \(\Omega \), \(n\ge 2,\) in the sense that locality cannot be removed as a condition and independently \(\Sigma \) cannot be removed from the conclusion.     

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).     

Subseries of $$ \mathrm{\mathcal{I}}$$-convergent series

We say that an ideal \( \mathrm{\mathcal{I}}\) has property (T) if for every \( \mathrm{\mathcal{I}}\)-convergent series \( {\sum}_{n=1}^{\infty }{x}_n \), there exists a set A ∈ \( \mathrm{\mathcal{I}}\) such that ∑_n?∈??\Ax _n converges in the usual sense. The main aim of this paper is to focus on several different classes of ideals, such as summable ideals, F _σ ideals, and matrix summability ideals, and to show that they do not have the mentioned property.     

Rank two aCM bundles on general determinantal quartic surfaces in $${\mathbb {P}^{3}}$$

Let \(F\subseteq {\mathbb {P}^{3}}\) be a smooth determinantal quartic surface which is general in the Nöther–Lefschetz sense. In the present paper we give a complete classification of locally free sheaves \({\mathcal E}\) of rank 2 on F such that \(h^1(F,{\mathcal E}(th))=0\) for \(t\in \mathbb {Z}\).