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In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrodinger equation $$i\partial _{t}u+\Delta u=\lambda |x|^{-\alpha }|u|^{\beta }u$$ in $$H^1$$ . The well-posedness theory in $$H^1$$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $$\beta =(4-2\alpha )/(n-2)$$ . It is still an open problem. The main contribution of this paper is to develop the theory in this case.

In this paper, we study the following nonlinear elliptic systems: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1=\partial _{u_1}F(x,u)&{}\quad x\in {\mathbb {R}}^N,\\ -\Delta u_2+V_2(x)u_2=\partial _{u_2}F(x,u)&{}\quad x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$ - Δ u 1 + V 1 ( x ) u 1 = ∂ u 1 F ( x , u ) x ∈ R N , - Δ u 2 + V 2 ( x ) u 2 = ∂ u 2 F ( x , u ) x ∈ R N , where $$u=(u_1,u_2):{\mathbb {R}}^N\rightarrow {\mathbb {R}}^2$$ u = ( u 1 , u 2 ) : R N → R 2 , F and $$V_i$$ V i are periodic in $$x_1,\ldots ,x_N$$ x 1 , … , x N and $$0\notin \sigma (-\,\Delta +V_i)$$ 0 ∉ σ ( - Δ + V i ) for $$i=1,2$$ i = 1 , 2 , where $$\sigma (-\,\Delta +V_i)$$ σ ( - Δ + V i ) stands for the spectrum of the Schrödinger operator $$-\,\Delta +V_i$$ - Δ + V i . Under some suitable assumptions on F and $$V_i$$ V i , we obtain the existence of infinitely many geometrically distinct solutions. The result presented in this paper generalizes the result in Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009).

Recently, the authors have established $$L^p$$ -boundedness of vector-valued q-variational inequalities for averaging operators which take values in the Banach space satisfying the martingale cotype q property in Hong and Ma (Math Z 286(1–2):89–120, 2017). In this paper, we prove that the martingale cotype q property is also necessary for the vector-valued q-variational inequalities, which was a question left open in the previous paper. Moreover, we also prove that the UMD property and the martingale cotype q property can be characterized in terms of vector valued q-variational inequalities for the Hilbert transform.

Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

The paper contains the identification of the $$L_p$$ L p norms of paraproducts, defined on general measure spaces equipped with a dyadic-like structure. The proof exploits the Bellman function method.

In 1947, M.S. Macphail constructed a series in $$\ell _{1}$$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach space theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky–Rogers theorem asserts that in every infinite-dimensional Banach space E, there exists an unconditionally convergent series $$\sum x^{\left( j\right) }$$ such that $$\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty $$ for all $$\varepsilon >0$$ . Their proof is non-constructive and Macphail’s result for $$E=\ell _{1}$$ provides a constructive proof just for $$\varepsilon \ge 1$$ . In this note, we revisit Macphail’s paper and present two alternative constructions that work for all $$\varepsilon >0.$$

A natural variant of Lehmer’s conjecture that the Ramanujan $$\tau $$ -function never vanishes asks whether, for any given integer $$\alpha $$ , there exist any $$n \in \mathbb {Z}^+$$ such that $$\tau (n) = \alpha $$ . A series of recent papers excludes many integers as possible values of the $$\tau $$ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $$\tau (n)$$ . We synthesize these results and methods to prove that if $$0< \left| \alpha \right| < 100$$ and $$\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$$ , then $$\tau (n) \ne \alpha $$ for all $$n > 1$$ . Moreover, if $$\alpha \in T$$ and $$\tau (n) = \alpha $$ , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that $$\left| \tau (n) \right| > 100$$ for all $$n > 2$$ .

Foxby (Math Scand 2:175–186, 1971–1972) showed that a Cohen-Macaulay module over a Gorenstein local ring has finite projective dimension if and only if its canonical module has finite injective dimension. In this paper, we establish the result given by Foxby in a general setting. As a byproduct, some criteria to detect the Cohen-Macaulay property of a ring are provided in terms of intrinsic properties of certain local cohomology modules. Also, as an application, we show that any Cohen-Macaulay module that has a canonical module with finite injective dimension satisfies the Auslander–Reiten conjecture.

We deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke., 16 pages

Recently, the different types of unbounded convergences (uo, un, uaw, uaw*) in Banach lattices were studied. In this paper, we study the continuous functionals with respect to unbounded convergences. We first characterize the continuity of linear functionals for these convergences. Then we define the corresponding unbounded dual spaces and get their exact form. Based on these results, we discuss order continuity and reflexivity of Banach lattices. Some related results are obtained as well., Comment: 9 pages

The paper contains a study of weighted exponential inequalities for differentially subordinate martingales, under the assumption that the underlying weight satisfies Muckenhoupt’s condition$$A_{\infty }$$A∞. The proof exploits certain functions enjoying appropriate size conditions and concavity. The martingales are adapted, uniformly integrable, and càdlàg - we do not assume any path-continuity restrictions. Because of this generality, we need to handle jump parts of processes which forces us to construct a Bellman function satisfying a stronger condition than local concavity. As a corollary, we will establish some new weighted$$L^p$$Lpestimates for differential subordinates of bounded martingales.

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal $I$ in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound $��(I^2)\ge 9$ for the number of minimal generators of $I^2$ with $��(I)\geq 6$. Recently, Gasanova constructed monomial ideals such that $��(I)>��(I^n)$ for any positive integer $n$. In reference to them, we construct a certain class of monomial ideals such that $��(I)>��(I^2)>\cdots >��(I^n)=(n+1)^2$ for any positive integer $n$, which provides one of the most unexpected behaviors of the function $��(I^k)$. The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of $R/I^n$ descends., 10 pages

We revive the concept of Lambda-freeness of Mlotkowski, which describes a mixture of classical and free independence between algebras of random variables. In particular, we give a description of this in terms of cumulants; this will be instrumental in the subsequent paper [SW] where the quantum symmetries underlying these mixtures of classical and free independences will be considered., Comment: We rewrote and shortened the earlier version. The third version contains mainly the results which are new compared to the paper of Mlotkowski

We investigate the possible values for geometric genera of subvarieties in a smooth projective variety. Values which are not attained are called gaps. For curves on a very general surface in $\mathbb{P}^3$, the initial gap interval was found by Xu (see [7] in References), and the next one in our previous paper (see [4] in References), where also the finiteness of the set of gaps was established and an asymptotic upper bound of this set was found. In the present paper we extend some of these results to smooth projective varieties of arbitrary dimension using a different approach., 9 pages, submitted preprint

In a recent paper, we introduced sine functions on commutative hypergroups. These functions are natural generalizations of those functions on groups which are products of additive and multiplicative homomorphisms. In this paper, we describe sine functions on different types of hypergroups, including polynomial hypergroups, Sturm–Liouville hypergroups, etc. A non-commutative hypergroup is also considered.

The Liouville–von Neumann equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not as an equation for density functions. This setting leads to an extended form of the Schrodinger wave equation governing the motion of a quantum particle. In this paper, we obtain the integrability of the wave propagator arising from the Liouville–von Neumann equation in this setting.

Let $\mathcal{G}$ be the class of all finite groups and consider the function $\psi'':\mathcal{G}\longrightarrow(0,1]$, given by $\psi''(G)=\frac{\psi(G)}{|G|^2}$, where $\psi(G)$ is the sum of element orders of a finite group $G$. In this paper, we show that the image of $\psi''$ is a dense set in $[0, 1]$. Also, we study the injectivity and the surjectivity of $\psi''$., Comment: 7 pages

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).

Let $G$ be a finite group and $\psi(G)=\sum_{g\in{G}}{o(g)}$. There are some results about the relation between $\psi(G)$ and the structure of $G$. For instance, it is proved that if $G$ is a group of order $n$ and $\psi(G)>\dfrac{211}{1617}\psi(C_n)$, then $G$ is solvable. Herzog {\it{et al.}} in [Herzog {\it{et al.}}, Two new criteria for solvability of finite groups, J. Algebra, 2018] put forward the following conjecture: \noindent{\bf Conjecture.} {\it {If $G$ is a non-solvable group of order $n$, then $${\psi(G)}\,{\leq}\,{{\dfrac{211}{1617}}{\psi(C_n)}}$$ with equality if and only if $G=A_5$. In particular, this inequality holds for all non-abelian simple groups.} } In this paper, we prove a modified version of Herzog's Conjecture., Comment: 9 pages

We determine the set of primitive integral solutions to the generalised Fermat equation x^2 + y^3 = z^15. As expected, the only solutions are the trivial ones with xyz = 0 and the non-trivial pair (x,y,z) = (+-3, -2, 1)., The paper is slightly shorter. Specifically, in the notation of the paper, we are now able to carry out Chabauty on the curve C_{3,0}. This allows us to eliminate a lengthy elliptic curve Chabauty computation

This erratum addresses an error in the application of a theorem of Hwang and To in the cited paper.

We consider the Cauchy problem for the nonlinear wave equation $$u_{tt} - \Delta _x u +q(t, x) u + u^3 = 0$$ with smooth potential $$q(t, x) \ge 0$$ having compact support with respect to x. The linear equation without the nonlinear term $$u^3$$ and potential periodic in t may have solutions with exponentially increasing $$H^1(\mathbb {R}^{3}_{x})$$ norm as $$t\rightarrow \infty $$. In Petkov and Tzvetkov (IMRN, https://doi.org/10.1093/imrn/rnz014), it was established that by adding the nonlinear term $$u^3$$, the $$H^1(\mathbb {R}^{3}_{x})$$ norm of the solution is polynomially bounded for every choice of q. In this paper, we show that the $$H^k({{\mathbb {R}}}^3_x)$$ norm of this global solution is also polynomially bounded. To prove this, we apply a different argument based on the analysis of a sequence $$\{Y_k(n\tau _k)\}_{n = 0}^{\infty }$$ with suitably defined energy norm $$Y_k(t)$$ and $$0< \tau _k

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category \(\textsf {D} _{\textsf {sg} }(R)\) of a hypersurface R of countable representation type. For a thick subcategory \({\mathcal {T}}\) of \(\textsf {D} _{\textsf {sg} }(R)\) and a full subcategory \(\mathcal {X}\) of \({\mathcal {T}}\), we calculate the Rouquier dimension of \({\mathcal {T}}\) with respect to \(\mathcal {X}\). Furthermore, we prove that the level in \(\textsf {D} _{\textsf {sg} }(R)\) of the residue field of R with respect to each nonzero object is at most one.

A collection of disjoint subsets ${\cal A}=\{A_1,A_2,\dotsc,A_m\}$ of a finite abelian group is said to have the \emph{bimodal} property if, for any non-zero group element $\delta$, either $\delta$ never occurs as a difference between an element of $A_i$ and an element of some other set $A_j$, or else for every element $a_i$ in $A_i$ there is an element $a_j\in A_j$ for some $j\neq i$ such that $a_i-a_j=\delta$. This property arises in various familiar situations, such as the cosets of a fixed subgroup or in a group partition, and has applications to the construction of optimal algebraic manipulation detection (AMD) codes. In this paper, we obtain a structural characterisation for bimodal collections of sets.

The works of V.A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm–Liouville problems are analytic in potentials, considered as mappings from the Lebesgue space to the space of real numbers and the Banach space of continuous functions respectively. Moreover, the first-order Frechet derivatives are known and play an important role in many problems. In this paper, we will find the second-order Frechet derivatives of eigenvalues in potentials, which are also proved to be negative definite quadratic forms for some cases.

In this paper, we give a nontrivial lower bound for the fundamental unit of norm $$-1$$ of a real quadratic field of class number 1.

The Nekrasov–Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov–Okounkov formula involving hooks with trivial legs. We give a proof of the conjecture. The second conjecture is on properties of the roots of the underlying D’Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We confirm the conjecture up to the polynomial degree 1000.

We study the class Mp of Schur multipliers on the Schatten-von Neumann class Sp with 1 ≤ p≤ ∞ as well as the class of completely bounded Schur multipliers Mpcb. We first show that for 2 ≤ pl q≤ ∞ there exists m∈Mpcb with m∉ Mq, so in particular the following inclusions that follow from interpolation are strict: Mq⊊ Mp and Mqcb⊊Mpcb. In the remainder of the paper we collect computational evidence that for p≠ 1 , 2 , ∞ we have Mp=Mpcb, moreover with equality of bounds and complete bounds. This would suggest that a conjecture raised by Pisier (Asterisque 247:vi+131, 1998) is false.

The paper contains a transference theorem which allows to extend a large class of unweighted inequalities for the dyadic maximal operator to their weighted Fefferman–Stein counterparts on general probability spaces.

The groups having exactly one normalizer are Dedekind groups. All finite groups with exactly two normalizers were classified by Perez-Ramos in 1988. In this paper we prove that every finite group with at most 26 normalizers of $$\{2,3,5\}$$ -subgroups is soluble and we also show that every finite group with at most 21 normalizers of cyclic $$\{2,3,5\}$$ -subgroups is soluble. These confirm Conjecture 3.7 of Zarrin (Bull Aust Math Soc 86:416–423, 2012).

An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a “local” version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014).

In this paper we consider the following quasilinear Schrodinger–Poisson system in a bounded domain in $${\mathbb {R}}^{2}$$ : $$\begin{aligned} \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &{}\ \text{ in } \Omega , \\ -\Delta \phi - \varepsilon ^{4}\Delta _4 \phi = u^{2} &{} \ \text{ in } \Omega ,\\ u=\phi =0 &{} \ \text{ on } \partial \Omega \end{array} \right. \end{aligned}$$ depending on the parameter $$\varepsilon >0$$ . The nonlinearity f is assumed to have critical exponential growth. We first prove existence of nontrivial solutions $$(u_{\varepsilon }, \phi _{\varepsilon })$$ and then we show that as $$\varepsilon \rightarrow 0^{+}$$ , these solutions converge to a nontrivial solution of the associated Schrodinger–Poisson system, that is, by making $$\varepsilon =0$$ in the system above.

For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being σ(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) − 2)k(H) + 1/2^n for all n ∈ N. Consequently, this yields also the following formula k(coH) ≤ (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions., The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.

In this paper, we obtain bounds on the $$L^1$$ -norm of the sum $$\sum _{n\le x}\tau (n)e(\alpha n)$$ where $$\tau (n)$$ is the divisor function.

In this paper we connect algebraic properties of the pair-of-pants product in local Floer homology and Hamiltonian dynamics. We show that for an isolated periodic orbit the product is non-uniformly nilpotent and use this fact to give a simple proof of the Conley conjecture for closed manifolds with aspherical symplectic form. More precisely, we prove that on a closed symplectic manifold the mean action spectrum of a Hamiltonian diffeomorphism with isolated periodic orbits is infinite., Comment: 9 pages, revised version, main results unchanged, Archiv der Mathematik, 2018

Clifford indices for semistable vector bundles on a smooth projective curve of genus at least four were defined in a previous paper of the authors. The present paper studies bundles which compute these Clifford indices. We show that under certain conditions on the curve all such bundles and their Serre duals are generated.

In this paper we consider generalizations of classical results on chains of tangent spheres to higher dimensions., 9 pages, 2 figures

Lin and Xi introduced Auslander–Dlab–Ringel (ADR) algebras of semilocal modules as a generalization of original ADR algebras and showed that they are quasi-hereditary. In this paper, we prove that such algebras are always left-strongly quasi-hereditary. As an application, we give a better upper bound for the global dimension of ADR algebras of semilocal modules. Moreover, we describe characterizations of original ADR algebras to be strongly quasi-hereditary.

In this paper, we show that if I is a matroidal ideal, then the ideal generated by the i-th multigraded shifts is also a matroidal ideal for every $$i=0,\ldots ,{\text {proj dim}}(I)$$ .

Cuckovic and Paudyal recently characterized the lattice of invariant subspaces of the shift plus a complex Volterra operator on the Hilbert space \(H^2\) on the unit disk. Motivated by the idea of Ong, in this paper, we give a complete characterization of the lattice of invariant subspaces of the shift operator plus a positive integer multiple of the Volterra operator on Hardy spaces \(H^p\), which essentially extends their works to the more general cases when \(1\le p

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by $$S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}$$ . In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space $$\mathcal {L}^2[0,2\pi ]$$ is a simple point set $$\{0\}$$ .

For $$\mu \in L^{\infty }(\Delta )$$ , the vector fields on the unit circle determined by $$\mu $$ play an important role in the theory of the universal Teichmuller space. The aim of this paper is to give some characterizations of the vector fields induced by dynamically invariant $$\mu $$ . We show that those vector fields are not contained in the Sobolev class $$H^{3/2}$$ . At last, we give some results on dynamically invariant vectors to show that the vector fields, the quasi-symmetric homeomorphisms, and the quasi-circles are closely related.

The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit examples of real Riemann surfaces which cannot be defined over the field of moduli., In this new version, Eslam Badr has been added as a coauthor and the examples provided in the previous version has been generalised

In this paper we explore the structure of certain generalized isometries of the special orthogonal group SO(n) which are transformations that leave any member of a large class of generalized distance measures invariant.

D e f i n i t i o n . A symmetric with respect to 0 bounded closed convex set A in a finite dimensional normed space X is called a sufficient enlargement for X (or of B(X)) if for arbitrary isometric embedding of X into a Banach space Y there exists a projection \(P:Y\to X\) such that P(B(Y)) \(\subset\) A (by B we denote the unit ball). ¶The notion of sufficient enlargement is implicit in the paper: B. Grunbaum, Projection constants, Trans. Amer. Math. Soc. 95, 451 - 465 (1960). It was explicitly introduced by the author in: M. I. Ostrovskii, Generalization of projection constants: sufficient enlargements, Extracta Math. 11, 466 - 474 (1996). ¶The main purpose of the present paper is to continue investigation of sufficient enlargements started in the papers cited above. In particular the author investigate sufficient enlargements whose support functions are in some directions close to those of the unit ball of the space, sufficient enlargements of minimal volume, sufficient enlargements for euclidean spaces.

The authors would like to correct the errors in the publication of the original article, and one of the authors also wants to update his present affiliation. The present affiliation and corrected details are given below for your reading: Unfortunately, Corollary 2.3 in the original paper was stated incorrectly; it should be noted that all other results in the original paper are correct as stated. The correct statement is as follows. Note that the only difference is the addition of the hypothesis Δy(0) ≥ 0. The exclusion of this hypothesis is the error Corollary 2.3 in the original paper. Corollary 2.3. Let y : N0 → R be a nonnegative function. Fix ν ∈ (1, 2) and suppose that Δ0y(t) ≥ 0 for each t ∈ N2−ν . If, in addition, it holds that Δy(0) ≥ 0, then y is increasing on N0. Due to this change, we should also point out that some of the results in Goodrich [1] must be slightly changed as well, namely if we wish to argue that ΔN−1y(t) ≥ 0 for all t ∈ N0, then we must require ΔN−1y(0) ≥ 0, and this was omitted in certain of the results—see, for example, [1, Theorem 2.6, Example 2.9, Corollaries 2.8, 2.10, and 2.11].

In this paper, we define and, then, we characterize constant angle spacelike and timelike surfaces in the three-dimensional Heisenberg group, equipped with a 1-parameter family of Lorentzian metrics. In particular, we give an explicit local parametrization of these surfaces and we produce some examples., 13 pages, 8 figures

In this paper we show that an immersed nontrivial translating soliton for a mean curvature flow in $$\mathbb {R}^{n+1}$$ ( $$n=2,3)$$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $$\varSigma $$ with nonnegative scalar curvature, we prove that if the mean curvature of $$\varSigma $$ does not change signs on each end, then $$\varSigma $$ must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.