We obtain a complex interpolation theorem between weighted Calderón-Hardy spaces for weights in a Sawyer class. The technique used is based on the method obtained by J.-O. Strömberg and A. Torchinsky; however, we must overcome several technical difficulties associated with considering one-sided Calderón-Hardy spaces. Interpolation results of this type are useful in the study of weighted weak type inequalities of strongly singular integral operators.

Let $S(H)$ be the unit sphere of a Hilbert space $H$ and $U_p(H)$ the group of unitary operators in $H$ such that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. This group acts smoothly and transitively in $S(H)$ and endows it with a natural Finsler metric induced by the $p$-norm $\Vert z \Vert_p= \operatorname{tr}\left((z z^*)^{p/2}\right)^{1/p}$. This metric is given by \[ \Vert v \Vert_{x,p} = \min \lbrace \Vert z-y \Vert_p: y \in \mathfrak{g} _x \rbrace, \] where $z \in \mathcal{B} _p(H)_ {ah}$ satisfies that $(d \pi_x)_1(z)=z \cdot x = v$ and $ \mathfrak{g} _x$ denotes the Lie algebra of the subgroup of unitaries which fix $x$. We call $z$ a lifting of $v$. A lifting $z_0$ is called a minimal lifting if additionally $ \Vert v \Vert_{x,p} = \Vert z_0 \Vert_p$. In this paper we show properties of minimal liftings and we treat the problem of finding short curves $ \alpha$ such that $ \alpha(0)=x $ and $ \dot{\alpha} (0)= v$ with $x \in S(H)$ and $v \in T_xS(H)$ given. Also we consider the problem of finding short curves which join two given endpoints $x,y \in S(H)$.

We introduce a new structure, called affine Szabó connection. We prove that, on $2$-dimensional affine manifolds, the affine Szabó structure is equivalent to one of the cyclic parallelisms of the Ricci tensor. A characterization for locally homogeneous affine Szabó surfaces is obtained. Examples of two- and three-dimensional affine Szabó manifolds are also given.

Let $M\subset \mathbb{S}^{n+1}\subset $ $\mathbb{R}^{n+2}$ be a homogeneous isoparametric hypersurface and consider the algebraic set of unit tangent vectors generating planar normal sections at a point $E\in M$ (denoted by $\widehat{X}_{E}[M] \subset T_{E}(M)$). The present paper is devoted to prove that $\widehat{X}_{E}[M]$ is connected by arcs. This in turn proves that its projective image $X[M] \subset \mathbb{RP}(T_{E}(M))$ also has this property.

Let $(R,m)$ be a commutative Noetherian regular local ring and $I$ be a proper ideal of $R$. It is shown that $H^ {d-1} _ \mathfrak{p} (R)=0$ for any prime ideal $ \mathfrak{p} $ of $R$ with $ \dim(R/ \mathfrak{p})=2$, whenever the set $ \{ n \in \mathbb{N} : R/ \mathfrak{p} ^ {(n)}$ is Cohen-Macaulay$\}$ is infinite. Now, let $(R,m)$ be a commutative Noetherian unique factorization local domain of dimension $d$, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. It is shown that the Bass numbers of the $R$-module $H^i_I(M)$ are finite, for all integers $i \geq 0$, whenever $ \operatorname{height} (I)=1$ or $d \leq 3$.

The notions of soft left and soft right ideals, soft quasi-ideal and soft bi-ideal in ordered semigroups are introduced. We show here that in ordered groupoids the soft right and soft left ideals are soft quasi-ideals, and in ordered semigroups the soft quasi-ideals are soft bi-ideals. Moreover, we prove that in regular ordered semigroups the soft quasi-ideals and the soft bi-ideals coincide. We finally show that in an ordered semigroup the soft quasi-ideals are just intersections of soft right and soft left ideals.

An $n$-dimensional Lie algebra $ \mathfrak{g} $ over a field $ \mathbb{F} $ of characteristic two is said to be of Vergne type if there is a basis $e_1, \dots,e_n$ such that $ [e_1,e_i] =e_ {i+1} $ for all $2 \leq i \leq[e_i,e_j] n-1$ and $ = c_ {i,j} e_ {i+j} $ for some $c_ {i,j} \in \mathbb{F} $ for all $i,j \ge 2$ with $i+j \le n$. We define the algebra $ \mathfrak{m} _0$ by its nontrivial bracket relations: $ [e_1,e_i] =e_ {i+1} $, $2 \leq i \leq n-1$, and the algebra $ \mathfrak{m} _2$: $ [e_1,e_i ] =e_ {i+1} $, $2 \le i \le[e_2, e_j ] n-1$, $ =e_ {j+2} $, $3 \le j \le n-2$.

We show that, in contrast to the corresponding real and complex cases, $ \mathfrak{m} _0(n)$ and $ \mathfrak{m} _2(n)$ have the same Betti numbers. We also prove that for any Lie algebra of Vergne type of dimension at least $5$, there exists a non-isomorphic algebra of Vergne type with the same Betti numbers.

The object of the present paper is to characterize three-dimensional trans-Sasakian generalized Sasakian space forms admitting biharmonic almost contact curves with respect to generalized Tanaka Webster Okumura (gTWO) connections and to give illustrative examples. The mean curvature vector of almost contact curves has been analyzed on trans-Sasakian manifolds with gTWO connections. Some properties of slant curves on the same manifolds have been established. Finally curvature and torsion, with respect to gTWO connections, of C-parallel and C-proper slant curves in three-dimensional almost contact metric manifolds have been deduced.

It is shown that an element $a$ in a ring is Drazin invertible if and only if so is $a^ {n} $; the Drazin inverse of $a$ is given by that of $a^ {n} $, and vice versa. Using this result, we prove that, in the presence of $aba=aca$, for any natural numbers $n$ and $m$, $(ac)^ {n} $ is Drazin invertible in a ring if and only if so is $(ba)^ {m} $; the Drazin inverse of $(ac)^ {n} $ is expressed by that of $(ba)^ {m} $, and vice versa. Also, analogous results for the pseudo Drazin inverse and the generalized Drazin inverse are established on Banach algebras.

We consider a new differential-difference operator $\Lambda$ on the real line. We study the harmonic analysis associated with this operator. Next, we establish the Paley-Wiener theorems for its Hartley transform on $\mathbb{R}$.

In 1965, Adler, Konheim and McAndrew introduced the topological entropy of a given dynamical system, which consists of a real number that explains part of the complexity of the dynamics of the system. In this context, a good question could be if the topological entropy $H_{\mathrm{top}} (f)$ changes continuously with $f$. For continuous maps this problem was studied by Misisurewicz, Slenk and Urbański. Recently, and related with the lexicographic and the Milnor-Thurston worlds, this problem was studied by Labarca and others. In this paper we will prove, by elementary methods, the continuity of the topological entropy in a maximal periodic orbit ($ \theta= \underline{1001} $) in the Milnor-Thurston world. Moreover, by using dynamical methods, we obtain interesting relations and results concerning the largest eigenvalue of a sequence of square matrices whose lengths grow up to infinity.

Two geometric inequalities are established for Einstein totally real submanifolds in a complex space form. As immediate applications of these inequalities, some non-existence results are obtained.

Srivastava and Gupta proposed in 2003 a general family of linear positive operators which include several well known operators as its special cases and investigated the rate of convergence of these operators for functions of bounded variation by using the decomposition techniques. Subsequently, researchers proposed several modifications of these operators and studied their various approximation properties. Yadav, in 2014, proposed a modification of these operators and studied a Voronovskaya-type approximation theorem and statistical convergence. In this paper, we introduce the Bézier variant of the operators defined by Yadav and give a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. Furthermore, we show the comparisons of the rate of convergence of the Srivastava–Gupta operators vis-à-vis its Bézier variant to a certain function by illustrative graphics using Maple algorithms.

We show that metallic shaped hypersurfaces in Lorentzian space forms are isoparametric and obtain their full classification.

We establish a link between the values of a frieze of type $\mathbb{D}_{n}$ and some values of a particular frieze of type $\mathbb{A}_{2n-1}$. This link allows us to compute, independently of each other, all the cluster variables in the cluster algebra associated with a quiver $Q$ of type $\mathbb{D}_{n}$.

Given a reducible Galois representation $\overline{\rho}: G_{\mathbb{Q}} \rightarrow GL_2( \mathbb{F}_q)$ we show that there exists an irreducible deformation $\rho : G_{\mathbb{Q}} \rightarrow GL_2 (\mathbb{W} [[T_1, T_2,\dots, T_r,\dots]])$ of $\overline{\rho}$ ramified at infinitely many primes, where $\mathbb{W}$ denotes the ring of Witt vectors of $\mathbb{F}_q$. This is a modification of Ramakrishna's result for the irreducible case.

We introduce a new dyadic one-sided maximal operator $M_d^ {+\dots +} $ in $ \mathbb{R} ^n$ that allows us to obtain good weights for the $L^p$-boundedness of a one-sided maximal operator $N^ {+\dots +} $ in $ \mathbb{R} ^n$, which is equivalent to the classical one-sided Hardy–Littlewood maximal operator in the case $n=1$, but not in the case $n > 1$. In order to do this, we characterize the good pairs of weights for the weak and strong type inequalities for $M_d^ {+\dots +} $ and we use a Fefferman–Stein type inequality which gives that, in a certain sense, $M_d^ {+\dots +} $ controls $N^ {+\dots +} $.

We consider $2m^ \text{th} $ order elliptic equations with Hölder coefficients in half space. We solve the Dirichlet problem with $m$ conditions on the boundary of the upper half space. We first analyze the constant coefficient case, finding the Green function and a representation formula, and then prove Schauder estimates.

We study the boundedness on the variable Lebesgue spaces of the operators of variable order \[ T_ {\Omega} f(x) = \int_{\Omega} |y-A_ {1} x|^ {-\alpha_{1} (y)} \dots{m} |y-A_ x|^ {-\alpha_{m} (y)} f(y) \, dy \] and \[ U_ {\Omega} f(x) = \int_{\Omega} |y-A_ {1} x|^ {-\alpha_{1} (x)} \dots{m} |y-A_ x|^ {-\alpha_{m} (x)} f(y) \, dy, \] where $ \alpha_{i} : \Omega \subset \mathbb{R} ^ {n} \rightarrow(0,n)$ are positive measurable functions, for $i=1, \dots, m$ $(m \geq2)$, such that $ \alpha_{1} (x) + \dots + \alpha_{m} (x) = n - \alpha(x)$ with $0 \leq \alpha(x) < n$ for all $x \in \Omega$, and the $A_{i}$’s are $n \times n$ invertible real matrices such that $A_ {i} - A_ {j} $ is invertible if $i \neq j$.

Let $L$ be the one-dimensional Schrödinger operator defined by $Ly=-y''+qy$. We investigate the existence of a solution to the initial value problem for the differential equation $(L- \lambda)y=g$, when $q$ and $g$ are Henstock–Kurzweil integrable functions on $ [a,b] $. Results presented in this article are generalizations of classical results for the Lebesgue integral.

Let $M^3$ be a three-dimensional almost co-Kähler manifold whose Reeb vector field is harmonic. We obtain some local classification results of $M^3$ under some additional conditions related to the Ricci tensor.

Let $ \mathcal{A} $ be a $C^*$-algebra with a faithful state $ \varphi$. It is proved that the projective unitary group $ \mathbb{P} \, \mathcal{U} _ \mathcal{A} $ of $ \mathcal{A} $, \[ \mathbb{P} \, \mathcal{U} _ \mathcal{A} = \mathcal{U} _ \mathcal{A} / \mathbb{T} .1, \] ($ \mathcal{U} _ \mathcal{A} $ denotes the unitary group of $ \mathcal{A} $) is a $C^ \infty$-submanifold of the Banach space $ \mathcal{B} _s( \mathcal{A})$ of bounded operators acting in $ \mathcal{A} $, which are symmetric for the $ \varphi$-inner product, and are usually called symmetrizable linear operators in $ \mathcal{A} $.

A quotient Finsler metric is introduced in $ \mathbb{P} \, \mathcal{U} _ \mathcal{A} $, following the theory of homogeneous spaces of the unitary group of a $C^*$-algebra. Curves of minimal length with any given initial conditions are exhibited. Also it is proved that if $ \mathcal{A} $ is a von Neumann algebra (or more generally, an algebra where the unitary group is exponential) two elements in $ \mathbb{P} \, \mathcal{U} _ \mathcal{A} $ can be joined by a minimal curve.

In the case when $ \mathcal{A} $ is a von Neumann algebra with a finite trace, these minimality results hold for the quotient of the metric induced by the $p$-norm of the trace ($p \ge 2$), which metrizes the strong operator topology of $ \mathbb{P} \, \mathcal{U} _ \mathcal{A} $.

We introduce two classes of functions, one containing the class of $L^2$ differentiable functions, and another containing the class of $L^2$ lateral differentiable functions. For functions in these new classes we prove existence of best local approximation at several points. Moreover, we get results about the asymptotic behavior of the derivatives of the net of best approximations, which are unknown even for $L^2$ differentiable functions.