This paper gives an explicit formula for the $\mathbb Z_2$-cup-length of the rotation group $\mathrm{SO}(n)$.

This paper deals with the singularly perturbed nonlocal boundary value problem for a linear first order differential equation. For the numerical solution of this problem, we use a fitted difference scheme on a piecewise uniform Shishkin mesh. An error analysis shows that the method is almost first order convergent, in the discrete maximum norm, independently of the perturbation parameter. Numerical results are presented which illustrate the theoretical results.

In the present article we prove direct, simultaneous and converse approximation theorems by trigonometric polynomials for functions $f$ and $ \left( \psi ,\beta \right) $-derivatives of $f$ in weighted Lorentz spaces.

We obtain a characterization of the uniform exponential stability for the continuous-time skew-product three-parameter semiflows in Banach spaces, using a discrete-time approach. Our technique is based on the classical "test-function" method of O. Perron and Ta Li.

A Banach space operator $A\in B({\cal X})$ is polaroid, $A\in(\cal P)$, if the isolated points of the spectrum $\sigma(A)$ are poles of the operator; $A$ is hereditarily polaroid, $A\in(\cal {HP})$, if every restriction of $A$ to a closed invariant subspace is polaroid. It is seen that operators $A\in(\cal {HP})$ have SVEP - the single-valued extension property - on $\Phi_{sf}(A)=\{\lambda: A-\lambda$ is semi Fredholm $\}$. Hence $\Phi^+_{sf}(A)=\{\lambda\in\Phi_{sf}(A), \ind(A-\lambda)>0\}=\emptyset$ for operators $A\in(\cal {HP})$, and a necessary and sufficient condition for the perturbation $A+K$ of an operator $A\in B({\cal X})$ by a compact operator $K\inB({\cal X})$ to be hereditarily polaroid is that $\Phi_{sf}^+(A)=\emptyset$. A sufficient condition for $A\in B({\cal X})$ to have SVEP on $\Phi_{sf}(A)$ is that its component $\Omega_a(A)=\{\lambda\in\Phi_{sf}(A): \ind(A-\lambda)\leq 0\}$ is connected. We prove: If $A\in B({\cal H})$ is a Hilbert space operator, then a necessary and sufficient condition for there to exist a compact operator $K\in B({\cal H})$ such that $A+K\in(\cal {HP})$ is that $\Omega_a(A)$ is connected.

In this paper, we have generalized the Wijsman statistical convergence of closed sets in metric space by introducing the $f$-Wijsman statistical convergence of these sets, where $f$ is an unbounded modulus. It is shown that the Wijsman convergent sequences are precisely those sequences which are $f$-Wijsman statistically convergent for every unbounded modulus $f$. We have also introduced a new concept of Wijsman strong Cesàro summability with respect to a modulus $f$, and investigate the relationship between the $f$-Wijsman statistically convergent sequences and the Wijsman strongly Cesàro summable sequences with respect to $f$.

We extend the Gromov geometric definition of CAT(0) spaces to the case where the comparison triangles are not in the Euclidean plane but belong to a general Banach space. In particular, we study the case where the Banach space is $\ell_p$, for $p > 2$.

Let $\mathcal{A}(p)$ be the class consisting of functions $f$ that are holomorphic in $\mathbb D\setminus \{p\}$, $p\in (0,1)$ possessing a simple pole at the point $z=p$ with nonzero residue and normalized by the condition $f(0)=0=f'(0)-1$. In this article, we first prove a sufficient condition for univalency for functions in $\mathcal{A}(p)$. Thereafter, we consider the class denoted by $\Sigma(p)$ that consists of functions $f \in \mathcal{A}(p)$ that are univalent in $\mathbb D$. We obtain the exact value for $\displaystyle\max_ {f\in \Sigma(p)}\Delta(r,z/f)$, where the Dirichlet integral $\Delta(r,z/f)$ is given by $$ \Delta(r,z/f)=\displaystyle\iint_{|z|<r} |\left(z/f(z)\right)'|^2 \,dx\, dy, \quad(z=x+iy),~0<r\leq 1. $$ We also obtain a sharp estimate for $\Delta(r,z/f)$ whenever $f$ belongs to certain subclasses of $\Sigma(p)$. Furthermore, we obtain sharp estimates of the integral means for the aforementioned classes of functions.

We study the property of asymptotic midpoint uniform convexity for infinite direct sums of Banach spaces, where the norm of the sum is defined by a Banach space $E$ with a 1-unconditional basis. We show that a sum $(\sum_{n=1}^\infty X_n)_E$ is asymptotically midpoint uniformly convex (AMUC) if and only if the spaces $X_n$ are uniformly AMUC and $E$ is uniformly monotone. We also show that $L_p(X)$ is AMUC if and only if $X$ is uniformly convex.

Let $X$ be a hyperkähler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology. We study this conjecture for the Fano variety of lines on a very general cubic fourfold.

In this paper, the nonlinear singular Thomas-Fermi differential equation on a semi-infinite domain for neutral atoms is solved by using the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind. First, this collocation method reduces the solution of this problem to the solution of a system of nonlinear algebraic equations. Second, using solve a system of nonlinear equations, the initial value for the unknown parameter $L$ is calculated, and finally, the value of $L$ to increase the accuracy of the initial slope is improved and the value of $y'(0)=-1.588071022611375312718684509$ is calculated. The comparison with some numerical solutions shows that the present solution is highly accurate.