We present a way to turn an arbitrary (unbounded) metric space $\mathcal{M}$ into bounded $\mathcal{B}$ in such that the corresponding Lipschitz-free spaces $\mathcal{F}(\mathcal{M})$ and $\mathcal{F}(\mathcal{B})$ are isomorphic. The construction we provide is functorial weak sense has advantage of being explicit. Apart from its intrinsic theoretical interest, it many applications allows trans...

A compact metric space $(X, \rho)$ is given. Let $\mu$ be a Borel measure on $X$. By $r$-cluster we mean a measurable subset of $X$ with diameter at most $r$. A family of $k$ $2r$-clusters is called a $r$-cluster structure of order $k$ if any two clusters from the family are separated by a distance at least $r$. By measure of a cluster structure we mean a sum of clusters measures from the clust...

Previous computer-assisted problem-solving systems have incorporated all the problem-solving steps within a single stage, making it difficult to diagnose stages at which errors occurred when a student encounters difficulties, and imposing a too-high cognitive load on students in their problem solving. This study proposes a computer-assisted system named MathCAL, whose design is based on four pr...

In the present paper, we consider generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable on a neighborhood $\Omega$ of point $\bar{x}$ in $\mathcal X$, $g:\mathcal at and linear as well F$ set-valued mapping with closed graph acting between two Banach spaces X$ Y$. We study above help extended Newton-type method, introduced [ M. Z. ...

We develop the theory of trace modules up to isomorphism and explore relationship between preenveloping classes property being a module, guided by question whether given module is in preenvelope. As consequence we identify new examples ideals modules, characterize several rings with focus on Gorenstein regular properties.

The Hardy space $H^1$ consists of the integrable functions $f$ on unit circle whose Fourier coefficients $\widehat f(k)$ vanish for $k<0$. We are concerned with that have some additional (finitely many) holes in spectrum, so we fix a finite set $\mathcal K$ positive integers and consider "punctured" $$H^1_{\mathcal K}:=\{f\in H^1:\,\widehat f(k)=0\,\,\,\text{for all }\, k\in\mathcal K\}.$$ then...

In this paper, we consider static self-gravitating spherical spacetime and determine various anisotropic solutions through the extended gravitational decoupling technique in $f(\mathcal{R},\mathcal{T},\mathcal{R}_{\lambda\xi}\mathcal{T}^{\lambda\xi})$ gravity to analyze influence of electromagnetic field on them. We construct two different sets modified equations by employing transformations bo...

The idea of difference sequences of real (or complex) numbers was introducedby Ki zmaz cite{Kizmaz}. In this paper, using the difference operator and alacunary sequence, we introduce and examine the class of sequence $bv_{theta}left(  Delta,mathcal{F}right)  .$ We study some of its properties likesolidity, symmetricity, etc.

We consider cyclic codes $\mathcal{C}_\mathcal{L}$ associated to quadratic trace forms in $m$ variables $Q_R(x) = \operatorname{Tr}_{q^m/q}(xR(x))$ determined by a family $\mathcal{L}$ of $q$-linearized polynomials $R$ over $\mathbb{F}_{q^m}$, and three related $\mathcal{C}_{\mathcal{L},0}$, $\mathcal{C}_{\mathcal{L},1}$ $\mathcal{C}_{\mathcal{L},2}$. describe the spectra for all these when is ...