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Materials and systems that exhibit persistent spin texture provide a platform for creating robust spin states that can be used in quantum computing, memory storage, and other advanced technological applications. In this paper we show that persistent spin-texture in single-layer phosphorene electrons close to the $\Gamma$ point, subjected to the finite perpendicular electric field, can be achieved by appropriately tuning the extrinsic spin-orbit coupling strength using the tensile strain of about $1.2\%$ in the zigzag direction. This is confirmed by detailed numerical investigations of the effects of strain on the intrinsic and extrinsic spin-orbit coupling, and by the effective spin-orbit Hamiltonian of phosphorene electrons and holes around the $\Gamma$ point, assuming the presence of the perpendicular electric field. Furthermore, the calculated spin relaxation rates due to the Dyakonov-Perel mechanism indicate a giant anisotropy of the in-plane spin, up to $10^5$, which is directly related to the discovered persistent spin texture of phosphorene electrons close to the $\Gamma$ point. %%%%%%%% We also show, that strain can reverse the anisotropy of spin mixing parameter $b^2$ connected to the Elliott-Yafet spin relaxation mechanism which dominates spin relaxation in phosphorene. We find the conditions under which Elliott-Yafet spin lifetime anisotropy can be largely enhanced due to synergy of spin mixing and g-factor anisotropy. %%%%%%%%%% Our results suggest that spin texture in phosphorene can be modulated by strain, enabling its potential usage in the field of spintronics., Comment: 8 pages, 3 figures

The main result of this paper is to show that for any compact space $X$ and any family $\{f_\alpha \colon \alpha< 2^\kappa, f_\alpha \colon X \stackrel{onto}{\longrightarrow} \beta \kappa\}$ of open mappings there exists a point $x \in X$ such that for each $\alpha< 2^\kappa$ either $f_\alpha(x)$ belongs to $\kappa$ or $f_\alpha(x)$ is a weak $P_\kappa$-point in $\kappa^*$, whenever $\kappa$ is regular., Comment: arXiv admin note: text overlap with arXiv:2305.02794

We prove under ZFC that in each extremally disconnected compact space there exists a non-limit point of any countable discrete subset., Comment: arXiv admin note: substantial text overlap with arXiv:2305.04998

The paper deals with two notions: polarized partition relations and product of generalized strong sequences. Strong sequences were introduced by Efimov in 1965 as a usefull tool for proving famous theorems in dyadic spaces, i.e. continuous images of Cantor cube. In this paper we introduce the notion of product of generalized strong sequences and give pure combinatorial proof that existence of product of generalized strong sequences is equivalent to polarized partition relations.

This paper deals with two notions: a polarized partition relations $\left( \begin{array}{c} \alpha \beta \end{array} \right) \to \left( \begin{array}{cc} \gamma & \eta \delta & \lambda \end{array} \right)$ and product of generalized strong sequences. Strong sequences were introduced by Efimov in 1965 as a usefull tool for proving famous theorems in dyadic spaces, i.e. continuous images of Cantor cube. In this paper we introduce the notion of product of generalized strong sequences and give the pure combinatorial proof that $\left( \begin{array}{c} \alpha \beta \end{array} \right) \to \left( \begin{array}{cc} \gamma & \eta \delta & \lambda \end{array} \right)$ is a consequence of the existence of product of generalized strong sequences., Comment: arXiv admin note: substantial text overlap with arXiv:2305.02417

The aim of this paper is to construct ultrafilters without immediate predecessors in the Rudin-Frolik order in $\beta \kappa\setminus \kappa$, where $\kappa$ is a regular cardinal. This generalizes the problem posed by Peter Simon more than 40 years ago., Comment: arXiv admin note: substantial text overlap with arXiv:2304.02143, arXiv:2304.00097, arXiv:2304.01398

The aim of this paper is to prove that for there exist a chain in the Rudin-Frol\'ik order of $\beta\kappa\setminus \kappa$ of length $\mu$ with $\kappa \leqslant \mu \leqslant 2^\kappa$ for regular $\kappa> \omega$ without a lower bound., Comment: arXiv admin note: substantial text overlap with arXiv:2304.00097

The aim of this paper is to construct chains of length $(2^\kappa)^+$ in the sense of Rudin-Frol\'ik order in $\beta \kappa$ for $\kappa$ regular.

We show that there is a set of $2^{2^{\kappa}}$ ultrafilters incomparable in Rudin-Frol\'ik order of $\beta \kappa \setminus \kappa$, where $\kappa$ is regular, for which no subset with more than one element has an infimum., Comment: arXiv admin note: substantial text overlap with arXiv:2304.00097, arXiv:2304.01398

The aim of this paper is to provide the results that answer the Kuratowski problem posed in 1935 concerning the existence of nonmeasurable sets. The Kuratowski problem was considered for partitions, here we provide a generalization to point-finite families for tree structures and structures in Ellentuck space. The main result of this paper generalizes previous results in the topic of nonmeasurable sets., Comment: arXiv admin note: text overlap with arXiv:2012.10753

In this paper we show that the existence of Kuratowski partitions of Hausdorff Baire space is equivalent to the existence of point-finite covers of the same space.

We prove that the existence of a complete metric space of cardinality at most $2^{\kappa}$ admitting Kuratowski partition is a consequence of $\kappa$ being the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$.

In this paper, several generalizations of the classical Halpern-L\"{a}uchli Theorem are proven for Marczewski and Ellentuck structures using only combinatorial methods.

The immunity of Ising superconductors to external magnetic fields originates from a spin locking of the paired electrons to an intrinsic Zeeman-like field. The spin-momentum locking in non-centrosymmetric crystalline materials leads to type-I Ising pairing in which the direction of the intrinsic field can be deduced from the spin expectation values. Conversely, in centrosymmetric crystals the electron spins locked to the orbitals can form Ising type-II pairs consisting of spin-orbit split doublets. Due to time-reversal symmetry, the doublets are spin degenerate, making it difficult to read the spin polarization of bands and the direction of spin-orbit fields. Here we present an efficient approach to determine the direction of the intrinsic field using the spin-mixing parameter $b^2$. Using first principles calculations based on the density functional theory, we study monolayer transition metal dichalcogenide superconductors PdTe$_2$, NbTe$_2$, and TiSe$_2$ with the 1T structure. We calculate $b^2$ for individual Fermi pockets and provide a general picture of possible Ising type-II pairing within the full Brillouin zone. In order to complement our first principles results, we use group theory to provide a detailed picture of spin-orbit coupling and spin mixing in the relevant bands forming Fermi pockets. We demonstrate that contrary to the anticipated effects of spin-orbit locking, not every spin-orbit split spin doublet actively participates in Ising pairing. Finally, by connecting the spin-mixing parameter $b^2$ with the intrinsic out-of-plane Zeeman field we estimate the upper in-plane critical magnetic field.

A classical theorem from measure theory that gives a sufficient condition for a multifunction to have a measurable selection is Kuratowski and Ryll-Nardzewski Selection Theorem. The aim of this paper is to show some generalizations of this result.

It is proved that $P(\omega)/\triangle_d$, where $\triangle_d$ is the ideal of sets of asymptotic density zero, is universal in the sense of embeddings.

In this paper we present the generalizations of results, given in the paper published in Georgian J. Math. 26(2019) no. 4, pp 591-598, towards point-finite families.

In this note we give equivalence of the existence of K-partitions with the existence of the precipitous ideal which is essentially topological. This way we strengthen the main result of Frankiewczi and Kunen (1987)., Comment: arXiv admin note: text overlap with arXiv:2003.10307, arXiv:2003.11017

We investigate the properties of ideals associated with Kuratowski partitions of non-complete Baire metric spaces. We show that such an ideal can be precipitous.

The main result of this paper is to show that, if $\kappa$ is the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$, then there exists a complete metric space of cardinality not greater than $ 2^{\kappa}$ admitting a Kuratowski partition., Comment: arXiv admin note: text overlap with arXiv:2003.10307

In 1935 K. Kuratowski posed the problem whether a function f:X ! Y , (X is completely metrizable and Y is metrizable), with the property that a preimage of each open has the Baire property, is continuous apart from a meager set. This paper is a selection of older results related to the question posed by Kuratowski in 1935, coming from among others Solovay and Bukovsk?y, and quite new ones concerning considerations in Ellentuck topology and some propertiesof K-ideals, (i.e. ideals associated with Kuratowski partitions).

In this paper there is proved without any metamathematical techniques that the existence of precipitous ideals immediately follows from Kuratowski partitions.

We show that large sets in Ellentuck topology (i.e. sets which are not nowhere Ramsey) do not admit Kuratowski's partition. The similar result is true for the Sacks real forcing.

We introduce the notion of $K$-ideals associated with Kuratowski partitions and we prove that each $\kappa$-complete ideal on a measurable cardinal $\kappa$ can be represented as a $K$-ideal. Moreover, we show some results concerning precipitous and Fr\'echet ideals.