On Strong Small Loop Transfer Spaces Relative to Subgroups of Fundamental Groups

Let $H$ be a subgroup of the fundamental group $\pi_{1}(X,x_{0})$. By extending the concept of strong SLT space to a relative version with respect to $H$, strong $H$-SLT space, first, we investigate the existence of a covering map for strong $H$-SLT spaces. Moreover, we show that a semicovering map is a covering map in the presence of strong $H$-SLT property. Second, we present conditions under which the whisker topology agrees with the lasso topology on $\widetilde{X}_{H}$. Also, we study the relationship between open subsets of $\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{l}(X,x_{0})$. Finally, we give some examples to justify the definition and study of strong $H$-SLT spaces.Comment: 16 page

On the Order of the Schur Multiplier of a Pair of Finite p-Groups

In 1998, G. Ellis defined the Schur multiplier of a pair $(G,N)$ of groups and mentioned that this notion is a useful tool for studying pairs of groups. In this paper, we characterize the structure of a pair of finite $p$-groups $(G,N)$ in terms of the order of the Schur multiplier of $(G,N)$ under some conditions.Comment: 11 pages, to appear in Journal of Advanced Research in Pure Mathematic

A fast ultra performance supercritical fluid chromatography-tandem mass spectrometric method for profiling of targeted phytosterols

Phytosterols are essential structural components of plant cell membranes and possess health-related benefits, including lowering blood cholesterol levels in humans. Numerous analytical methods are being used to profile plant and animal sterols. Chromatography hyphenated to tandem mass spectrometry, is a better option due to its specificity, selectivity, and sensitivity. An ultra-performance supercritical fluid chromatography hyphenated with atmospheric pressure chemical ionization (APCI) tandem mass spectrometric method was developed and eval-uated for fingerprint analysis of seven phytosterols. Mass spectrometry fragmentation behavior was used for phytosterol identification, and multiple reaction monitoring scanning was utilized for phytosterol confirmation, where APCI outperformed superiority in terms of ion intensity, particularly in the production of [M + H-H2O]+ ions rather than [M + H]+ ions. The chromatographic conditions were thoroughly evaluated, and the ionization parameters were optimized as well. In a 3 min. run, the seven phytosterols were separated concurrently. The calibration and repeatability tests were conducted to check the instrument's performance, and the results indicated that all of the phytosterols tested had correlation coefficients (r2) greater than 0.9911 over the con-centration range of 5-5000 ng/mL. The limit of quantification was below 20 ng/mL for all the tested analytes except for stigmasterol and campesterol. The partially validated method was applied for the evaluation of phytosterols in pure coconut oil and palm oil in order to demonstrate its applicability. Total sterols in coconut and palm oils were 126.77 ng/mL and 101.73 ng/mL, respectively. In comparison to earlier methods of phytosterol analysis, the novel method offers a far faster, more sensitive, and more selective analytical process