• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.1137-1150
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• 2022

Consider the quantum algebra U_q(sl₂) over field 𝓕 (char(𝓕) = 0) with equitable generators x^±1, y and z, where q is fixed nonzero, not root of unity scalar in 𝓕. Let V denote a finite dimensional irreducible module for this algebra. Let Λ ∈ End(V), and let {A₁, A₂, A₃} = {x, y, z}. First we show that if Λ, A₁ is a Leonard pair, then this Leonard pair have four types, and we show that for each type there exists a Leonard pair Λ, A₁ in which Λ is a linear combination of 1, A₂, A₃, A₂A₃. Moreover, we use Λ to construct 𝚼 ∈ U_q(sl₂) such that 𝚼, A^-1₁ is a Leonard pair, and show that 𝚼 = I + A₁Φ + A₁ΨA₁ where Φ and Ψ are linear combination of 1, A₂, A₃.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.401-414
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• 2019

Let ${\mathcal{F}}$ denote an algebraically closed field with characteristic not two. Fix an integer $d{\geq}3$, let $Mat_{d+1}({\mathcal{F}})$ denote the ${\mathcal{F}}$-algebra of $(d+1){\times}(d+1)$ matrices with entries in ${\mathcal{F}}$. An ordered pair of matrices A, $A^*$ in $Mat_{d+1}({\mathcal{F}})$ is said to be LB-TD form whenever A is lower bidiagonal with subdiagonal entries all 1 and $A^*$ is irreducible tridiagonal. Let A, $A^*$ be a Leonard pair in $Mat_{d+1}({\mathcal{F}})$ with fundamental parameter ${\beta}=2$, with this assumption there are four families of Leonard pairs, Racah, Hahn, dual Hahn, Krawtchouk type. In this paper we show from these four families only Racah and Krawtchouk have LB-TD form.

• Journal of Cardiovascular Imaging
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• v.30 no.3
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• pp.153-168
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• 2022

Recent studies have shown that sodium/glucose cotransporter 2 (SGLT2) inhibitors might exert favourable changes on cardiac parameters as observed on cardiovascular imaging. We conducted a systematic review and meta-analysis to determine the effects of SGLT2 inhibitors on cardiac imaging parameters. Four electronic databases (PubMed, Embase, Cochrane, Scopus) were searched for studies in which the effects of SGLT2 inhibitors on cardiac imaging parameters were examined. Studies in which a population was administered SGLT2 inhibitors and analysed by echocardiography and/or cardiac magnetic resonance (CMR) imaging were included. Random-effects pair-wise meta-analysis models were utilized to summarize the studies. A total of 11 randomized controlled trials was included with a combined cohort of 910 patients. Comparing patients receiving SGLT2 inhibitors with subjects receiving placebo, the mean change in CMR-measured left ventricular mass (LVM) was -3.87 g (95% confidence interval [CI], -7.77 to 0.04), that in left ventricular end-systolic volume (LVESV) was -5.96 mL (95% CI, -10.52 to -1.41) for combined LVESV outcomes, that in left atrial volume index (LAVi) was -1.78 mL/m² (95% CI, -3.01 to -0.55) for combined LAVi outcomes, and that in echocardiography-measured E/e' was -0.73 (95% CI, -1.43 to -0.03). Between-group differences were not observed in LVM and LVESV after indexation. The only between-group difference that persisted was for LAVi. Treatment with SGLT2 inhibitors resulted in reduction in LAVi and E/e' on imaging, indicating they might have an effect on outcomes associated with LV diastolic function.