JIMS Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 Iranian Mathematical Society 110856 10.30504/jims.2020.110856 Research Article Sanov's theorem on Lie relators in groups of exponent \$p\$ Sanov's theorem on Lie relators in groups of exponent \$p\$ Vaughan-Lee M. Christ Church, University of Oxford, Oxford, OX1 1DP, England. 01 03 2021 2 1 1 16 15 05 2020 20 07 2020 Copyright © 2021, Iranian Mathematical Society. 2021 http://jims.ims.ir/article_110856.html

‎I give a proof of Sanov's theorem that the Lie relators of weight at most‎ ‎\$2p-2\$ in groups of exponent \$p\$ are consequences of the identity \$px=0\$ and‎ ‎the \$(p-1)\$-Engel identity‎. ‎This implies that the order of the class \$2p-2\$‎ ‎quotient of the Burnside group \$B(m,p)\$ is the same as the order of the class‎ ‎\$2p-2\$ quotient of the free \$m\$ generator \$(p-1)\$-Engel Lie algebra over‎ ‎GF\$(p)\$‎. ‎To make the proof self-contained I have also included a derivation of‎ ‎Hausdorff's formulation of the Baker Campbell Hausdorff formula‎.

‎Sanov's theorem Lie relators Groups of exponent \$p\$
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JIMS Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 Iranian Mathematical Society 138341 10.30504/jims.2021.299742.1038 46-XX Functional Analysis Generalized trapezoid type inequalities for functions with values in Banach spaces Generalized trapezoid type inequalities for functions with values in Banach spaces Dragomir S. S. Victoria University, Melbourne, Australi 01 03 2021 2 1 17 38 14 08 2021 06 10 2021 Copyright © 2021, Iranian Mathematical Society. 2021 http://jims.ims.ir/article_138341.html

Let E be a complex Banach space. In this paper we show among others that, if α:[a,b]→C is continuous and Y:[a,b]→E is strongly differentiable on the interval (a,b), then for all u∈[a,b],‖(∫_{u}^{b}α(s)ds)Y(b)+(∫_{a}^{u}α(s)ds)Y(a)-∫_{a}^{b}α(t)Y(t)dt‖ ≤{┊max{∫_{u}^{b}|α(s)|ds,∫_{a}^{u}|α(s)|ds}∫_{a}^{b}‖Y′(t)‖dt, [∫_{u}^{b}(b-t)|α(t)|dt+∫_{a}^{u}(t-a)|α(t)|dt]sup_{t∈[a,b]}‖Y′(t)‖,  ≤(b-a)^{1/p}[(∫_{u}^{b}|α(s)|ds)^{p}+(∫_{a}^{u}|α(s)|ds)^{p}]^{1/p} ×(∫_{a}^{b}‖Y′(t)‖^{q}dt)^{1/q}for p, q>1 with (1/p)+(1/q)=1. Applications for operator monotone functions with examples for power and logarithmic functions are also given.

Banach spaces integral inequalities Operator monotone functions
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JIMS Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 Iranian Mathematical Society 141199 10.30504/jims.2021.309063.1044 26-XX Real Functions On a Hilbert-type integral inequality in the whole plane On a Hilbert-type integral inequality in the whole plane Rassias M. Th. Department of Mathematics and Engineering Sciences, Hellenic Military Academy, 16673 Vari Attikis, Greece Yang B. Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P. R. China Meletiou G. C. University of Ioannina‎, ‎Ioannina‎, ‎Greece. 01 03 2021 2 1 39 51 05 10 2021 04 12 2021 Copyright © 2021, Iranian Mathematical Society. 2021 http://jims.ims.ir/article_141199.html

Using weight functions and techniques of real analysis, a new Hilbert-type integral inequality in the whole plane with nonhomogeneous kernel and a best possible constant factor is proved. Equivalent forms, several particular inequalities and operator expressions are considered.

Hilbert-type integral inequality weight function parameter equivalent form operator
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JIMS Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 Iranian Mathematical Society 141490 10.30504/jims.2021.263742.1026 53-XX Differential geometry The first eigenvalue of \$left(p,qright)\$-elliptic quasilinear system along the Ricci flow The first eigenvalue of \$left(p,qright)\$-elliptic quasilinear system along the Ricci flow Azami S. Department of pure mathematics, Faculty of mathematical science, Imam Khomeini international university, Qazvin, Iran Habibi Vosta Kolaei M. Department of pure mathematics, Faculty of science, Imam Khomeini international university, Qazvin, Iran 01 03 2021 2 1 53 70 25 12 2020 11 12 2021 Copyright © 2021, Iranian Mathematical Society. 2021 http://jims.ims.ir/article_141490.html

In this paper we investigate the monotonicity of the first eigenvalue of \$left(p,qright)\$-elliptic quasilinear systems along the Ricci flow in both normalized and unnormalized conditions. In particular, we study the eigenvalue problem for this system in the case of Bianchi classes for \$3\$-homogeneous manifolds.

Ricci flow \$(p q)\$-elliptic quasilinear system Eigenvalue
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