Bures distance and transition probability for α\alpha-CPD-kernels

If the symmetry (fixed invertible self adjoint map) of Krein spaces is replaced by a fixed unitary, then we obtain the notion of S-spaces which was introduced by Szafraniec. Assume $\alpha$ to be an automorphism on a $C^*$-algebra. In this article, we obtain the Kolmogorov decomposition of $\alpha$-completely positive definite (or $\alpha$-CPD-kernels for short) and investigate the Bures distance between $\alpha$-CPD-kernels. We also define transition probability for these kernels and find a characterization of the transition probability.Comment: 18page

The Cancer Treatment by Using Liposome as Drug Delivery Systems and Its Applications: An Extensive Review

Cancer is a leading cause of death in many countries around the world. However, the efficacy of current standard treatments for a variety of cancers is suboptimal. In the development of cancer chemotherapy, besides the discovery of new anticancer drugs, a variety of nanocarrier systems for the delivery of previously developed and new chemotherapeutic drugs have currently been explored. Liposome is one of the most studied nanocarrier systems because of its biodegradability, simple preparation method, high efficacy and low toxicity. To make the best use of this vehicle, a number of multifunctionalized liposomal formulations have been investigated. This review will be including the development of liposomes for cancer therapy, liposomal formulation of various anticancer drugs that are commercially available, recent progress in liposome technology for the treatment of cancer, and the next generation of lipid-based nanoparticles., recent progress in liposome technology for the treatment of cancer, and the next generation of lipid-based nanoparticles

Regular covariant representations and their Wold-type decomposition

Olofsson introduced a growth condition regarding elements of an orbit for an expansive operator and generalized Richter's wandering subspace theorem. Later on, using the Moore-Penrose inverse, Ezzahraoui, Mbekhta, and Zerouali extended the growth condition and obtained a Shimorin-Wold-type decomposition. Shimorin-Wold-type decomposition for completely bounded covariant representations, which are close to isometric representations, is obtained in \cite{HV19}. This paper extends this decomposition for regular, completely bounded covariant representation having reduced minimum modulus $\geq 1$ that satisfies the growth condition. To prove the decomposition, we introduce the terms regular, algebraic core, and reduced minimum modulus in the completely bounded covariant representation setting and work out several fundamental results. Consequently, we shall analyze the weighted unilateral shift introduced by Muhly and Solel and introduce and explore a non-commutative weighted bilateral shift.Comment: 35 page

Cyclodextrins as a Drug Delivery Carrier for Anti-Cancer Drugs

Cyclodextrin “molecules are large with a number of hydrogen donors. Cyclodextrins are widely used as "molecular cages" in the pharmaceutical, agrochemical, food and cosmetic industries. In the pharmaceutical industry they are used as complexing agents to increase the aqueous solubility of poorly soluble drugs and to increase their bioavailability and stability. Chemotherapeutic treatment for cancer has limitations such as poor drug solubility, non-specificity, poor bioavailability and poor survival rate. The lack of efficient treatment has created the need to develop and implement novel technology based on combination strategy of cyclodextrin complexation and nanotechnology with a view to make the therapy more useful and acceptable. The review deals with cyclodextrin and its applications in anti-cancer drug delivery for cancer treatment