Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

Ends,tangles and critical vertex sets

We show that an arbitrary infinite graph G can be compactified by its ends plus its critical vertex sets, where a finite set X of vertices of an infinite graph is critical if its deletion leaves some infinitely many components each with neighbourhood precisely equal to X. We further provide a concrete separation system whose ?₀‐tangles are precisely the ends plus critical vertex sets. Our tangle compactification is a quotient of Diestel's (denoted by ), and both use tangles to compactify a graph in much the same way as the ends of a locally finite and connected graph compactify it in its Freudenthal compactification. Finally, generalising both Diestel's construction of and our construction of , we show that G can be compactified by every inverse limit of compactifications of the sets of components obtained by deleting a finite set of vertices. Diestel's is the finest such compactification, and our is the coarsest one. Both coincide if and only if all tangles are ends. This answers two questions of Diestel.     

Jordan operator algebras revisited

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.     

C‐S cross‐coupling reaction using novel and green synthesized CuO nanoparticles assisted by Euphorbia maculata extract

In the present study, biosynthesis of CuO nanoparticles using a rapid, eco‐friendly, cost‐effective and efficient method has been reported employing aqueous Euphorbia maculata extract as mild, renewable and non‐toxic reducing and capping agents without adding any surfactants. The biogenic and green method has some benefits compared to conventional physical and chemical methods. It is simple, cheap and environmentally friendly. The biosynthesized CuO NP displayed a color change pattern (from sky blue to black) on preparation and presented its respective broad peak at 365 nm, which was analyzed by UV–Vis spectroscopy. Using the FT‐IR analysis, biomolecules in E. maculata extract which are responsible for bioreduction activity and synthesize of CuO NP, were identified. The XRD, EDX and FESEM results confirmed the successful synthesis of CuO nanoparticles of 18 nm sizes, with spherical and sponge crystal structure. The catalytic activity of biosynthesized CuO NPs was studied in C‐S cross‐coupling reaction. This method has the advantages of high yields, easy work‐up, and simple reusability. The recovered CuO NP can be reused four times without any considerable loss of its catalytic activity.     

σ2 Hessian方程的Pogorelov型C2 内估计及应用

提出利用拉格朗日乘子法重新证明σ2算子的最优凹性,并定义了一个凸锥Γ3?=λ=(λ1,λ2,?,λn)∈Rn:σ1(λ)>0,σ2(λ|i)>0,1≤i≤n。利用σ2算子的最优凹性，给出了σ2Hessian方程Pogorelov型C2内估计，进而证明了σ2(D2u(x))=1,x∈Rn的满足二次多项式增长条件的Γ3?-凸整解为二次多项式。     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

Minimal kernels of Dirac operators along maps

Let M be a closed spin manifold and let N be a closed manifold. For maps and Riemannian metrics g on M and h on N, we consider the Dirac operator of the twisted Dirac bundle . To this Dirac operator one can associate an index in . If M is 2‐dimensional, one gets a lower bound for the dimension of the kernel of out of this index. We investigate the question whether this lower bound is obtained for generic tupels .     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

Biological and catalytic evaluation of Ru(II)‐p‐cymene complexes of Schiff base ligands: Impact of ligand appended moiety on photo‐induced DNA and protein cleavage,cytotoxicity and C‐H activation

Two organometallic Ru(II)‐p‐cymene complexes of the type [Ru(η⁶‐p‐cymene)(L)Cl]PF₆ 1 and 2 , where L is N,N‐bis(4‐isopropylbenzylidene)ethane‐1,2‐diamine (bien, L1 ) or N,N‐bis (pyren‐2‐ylmethylene)ethane‐1,2‐diamine (bpen, L2 ) have been prepared and characterized well. Because of appended pyrenyl groups in coordinated bpen ligand, the complex 2 exhibits higher DNA and protein binding than complex 1 in which isopropylbenzyl groups are incorporated. Interestingly, the luminescent characteristic complex 2 is unique in displaying DNA cleavage after light activation by UVA light at 365 nm through oxygen dependent mechanism. AFM analysis attests the photo‐induced DNA fragmentation ability of complex 2 . Also, the complex 2 cleaves the protein after light exposure in a non‐specific manner suggesting that it can act as a protein photo cleaving agent. In contrast to the trend of DNA and protein interaction of complexes, the complex 1 exhibits cytotoxic activity against human breast carcinoma ( MCF‐7 ) and liver carcinoma ( HepG2 ) with potency higher than that of complex 2 due to enhanced hydrophobicity of isopropyl groups present in p‐cymene and bien ligands. Indeed, complex 2 is inactive against MCF‐7 and HepG2 cancer cell lines even up to 200 μM concentration. The AO/EB staining assay reveals that the complex 1 is able to induce late apoptotic mode of cell death in breast cancer cells, which is further confirmed by inter‐nucleosomal DNA cleavage. Furthermore, the complexes 1 and 2 are evaluated for their catalytic activities and found to be working well for the β‐carboline directed C–H arylation to afford the desired products in good yield (40–47%).