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Authors: Quan-Guo Chen, Yang-Lei Pang and Ding-Guo Wang

In this paper, we introduce comatrix group corings and define the generalized Galois group corings. Then we give the generalized Galois group coring Structure Theorem.

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In present paper, we give some new reverses of the Young type inequalities which were established by X. Hu and J. Xue [7] via Kantorovich constant. Then we apply these inequalities to establish corresponding inequalities for the Hilbert-Schmidt norm and the trace norm.

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Let L(f, r) denote the length of the closed curve which is the image of |z| = r < 1 under the mapping w = f(z). We establish some sufficient conditions for L(f, r) to be bounded and for f(z) to in the classes of strongly close-to-convex function of order α and to be strongly Bazilevič function of type β of order α. Moreover, we prove an inequality connected with the Fejér-Riesz's inequality.

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We characterize the normal distribution based on the Q-independence of linear forms based on infinite sequences of Q-independent random variables.

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In this paper we obtain some new power and hölder type trace inequalities for positive operators in Hilbert spaces. As tools, we use some recent reverses and refinements of Young inequality obtained by several authors.

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Let R be a commutative Noetherian ring, M a finitely generated R-module, I an ideal of R and N a submodule of M such that IMN. In this paper, the primary decomposition and irreducible decomposition of ideal I × N in the idealization of module R ⋉ M are given. From theses we get the formula for associated primes of R ⋉ M and the index of irreducibility of 0R ⋉ M.

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Authors: Yuefeng Gao, Jianlong Chen, Pedro Patrício and Dingguo Wang

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index 1. Recently, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in *-rings; meanwhile, it characterized the core-EP inverse introduced by Manjunatha Prasad and Mohana for complex matrices, in terms of three equations. Many works have been done on classical generalized inverses of companion matrices and Toeplitz matrices. In this paper, we discuss existence criteria and formulae of the pseudo core inverse of a companion matrix over a *-ring. A {1,3}-inverse of a Toeplitz matrix plays an important role in that process.

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If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a re exive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint.

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Some Erdős-Ko-Rado type extremal properties of families of vectors from {-1; 0; 1}n are considered.

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We find upper and lower bounds for the probability of a union of events which generalize the well-known Chung-Erdős inequality. Moreover, we will show monotonicity of the bounds.

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