GENUS MINIMAL k-NOIDS AND SADDLE TOWERS IN

For each $k\geq 3$ , we construct a $1$ -parameter family of complete properly Alexandrov-embedded minimal surfaces in the Riemannian product space $\mathbb {H}^2\times \mathbb {R}$ with genus $1$ and k embedded ends asymptotic to vertical planes. We also obtain complete minimal surfaces with genus $1$ and $2k$ ends in the quotient of $\mathbb {H}^2\times \mathbb {R}$ by an arbitrary vertical translation. They all have dihedral symmetry with respect to k vertical planes, as well as finite total curvature $-4k\pi $ . Finally, we provide examples of complete properly Alexandrov-embedded minimal surfaces with finite total curvature with genus $1$ in quotients of $\mathbb {H}^2\times \mathbb {R}$ by the action of a hyperbolic or parabolic translation.

Product Space Singular Integrals with Mild Kernel Regularity

We develop product space theory of singular integrals with mild kernel regularity. We study these kernel regularity questions specifically in situations that are very tied to the T1 type arguments and the corresponding structural theory. In addition, our results are multilinear.

Some Properties of Cone Inner Product Spaces over Banach Algebra

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

Complete Hausdorffness and Complete Regularity on Supra Topological Spaces

The supra topological topic is of great importance in preserving some topological properties under conditions weaker than topology and constructing a suitable framework to describe many real-life problems. Herein, we introduce the version of complete Hausdorffness and complete regularity on supra topological spaces and discuss their fundamental properties. We show the relationships between them with the help of examples. In general, we study them in terms of hereditary and topological properties and prove that they are closed under the finite product space. One of the issues we are interested in is showing the easiness and diversity of constructing examples that satisfy supra T i spaces compared with their counterparts on general topology.

An Inner Product Space-Based Hierarchical Key Assignment Scheme for Access Control

An inner product space-based hierarchical key assignment/access control scheme is presented in this work. The proposed scheme can be utilized in any cloud delivery model where the data controller implements a hierarchical access control policy. In other words, the scheme adjusts any hierarchical access control policy to a digital medium. The scheme is based on inner product spaces and the method of orthogonal projection. While distributing a basis for each class by the data controller, the left-to-right and bottom-up policy can ensure much more flexibility and efficiency, especially during any change in the structure. For each class, the secret keys can be derived only when a predetermined subspace is available. The parent class can obtain the keys of the child class, which means a one-way function, and the opposite direction is not allowed. Our scheme is collusion attack and privilege creep problem resistant, as well as key recovery and indistinguishability secure. The performance analysis shows that the data storage overhead is much more tolerable than other schemes in the literature. In addition, the other advantage of our scheme over many others in the literature is that it needs only one operation for the derivation of the key of child classes.

An Inner Product Space-Based Hierarchical Key Assignment Scheme for Access Control

An inner product space-based hierarchical key assignment/access control scheme is presented in this work. The proposed scheme can be utilized in any cloud delivery model where the data controller implements a hierarchical access control policy. In other words, the scheme adjusts any hierarchical access control policy to a digital medium. The scheme is based on inner product spaces and the method of orthogonal projection. While distributing a basis for each class by the data controller, the left-to-right and bottom-up policy can ensure much more flexibility and efficiency, especially during any change in the structure. For each class, the secret keys can be derived only when a predetermined subspace is available. The parent class can obtain the keys of the child class, which means a one-way function, and the opposite direction is not allowed. Our scheme is collusion attack and privilege creep problem resistant, as well as key recovery and indistinguishability secure. The performance analysis shows that the data storage overhead is much more tolerable than other schemes in the literature. In addition, the other advantage of our scheme over many others in the literature is that it needs only one operation for the derivation of the key of child classes.

Norms on Quotient Spaces of The 2-Inner Product Space

This paper discussed about construction of some quotients spaces of the 2-inner product spaces. On those quotient spaces, we defined an inner product with respect to a linear independent set. These inner products was derived from the -inner product. We then defined a norm which induced by the inner product in these quotient spaces.