in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relat...

Augmented reality (AR) refers to the introduction of virtual elements in the real world. That is, the person is seeing an image composed of a visualization of the real world, and a series of virtual elements that, at that same moment, are super-imposed on the real world. The most important aspect of AR is that the virtual elements supply to the person relevant and useful information that is not...

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real ...

We use the following convention: m, n denote non empty elements of N, i, j denote elements of N, and Z denotes a set. One can prove the following propositions: (1) Let S, T be real normed spaces, f be a point of the real norm space of bounded linear operators from S into T , and r be a real number. Suppose 0 ≤ r and for every point x of S such that ‖x‖ ≤ 1 holds ‖f(x)‖ ≤ r · ‖x‖. Then ‖f‖ ≤ r. ...

Differential fields of germs of continuous real valued functions of one real variable (Hardy fields) have the property that all elements have limits in the extended real numbers and thus have a canonical valuation. For differential fields of holomorphic germs this is not generally the case. We provide a criterion for differential fields of holomorphic germs for its elements to have uniform limi...

Rule based systems or production rule languages have become an established means of encoding expertise 2]. The evaluation of satisfaction of a set of production rules (match phase in the inference cycle) is computationally expensive 2]. RETE 1] and TREAT 3] are two algorithms that perform this evaluation eeciently. The complexity of these algorithms is O(W C), where W is the number of working m...