we consider the fundamental relation  on a semihypergroup  to interpret the lower and upper approximations as subsets of the fundamental semigroup  and we give some results in this connection. also, we introduce the notion of a bi-hyperideal to study the relationship between approximations and bi-hyperideals.

In his famous "deathbed" letter, Ramanujan "defined" the notion of a mock theta function and offered some examples of functions he believed satisfied his definition. Very recently, Griffin et al. established for the first time that Ramanujan's mock theta functions actually satisfy his own definition. On the other hand, Zwegers' 2002 doctoral thesis [Zwegers S (2002) Mock theta functions. PhD th...

We give a local deformation theoretic proof of Farkas’ conjecture that a principally polarized complex abelian variety of dimension 4 whose theta divisor has an isolated double point of rank 3 at a point of order two is a Jacobian. The argument yields an explicit local normal form for the theta function near such a point. The proof depends only on the facts that the theta function is even, a ge...

We investigate existence and properties of discrete mixture representations $P_{\theta }=\sum _{i\in E}w_{\theta }(i)\,Q_{i}$ for a given family }$, $\theta \in \Theta $, probability measures. The noncentral chi-squared distributions provide classical example. obtain results about geometric statistical aspects the problem, latter including loss Fisher information, Rao-Blackwellization, asymptot...

The alpha-like globin gene cluster in rabbits contains embryonic zeta-globin genes, an adult alpha-globin gene, and theta-globin genes of undetermined function. The basic arrangement of genes, deduced from analysis of cloned DNA fragments, is 5'-zeta 0-zeta 1-alpha 1-theta 1-zeta 2-zeta 3-theta 2-3'. However, the pattern of restriction fragments containing zeta- and theta-globin genes varies am...

Protein kinase C (PKC) is a family of enzymes involved in synapse formation and signal transduction at the neuromuscular junction. Two PKC isoforms, classical PKC alpha and novel PKC theta, have been shown to be enriched in skeletal muscle or localized to the endplate. We examined the role of nerve in regulating the expression of these PKC isoforms in rat skeletal muscle by denervating diaphrag...

Theta characteristics appeared for the first time in the context of characteristic theory of odd and even theta functions in the papers of Göpel [Go] and Rosenhain [Ro] on Jacobi’s inversion formula for genus 2. They were initially considered in connection with Riemann’s bilinear addition relation between the degree two monomials in theta functions with characteristics. Later, in order to syste...

Theta frequency oscillations in the 6- to 10-Hz range dominate the rodent hippocampal local field potential during translational movement, suggesting that theta encodes self-motion. Increases in theta power have also been identified in the human hippocampus during both real and virtual movement but appear as transient bursts in distinct high- and low-frequency bands, and it is not yet clear how...

Field potential recordings from the rat hippocampus in vivo contain distinct frequency bands of activity, including delta (0.5-2 Hz), theta (4-12 Hz), and gamma (30-80 Hz), that are correlated with the behavioral state of the animal. The cholinergic agonist carbachol (CCH) induces oscillations in the delta (CCH-delta), theta (CCH-theta), and gamma (CCH-gamma) frequency ranges in the hippocampal...

The generation of EEG theta rhythm in the mammalian limbic cortex is a prime example of rhythmic activity that involves central mechanisms of oscillations and synchrony. This EEG pattern has been extensively studied since 1938, when Jüng and Kornmüller (1938) demonstrated the first theta recordings in the hippocampal formation of rabbits. In 1986 in collaboration with Drs. B.H. Bland, S.H. Roth...