The purpose of the present notes is to examine following issues related Chernoff estimate: (1) For contractions on a Banach space we modify $\sqrt{n}$-estimate and apply it in proof product formula for $C_0$-semigroups \textit{strong} operator topology. (2) We use idea {probabilistic} approach, proving estimate strong topology, uplift \textit{operator-norm} \textit{quasi-sectorial} contraction ...

which is part of a certain 2d Hamiltonian cohomological field theory [11]. Here we use this map to study the topology of ΩHam(M, ω) and its Hofer geometric properties via a kind of virtual Morse theory for the positive Hofer length functional, L : ΩHam(M, ω) → R . We also use Ψ to prove a kind of index theorem for some critical points of L . We apply this theory to get some concrete application...

Since Chang [2] introduced fuzzy theory into topology, many authors have discussed various aspects of fuzzy topology. It is well known that weakly induced and induced topological spaces play an important role in L-topological spaces (see book [8]). According to their value ranges, L-topological spaces form different categories. Clearly, the investigation on their relationships is certainly impo...

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...

In pointfree topology, a continuous real function on a frame L is a map L(R) → L from the frame of reals into L. The discussion of continuous real functions with possibly infinite values can be easily brought to pointfree topology by replacing the frame L(R) with the frame of extended reals L ( R ) (i.e. the pointfree counterpart of the extended real line R = R ∪ {±∞}). One can even deal with a...

Introduction. The question of ergodicity of a semigroup of bounded linear operators on a Banach space has been reduced, by Alaoglu and Birkhoff [l],1 Day [2, 3], and Eberlein [4], to the study firstly of the ergodicity of the semigroup itself and secondly, of the ergodicity of each element of the Banach space with respect to this ergodic semigroup. In the case of a bounded and commutative semig...

References 1. C. E. Capel, Inverse limit spaces, Duke Math. J. vol. 21 (1954) pp. 233-246. 2. H. Hahn, Mengentheoretische Charakterisierung der stetigen Kurve, Sitzungsber. Akad. Wiss., Wien, vol. 123 (1914) pp. 2433-2489. 3. J. L. Kelley, General topology, New York, van Nostrand Co., 1955. 4. S. Mazurkiewicz, Sur les lignes de Jordan, Fund. Math. vol. 1 (1920) pp. 166209. 5. R. L. Wilder, Topo...

Modern axiomatics of elementary geometry has steadily followed Hubert's program of elimination of continuity arguments. A brilliant example of the success of this method is to be found in the book [l ] by F. Bachmann which gives a unified treatment for geometries over arbitrary fields of characteristic y¿2. On the other hand the author has shown [2 ] that the plane axioms of incidence and order...

In this paper, some characterizations of fuzzifying strong compactness are given, including characterizations in terms of nets and pre -subbases. Several characterizations of locally strong compactness in the framework of fuzzifying topology are introduced and the mapping theorems are obtained.