6. [This is simply an adapted proof of Theorem 2.4.2.] First observe that, the map φ(g) [for a fixed g ∈ G] is well defined: Hx = Hy iff xy−1 ∈ H iff xg−1gy−1 ∈ H iff (xg−1)(yg(−1)−1 ∈ H iff Hxg−1 = Hyg−1 iff φ(g)(Hx) = φ(g)(Hy). Note also, that since we have equivalencies in all steps above, we also have proved that φ(g) is one-to-one. Also, for a fixed g ∈ G, the map φ(g) : Hx 7→ Hxg is indee...

Langlands recently constructed a map φ → |x(φ)〉 that factorizes the partition function of a free boson on a cylinder with boundary condition given by two arbitrary functions φB1 and φB2 in the form 〈x(φB1)|q00|x(φB2)〉. We rewrite |x(φ)〉 in a compact form, getting rid of technical assumptions necessary in his construction. This simpler form allows us to show explicitly that the map φ → |x(φ)〉 co...

The notion of n-type was introduced by J.H.C. Whitehead ([22, 23]) where its clear geometric meaning was presented. Following J.L. Hernandez and T. Porter ([12, 13]) we use the term weak n-equivalence for a map f : X → Y of path-connected spaces which induces isomorphisms πk(f) : πk(X)→ πk(Y ) on homotopy groups for k ≤ n. Certainly, weak n-equivalence of a map determines its n-connectedness bu...

To study the effects of contractile activity on mitogen-activated protein kinase (MAP kinase), p70 S6 kinase (p70S6K), and Akt kinase signaling in rat skeletal muscle, hindlimb muscles were contracted by electrical stimulation of the sciatic nerve for periods of 15 s to 60 min. Contraction resulted in a rapid and transient activation of Raf-1 and MAP kinase kinase 1, a rapid and more sustained ...

In this paper, we give a generalization of Hicks type contractions and Golet type contractions on fuzzy metric spaces. We prove some fixed point theorems for this new type contraction mappings on fuzzy metric spaces.

We present a weak lensing analysis and comparison to optical and Xray maps of the close pair of massive clusters A222/223. Indications for a filamentary connection between the clusters are found and discussed.

A right-sided, nearest neighbour cellular automaton (RNNCA) is a continuous transformation Φ : A−→A determined by a local rule φ : A{0,1}−→A so that, for any a ∈ A and any z ∈ Z, Φ(a)z = φ(az, az+1). We say that Φ is bipermutative if, for any choice of a ∈ A, the map A ∋ b 7→ φ(a, b) ∈ A is bijective, and also, for any choice of b ∈ A, the map A ∋ a 7→ φ(a, b) ∈ A is bijective. We characterize ...

Let H be a complex Hilbert space. Denote by B(H)+ the set of all positive bounded linear operators on H. A bijective map φ : B(H)+ → B(H)+ is said to preserve Lebesgue decompositions in both directions if for any quadruple A,B,C,D of positive operators, B = C +D is an A-Lebesgue decomposition of B if and only if φ(B) = φ(C)+φ(D) is a φ(A)-Lebesgue decomposition of φ(B). It is proved that every ...