In this paper x, r are real numbers. Let A be a set. We say that A is symmetrical if and only if: (Def. 1) For every complex number x such that x ∈ A holds −x ∈ A. Let us note that there exists a subset of C which is symmetrical. Let us observe that there exists a subset of R which is symmetrical. In the sequel A denotes a symmetrical subset of C. Let R be a binary relation. We say that R has s...

ly, a watermarking scheme is the six-tuple (X, W, K, G, E, D) where [13] 1. X represents the set of original or host images. 2. W is the set of all watermarks w such that ∃x ∈ X, k ∈ K, w = G(x, k). 3. K is the set of numbers called watermark keys. 4. G is the algorithm that generates W using K and X: G: X × K → W, w = G(x, k) 5. E denotes the encoding algorithm that embeds a watermark w in an ...

It can be easily verified that the system (1.1) by virtue of the condition (1.2) is hyperbolic. Let D be the conic domain {(x, t) ∈Rn+1 : |x|g(x/|x|) < t < +∞} lying in the half-space t > 0, and bounded by the conic manifold S= {(x, t) ∈Rn+1 : t = |x|g(x/|x|)}, where g is an entirely definite, positive, continuous, piecewise smooth function given on the unit sphere of the space Rn. For τ > 0, b...

This paper focuses on the relationship between an $L$-subset and the system of level-elements induced by it, where the underlying lattice $L$ is a complete residuated lattice and the domain set of $L$-subset is an $L$-partially ordered set $(X,P)$. Firstly, we obtain the sufficient and necessary condition that an $L$-subset is represented by its system of level-elements. Then, a new representat...

Let X be a dcpo and let L be a complete lattice. The family σL(X) of all Scott continuous mappings from X to L is a complete lattice under pointwise order, we call it the L-fuzzy Scott structure on X. Let E be a dcpo. A mapping g : σL(E) −> M is called an LM-fuzzy possibility valuation of E if it preserves arbitrary unions. Denote by πLM(E) the set of all LM-fuzzy possibility valuations of E. T...