Casimir effect of a topologically nontrivial two-dimensional space-time, through Krein space quantization [1, 2], has been calculated. In other words, auxiliary negative norm states have been utilized here. Presence of negative norm states play the role of an automatic renormalization device for the theory. The negative norm states (which do not interact with the physical world) could be chosen...

We prove that every separable polyhedral Banach space X is isomorphic to a polyhedral Banach space Y such that, the set ext BY ∗ cannot be covered by a sequence of balls B(yi, 2i) with 0 < 2i < 1 and 2i → 0. In particular ext BY ∗ cannot be covered by a sequence of norm compact sets. This generalizes a result from [7] where an equivalent polyhedral norm ||| · ||| on c0 was constructed such that...

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for onedimensional time-harmonic wave propagation problems. The method is constructed within the framework of the Discontinuous Petrov-Gale...

The space l^p with 1≤p∞ is the set of real numbers that satisfy _(n=1)^∞▒〖|x_n |^p∞〗.The function in vector X which has value fulfills norm-2 properties denoted by ,⋅‖ and pair (X,‖⋅,⋅‖) called space.A said to be complete or a Banach-2 if every Cauchy sequence converges an element space.This research was conducted prove norm-2.The first step completeness norm contained satisfies norm-2.Next, de...

Each space C N = {f ∈ C c (R) : sptf ⊂ [−N,N ]} ⊂ C[−N,N ] is strictly smaller than the space C[−N,N ] of all continuous functions on the interval [−N,N ], since functions in C N must vanish at the endpoints. Still, C o N is a closed subspace of the Banach space C [−N,N ] (with sup norm), since a sup-norm limit of functions vanishing at ±N must also vanish there. Thus, each individual C N is a ...

Fuzzy set theory is a useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situation by attributing a degree to which a certain object belongs to a set. The idea of fuzzy norm was initiated by Katsaras in [1984]. Felbin [1992] defined a fuzzy norm on a linear space whose associated fuzzy metric is of Kaleva and Seikkala type [1984]. Cheng and Morde...

• Positivity: N(v) ≥ 0 with equality if and only if v = 0. • Positive Homogeneity: N(αv) = |α|N(v). • Triangle Inequality: N(x1 + x2) ≤ N(x1) +N(x2). If N is a norm for V then we call ρ N (x1, x2) := N(x1−x2) the associated distance function (or metric) for V . A vector space V together with some a choice of norm is called a normed space, and the norm is usually denoted by ‖ ‖. If V is complete...

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the highfrequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method...

Let Φ be a super-operator, i.e., a linear mapping of the form Φ : L(F) → L(G) for finite dimensional Hilbert spaces F and G. This paper considers basic properties of the super-operator norms defined by ‖Φ‖q→p = sup{‖Φ(X)‖p/‖X‖q : X 6= 0}, induced by Schatten norms for 1 ≤ p, q ≤ ∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is prov...