The starting point of this paper are the works of Hájek and Vychodil on the axiomatization of truthstressing and -depressing hedges as expansions of Hájek’s BL logic by new unary connectives. They showed that their logics are chain-complete, but standard completeness was only proved for the expansions over Gödel logic. We propose weaker axiomatizations over an arbitrary core fuzzy logic which h...

The starting point of this paper are the works of Hájek and Vychodil on the axiomatization of truth-stressing and depressing hedges as expansions of Hájek's BL logic by new unary connectives. They showed that their logics are chain-complete, but standard completeness was only proved for the expansions over Gödel logic. We propose weaker axiomatizations over an arbitrary core fuzzy logic which h...

The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all homotopically self-contained. The left half of this statement essentially means that any functor that looks like it could be a tensor product (or product, or sma...

We obtain some methods to construct a (strongly) proper resolution (resp. coproper coresolution) of one end term in a short exact sequence from that of the other two terms. By using this method, we prove that for a left and right Noetherian ring R, RR satisfies the Auslander condition if and only if so does every flat left R-module, if and only if the injective dimension of the ith term in a mi...

Traditionally, membership to the fuzzy-rough lower, resp. upper approximation is determined by looking only at the worst, resp. best performing object. Consequently, when applied to data analysis problems, these approximations are sensitive to noisy and/or outlying samples. In this paper, we advocate a mitigated approach, in which membership to the lower and upper approximation is determined by...

X(n) = X1(n1) + · · ·+XN (nN ), n ∈ N0 . Here, nj denotes the jth coordinate of n ∈ N0 . We are following the standard convention of starting (ordinary) random walks at the origin. That is, Xj(0) = 0 for all j = 1, . . . , N . From now on, N will always denote the temporal dimension and d the spatial one, to use the standard language of stochastic processes. In order to describe a natural parti...

— Let L : TM → R be a Tonelli Lagrangian function (with M compact and connected and dimM > 2). The tiered Aubry set (resp. Mañé set) AT (L) (resp. NT (L)) is the union of the Aubry sets (resp. Mañé sets) A(L + λ) (resp. N (L+ λ)) for λ closed 1-form. Then 1. the set NT (L) is closed, connected and if dimH1(M) > 2, its intersection with any energy level is connected and chain transitive; 2. for ...

• Using the ladder operators, Jx = 1 2(J+ + J−) and Jy = 1 2i(J+ − J−). When Jx (resp. Jy) acts on | j,m〉, it produces a linear combination of two ket states | j,m+ 1〉 and | j,m− 1〉, both of which are orthogonal to | j,m〉. So if we put the bra 〈j,m | with the ket Jx| j,m〉 (resp. Jy| j,m〉) together, the bracket must vanish: that is, the expectation value of Jx (resp. Jy) in the state | j,m〉 is 0...

Algebraic structures especially an ordered semigroups play a prominent role in mathematics with wide ranging applications in many disciplines such as control engineering, computer arithmetics, coding theory, sequential machines and formal languages. A theory of fuzzy sets in terms of fuzzy points on ordered semigroups can be developed. In this paper, we generalize the concept of (α, β)-fuzzy le...