R. Meyer and E. Mares [10] studied the logic CNR, a classical relevant logic with a modal connective. Such logic is of type S4. As it has been noted in [10], the addition of a modal operator in the language of classical relevant logic and the axioms φ ∧ ψ → (φ ∧ ψ) and (φ ∧ ψ) → ( φ ∧ ψ) do not necessarily produce theorems such that as (φ→ ψ) → φ → ψ or (♦φ→ ψ) → (φ→ ψ). These formulas are well...

The study of fixed points on the maps fulfilling certain contraction requirements has several applications and been focus numerous research endeavors. On other hand, as an extension idea best approximation, proximity point (ƁƤƤ) emerges. approximation theorem ensures existence approximate solution; is considered for addressing problem in order to arrive at optimum solution. This paper introduce...

1 2 ∆ φ t = φ t − φ 1 , ∆ ψ t = ψ t − ψ 1 , ∆φ t = φ t − φ 1 , ∆ψ t = ψ t − ψ 1 Further, we define cos x (u, v) = u Sxv uxvx , the cosine of the angle between two vectors induced by the inner product u, v = u S x v. Similarly, we define cos y (u, v) = u Syv uyvy. To prove the theorem, we will repeatedly use the following two lemmas. Proof of Lemma 1. The proof is in the main paper. Lemma 2. ∆φ ...

– For the system of d-dim stochastic differential equations, dXε(t)= b(X(t))dt + εσ(X(t))dW(t), t ∈ [0,1], Xε(0)= x0 ∈Rd, where b(x) and σ(x) are smooth except possibly along the hyperplane {(x1, . . . , xd);x1 = 0}, we shall demonstrate that the natural setup of its large deviation principle is to consider the probability ε2 logP(‖Xε − φ‖< δ, ‖uε − ψ‖ < δ, ‖ ε − η‖< δ)∼−I (φ,ψ,η) of the triple...

We investigate modal logical aspects of provability predicates PrT(x) satisfying the following condition: M: If T⊢φ→ψ, then T⊢PrT(⌜φ⌝)→PrT(⌜ψ⌝). prove arithmetical completeness theorems for monotonic logics MN, MN4, MNP, MNP4, and MND with respect to condition M. That is, we that each logic L them, there exists a Σ1 predicate M such is exactly L. In particular, formulas P: ¬□⊥ D: ¬(□A∧□¬A) are ...

where φt x t−1φ x/t , ψt x t−1ψ x/t , and ∗ denotes the convolution on R. The Calderón reproducing formula is a useful tool in pure and applied mathematics see 1– 4 , particularly in wavelet theory see 5, 6 . We always call 1.1 an inverse formula of wavelet transform. In 7 , the authors generalized 1.1 to R when φ and ψ are sufficiently nice normalized radial wavelet functions. The generalizati...

We will use K̂iφ as an abbreviation for ¬Ki¬φ and 〈φ〉ψ as an abbreviation for ¬[φ]¬ψ. Definition 2 (Semantics). An epistemic model (or model for short) is a triple M = (S,∼n, V ) with S a non-empty set of states, ∼i an equivalence relation for i < n, and V a function assigning a subset V (p) ⊆ S with each p ∈ PL. Given a modelM = (S,∼n, V ) and an s ∈ S, we define: M, s |= p iff s ∈ V (p) M, s |...

Ex. 17 Let us call the new system L, i.e. its axioms are all propositional tautologies (Axiom 1) plus the axioms 2> (Axiom 2) and 2φ∧2ψ → 2(φ∧ψ) (Axiom 3), and the rules modus ponens and φ→ ψ 2φ→ 2ψ We have to show that for all formulas φ `K φ ⇔ `L φ. ⇒: For this direction we have to show that L derives all axioms of K and all its rules. Axiom 1 of K is the same as Axiom 1 in L, thus we have no...

Hence, there is an eigenstate |ψ〉 = A|φ〉 of H with this same energy eigenvalue. We could equally well have begun with the statement that H|ψ〉 = E|ψ〉 and then found a state of H given by |φ〉 = A†|ψ〉 with the same energy. Note that a Hamiltonian of the two given forms will have only non-negative eigenvalues: 〈ψ|A†A|ψ〉 = ∣∣|Aψ〉∣∣2 ≥ 0. Equality holds only if the state is annihilated by A. Analogou...