The Lebesgue Monotone Convergence Theorem
In this article we prove the Monotone Convergence Theorem [16].MML identifier: MESFUNC9, version: 7.8.10 4.100.1011
The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space
In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10],...
The logic of neural networks.
This paper establishes the equivalence between multilayer feedforward networks and linear combinations of Lukasiewicz propositions. In this sense, multilayer forward networks have a logic interpretation, which should permit to apply logical techniques in the neural networks framework.
The maximality of the typed lambda calculus and of cartesian closed categories
The Measurability of Complex-Valued Functional Sequences
In this article, we formalized the measurability of complex-valued functional sequences. First, we proved the measurability of the limits of real-valued functional sequences. Next, we defined complex-valued functional sequences dividing real part into imaginary part. Then using the former theorems, we proved the measurability of each part. Lastly, we proved the measurability of the limits of complex-valued functional sequences. We also showed several properties of complex-valued measurable functions....
The monadic second-order logic of graphs III : tree-decompositions, minors and complexity issues
The -propositional calculus
T. Almada and J. Vaz de Carvalho (2001) stated the problem to investigate if these Łukasiewicz algebras are algebras of some logic system. In this article an affirmative answer is given and the -propositional calculus, denoted by , is introduced in terms of the binary connectives (implication), (standard implication), (conjunction), (disjunction) and the unary ones (negation) and , (generalized Moisil operators). It is proved that belongs to the class of standard systems of implicative...
The Neg.-propositional Calculus
The non-axiomatizability of the observational predicate calculus (generalized Trachtenbrot's theorems)
The observational predicate calculus and complexity of computations (Preliminary communication)
The Orthogonal Projection and the Riesz Representation Theorem
In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals...
The positivity problem for fourth order linear recurrence sequences is decidable
The problem whether each element of a sequence satisfying a fourth order linear recurrence with integer coefficients is nonnegative, referred to as the Positivity Problem for fourth order linear recurrence sequence, is shown to be decidable.
The prime and maximal spectra and the reticulation of BL-algebras
In this paper we study the prime and maximal spectra of a BL-algebra, proving that the prime spectrum is a compact T 0 topological space and that the maximal spectrum is a compact Hausdorff topological space. We also define and study the reticulation of a BL-algebra.
The problem of the additive generation of finitely-valued t-conorms.
The process of induction as a non-classical logic's double negation: evidence from classical scientific theories.
The Properties of Sets of Temporal Logic Subformulas
This is a second preliminary article to prove the completeness theorem of an extension of basic propositional temporal logic. We base it on the proof of completeness for basic propositional temporal logic given in [17]. We introduce two modified definitions of a subformula. In the former one we treat until-formula as indivisible. In the latter one, we extend the set of subformulas of until-formulas by a special disjunctive formula. This is needed to construct a temporal model. We also define an...
The pseudovariety is hyperdecidable
The recursive sets in certain monadic second order fragments of arithmetic.
The resolution tableau for logics of likelihood
The second-order monadic theory of the free inverse monoid is undecidable. (La théorie monadique du second ordre du monoïde inversif libre est indécidable.)