The Hahn-Banach theorem and a restricted inductive definition

1981 ◽  
pp. 359-394
Author(s):  
Mariko Yasugi
Keyword(s):  
Inductive Definition ◽  
Banach Theorem

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Małgorzata Klimek ◽  
Marek Błasik
Keyword(s):  
Differential Equations ◽  
Arbitrary Order ◽  
Continuous Solution ◽  
Initial Value ◽  
Uniqueness Results ◽  
Banach Theorem ◽  
Arbitrary Partition ◽  
Particular Solution ◽  
Stationary Function ◽  
Fractional Differential

AbstractTwo-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.

Left amenability and left contractibility of Lau algebras

2014 ◽  
Vol 51 (3) ◽  
pp. 407-427
Author(s):  
Ali Jabbari
Keyword(s):  
Topological Semigroups ◽  
Banach Theorem

In this paper we study left amenability of Lau algebras by introducing left approximate diagonal and virtual diagonal for Lau algebras. Some results related to Hahn-Banach theorem property on foundation topological semigroups are obtained. We introduce the left contractibility of Lau algebras. Some examples for clarifying that left contractibility of Lau algebras is stronger than left amenability of them are given.

μ-definable sets of integers

1993 ◽  
Vol 58 (1) ◽  
pp. 291-313 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Fixed Point ◽  
Free Variable ◽  
Order Logic ◽  
First Order Logic ◽  
Inductive Definition ◽  
Natural Class ◽  
First Order ◽  
Inductive Definitions ◽  
Definable Sets ◽  
The Universe

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

Hahn–Banach Theorem

2020 ◽  
pp. 55-76
Author(s):  
P.N. Natarajan
Keyword(s):  
Banach Theorem
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