On a class of inequalities

1963 ◽  
Vol 3 (4) ◽  
pp. 442-448 ◽  
Author(s):  
B. C. Rennie
Keyword(s):  
General Theory ◽  
Geometric Mean ◽  
Schwarz Inequality ◽  
Image Position

First consider some familiar results, the inequality of the arithmetic and geometric mean is: Kantorovich's inequality (reference [1]) asserts that if 0 < A ≦f(x) ≦ B then: The Cauchy-Schwarz inequality is: This paper discusses a certain class of inequalities which includes the three above. Three theorems are proved which apply to any inequality of this class; then follow some examples. They are mainly to show how the general theory helps in the finding of inequalities, but the result of Example 1 seems worth reporting for its own sake.

On the centres of hereditary JBW-subalgebras of a JBW-algebra

1979 ◽  
Vol 85 (2) ◽  
pp. 317-324 ◽  
Author(s):  
C. M. Edwards
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Jordan Algebra ◽  
Identity Element ◽  
Image Position

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions thatandfor all elements a and b in A. It follows from (1.1) and (l.2) thatfor all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition thatfor all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

SUBSTRUCTURAL INQUISITIVE LOGICS

2019 ◽  
Vol 12 (2) ◽  
pp. 296-330 ◽  
Author(s):  
VÍT PUNČOCHÁŘ
Keyword(s):  
Fuzzy Logic ◽  
General Theory ◽  
Propositional Logic ◽  
Substructural Logics ◽  
Substructural Logic ◽  
Fuzzy Logics ◽  
Inquisitive Semantics ◽  
Semantic Framework ◽  
Noncommutative Version ◽  
Image Position

AbstractThis paper shows that any propositional logic that extends a basic substructural logic BSL (a weak, nondistributive, nonassociative, and noncommutative version of Full Lambek logic with a paraconsistent negation) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as ${\lambda ^?}$, for any logic λ that is at least as strong as BSL. A general theory of these “inquisitive extensions” is worked out. In particular, it is shown how to axiomatize ${\lambda ^?}$, given the axiomatization of λ. Furthermore, the general theory is applied to some prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, and t-norm based fuzzy logics, including for example Łukasiewicz fuzzy logic Ł. For the inquisitive extensions of these logics, axiomatization is provided and a suitable semantics found.

Instability theorems for certain fifth-order differential equations

1978 ◽  
Vol 84 (2) ◽  
pp. 343-350 ◽  
Author(s):  
J. O. C. Ezeilo
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Trivial Solution ◽  
Real Root ◽  
Order Differential Equation ◽  
Positive Real Part ◽  
Auxiliary Equation ◽  
Positive Real ◽  
Image Position ◽  
Fifth Order

1. Consider the constant-coefficient fifth-order differential equation:It is known from the general theory that the trivial solution of (1·1) is unstable if, and only if, the associated (auxiliary) equation:has at least one root with a positive real part. The existence of such a root naturally depends on (though not always all of) the coefficients a1, a2,…, a5. For example, ifit is clear from a consideration of the fact that the sum of the roots of (1·2) equals ( – a1) that at least one root of (1·2) has a positive real part for arbitrary values of a2,…, a5. A similar consideration, combined with the fact that the product of the roots of (1·2) equals ( – a5) will show that at least one root of (1·2) has a positive real part iffor arbitrary a2, a3 and a4. The condition a1 = 0 here in (1·4) is however superfluous whenfor then X(0) = a5 < 0 and X(R) > 0 if R > 0 is sufficiently large thus showing that there is a positive real root of (1·2) subject to (1·5) and for arbitrary a1, a2, a3 and a4.

An Inequality for Elementary Symmetric Functions

1972 ◽  
Vol 15 (1) ◽  
pp. 133-135 ◽  
Author(s):  
K. V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Geometric Mean ◽  
Binomial Coefficient ◽  
Elementary Symmetric Function ◽  
Elementary Symmetric Functions ◽  
Image Position

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by1E0 = 1 and Er=0(r>m).We define the rth symmetric mean by2where denote the binomial coefficient. If α1 α2,…,αm are positive reals thenwe have two well-known inequalities3and4In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

scholarly journals On the General Theory of Differentiable Manifolds with Almost Tangent Structure

1965 ◽  
Vol 8 (6) ◽  
pp. 721-748 ◽  
Author(s):  
Hermes A. Eliopoulos
Keyword(s):  
General Theory ◽  
Complex Structure ◽  
Differentiable Manifold ◽  
Linear Operators ◽  
Identity Operator ◽  
Almost Complex Structure ◽  
Differentiable Manifolds ◽  
Almost Complex ◽  
Image Position ◽  
Almost Tangent Structure

Some of the most important G-Structures of the first kind [1] are those defined by linear operators satisfying algebraic relations. If the linear operator J acting on the complexified space of a differentiable manifold V satisfies a relation of the formwhere I is the identity operator, the manifold has an almost complex structure ([2] [3]). The structures defined byare the almost product structures ([3] [4]).

The Norm of the Lp-Fourier Transform, II

1976 ◽  
Vol 28 (6) ◽  
pp. 1121-1131 ◽  
Author(s):  
Bernard Russo
Keyword(s):  
Fourier Transform ◽  
General Theory ◽  
Measure Space ◽  
Locally Compact ◽  
Unimodular Group ◽  
Image Position

Let G be a locally compact separable unimodular group. The general theory [18] assigns to G a measure space (Λ, μ) whose points ƛ index a family of unitary factor representations of G in such a way that if U ƛ corresponds to ƛ and thenfor all .

An Inequality Characterizes the Trace

1979 ◽  
Vol 31 (6) ◽  
pp. 1322-1328 ◽  
Author(s):  
L. Terrell Gardner
Keyword(s):  
Triangle Inequality ◽  
Operator Algebras ◽  
Schwarz Inequality ◽  
Unit Element ◽  
Integration Theory ◽  
List Type ◽  
Bounded Operators ◽  
Linear Maps ◽  
Image Position ◽  
Equivalent Conditions

1. Introduction. While analogues of the Schwarz inequality have been much studied in the context of positive linear maps of operator algebras ([1], [2], [6], [7], [10]) the simpler triangle inequality |ϕ(x)| ≦ (|x|) has been neglected, outside of (possibly non-commutative) integration theory—perhaps partly because except for the important and familiar example of traces, scalar maps satisfying the triangle inequality are rarely encountered. In fact we here prove that they are never encountered: every such map is a trace.For C*-algebras (norm-closed self-ad joint algebras of bounded operators on a Hilbert space) this means, for instance, that if the linear functional ϕ on the C*-algebra satisfies(†)then ϕ satisfies also the equivalent conditions (i) ϕ(xy) = ϕ(yx) for all x, y in ;(ii) ϕ(x*x) = ϕ(xx*) for all x in ;(iii) ϕ(x) = ϕ(uxu*) for all x in and all unitary u in Ae, the C*-lgebra formed from by adjunction of a unit element.

scholarly journals On limit problems associated with some inequalities

1973 ◽  
Vol 14 (2) ◽  
pp. 123-127
Author(s):  
P. H. Diananda
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Limit Problems ◽  
Generalized Mean ◽  
Image Position ◽  
Necessary And Sufficient

Let {an} be a sequence of non-negative real numbers. Suppose thatThen M1,n is the arithmetic mean, MO,n the geometric mean, and Mr,n the generalized mean of order r, of a1, a2, …, an. By a result of Everitt [1] and McLaughlin and Metcalf [5], {n(Mr,n–Ms,n)}, where r ≧ l ≧ s, is a monotonic increasing sequence. It follows that this sequence tends to a finite or an infinite limit as n → ∞. Everitt [2, 3] found a necessary and sufficient condition for the finiteness of this limit in the cases r, s = 1, 0 and r ≧ 1 > s > 0. His results are included in the following theorem.

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