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On the quality of regular correlated pairs (R(K), R0(K)), measured by the value of the coefficient R2(k)

100%

Plebaniak J.

Econometrics. Ekonometria. Advances in Applied Data Analytics

2013

nr 3(41)

9-14

If a correlated pair (R(k), R0(k)) is a regular correlated pair, then the coefficient measuring the quality of such a pair satisfies the inequality: r2(k) ≥ F(k)...

Boolean matrices ... neither Boolean nor matrices

100%

Ricci G.

Discussiones Mathematicae - General Algebra and Applications

2000

tom 20

nr 1

141-151

Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we "show" that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A minor modification of this standard definition might fix it.

A semantic construction of two-ary integers

88%

Ricci G.

Discussiones Mathematicae - General Algebra and Applications

2005

tom 25

nr 2

165-219

To binary trees, two-ary integers are what usual integers are to natural numbers, seen as unary trees. We can represent two-ary integers as binary trees too, yet with leaves labelled by binary words and with a structural restriction. In a sense, they are simpler than the binary trees, they relativize. Hence, contrary to the extensions known from Arithmetic and Algebra, this integer extension does not make the starting objects more complex. We use a semantic construction to get this extension. This method differs from the algebraic ones, mainly because it is able to find equational features of the extended objects. Two-ary integers turn out to form the free algebra corresponding to the Jónsson-Tarski's "paradoxical" equations. This entails that they have a "sum" operation as well as other operations of higher dimensions. Two-ary integers can provide LISP memories with convenient direct access jumps and the above low complexity hints at feasible hardware implementations.

Sameness between based universal algebras

75%

Ricci G.

2009

tom Vol. 42, nr 1

3-22

This is the continuation of the paper "Transformations between Menger systems". To define when two universal algebras with bases "are the same", here we propose a universal notion of transformation that comes from a triple characterization concerning three representation facets: the determinations of the Menger system, analytic monoid and endomorphism representation corresponding to a basis. Hence, this notion consists of three equivalent definitions. It characterizes another technical variant and also the universal version of the very semi-linear transformations that were coordinate-free. Universal transformations allow us to check the actual invariance of general algebraic constructions, contrary to the seeming invariance of representation-free thinking. They propose a new interpretation of free algebras as superpositions of "analytic spaces" and deny that our algebras differ from vector spaces at fundamental stages. Contrary to present beliefs, even the foundation of abstract Linear Algebra turns out to be incomplete.

Konsekwencje majoryzowania macierzy korelacji przez macierz uniwersalną na tle nierównoci Hellwiga

75%

Kolupa M. , Plebaniak J.

2011

tom 58

nr 3-4

224-230

Transformations between Menger system

75%

Ricci G.

2008

tom Vol. 41, nr 4

743-762

To define transformations between based universal algebras we must introduce representations that depend on the bases, contrary to what was possible for general vector spaces and believed possible for universal algebras. In fact, a counterex-ample shows that by representation-free transformations alone one cannot even ascertain whether a universal algebra has any dimension or not. A transformation notion, which can do, concerns basis dependent Menger systems. It enjoys a basic geometric property of universal algebras, the preservation of reference flocks, and generalizes the transformation groups of Linear Algebra into groupoids.