scholarly journals On the Lojasiewicz exponents of quasi-homogeneous functions

2015 ◽  
Author(s):  
Alain Haraux ◽  
Tien Son Pham
Keyword(s):  
Homogeneous Functions ◽  
Łojasiewicz Exponents

scholarly journals Robust Stabilization of Control Affine Systems with Homogeneous Functions

2020 ◽  
Vol 53 (2) ◽  
pp. 6311-6316
Author(s):  
Konstantin Zimenko ◽  
Andrey Polyakov ◽  
Denis Efimov
Keyword(s):  
Robust Stabilization ◽  
Affine Systems ◽  
Homogeneous Functions

scholarly journals XXV.—The Generating Function of the Reciprocal of a Determinant

1905 ◽  
Vol 40 (3) ◽  
pp. 615-629
Author(s):  
Thomas Muir
Keyword(s):  
Generating Function ◽  
Specific Character ◽  
Partial Fraction ◽  
Homogeneous Functions ◽  
Series Form ◽  
General Proposition ◽  
In Series ◽  
Image Position ◽  
The Given

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

scholarly journals THE THEORY OF SPECTRUM EXCHANGEABILITY

2014 ◽  
Vol 8 (1) ◽  
pp. 108-130
Author(s):  
E. HOWARTH ◽  
J. B. PARIS
Keyword(s):  
Probability Function ◽  
Inductive Logic ◽  
Probability Functions ◽  
Homogeneous Functions ◽  
Finite Structures ◽  
Spectrum Exchangeability

AbstractSpectrum Exchangeability, Sx, is an irrelevance principle of Pure Inductive Logic, and arguably the most natural (but not the only) extension of Atom Exchangeability to polyadic languages. It has been shown1 that all probability functions which satisfy Sx are comprised of a mixture of two essential types of probability functions; heterogeneous and homogeneous functions. We determine the theory of Spectrum Exchangeability, which for a fixed language L is the set of sentences of L which must be assigned probability 1 by every probability function satisfying Sx, by examining separately the theories of heterogeneity and homogeneity. We find that the theory of Sx is equal to the theory of finite structures, i.e., those sentences true in all finite structures for L, and it emerges that Sx is inconsistent with the principle of Super-Regularity (Universal Certainty). As a further consequence we are able to characterize those probability functions which satisfy Sx and the Finite Values Property.

scholarly journals NEW INEQUALITIES ON THE HOMOGENEOUS FUNCTIONS

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
V. Lokesha ◽  
K. M. Nagaraja ◽  
Y. Simsek
Keyword(s):  
Homogeneous Functions ◽  
New Inequalities
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