Object and Meaning
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(Images and Text by Emilio Charles Mazzie)Sun, 18 Nov 2018 14:59:11 +0000en-UShourly1http://foliotwist.com/?v=4.9.9Chapter One: Ode to Richie
https://www.gallerymazzie.com/2016/10/02/chapter-one-ode-to-richie/
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Hell is people! I should live in seclusion in the middle of nowhere and with only my few possessions. And it should be in the desert or the plains or the mountains where the sounds of the cicadas and the coyotes and the wolves are the only true sounds, those which can be felt and heard deep within the soul.
 
But too late. I am only food for flies now. Those tiny black truculent, beleaguering ones that hover about relentlessly as I try to reason and link the arguements together, over and over in my mind, again and again. “A choo-choo train, a choo-choo train”, my cousin Richie would whisper in concatenations ever so softly beneath his breath. How strange, I thought. It was barely perceptible and as though when he began to concentrate on the toy trains, it somehow, his mind, set up a terrible field all around him pulling the ananthema of the flies toward him. And the harder he concentrated, the stronger the field became.
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He was a valiant serpent too wise for the layers of his skins and yearned to shed the curse of them, scraping and twisting his writhing body against the grates of the tracks, as he struggled to go forward and tear the skins away from him.
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https://www.gallerymazzie.com/2016/10/02/chapter-one-ode-to-richie/feed/0Section 3. The Operant and Modes of Perception
https://www.gallerymazzie.com/2016/07/16/section-3-the-operant-and-modes-of-perception/
https://www.gallerymazzie.com/2016/07/16/section-3-the-operant-and-modes-of-perception/#respondSat, 16 Jul 2016 04:04:17 +0000http://gallerymazzie.com/2016/07/16/section-3-the-operant-and-modes-of-perception/read more]]>The proposition, section 2, asserts that there may be defined an infinite sequence of orthogonal polynomial functions of a single independent variable and of the classical kind, satisfying the boundary conditions that the functions vanish at the endpoints of the interval of their domain of the real line. The general expression describing any one polynomial of the sequence ascertained from the knowledge of the first five functions which satisfy their linear second-order differential equation in x of the Sturm-Liouville type. These, the principal generating functions, posited as deriving a priori in the mind from its operant and modes of perception.
 
For simplicity, let the domain of the functions y_n(x), n=1, 2, 3, 4, 5, correspond to that of the unit interval [0,1]. The first function, n=1, being a polynomial of degree two with two real zeros (one at each endpoint of the domain at zero and one) and one relative extremum with either the downward or upward concavity of a local maximum or minimum (Fig. 4):
 
y_1(x)=a_11X^2+a_12x+a_13.
 
The second function, n=2, being of degree three with three real zeros (one at each endpoint of the domain and one at one-half at the center ) and two alternating relative extrema beginning in the same way as the previous function (from left to right) with either the downward or upward concavity of a local maximum or minimum:
 
y_2(x)=a_21x^3+a_22x^2+a_23x+a_24.
 
The third function, n=3, being of degree four with four real zeros (one at each endpoint of the domain and one at one-third and two-thirds) and three alternating relative extrema beginning in the same way as the previous functions:
 
y_3(x)=a_31x^4+a_32x^3+a_33x^2+a_34x+a_35.
 
The symmetry pattern thus repeating for the remaining two functions, n=4, 5.
 
Aesthetically, the graphs of the functions which exhibit the following two properties:
 
(i) The zeros divide the unit interval into equal lengths.
 
(ii) The shapes of the graphs are symmetric with respect to the unit interval.
 
The boundary conditions, y_n(0)=y_n(1)=0, requiring that the constant terms and the sums of the coefficients of the polynomials vanish:
 
a_11+a_12=0,
a_21+a_22+a_23=0,
a_31+a_32+a_33+a_34=0,
a_41+a_42+a_43+a_44+a_45=0,
a_51+a_52+a_53+a_54+a_55+a_56=0.
 
The problem being to solve for the coefficients so that the polynomials are forced to assume the symmetry properties (i) and (ii). The characteristics of which manifest in analogy to the physical constraints of an object bound to the unit interval. The coefficients which then characterize the polynomials as representations of the object, their values encoded, through the modes of the object’s perception, in the operant set of two elements identified as the boundary points of zero and one. The modes which are applied under seven lines of reasoning to yield the first five orthogonal polynomials as the principal generating functions.
 
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https://www.gallerymazzie.com/2016/07/16/section-3-the-operant-and-modes-of-perception/feed/0Section 1. The Metakinetic and Graphical Image
https://www.gallerymazzie.com/2016/07/04/foreword/
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(i) The Metakinetic Image: the transcendent extension of the intellectual content; its purpose being to inspire the contemplation of its concept which derives a priori in the mind from its operant and modes of perception. It is, then, in relation to the content, an artifact of its literal representation which cannot be expressed in words or encoded in logical symbols and therefore conveyed through the stimulus of its kinetic, metaphysical imagery.
 
(ii) The Graphical Image: the literal extension of the intellectual content; its purpose being to illuminate its concept which derives a priori in the mind from its operant and modes of perception. It is, then, in relation to the content, the logical manifestation of its meaning encoded in the geometry and or mathematical symbols of its diagram.
 
 
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https://www.gallerymazzie.com/2016/07/04/foreword/feed/0Section 2. The Proposition
https://www.gallerymazzie.com/2016/06/23/1-introduction/
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Let the transformation of my soul lie within its infinite sequence of orthogonal polynomial functions. Those satisfying the boundary conditions that the functions vanish at the endpoints of the unit interval of their domain, and in being of the class of Legendre, Laguerre and Hermite, satisfy their second-order, ordinary differential equation as the characteristic functions of their boundary value problem:
A(x)y”+B(x)y’+k_ny=0.
 
Where A(x) is at most quadratic (e.g., ax^2+bx+c), B(x) is at most linear (e.g., mx+t) and k_n is the nth term of the sequence of characteristic values corresponding to the characteristic functions y=y_n(x), n=1,2, etc. The second and first derivatives of the functions denoted by y” and y’, respectively.
 
Knowing the first five polynomials of the sequence, y_1(x), …, y_5(x), these and their derivatives may be substituted into the differential equation in order to solve for the unknowns A(x), B(x) and k_n. The results of which give the exact form of the differential equation which may be solved to obtain the general expression describing any one polynomial of the infinite sequence.
 
The exact form of the differential equation and its solution which provide the key for determing all further analytical and metaphysical properties of the proposed functions as the constituents of the metakinetics of my soul. The first five polynomials posited as deriving a priori in the mind from its operant and modes of perception as the principal generating functions of the boundary value problem.
 
The operant being that which is given to the mind as the seed and catalyst of the transformation; the modes of perception being the faculties of the mind which instantiate the transformation.
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