Greed followed by avarice….We consider the variation in which events occur at a rate equal to the difference in capital of the two traders. That is, an individual is more likely to take capital from a much poorer person rather than from someone of slightly less wealth. For this “avaricious” exchange, the corresponding rate equations are
dck/dt = ck-1∑j=1k-1(k – 1 – j)cj + ck+1∑j=k+1∞(j – k – 1)cj – ck∑j=1∞|k – j|cj —– (1)
while the total density obeys,
dN/dt = -c1(1 – N) —– (2)
under the assumption that the total wealth density is set equal to one, ∑kck = 1
These equations can be solved by again applying scaling. For this purpose, it is first expedient to rewrite the rate equation as,
dck/dt = (ck-1 – ck)∑j=1k-1(k – j)cj – ck-1∑j=1k-1cj + (ck+1 – ck)∑j=k+1∞(j – k)cj – ck+1∑j=k+1∞cj —– (3)
taking the continuum limits
∂c/∂t = ∂c/∂k – N∂/∂k(kc) —— (3)
We now substitute the scaling ansatz,
ck(t) ≅ N2C(x), with x = kN to yield
C(0)[2C + xC′] = (x − 1)C′ + C —– (4)
dN/dt = -C(0)N2 —– (5)
Solving the above equations gives N ≅ [C(0)t]−1 and
C(x) = (1 + μ)(1 + μx)−2−1/μ —– (6)
with μ = C(0) − 1. The scaling approach has thus found a family of solutions which are parameterized by μ, and additional information is needed to determine which of these solutions is appropriate for our system. For this purpose, note that equation (6) exhibits different behaviors depending on the sign of μ. When μ > 0, there is an extended non-universal power-law distribution, while for μ = 0 the solution is the pure exponential, C(x) = e−x. These solutions may be rejected because the wealth distribution cannot extend over an unbounded domain if the initial wealth extends over a finite range.
The accessible solutions therefore correspond to −1 < μ < 0, where the distribution is compact and finite, with C(x) ≡ 0 for x ≥ xf = −μ−1. To determine the true solution, let us re-examine the continuum form of the rate equation, equation (3). From naive power counting, the first two terms are asymptotically dominant and they give a propagating front with kf exactly equal to t. Consequently, the scaled location of the front is given by xf = Nkf. Now the result N ≃ [C(0)t]−1 gives xf = 1/C(0). Comparing this expression with the corresponding value from the scaling approach, xf = [1 − C(0)]−1, selects the value C(0) = 1/2. Remarkably, this scaling solution coincides with the Fermi distribution that found for the case of constant interaction rate. Finally, in terms of the unscaled variables k and t, the wealth distribution is
ck(t) = 4/t2, k < t
= 0, k ≥ 0 —– (7)
This discontinuity is smoothed out by diffusive spreading. Another interesting feature is that if the interaction rate is sufficiently greedy, “gelation” occurs, whereby a finite fraction of the total capital is possessed by a single individual. For interaction rates, or kernels K(j, k) between individuals of capital j and k which do not give rise to gelation, the total density typically varies as a power law in time, while for gelling kernels N(t) goes to zero at some finite time. At the border between these regimes N(t) typically decays exponentially in time. We seek a similar transition in behavior for the capital exchange model by considering the rate equation for the density
dN/dt = -c1∑k=1∞k(1, k)ck —– (8)
For the family of kernels with K(1, k) ∼ kν as k → ∞, substitution of the scaling ansatz gives N ̇ ∼ −N3−ν. Thus N(t) exhibits a power-law behavior N ∼ t1/(2−ν) for ν < 2 and an exponential behavior for ν = 2. Thus gelation should arise for ν > 2.
आत्मा त्वं गिरिजा मतिः सहचराः प्राणाः शरीरं गृहं पूजा ते विषयोपभोगरचना निद्रा समाधिस्थितिः।
सञ्चारः पदयोः प्रदक्षिणविधिः स्तोत्राणि सर्वा गिरो यद्यत्कर्म करोमि तत्तदखिलं शम्भो तवाराधनम्॥
Ātmā tvaṃ Girijā matiḥ sahacarāḥ prāṇāḥ śarīraṃ gṛham
Pūjā te viṣayopabhoga-racanā nidrā samādhi-sthitiḥ |
Sañcāraḥ padayoḥ pradakṣiṇa-vidhiḥ stotrāṇi sarvā giraḥ
Yad-yat karma karomi tat-tad-akhilaṁ Śambho tavārādhanam ||
You (tvam) (are) the Self (ātmā) and Girijā –an epithet of Pārvatī, Śiva’s wife, meaning “mountain-born”– (girijā) (is) the intelligence (matiḥ). The vital energies (prāṇāḥ) (are Your)companions (sahacarāḥ). The body (śarīram) (is Your) house (gṛham). Worship (pūjā) of You (te) is prepared (racanā) with the objects (viṣaya) (known as sensual) enjoyments (upabhoga). Sleep (nidrā) (is Your) state (sthitiḥ) of Samādhi –i.e. perfect concentration or absorption– (samādhi). (My) wandering (sañcāraḥ) (is) the ceremony (vidhiḥ) of circumambulation from left to right (pradakṣiṇa) of (Your) feet (padayoḥ) –this act is generally done as a token of respect–. All (sarvāḥ) (my) words (giraḥ) (are) hymns of praise (of You) (stotrāṇi). Whatever (yad yad) action (karma) I do (karomi), all (akhilam) that (tad tad) is adoration (ārādhanam) of You (tava), oh Śambhu — an epithet of Śiva meaning “beneficent, benevolent”.
This Self is an embodiment of the Light of Consciousness; it is Śiva, free and autonomous. As an independent play of intense joy, the Divine conceals its own true nature [by manifesting plurality], and may also choose to reveal its fullness once again at any time. All that exists, throughout all time and beyond, is one infinite divine Consciousness, free and blissful, which projects within the field of its awareness a vast multiplicity of apparently differentiated subjects and objects: each object an actualization of a timeless potentiality inherent in the Light of Consciousness, and each subject the same plus a contracted locus of self-awareness. This creation, a divine play, is the result of the natural impulse within Consciousness to express the totality of its self-knowledge in action, an impulse arising from love. The unbounded Light of Consciousness contracts into finite embodied loci of awareness out of its own free will. When those finite subjects then identify with the limited and circumscribed cognitions and circumstances that make up this phase of their existence, instead of identifying with the transindividual overarching pulsation of pure Awareness that is their true nature, they experience what they call “suffering.” To rectify this, some feel an inner urge to take up the path of spiritual gnosis and yogic practice, the purpose of which is to undermine their misidentification and directly reveal within the immediacy of awareness the fact that the divine powers of Consciousness, Bliss, Willing, Knowing, and Acting comprise the totality of individual experience as well – thereby triggering a recognition that one’s real identity is that of the highest Divinity, the Whole in every part. This experiential gnosis is repeated and reinforced through various means until it becomes the nonconceptual ground of every moment of experience, and one’s contracted sense of self and separation from the Whole is finally annihilated in the incandescent radiance of the complete expansion into perfect wholeness. Then one’s perception fully encompasses the reality of a universe dancing ecstatically in the animation of its completely perfect divinity.”
Thus spoke André Weil,
Nothing is more fruitful – all mathematicians know it – than those obscure analogies, those disturbing reflections of one theory in another; those furtive caresses, those inexplicable discords; nothing also gives more pleasure to the researcher. The day comes when the illusion dissolves; the yoked theories reveal their common source before disappearing. As the Gita teaches, one achieves knowledge and indifference at the same time.
The notion of Weil algebra is ordinarily defined for a Lie algebra g. In mathematics, the Weil algebra of a Lie algebra g, introduced by Cartan based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ (g*) ⊗ S(g*) of its dual g*.
- how to map the real line smoothly into it,
- and how to map out of the space smoothly to the real line.
In the general context of space and quantity, Frölicher spaces take an intermediate symmetric position: they are both presheaves as well as copresheaves on their test domain (which here is the full subcategory of manifolds on the real line) and both of these structures determine each other.
After assigning, to each pair (X, W ) of a Frölicher space X and a Weil algebra W , another Frölicher space X ⊗ W , called the Weil prolongation of X with respect to W, which naturally extends to a bifunctor FS × W → FS, where FS is the category of Frölicher spaces and smooth mappings, and W is the category of Weil algebras. We also know
The functor · ⊗ W : FS → FS is product-preserving for any Weil algebra W.
A Frölicher space X is called Weil exponentiable if (X ⊗ (W1 ⊗∞ W2))Y = (X ⊗ W1)Y ⊗ W2 —– (1)
holds naturally for any Frölicher space Y and any Weil algebras W1 and W2. If Y = 1, then (1) degenerates into
X ⊗ (W1 ⊗∞ W2) = (X ⊗ W1) ⊗ W2 —– (2)
If W1 = R, then (1) degenerates into
(X ⊗ W2)Y = XY ⊗ W2 —– (3)
Proposition: Convenient vector spaces are Weil exponentiable.
Corollary: C∞-manifolds are Weil exponentiable.
Proposition: If X is a Weil exponentiable Frölicher space, then so is X ⊗ W for any Weil algebra W.
Proposition: If X and Y are Weil exponentiable Frölicher spaces, then so is X × Y.
Proposition: If X is a Weil exponentiable Frölicher space, then so is XY for any Frölicher space Y .
Theorem: Weil exponentiable Frölicher spaces, together with smooth mappings among them, form a Cartesian closed subcategory FSWE of the category FS.
Generally speaking, limits in the category FS are bamboozling. The notion of limit in FS should be elaborated geometrically.
A finite cone D in FS is called a transversal limit diagram providing that D ⊗ W is a limit diagram in FS for any Weil algebra W , where the diagram D ⊗ W is obtained from D by putting ⊗ W to the right of every object and every morphism in D. By taking W = R, we see that a transversal limit diagram is always a limit diagram. The limit of a finite diagram of Frölicher spaces is said to be transversal providing that its limit diagram is a transversal limit diagram.
Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then DX is also a transversal limit diagram for any Frölicher space X, where DX is obtained from D by putting X as the exponential over every object and every morphism in D.
Proof: Since the functor ·X : FS → FS preserves limits, we have DX ⊗ W = (D ⊗ W)X
for any Weil algebra W , so that we have the desired result.
Lemma: If D is a transversal limit diagram whose objects are all Weil exponentiable, then D ⊗ W is also a transversal limit diagram for any Weil algebra W.
Proof: Since the functor W ⊗∞ · : W → W preserves finite limits, we have (D ⊗ W) ⊗ W′ = D ⊗ (W ⊗∞ W′)
for any Weil algebra W′, so that we have the desired result.