Philosophizing Twistors via Fibration

The basic issue, is a question of so called time arrow. This issue is an important subject of examination in mathematical physics as well as ontology of spacetime and philosophical anthropology. It reveals crucial contradiction between the knowledge about time, provided by mathematical models of spacetime in physics and psychology of time and its ontology. The essence of the contradiction lies in the invariance of the majority of fundamental equations in physics with regard to the reversal of the direction of the time arrow (i. e. the change of a variable t to -t in equations). Neither metric continuum, constituted by the spaces of concurrency in the spacetime of the classical mechanics before the formulation of the Particular Theory of Relativity, the spacetime not having metric but only affine structure, nor Minkowski’s spacetime nor the GTR spacetime (pseudo-Riemannian), both of which have metric structure, distinguish the categories of past, present and future as the ones that are meaningful in physics. Every event may be located with the use of four coordinates with regard to any curvilinear coordinate system. That is what clashes remarkably with the human perception of time and space. Penrose realizes and understands the necessity to formulate such theory of spacetime that would remove this discrepancy. He remarked that although we feel the passage of time, we do not perceive the “passage” of any of the space dimensions. Theories of spacetime in mathematical physics, while considering continua and metric manifolds, cannot explain the difference between time dimension and space dimensions, they are also unable to explain by means of geometry the unidirection of the passage of time, which can be comprehended only by means of thermodynamics. The theory of spaces of twistors is aimed at better and crucial for the ontology of nature understanding of the problem of the uniqueness of time dimension and the question of time arrow. There are some hypotheses that the question of time arrow would be easier to solve thanks to the examination of so called spacetime singularities and the formulation of the asymmetric in time quantum theory of gravitation — or the theory of spacetime in microscale.

Although Lorentzian geometry is the mathematical framework of classical general relativity and can be seen as a good model of the world we live in, the theoretical-physics community has developed instead many models based on a complex space-time picture.

(1) When one tries to make sense of quantum field theory in flat space-time, one finds it very convenient to study the Wick-rotated version of Green functions, since this leads to well defined mathematical calculations and elliptic boundary-value problems. At the end, quantities of physical interest are evaluated by analytic continuation back to real time in Minkowski space-time.

(2) The singularity at r = 0 of the Lorentzian Schwarzschild solution disappears on the real Riemannian section of the corresponding complexified space-time, since r = 0 no longer belongs to this manifold. Hence there are real Riemannian four-manifolds which are singularity-free, and it remains to be seen whether they are the most fundamental in modern theoretical physics.

(3) Gravitational instantons shed some light on possible boundary conditions relevant for path-integral quantum gravity and quantum cosmology.  Unprimed and primed spin-spaces are not (anti-)isomorphic if Lorentzian space-time is replaced by a complex or real Riemannian manifold. Thus, for example, the Maxwell field strength is represented by two independent symmetric spinor fields, and the Weyl curvature is also represented by two independent symmetric spinor fields and since such spinor fields are no longer related by complex conjugation (i.e. the (anti-)isomorphism between the two spin-spaces), one of them may vanish without the other one having to vanish as well. This property gives rise to the so-called self-dual or anti-self-dual gauge fields, as well as to self-dual or anti-self-dual space-times.

(5) The geometric study of this special class of space-time models has made substantial progress by using twistor-theory techniques. The underlying idea is that conformally invariant concepts such as null lines and null surfaces are the basic building blocks of the world we live in, whereas space-time points should only appear as a derived concept. By using complex-manifold theory, twistor theory provides an appropriate mathematical description of this key idea.

A possible mathematical motivation for twistors can be described as follows.  In two real dimensions, many interesting problems are best tackled by using complex-variable methods. In four real dimensions, however, the introduction of two complex coordinates is not, by itself, sufficient, since no preferred choice exists. In other words, if we define the complex variables

z₁ ≡ x₁ + ix₂ —– (1)

z₂ ≡ x₃ + ix₄ —– (2)

we rely too much on this particular coordinate system, and a permutation of the four real coordinates x₁, x₂, x₃, x₄ would lead to new complex variables not well related to the first choice. One is thus led to introduce three complex variables u, z₁^u, z₂^u : the first variable u tells us which complex structure to use, and the next two are the

complex coordinates themselves. In geometric language, we start with the complex projective three-space P₃(C) with complex homogeneous coordinates (x, y, u, v), and we remove the complex projective line given by u = v = 0. Any line in P₃(C) − P₁(C) is thus given by a pair of equations

x = au + bv —– (3)

y = cu + dv —– (4)

In particular, we are interested in those lines for which c = −b^–, d = a^–. The determinant ∆ of (3) and (4) is thus given by

∆ = aa^– +bb^– + |a|² + |b|² —– (5)

which implies that the line given above never intersects the line x = y = 0, with the obvious exception of the case when they coincide. Moreover, no two lines intersect, and they fill out the whole of P₃(C) − P₁(C). This leads to the fibration P₃(C) − P₁(C) → R⁴ by assigning to each point of P₃(C) − P₁(C) the four coordinates Re(a), Im(a), Re(b), Im(b). Restriction of this fibration to a plane of the form

αu + βv = 0 —— (6)

yields an isomorphism C² ≅ R⁴, which depends on the ratio (α,β) ∈ P₁(C). This is why the picture embodies the idea of introducing complex coordinates.

∆=a

Such a fibration depends on the conformal structure of R⁴. Hence, it can be extended to the one-point compactification S⁴ of R⁴, so that we get a fibration P₃(C) → S⁴ where the line u = v = 0, previously excluded, sits over the point at ∞ of S⁴ = R⁴ ∪ ∞ . This fibration is naturally obtained if we use the quaternions H to identify C⁴ with H² and the four-sphere S⁴ with P₁(H), the quaternion projective line. We should now recall that the quaternions H are obtained from the vector space R of real numbers by adjoining three symbols i, j, k such that

i² = j² = k² =−1 —– (7)

ij = −ji = k,  jk = −kj =i,  ki = −ik = j —– (8)

Thus, a general quaternion ∈ H is defined by

x^– ≡ x₁ + x₂i + x₃j + x₄k —– (9)

where x₁, x₂, x₃, x₄ ∈ R⁴, whereas the conjugate quaternion x^– is given by

x^– ≡ x₁ – x₂i – x₃j – x₄k —– (10)

Note that conjugation obeys the identities

(xy)^– = y^– x^– —– (11)

xx^– = x^–x = ∑_μ=1⁴ x²_μ ≡ |x|² —– (12)

If a quaternion does not vanish, it has a unique inverse given by

x^-1 ≡ x^–/|x|² —– (13)

Interestingly, if we identify i with √−1, we may view the complex numbers C as contained in H taking x₃ = x₄ = 0. Moreover, every quaternion x as in (9) has a unique decomposition

x = z₁ + z₂j —– (14)

where z₁ ≡ x₁ + x₂i, z₂ ≡ x₃ + x₄i, by virtue of (8). This property enables one to identify H with C², and finally H² with C⁴, as we said following (6)

The map σ : P₃(C) → P₃(C) defined by

σ(x, y, u, v) = (−y^–, x^–, −v^–, u^–) —– (15)

preserves the fibration because c = −b^–, d = a^–, and induces the antipodal map on each fibre.

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Cobordism. Note Quote.

Objects and arrows in general categories can be very different from sets and functions. The category which quantum field theory concerns itself with is called nCob, the ‘n dimensional cobordism category’, and the rough definition is as follows. Objects are oriented closed (that is, compact and without boundary) (n − 1)-manifolds Σ, and arrows M : Σ → Σ′ are compact oriented n-manifolds M which are cobordisms from Σ to Σ′. Composition of cobordisms M : Σ → Σ′ and N : Σ′ → Σ′′ is defined by gluing M to N along Σ′.

Let M be an oriented n-manifold with boundary ∂M. Then one assigns an induced orientation to the connected components Σ of ∂M by the following procedure. For x ∈ Σ, let (v₁,…,v_n−1,v_n) be a positive basis for T_xM chosen in such a way that (v₁,…,v_n−1) ∈ T_xΣ. It makes sense to ask whether v_n points inward or outward from M. If it points inward, then an orientation for Σ is defined by specifying that (v₁, . . . , v_n−1) is a positive basis for T_xΣ. If M is one dimensional, then x ∈ ∂M is defined to have positive orientation if a positive vector in T_xM points into M, otherwise it is defined to have negative orientation.

Let Σ and Σ′ be closed oriented (n − 1)-manifolds. An cobordism from Σ to Σ′ is a compact oriented n-manifold M together with smooth maps

∑ →ⁱ M ^i’←∑’

where i is a orientation preserving diffeomorphism of Σ onto i(Σ) ⊂ ∂M, i′ is an orientation reversing diffeomorphism of Σ′ onto i′(Σ′) ⊂ ∂M, such that i(Σ) and i′(Σ′) (called the in- and out-boundaries respectively) are disjoint and exhaust ∂M. Observe that the empty set φ can be considered as an (n − 1)-manifold.

One can view M as interpolating from Σ to Σ′. An important property of cobordisms is that they can be glued together. Let M : Σ₀ → Σ1 and M′ : Σ₁ → Σ₂ be cobordisms,

∑₀ →^i₀ M ^i₁ ← ∑₁             ∑₁ →^i’₁ M ^i₂ ← ∑₂

Then we can form a composite cobordism M′ ◦ M : Σ → Σ by gluing M to M′ using i′ ◦ i − 1 : ∂M → ∂M′:

Austrian Economics. Ruminations. End Part.

Mainstream economics originates from Jevons’ and Menger’s marginal utility and Walras’ and Marshall’s equilibrium approach. While their foundations are similar, their presentation looks quite different, according to the two schools which typically represent these two approaches: the Austrian school initiated by Menger and the general equilibrium theory initiated by Walras. An important, albeit only formal, difference is that the former presents economic theory mainly in a literary form using ordinary logic, while the latter prefers mathematical expressions and logic.

Lachmann, who excludes determinism from economics since acts of mind are concerned, connects determinism with the equilibrium approach. However, equilibrium theory is not necessarily deterministic, also because it does not establish relationships of succession, but only relationships of coexistence. In this respect, equilibrium theory is not more deterministic than the theory of the Austrian school. Even though the Austrian school does not comprehensively analyze equilibrium, all its main results strictly depend on the assumption that the economy is in equilibrium (intended as a state everybody prefers not to unilaterally deviate from, not necessarily a competitive equilibrium). Considering both competition and monopoly, Menger examines the market for only two commodities in a barter economy. His analysis is the best to be obtained without using mathematics, but it is too limited for determining all the implications of the theory. For instance, it is unclear how the market for a specific commodity is affected by the conditions of the markets for other commodities. However, interdependence is not excluded by the Austrian school. For instance, Böhm-Bawerk examines at length the interdependence between the markets for labor and capital. Despite the incomplete analysis of equilibrium carried out by the Austrian school, many of its results imply that the economy is in equilibrium, as shown by the following examples.

a) The Gossen-Menger loss principle. This principle states that the price of a good can be determined by analyzing the effect of the loss (or the acquisition) of a small quantity of the same good.

b) Wieser’s theory of imputation. Wieser’s theory of imputation attempts to determine the value of the goods used for production in terms of the value (marginal utility) of the consumption goods produced.

c) Böhm-Bawerk’s theory of capital. Böhm-Bawerk proposed a longitudinal theory of capital, where production consists of a time process. A sequence of inputs of labor is employed in order to obtain, at the final stage, a given consumption good. Capital goods, which are the products obtained in the intermediate stages, are seen as a kind of consumption goods in the process of maturing.

A historically specific theory of capital inspired by the Austrian school focuses on the way profit-oriented enterprises organize the allocation of goods and resources in capitalism. One major issue is the relationship between acquisition and production. How does the homogeneity of money figures that entrepreneurs employ in their acquisitive plans connect to the unquestionable heterogeneity of the capital goods in production that these monetary figures depict? The differentiation between acquisition and production distinguishes this theory from the neoclassical approach to capital. The homogeneity of the money figures on the level of acquisition that is important to such a historically specific theory is not due to the assumption of equilibrium, but simply to the existence of money prices. It is real-life homogeneity, so to speak. It does not imply any homogeneity on the level of production, but rather explains the principle according to which the production process is conducted.

In neoclassical economics, in contrast, production and acquisition, the two different levels of analysis, are not separated but are amalgamated by means of the vague term “value”. In equilibrium, assets are valued according to their marginal productivity, and therefore their “value” signifies both their price and their importance to the production process. Capital understood in this way, i.e., as the value of capital goods, can take on the “double meaning of money or goods”. By concentrating on the value of capital goods, the neoclassical approach assumes homogeneity not only on the level of acquisition with its input and output prices, but also on the level of production. The neoclassical approach to capital assumes that the valuation process has already been accomplished. It does not explain how assets come to be valued originally according to their marginal product. In this, an elaborated historically specific theory of capital would provide the necessary tools. In capitalism, inputs and outputs are interrelated by entrepreneurs who are guided by price signals. In their efforts to maximize their monetary profits, they aim to benefit from the spread between input and output prices. Therefore, money tends to be invested where this spread appears to be wide enough to be worth the risk. In other words, business capital flows to those industries and businesses where it yields the largest profit. Competition among entrepreneurs brings about a tendency for price spreads to diminish. The prices of the factors of production are bid up and the prices of the output are bid down until, in the hypothetical state of equilibrium, the factor prices sum up to the price of the product. A historically specific theory of capital is able to describe and analyze the market process that results – or tends to result – in marginal productivity prices, and can therefore also formulate positions concerning endogenous and exogenous misdirections of this process which lead to disequilibrium prices. Consider Mises,

In balance sheets and in profit-and-loss statements, […] it is necessary to enter the estimated money equivalent of all assets and liabilities other than cash. These items should be appraised according to the prices at which they could probably be sold in the future or, as is especially the case with equipment for production processes, in reference to the prices to be expected in the sale of merchandise manufactured with their aid.

According to this, not the monetary costs of the assets, which can be verified unambiguously, but their values are supposed to be the basis of entrepreneurial calculation. As the words indicate, this procedure involves a tremendous amount of uncertainty and can therefore only lead to fair values if equilibrium conditions are assumed.

Deanonymyzing ToR

My anonymity is maintained in Tor as long as no single entity can link me to my destination. If an attacker controls the entry and the exit of my circuit, her anonymity can be compromised, as the attacker is able to perform traffic or timing analysis to link my traffic to the destination. For hidden services, this implies that the attacker needs to control the two entry guards used for the communication between the client and the hidden service. This significantly limits the attacker, as the probability that both the client and the hidden service select a malicious entry guard is much lower than the probability that only one of them makes a bad choice.

Our goal is to show that it is possible for a local passive adversary to deanonymize users with hidden service activities without the need to perform end-to-end traffic analysis. We assume that the attacker is able to monitor the traffic between the user and the Tor network. The attacker’s goal is to identify that a user is either operating or connected to a hidden service. In addition, the attacker then aims to identify the hidden service associated with the user.

In order for our attack to work effectively, the attacker needs to be able to extract circuit-level details such as the lifetime, number of incoming and outgoing cells, sequences of packets, and timing information. We discuss the conditions under which our assumptions are true for the case of a network admin/ISP and an entry guard.

Network administrator or ISP: A network administrator (or ISP) may be interested in finding out who is accessing a specific hidden service, or if a hidden service is being run from the network. Under some conditions, such an attacker can extract circuit-level knowledge from the TCP traces by monitoring all the TCP connections between me and  my entry guards. For example, if only a single active circuit is used in every TCP connection to the guards, the TCP segments will be easily mapped to the corresponding Tor cells. While it is hard to estimate how often this condition happens in the live network, as users have different usage models, we argue that the probability of observing this condition increases over time.

Malicious entry guard: Entry guard status is bestowed upon relays in the Tor network that offer plenty of bandwidth and demonstrate reliable uptime for a few days or weeks. To become one an attacker only needs to join the network as a relay, keep their head down and wait. The attacker can now focus their efforts to deanonymise users and hidden services on a much smaller amount of traffic. The next step is to observe the traffic and identify what’s going on inside it – something the researchers achieved with technique called website fingerprinting. Because each web page is different the network traffic it generates as it’s downloaded is different too. Even if you can’t see the content inside the traffic you can identify the page from the way it passes through the network, if you’ve seen it before. Controlling entry guards allows the adversary to perform the attack more realistically and effectively. Entry guards are in a perfect position to perform our traffic analysis attacks since they have full visibility to Tor circuits. In today’s Tor network, each OP chooses 3 entry guards and uses them for 45 days on average, after which it switches to other guards. For circuit establishment, those entry guards are chosen with equal probability. Every entry guard thus relays on average 33.3% of a user’s traffic, and relays 50% of a user’s traffic if one entry guard is down. Note that Tor is currently considering using a single fast entry guard for each user. This will provide the attacker with even better circuit visibility which will exacerbate the effectiveness of our attack. This adversary is shown in the figure below:

The Tor project has responded to the coverage generated by the research with an article of its own written by Roger Dingledine, Tor’s project leader and one of the project’s original developers. Fingerprinting home pages is all well and good he suggests, but hidden services aren’t just home pages:

…is their website fingerprinting classifier actually accurate in practice? They consider a world of 1000 front pages, but ahmia.fi and other onion-space crawlers have found millions of pages by looking beyond front pages. Their 2.9% false positive rate becomes enormous in the face of this many pages – and the result is that the vast majority of the classification guesses will be mistakes.

Austrian Economics. Some More Further Ruminations. Part 3.

The dominant British tradition received its first serious challenge in many years when Carl Menger’s Principles of Economics was published in 1871. Menger, the founder of the Austrian School proper, resurrected the Scholastic-French approach to economics, and put it on firmer ground.

Menger spelled out the subjective basis of economic value, and fully explained, for the first time, the theory of marginal utility (the greater the number of units of a good that an individual possesses, the less he will value any given unit). In addition, Menger showed how money originates in a free market when the most marketable commodity is desired, not for consumption, but for use in trading for other goods. Menger restored economics as the science of human action based on deductive logic, and prepared the way for later theorists to counter the influence of socialist thought. Indeed, his student Friederich von Wieser strongly influenced Friedrich von Hayek’s later writings.

Menger’s admirer and follower at the University of Innsbruck, Eugen Böhm-Bawerk, took Menger’s exposition, reformulated it, and applied it to a host of new problems involving value, price, capital, and interest. His History and Critique of Interest Theories, appearing in 1884, is a sweeping account of fallacies in the history of thought and a firm defense of the idea that the interest rate is not an artificial construct but an inherent part of the market. It reflects the universal fact of “time preference,” the tendency of people to prefer satisfaction of wants sooner rather than later.

Böhm-Bawerk’s Positive Theory of Capital demonstrated that the normal rate of business profit is the interest rate. Capitalists save money, pay laborers, and wait until the final product is sold to receive profit. In addition, he demonstrated that capital is not homogeneous but an intricate and diverse structure that has a time dimension. A growing economy is not just a consequence of increased capital investment, but also of longer and longer processes of production.

Böhm-Bawerk favored policies that deferred to the ever-present reality of economic law. He regarded interventionism as an attack on market economic forces that cannot succeed in the long run. But one area where Böhm-Bawerk had not elaborated on the analysis of Menger was money, the institutional intersection of the “micro” and “macro” approach. A young Ludwig von Mises, economic advisor to the Austrian Chamber of Commerce, took on the challenge.

The result of Mises’s research was The Theory of Money and Credit, published in 1912. He spelled out how the theory of marginal utility applies to money, and laid out his “regression theorem,” showing that money not only originates in the market, but must always do so. Drawing on the British Currency School, Knut Wicksell’s theory of interest rates, and Böhm-Bawerk’s theory of the structure of production, Mises presented the broad outline of the Austrian theory of the business cycle. To note once again, his was not a theory of the physical capital, but a theory of interest. So, even if some of the economists of the school had covered through their writings the complexities of the structure of production, that wasn’t really their research object, but rather what their concentration really opted for was interest phenomenon, trade cycle or entrepreneurship.

Ludwig Lachmann in his Capital and its Structure is most serious about the complexities of the structure of production, especially on the heterogeneity of physical capital not only in relation to successive stages of production, but denying any possibility of systematically categorizing, measuring or aggregating capital goods. But, does that mean he is from a different camp? Evidently not, since much of his discussion contains an important contribution to the historically specificity of capital, in that the heterogenous is not itself the research object, but only a problem statement for the theory of the entrepreneur. Says he,

For most purposes capital goods have to be used jointly. complementarity is of the essence of capital use. but the heterogenous capital resources do not lend themselves to combination in any arbitrary fashion. For any given number of them only certain modes of complementarity are technically possible, and only a few of these are economically significant. It is among the latter that the entrepreneur has to find the ‘optimum combination’.

for him, the true function of the entrepreneur must remain hidden as long as we disregard the heterogeneity of capital. But, Peter Lewin’s Capital in Disequilibrium reads Lachmann revealingly. What makes it possible for entrepreneurs to make production plans comprising numerous heterogenous capital goods is a combination of the market process and the institution of money and financial accounting. There, you can see Lachmann slipping into the historical territory. Says Lewin,

Planning within firms proceeds against the necessary backdrop of the market. Planning within firms can occur precisely because “the market” furnishes it with the necessary prices for the factor inputs that would be absent in a fullblown state ownership situation.

Based on these prices, the institution of monetary calculation allows entrepreneurs to calculate retrospective and prospective profits. The calculation of profits, Lewin states, is “indispensable in that it provides the basis for discrimination between viable and non-viable production projects.” The approach is not concerned with the heterogeneity of capital goods as such but, to the contrary, with the way these goods are made homogeneous so that entrepreneurs can make the calculations their production plans are based on. Without this homogeneity of capital goods in relation to the goal of the entrepreneur – making monetary profit – it would be difficult, if not impossible, to combine them in a meaningful way.

 

 

Catastrophe Revisited. Note Quote.

Transversality and structural stability are the topics of Thom’s important transversality and isotopy theorems; the first one says that transversality is a stable property, the second one that transverse crossings are themselves stable. These theorems can be extended to families of functions: If f: Rⁿ x R^r–>R is equivalent to any family f + p: Rⁿ x R^r–>R, where p is a sufficiently small family Rⁿ x R^r–> R, then f is structurally stable. There may be individual functions with degenerate critical points in such a family, but these exceptions from the rule are in a sense “checked” by the other family members. Such families can be obtained e.g. by parametrizing the original function with one or several extra variables. Thom’s classification theorem, comes in at this level.

So, in a given state function, catastrophe theory separates between two kinds of functions: one “Morse” piece, containing the nondegenerate critical points, and one piece, where the (parametrized) family contains at least one degenerate critical point. The second piece has two sets of variables; the state variables (denoted x, y…) responsible for the critical points, and the control variables or parameters (denoted a, b, c…), capable of stabilizing a degenerate critical point or steering away from it to nondegenerate members of the same function family. Each control parameter can control the degenerate point only in one direction; the more degenerate a singular point is (the number of independent directions equal to the corank), the more control parameters will be needed. The number of control parameters needed to stabilize a degenerate point (“the universal unfolding of the singularity” with the same dimension as the number of control parameters) is called the codimension of the system. With these considerations in mind, keeping close to surveyable, four-dimensional spacetime, Thom defined an “elementary catastrophe theory” with seven elementary catastrophes, where the number of state variables is one or two: x, y, and the number of control parameters, equal to the codimension, at most four: a, b, c, d. (With five parameters there will be eleven catastrophes). The tool used here is the above mentioned classification theorem, which lists all possible organizing centers (quadratic, cubic forms etc.) in which there are stable unfoldings (by means of control parameters acting on state variables). 

Two elementary catastrophes: fold and cusp

1. In the first place the classification theorem points out the simple potential function y = x³ as a candidate for study. It has a degenerate critical point at {0, 0} and is always declining (with minus sign), needing an addition from the outside in order to grow locally. All possible perturbations of this function are essentially of type x³ + x or type x³ – x (more generally x³ + ax); which means that the critical point (y, x = 0) is of codimension one. Fig. below shows the universal unfolding of the organizing centre y = x³, the fold:

This catastrophe, says Thom, can be interpreted as “the start of something” or “the end of something”, in other words as a “limit”, temporal or spatial. In this particular case (and only in this case) the complete graph in internal (x) and external space (y) with the control parameter a running from positive to negative values can be shown in a three-dimensional graph (Fig. below); it is evident why this catastrophe is called “fold”:

One point should be stressed already at this stage, it will be repeated again later on. In “Topological models…”, Thom remarks on the “informational content” of the degenerate critical point: 

This notion of universal unfolding plays a central role in our biological models. To some extent, it replaces the vague and misused term of ‘information’, so frequently found in the writings of geneticists and molecular biologists. The ‘information’ symbolized by the degenerate singularity V(x) is ‘transcribed’, ‘decoded’ or ‘unfolded’ into the morphology appearing in the space of external variables which span the universal unfolding family of the singularity V(x). 

2. Next let us as organizing centre pick the second potential function pointed out by the classification theorem: y = x⁴. It has a unique minimum (0, 0), but it is not generic , since nearby potentials can be of a different qualitative type, e.g. they can have two minima. But the two-parameter function x⁴ + ax² + bx is generic and contains all possible unfoldings of y = x⁴. The graph of this function, with four variables: y, x, a, b, can not be shown, the display must be restricted to three dimensions. The obvious way out is to study the derivative f'(x) = 4x³ + 2ax + b for y = 0 and in the proximity of x = 0. It turns out, that this derivative has the qualities of the fold, shown in the Fig. below; the catastrophes are like Chinese boxes, one contained within the next of the hierarchy. 

Finally we look for the position of the degenerate critical points projected on (a,b)-space, this projection has given the catastrophe its name: the “cusp” (Fig. below). (An arrowhead or a spearhead is a cusp). The edges of the cusp, the bifurcation set, point out the catastrophe zone, above the area between these limits the potential has two Morse minima and one maximum, outside the cusp limits there is one single Morse minimum. With the given configuration (the parameter a perpendicular to the axis of the cusp) a is called the normal factor – since x will increase continuously with a if b < 0, while b is called the splitting factor because the fold surface is split into two sheets if b > 0. If the control axes are instead located on either side of the cusp (A = b + a and B = b – a) A and B are called conflicting factors; A tries to push the result to the upper sheet (attractor), B to the lower sheet of the fold. (Here is an “inlet” for truly external factors; it is well-known how e.g. shadow or excessive light affects the morphogenetic process of plants. 

Thom states: the cusp is a pleat, a fault, its temporal interpretation is “to separate, to unite, to capture, to generate, to change”. Countless attempts to model bimodal distributions are connected with the cusp, it is the most used (and maybe the most misused) of the elementary catastrophes. 

Zeeman has treated stock exchange and currency behaviour from one and the same model, namely what he terms the cusp catastrophe with a slow feedback. Here the rate of change of indexes (or currencies) is considered as dependent variable, while different buying patterns (“fundamental” /N in fig. below and “chartist” /S in fig. below) serve as normal and splitting parameters. Zeeman argues: the response time of X to changes in N and S is much faster than the feedback of X on N and S, so the flow lines will be almost vertical everywhere. If we fix N and S, X will seek a stable equilibrium position, an attractor surface (or: two attractor surfaces, separated by a repellor sheet and “connected” by catastrophes; one sheet is inflation/bull market, one sheet deflation/bear market, one catastrophe collapse of market or currency. Note that the second catastrophe is absent with the given flow direction. This is important, it tells us that the whole pattern can be manipulated, “adapted” by means of feedbacks/flow directions). Close to the attractor surface, N and S become increasingly important; there will be two horizontal components, representing the (slow) feedback effects of N and S on X. The whole sheet (the fold) is given by the equation X³ – (S – S_o)X – N = 0, the edge of the cusp by 3X² + S_o = S, which gives the equation 4(S – S_o)³ = 27 N² for the bifurcation curve. 

Figure: “Cusp with a slow feedback”, according to Zeeman (1977). X, the state variable, measures the rate of change of an index, N = normal parameter, S = splitting parameter, the catastrophic behaviour begins at S_o. On the back part of the upper sheet, N is assumed constant and dS/dt positive, on the fore part dN/dT is assumed to be negative and dS/dt positive; this gives the flow direction of the feedback. On the fore part of the lower sheet both dN/dt and dS/dt are assumed to be negative, on the back part dN/dt is assumed to be positive and dS/dt still negative, this gives the flow direction of feedback on this sheet. The cusp projection on the {N,S}-plane is shaded grey, the visible part of the repellor sheet black. (The reductionist character of these models must always be kept in mind; here two obvious key parameters are considered, while others of a weaker or more ephemeral kind – e.g. interest levels – are ignored.)