Catastrophe, Gestalt and Thom’s Natural Philosophy of 3-D Space as Underlying All Abstract Forms – Thought of the Day 157.0

The main result of mathematical catastrophe theory consists in the classification of unfoldings (= evolutions around the center (the germ) of a dynamic system after its destabilization). The classification depends on two sorts of variables:

(a) The set of internal variables (= variables already contained in the germ of the dynamic system). The cardinal of this set is called corank,

(b) the set of external variables (= variables governing the evolution of the system). Its cardinal is called codimension.

The table below shows the elementary catastrophes for Thom:

The A-unfoldings are called cuspoids, the D-unfoldings umbilics. Applications of the E-unfoldings have only been considered in A geometric model of anorexia and its treatment. By loosening the condition for topological equivalence of unfoldings, we can enlarge the list, taking in the family of double cusps (X₉) which has codimension 8. The unfoldings A₃(the cusp) and A₅ (the butterfly) have a positive and a negative variant A₊₃, A_-3, A₊₅, A_-5.

We obtain Thorn’s original list of seven “catastrophes” if we consider only unfoldings up to codimension 4 and without the negative variants of A₃ and A₅.

Thom argues that “gestalts” are locally con­stituted by maximally four disjoint constituents which have a common point of equilibrium, a common origin. This restriction is ultimately founded in Gibb’s law of phases, which states that in three-dimensional space maximally four independent systems can be in equilibrium. In Thom’s natural philosophy, three-dimensional space is underlying all abstract forms. He, therefore, presumes that the restriction to four constituents in a “gestalt” is a kind of cognitive universal. In spite of the plausibility of Thom’s arguments there is a weaker assumption that the number of constituents in a gestalt should be finite and small. All unfoldings with codimension (i.e. number of external variables) smaller than or equal to 5 have simple germs. The unfoldings with corank (i.e. number of internal variables) greater than two have moduli. As a matter of fact the most prominent semantic archetypes will come from those unfoldings considered by René Thom in his sketch of catastrophe theoretic semantics.

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Graviton Fields Under Helicity Rotations. Thought of the Day 156.0

Einstein described gravity as equivalent to curves in space and time, but physicists have long searched for a theory of gravitons, its putative quantum-scale source. Though gravitons are individually too weak to detect, most physicists believe the particles roam the quantum realm in droves, and that their behavior somehow collectively gives rise to the macroscopic force of gravity, just as light is a macroscopic effect of particles called photons. But every proposed theory of how gravity particles might behave faces the same problem: upon close inspection, it doesn’t make mathematical sense. Calculations of graviton interactions might seem to work at first, but when physicists attempt to make them more exact, they yield gibberish – an answer of “infinity.” This is the disease of quantized gravity. With regard to the exchange particles concept in the quantum electrodynamics theory and the existence of graviton, let’s consider a photon that is falling in the gravitational field, and revert back to the behavior of a photon in the gravitational field. But when we define the graviton relative to the photon, it is necessary to explain the properties and behavior of photon in the gravitational field. The fields around a “ray of light” are electromagnetic waves, not static fields. The electromagnetic field generated by a photon is much stronger than the associated gravitational field. When a photon is falling in the gravitational field, it goes from a low layer to a higher layer density of gravitons. We should assume that the graviton is not a solid sphere without any considerable effect. Graviton carries gravity force, so it is absorbable by other gravitons; in general; gravitons absorb each other and combine. This new view on graviton shows, identities of graviton changes, in fact it has mass with changeable spin.

When we derive various supermultiplets of states, at the noninteracting level, these states can easily be described in terms of local fields. But, at the interacting level, there are certain ambiguities that withdraw as a result of different field representations describing the same massless free states. So the proper choice of the field representation may be subtle. The supermultiplets can then be converted into supersymmetric actions, quadratic in the fields. For selfdual tensor fields, the action must be augmented by a duality constraint on the corresponding field strength. For the graviton field,

The linearized Einstein equation for g_μν = η_μν + κh_μν implies that (for D ≥ 3)

R_μν ∝ ∂²h_μν + ∂_μ∂_νh – ∂_μ∂^ρh_νρ – ∂_ν∂^ρh_ρμ = 0 —– (1)

where h ≡ h_μμ and R_μν is the Ricci tensor. To analyze the number of states implied by this equation, one may count the number of plane-wave solutions with given momentum q^μ. It then turns out that there are D arbitrary solutions, corresponding to the linearized gauge invariance h_μν → h_μν + ∂_μξ_ν + ∂_νξ_μ, which can be discarded. Many other components vanish and the only nonvanishing ones require the momentum to be lightlike. Thee reside in the fields h_ij, where the components i, j are in the transverse (D-2) dimensional subspace. In addition, the trace of h_ij must be zero. Hence, the relevant plane-wave solutions are massless and have polarizations (helicities) characterized by a symmetric traceless 2-rank tensor. This tensor comprises 1/2D(D-3), which transform irreducibly under the SO(D-2) helicity group of transverse rotations. For the special case of D = 6 spacetime dimensions, the helicity group is SO(4), which factorizes into two SU(2) groups. The symmetric traceless representation then transforms as a doublet under each of the SU(2) factors and it is thus denoted by (2,2). As for D = 3, there are obviously no dynamic degrees of freedom associated with the gravitational field. When D = 2 there are again no dynamic degrees of freedom, but here (1) should be replaced by R_μν = 1/2g_μνR.

Incomplete Markets and Calibrations for Coherence with Hedged Portfolios. Thought of the Day 154.0

 

In complete market models such as the Black-Scholes model, probability does not really matter: the “objective” evolution of the asset is only there to define the set of “impossible” events and serves to specify the class of equivalent measures. Thus, two statistical models P₁ ∼ P₂ with equivalent measures lead to the same option prices in a complete market setting.

This is not true anymore in incomplete markets: probabilities matter and model specification has to be taken seriously since it will affect hedging decisions. This situation is more realistic but also more challenging and calls for an integrated approach between option pricing methods and statistical modeling. In incomplete markets, not only does probability matter but attitudes to risk also matter: utility based methods explicitly incorporate these into the hedging problem via utility functions. While these methods are focused on hedging with the underlying asset, common practice is to use liquid call/put options to hedge exotic options. In incomplete markets, options are not redundant assets; therefore, if options are available as hedging instruments they can and should be used to improve hedging performance.

While the lack of liquidity in the options market prevents in practice from using dynamic hedges involving options, options are commonly used for static hedging: call options are frequently used for dealing with volatility or convexity exposures and for hedging barrier options.

What are the implications of hedging with options for the choice of a pricing rule? Consider a contingent claim H and assume that we have as hedging instruments a set of benchmark options with prices C_i^∗, i = 1 . . . n and terminal payoffs H_i, i = 1 . . . n. A static hedge of H is a portfolio composed from the options H_i, i = 1 . . . n and the numeraire, in order to match as closely as possible the terminal payoff of H:

H = V₀ + ∑_i=1ⁿ x_iH_i + ∫₀^T φdS + ε —– (1)

where ε is an hedging error representing the nonhedgeable risk. Typically H_i are payoffs of call or put options and are not possible to replicate using the underlying so adding them to the hedge portfolio increases the span of hedgeable claims and reduces residual risk.

Consider a pricing rule Q. Assume that E_Q[ε] = 0 (otherwise E_Q[ε] can be added to V₀). Then the claim H is valued under Q as:

e^-rTE_Q[H] = V₀ ∑_i=1ⁿ x_i e^-rTE_Q[H_i] —– (2)

since the stochastic integral term, being a Q-martingale, has zero expectation. On the other hand, the cost of setting up the hedging portfolio is:

V₀ + ∑_i=1ⁿ x_i C_i^∗ —– (3)

So the value of the claim given by the pricing rule Q corresponds to the cost of the hedging portfolio if the model prices of the benchmark options H_i correspond to their market prices C_i^∗:

∀i = 1, …, n

e^-rTE_Q[H_i] = C_i^∗ —– (4)

This condition is called calibration, where a pricing rule verifies the calibration of the option prices C_i^∗, i = 1, . . . , n. This condition is necessary to guarantee the coherence between model prices and the cost of hedging with portfolios and if the model is not calibrated then the model price for a claim H may have no relation with the effective cost of hedging it using the available options H_i. If a pricing rule Q is specified in an ad hoc way, the calibration conditions will not be verified, and thus one way to ensure them is to incorporate them as constraints in the choice of the pricing measure Q.

Self-Financing and Dynamically Hedged Portfolio – Robert Merton’s Option Pricing. Thought of the Day 153.0

As an alternative to the riskless hedging approach, Robert Merton derived the option pricing equation via the construction of a self-financing and dynamically hedged portfolio containing the risky asset, option and riskless asset (in the form of money market account). Let Q_S(t) and Q_V(t) denote the number of units of asset and option in the portfolio, respectively, and M_S(t) and M_V(t) denote the currency value of Q_S(t) units of asset and Q_V(t) units of option, respectively. The self-financing portfolio is set up with zero initial net investment cost and no additional funds are added or withdrawn afterwards. The additional units acquired for one security in the portfolio is completely financed by the sale of another security in the same portfolio. The portfolio is said to be dynamic since its composition is allowed to change over time. For notational convenience, dropping the subscript t for the asset price process S_t, the option value process V_t and the standard Brownian process Z_t. The portfolio value at time t can be expressed as

Π(t) = M_S(t) + M_V(t) + M(t) = Q_S(t)S + Q_V(t)V + M(t) —– (1)

where M(t) is the currency value of the riskless asset invested in a riskless money market account. Suppose the asset price process is governed by the stochastic differential equation (1) in here, we apply the Ito lemma to obtain the differential of the option value V as:

dV = ∂V/∂t dt + ∂V/∂S dS + σ²/2 S² ∂²V/∂S² dt = (∂V/∂t + μS ∂V/∂S σ²/2 S² ∂²V/∂S²)dt + σS ∂V/∂S dZ —– (2)

If we formally write the stochastic dynamics of V as

dV/V = μ_V dt + σ_V dZ —– (3)

then μ_V and σ_V are given by

μ_V = (∂V/∂t + ρS ∂V/∂S + σ²/2 S² ∂²V/∂S²)/V —– (4)

and

σ_V = (σS ∂V/∂S)/V —– (5)

The instantaneous currency return dΠ(t) of the above portfolio is attributed to the differential price changes of asset and option and interest accrued, and the differential changes in the amount of asset, option and money market account held. The differential of Π(t) is computed as:

dΠ(t) = [Q_S(t) dS + Q_V(t) dV + rM(t) dt] + [S dQ_S(t) + V dQ_V(t) + dM(t)] —– (6)

where rM(t)dt gives the interest amount earned from the money market account over dt and dM(t) represents the change in the money market account held due to net currency gained/lost from the sale of the underlying asset and option in the portfolio. And if the portfolio is self-financing, the sum of the last three terms in the above equation is zero. The instantaneous portfolio return dΠ(t) can then be expressed as:

dΠ(t) = Q_S(t) dS + Q_V(t) dV + rM(t) dt = M_S(t) dS/S + M_V(t) dV/V +  rM(t) dt —– (7)

Eliminating M(t) between (1) and (7) and expressing dS/S and dV/V in terms of their stochastic dynamics, we obtain

dΠ(t) = [(μ − r)M_S(t) + (μ_V − r)M_V(t)]dt + [σM_S(t) + σ_V M_V(t)]dZ —– (8)

How can we make the above self-financing portfolio instantaneously riskless so that its return is non-stochastic? This can be achieved by choosing an appropriate proportion of asset and option according to

σM_S(t) + σ_V M_V(t) = σS Q_S(t) + σS ∂V/∂S Q_V(t) = 0

that is, the number of units of asset and option in the self-financing portfolio must be in the ratio

Q_S(t)/Q_V(t) = -∂V/∂S —– (9)

at all times. The above ratio is time dependent, so continuous readjustment of the portfolio is necessary. We now have a dynamic replicating portfolio that is riskless and requires zero initial net investment, so the non-stochastic portfolio return dΠ(t) must be zero.

(8) becomes

0 = [(μ − r)M_S(t) + (μ_V − r)M_V(t)]dt

substituting the ratio factor in the above equation, we get

(μ − r)S ∂V/∂S = (μ_V − r)V —– (10)

Now substituting μ_V from (4) into the above equation, we get the black-Scholes equation for V,

∂V/∂t + σ²/2 S² ∂²V/∂S² + rS ∂V/∂S – rV = 0

Suppose we take Q_V(t) = −1 in the above dynamically hedged self-financing portfolio, that is, the portfolio always shorts one unit of the option. By the ratio factor, the number of units of risky asset held is always kept at the level of ∂V/∂S units, which is changing continuously over time. To maintain a self-financing hedged portfolio that constantly keeps shorting one unit of the option, we need to have both the underlying asset and the riskfree asset (money market account) in the portfolio. The net cash flow resulting in the buying/selling of the risky asset in the dynamic procedure of maintaining ∂V/∂S units of the risky asset is siphoned to the money market account.

Derivative Pricing Theory: Call, Put Options and “Black, Scholes'” Hedged Portfolio.Thought of the Day 152.0

Fischer Black and Myron Scholes revolutionized the pricing theory of options by showing how to hedge continuously the exposure on the short position of an option. Consider the writer of a call option on a risky asset. S/he is exposed to the risk of unlimited liability if the asset price rises above the strike price. To protect the writer’s short position in the call option, s/he should consider purchasing a certain amount of the underlying asset so that the loss in the short position in the call option is offset by the long position in the asset. In this way, the writer is adopting the hedging procedure. A hedged position combines an option with its underlying asset so as to achieve the goal that either the asset compensates the option against loss or otherwise. By adjusting the proportion of the underlying asset and option continuously in a portfolio, Black and Scholes demonstrated that investors can create a riskless hedging portfolio where the risk exposure associated with the stochastic asset price is eliminated. In an efficient market with no riskless arbitrage opportunity, a riskless portfolio must earn an expected rate of return equal to the riskless interest rate.

Black and Scholes made the following assumptions on the financial market.

1. Trading takes place continuously in time.
2. The riskless interest rate r is known and constant over time.
3. The asset pays no dividend.
4. There are no transaction costs in buying or selling the asset or the option, and no taxes.
5. The assets are perfectly divisible.
6. There are no penalties to short selling and the full use of proceeds is permitted.
7. There are no riskless arbitrage opportunities.

The stochastic process of the asset price S_t is assumed to follow the geometric Brownian motion

dS_t/S_t = μ dt + σ dZ_t —– (1)

where μ is the expected rate of return, σ is the volatility and Z_t is the standard Brownian process. Both μ and σ are assumed to be constant. Consider a portfolio that involves short selling of one unit of a call option and long holding of Δ_t units of the underlying asset. The portfolio value Π (S_t, t) at time t is given by

Π = −c + Δ_t S_t —– (2)

where c = c(S_t, t) denotes the call price. Note that Δ_t changes with time t, reflecting the dynamic nature of hedging. Since c is a stochastic function of S_t, we apply the Ito lemma to compute its differential as follows:

dc = ∂c/∂t dt + ∂c/∂S_t dS_t + σ²/2 S_t² ∂²c/∂S_t² dt

such that

-dc + Δ_t dS_t = (-∂c/∂t – σ²/2 S_t² ∂²c/∂S_t²)dt + (Δ_t – ∂c/∂S_t)dS_t

= [-∂c/∂t – σ²/2 S_t² ∂²c/∂S_t² + (Δ_t – ∂c/∂S_t)μS_t]dt + (Δ_t – ∂c/∂S_t)σS_t dZ_t

The cumulative financial gain on the portfolio at time t is given by

G(Π (S_t, t )) = ∫₀^t -dc + ∫₀^t Δ_u dS_u

= ∫₀^t [-∂c/∂u – σ²/2 S_u² ∂²c/∂S_u² + (Δ_u – ∂c/∂S_u)μS_u]du + ∫₀^t (Δ_u – ∂c/∂S_u)σS_u dZ_u —– (3)

The stochastic component of the portfolio gain stems from the last term, ∫₀^t (Δ_u – ∂c/∂S_u)σS_u dZ_u. Suppose we adopt the dynamic hedging strategy by choosing Δ_u = ∂c/∂S_u at all times u < t, then the financial gain becomes deterministic at all times. By virtue of no arbitrage, the financial gain should be the same as the gain from investing on the risk free asset with dynamic position whose value equals -c + S_u∂c/∂S_u. The deterministic gain from this dynamic position of riskless asset is given by

M_t = ∫₀^tr(-c + S_u∂c/∂S_u)du —– (4)

By equating these two deterministic gains, G(Π (S_t, t)) and M_t, we have

-∂c/∂u – σ²/2 S_u² ∂²c/∂S_u² = r(-c + S_u∂c/∂S_u), 0 < u < t

which is satisfied for any asset price S if c(S, t) satisfies the equation

∂c/∂t + σ²/2 S² ∂²c/∂S² + rS∂c/∂S – rc = 0 —– (5)

This parabolic partial differential equation is called the Black–Scholes equation. Strangely, the parameter μ, which is the expected rate of return of the asset, does not appear in the equation.

To complete the formulation of the option pricing model, let’s prescribe the auxiliary condition. The terminal payoff at time T of the call with strike price X is translated into the following terminal condition:

c(S, T ) = max(S − X, 0) —– (6)

for the differential equation.

Since both the equation and the auxiliary condition do not contain ρ, one concludes that the call price does not depend on the actual expected rate of return of the asset price. The option pricing model involves five parameters: S, T, X, r and σ. Except for the volatility σ, all others are directly observable parameters. The independence of the pricing model on μ is related to the concept of risk neutrality. In a risk neutral world, investors do not demand extra returns above the riskless interest rate for bearing risks. This is in contrast to usual risk averse investors who would demand extra returns above r for risks borne in their investment portfolios. Apparently, the option is priced as if the rates of return on the underlying asset and the option are both equal to the riskless interest rate. This risk neutral valuation approach is viable if the risks from holding the underlying asset and option are hedgeable.

The governing equation for a put option can be derived similarly and the same Black–Scholes equation is obtained. Let V (S, t) denote the price of a derivative security with dependence on S and t, it can be shown that V is governed by

∂V/∂t + σ²/2 S² ∂²V/∂S² + rS∂V/∂S – rV = 0 —– (7)

The price of a particular derivative security is obtained by solving the Black–Scholes equation subject to an appropriate set of auxiliary conditions that model the corresponding contractual specifications in the derivative security.

The original derivation of the governing partial differential equation by Black and Scholes focuses on the financial notion of riskless hedging but misses the precise analysis of the dynamic change in the value of the hedged portfolio. The inconsistencies in their derivation stem from the assumption of keeping the number of units of the underlying asset in the hedged portfolio to be instantaneously constant. They take the differential change of portfolio value Π to be

dΠ =−dc + Δ_t dS_t,

which misses the effect arising from the differential change in Δ_t. The ability to construct a perfectly hedged portfolio relies on the assumption of continuous trading and continuous asset price path. It has been commonly agreed that the assumed Geometric Brownian process of the asset price may not truly reflect the actual behavior of the asset price process. The asset price may exhibit jumps upon the arrival of a sudden news in the financial market. The interest rate is widely recognized to be fluctuating over time in an irregular manner rather than being constant. For an option on a risky asset, the interest rate appears only in the discount factor so that the assumption of constant/deterministic interest rate is quite acceptable for a short-lived option. The Black–Scholes pricing approach assumes continuous hedging at all times. In the real world of trading with transaction costs, this would lead to infinite transaction costs in the hedging procedure.

Network Theoretic of the Fermionic Quantum State – Epistemological Rumination. Thought of the Day 150.0

In quantum physics, fundamental particles are believed to be of two types: fermions or bosons, depending on the value of their spin (an intrinsic ‘angular moment’ of the particle). Fermions have half-integer spin and cannot occupy a quantum state (a configuration with specified microscopic degrees of freedom, or quantum numbers) that is already occupied. In other words, at most one fermion at a time can occupy one quantum state. The resulting probability that a quantum state is occupied is known as the Fermi-Dirac statistics.

Now, if we want to convert this into a model with maximum entropy, where the real movement is defined topologically, then we require a reproduction of heterogeneity that is observed. The starting recourse is network theory with an ensemble of networks where each vertex i has the same degree k_i as in the real network. This choice is justified by the fact that, being an entirely local topological property, the degree is expected to be directly affected by some intrinsic (non-topological) property of vertices. The caveat is that the real shouldn’t be compared with the randomized, which could otherwise lead to interpreting the observed as ‘unavoidable’ topological constraints, in the sense that the violation of the observed values would lead to an ‘impossible’, or at least very unrealistic values.

The resulting model is known as the Configuration Model, and is defined as a maximum-entropy ensemble of graphs with given degree sequence. The degree sequence, which is the constraint defining the model, is nothing but the ordered vector k of degrees of all vertices (where the i^th component k_i is the degree of vertex i). The ordering preserves the ‘identity’ of vertices: in the resulting network ensemble, the expected degree ⟨k_i⟩ of each vertex i is the same as the empirical value k_i for that vertex. In the Configuration Model, the graph probability is given by

P(A) = ∏_i<jq_ij(a_ij) =  ∏_i<jp_ij^a_ij (1 – p_ij)^1-a_ij —– (1)

where q_ij(a) = p_ij^{a (1 – p_ij)1-a} is the probability that particular entry of the adjacency matrix A takes the value a_ij = a, which is a Bernoulli process with different pairs of vertices characterized by different connection probabilities p_ij. A Bernoulli trial (or Bernoulli process) is the simplest random event, i.e. one characterized by only two possible outcomes. One of the two outcomes is referred to as the ‘success’ and is assigned a probability p. The other outcome is referred to as the ‘failure’, and is assigned the complementary probability 1 − p. These probabilities read

⟨a_ij⟩ = p_ij = (x_ix_j)/(1 + x_ix_j) —– (2)

where x_i is the Lagrange multiplier obtained by ensuring that the expected degree of the corresponding vertex i equals its observed value: ⟨k_i⟩ = k_i ∀ i. As always happens in maximum-entropy ensembles, the probabilistic nature of configurations implies that the constraints are valid only on average (the angular brackets indicate an average over the ensemble of realizable networks). Also note that p_ij is a monotonically increasing function of x_i and x_j. This implies that ⟨k_i⟩ is a monotonically increasing function of x_i. An important consequence is that two variables i and j with the same degree k_i = kj must have the same value x_i = x_j.

(2) provides an interesting connection with quantum physics, and in particular the statistical mechanics of fermions. The ‘selection rules’ of fermions dictate that only one particle at a time can occupy a single-particle state, exactly as each pair of vertices in binary networks can be either connected or disconnected. In this analogy, every pair i, j of vertices is a ‘quantum state’ identified by the ‘quantum numbers’ i and j. So each link of a binary network is like a fermion that can be in one of the available states, provided that no two objects are in the same state. (2) indicates the expected number of particles/links in the state specified by i and j. With no surprise, it has the same form of the so-called Fermi-Dirac statistics describing the expected number of fermions in a given quantum state. The probabilistic nature of links allows also for the presence of empty states, whose occurrence is now regulated by the probability coefficients (1 − p_ij). The Configuration Model allows the whole degree sequence of the observed network to be preserved (on average), while randomizing other (unconstrained) network properties. now, when one compares the higher-order (unconstrained) observed topological properties with their expected values calculated over the maximum-entropy ensemble, it should be indicative of the fact that the degree of sequence is informative in explaining the rest of the topology, which is a consequent via probabilities in (2). Colliding these into a scatter plot, the agreement between model and observations can be simply assessed as follows: the less scattered the cloud of points around the identity function, the better the agreement between model and reality. In principle, a broadly scattered cloud around the identity function would indicate the little effectiveness of the chosen constraints in reproducing the unconstrained properties, signaling the presence of genuine higher-order patterns of self-organization, not simply explainable in terms of the degree sequence alone. Thus, the ‘fermionic’ character of the binary model is the mere result of the restriction that no two binary links can be placed between any two vertices, leading to a mathematical result which is formally equivalent to the one of quantum statistics.

Lie-Dragging Sections Vectorially. Thought of the Day 149.0

Generalized vector fields over a bundle are not vector fields on the bundle in the standard sense; nevertheless, one can drag sections along them and thence define their Lie derivative. The formal Lie derivative on a bundle may be seen as a generalized vector field. Furthermore, generalized vector fields are objects suitable to describe generalized symmetries.

Let B = (B, M, π; F) be a bundle, with local fibered coordinates (x^μ; yⁱ). Let us consider the pull-back of the tangent bundle  τ_B: T_B → B along the map π^k₀: J^kB → B:

A generalized vector field of order k over B is a section Ξ of the fibre bundle π^* : π^k₀ : ^*T_B → J^kB, i.e.

for each section σ: M → B, one can define Ξ_σ = i ○ Ξ ○ j^kσ: M → TB, which is a vector field over the section σ. Generalized vector fields of order k = 0 are ordinary vector fields over B. Locally, Ξ(x^μ, yⁱ, …, yⁱ_μ1,…μk) is given the form:

Ξ = ξ^μ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_μ + ξⁱ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_i

which, for k ≠ 0, is not an ordinary vector field on B due to the dependence of the components (ξ^μ, ξⁱ) on the derivative of fields. Once one computes it on a section σ, then the pulled-back components depend just on the basic coordinates (x^μ) so that Ξ_σ is a vector field over the section σ, in the standard sense. Thus, generalized vector fields over B do not preserve the fiber structure of B.

A generalized projectable vector field of order k over the bundle B is a generalized vector field Ξ over B which projects on to an ordinary vector field ξ = ξ^μ(x)∂_μ on the base. Locally, a generalized projectable vector field over B is in the form:

Ξ = ξ^μ(x^μ)∂_μ + ξⁱ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_i

As a particular case, one can define generalized vertical vector fields (of order k) over B, which are locally of the form:

Ξ = ξⁱ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_i

In particular, for any section σ of B and any generalized vertical vector field Ξ over B, one can define a vertical vector field over σ given by:

Ξ_σ = ξⁱ(x^μ, σⁱ(x),…, ∂_{μ1,…, μk}σⁱ(x))∂_i

If Ξ = ξ^μ∂_μ + ξⁱ∂_i is a generalized projectable vector field, then Ξ_(v) = (ξⁱ – yⁱ_μξ^μ)∂_i = ξⁱ_(v)∂_i is a generalized vertical vector field, where Ξ_(v) is called the vertical part of Ξ.

If σ’: ℜ x M → B is a smooth map such that for any fixed s ∈ ℜ σ_s(x) = σ'(s, x): M → B is a global section of B. The map σ’ as well as the family {σ_s}, is then called a 1-parameter family of sections. In other words, a suitable restriction of the family σ_s, is a homotopic deformation with s ∈ ℜ of the central section σ = σ₀. Often one restricts it to a finite (open) interval, conventionally (- 1, 1) (or (-ε, ε) if “small” deformations are considered). Analogous definitions are given for the homotopic families of sections over a fixed open subset U ⊆ M or on some domain D ⊂ M (possibly with values fixed at the boundary ∂D, together with any number of their derivatives).

A 1-parameter family of sections σ_s is Lie-dragged along a generalized projectable vector field Ξ iff

(Ξ_(v))_σs = d/ds σ_s

thus dragging the section.

Symmetrical – Asymmetrical Dialectics Within Catastrophical Dynamics. Thought of the Day 148.0

Catastrophe theory has been developed as a deterministic theory for systems that may respond to continuous changes in control variables by a discontinuous change from one equilibrium state to another. A key idea is that system under study is driven towards an equilibrium state. The behavior of the dynamical systems under study is completely determined by a so-called potential function, which depends on behavioral and control variables. The behavioral, or state variable describes the state of the system, while control variables determine the behavior of the system. The dynamics under catastrophe models can become extremely complex, and according to the classification theory of Thom, there are seven different families based on the number of control and dependent variables.

Let us suppose that the process y_t evolves over t = 1,…, T as

dy_t = -dV(y_t; α, β)dt/dy_t —– (1)

where V (y_t; α, β) is the potential function describing the dynamics of the state variable y_t controlled by parameters α and β determining the system. When the right-hand side of (1) equals zero, −dV (y_t; α, β)/dy_t = 0, the system is in equilibrium. If the system is at a non-equilibrium point, it will move back to its equilibrium where the potential function takes the minimum values with respect to y_t. While the concept of potential function is very general, i.e. it can be quadratic yielding equilibrium of a simple flat response surface, one of the most applied potential functions in behavioral sciences, a cusp potential function is defined as

−V(y_t; α, β) = −1/4y_t⁴ + 1/2βy_t² + αy_t —– (2)

with equilibria at

-dV(y_t; α, β)dt/dy_t = -y_t³ + βy_t + α —– (3)

being equal to zero. The two dimensions of the control space, α and β, further depend on realizations from i = 1 . . . , n independent variables x_i,t. Thus it is convenient to think about them as functions

α_x = α₀ +α₁x_1,t +…+ α_nx_n,t —– (4)

β_x = β₀ + β₁x_1,t +…+ β_nx_n,t —– (5)

The control functions α_x and β_x are called normal and splitting factors, or asymmetry and bifurcation factors, respectively and they determine the predicted values of y_t given x_i,t. This means that for each combination of values of independent variables there might be up to three predicted values of the state variable given by roots of

-dV(y_t; α_x, β_x)dt/dy_t = -y_t³ + βy_t + α = 0 —– (6)

This equation has one solution if

δ_x = 1/4α_x² − 1/27β_x³ —– (7)

is greater than zero, δ_x > 0 and three solutions if δ_x < 0. This construction can serve as a statistic for bimodality, one of the catastrophe flags. The set of values for which the discriminant is equal to zero, δ_x = 0 is the bifurcation set which determines the set of singularity points in the system. In the case of three roots, the central root is called an “anti-prediction” and is least probable state of the system. Inside the bifurcation, when δ_x < 0, the surface predicts two possible values of the state variable which means that the state variable is bimodal in this case.

Most of the systems in behavioral sciences are subject to noise stemming from measurement errors or inherent stochastic nature of the system under study. Thus for a real-world applications, it is necessary to add non-deterministic behavior into the system. As catastrophe theory has primarily been developed to describe deterministic systems, it may not be obvious how to extend the theory to stochastic systems. An important bridge has been provided by the Itô stochastic differential equations to establish a link between the potential function of a deterministic catastrophe system and the stationary probability density function of the corresponding stochastic process. Adding a stochastic Gaussian white noise term to the system

dy_t = -dV(y_t; α_x, β_x)dt/dy_t + σ_ytdW_t —– (8)

where -dV(y_t; α_x, β_x)dt/dy_t is the deterministic term, or drift function representing the equilibrium state of the cusp catastrophe, σ_yt is the diffusion function and W_t is a Wiener process. When the diffusion function is constant, σ_yt = σ, and the current measurement scale is not to be nonlinearly transformed, the stochastic potential function is proportional to deterministic potential function and probability distribution function corresponding to the solution from (8) converges to a probability distribution function of a limiting stationary stochastic process as dynamics of y_t are assumed to be much faster than changes in x_i,t. The probability density that describes the distribution of the system’s states at any t is then

f_s(y|x) = ψ exp((−1/4)y⁴ + (β_x/2)y² + α_xy)/σ —– (9)

The constant ψ normalizes the probability distribution function so its integral over the entire range equals to one. As bifurcation factor β_x changes from negative to positive, the f_s(y|x) changes its shape from unimodal to bimodal. On the other hand, α_x causes asymmetry in f_s(y|x).

Morphism of Complexes Induces Corresponding Morphisms on Cohomology Objects – Thought of the Day 146.0

Let A = Mod(R) be an abelian category. A complex in A is a sequence of objects and morphisms in A

… → M^i-1 →^di-1 Mⁱ →^di → Mⁱ⁺¹ → …

such that dⁱ ◦ d^i-1 = 0 ∀ i. We denote such a complex by M.

A morphism of complexes f : M → N is a sequence of morphisms fⁱ : Mⁱ → Nⁱ in A, making the following diagram commute, where dⁱ_M, dⁱ_N denote the respective differentials:

We let C(A) denote the category whose objects are complexes in A and whose morphisms are morphisms of complexes.

Given a complex M of objects of A, the i^th cohomology object is the quotient

Hⁱ(M) = ker(dⁱ)/im(dⁱ⁻¹)

This operation of taking cohomology at the i^th place defines a functor

Hⁱ(−) : C(A) → A,

since a morphism of complexes induces corresponding morphisms on cohomology objects.

Put another way, an object of C(A) is a Z-graded object

M = ⊕_i Mⁱ

of A, equipped with a differential, in other words an endomorphism d: M → M satisfying d² = 0. The occurrence of differential graded objects in physics is well-known. In mathematics they are also extremely common. In topology one associates to a space X a complex of free abelian groups whose cohomology objects are the cohomology groups of X. In algebra it is often convenient to replace a module over a ring by resolutions of various kinds.

A topological space X may have many triangulations and these lead to different chain complexes. Associating to X a unique equivalence class of complexes, resolutions of a fixed module of a given type will not usually be unique and one would like to consider all these resolutions on an equal footing.

A morphism of complexes f: M → N is a quasi-isomorphism if the induced morphisms on cohomology

Hⁱ(f): Hⁱ(M) → Hⁱ(N) are isomorphisms ∀ i.

Two complexes M and N are said to be quasi-isomorphic if they are related by a chain of quasi-isomorphisms. In fact, it is sufficient to consider chains of length one, so that two complexes M and N are quasi-isomorphic iff there are quasi-isomorphisms

M ← P → N

For example, the chain complex of a topological space is well-defined up to quasi-isomorphism because any two triangulations have a common resolution. Similarly, all possible resolutions of a given module are quasi-isomorphic. Indeed, if

0 → S →^f M⁰ →^d0 M¹ →^d1 M² → …

is a resolution of a module S, then by definition the morphism of complexes

is a quasi-isomorphism.

The objects of the derived category D(A) of our abelian category A will just be complexes of objects of A, but morphisms will be such that quasi-isomorphic complexes become isomorphic in D(A). In fact we can formally invert the quasi-isomorphisms in C(A) as follows:

There is a category D(A) and a functor Q: C(A) → D(A)

with the following two properties:

(a) Q inverts quasi-isomorphisms: if s: a → b is a quasi-isomorphism, then Q(s): Q(a) → Q(b) is an isomorphism.

(b) Q is universal with this property: if Q′ : C(A) → D′ is another functor which inverts quasi-isomorphisms, then there is a functor F : D(A) → D′ and an isomorphism of functors Q′ ≅ F ◦ Q.

First, consider the category C(A) as an oriented graph Γ, with the objects lying at the vertices and the morphisms being directed edges. Let Γ∗ be the graph obtained from Γ by adding in one extra edge s⁻¹: b → a for each quasi-isomorphism s: a → b. Thus a finite path in Γ∗ is a sequence of the form f₁ · f₂ ·· · ·· f_r−1 · f_r where each f_i is either a morphism of C(A), or is of the form s⁻¹ for some quasi-isomorphism s of C(A). There is a unique minimal equivalence relation ∼ on the set of finite paths in Γ∗ generated by the following relations:

(a) s · s⁻¹ ∼ id_b and s⁻¹ · s ∼ id_a for each quasi-isomorphism s: a → b in C(A).

(b) g · f ∼ g ◦ f for composable morphisms f: a → b and g: b → c of C(A).

Define D(A) to be the category whose objects are the vertices of Γ∗ (these are the same as the objects of C(A)) and whose morphisms are given by equivalence classes of finite paths in Γ∗. Define a functor Q: C(A) → D(A) by using the identity morphism on objects, and by sending a morphism f of C(A) to the length one path in Γ∗ defined by f. The resulting functor Q satisfies the conditions of the above lemma.

The second property ensures that the category D(A) of the Lemma is unique up to equivalence of categories. We define the derived category of A to be any of these equivalent categories. The functor Q: C(A) → D(A) is called the localisation functor. Observe that there is a fully faithful functor

J: A → C(A)

which sends an object M to the trivial complex with M in the zeroth position, and a morphism F: M → N to the morphism of complexes

Composing with Q we obtain a functor A → D(A) which we denote by J. This functor J is fully faithful, and so defines an embedding A → D(A). By definition the functor Hⁱ(−): C(A) → A inverts quasi-isomorphisms and so descends to a functor

Hⁱ(−): D(A) → A

establishing that composite functor H⁰(−) ◦ J is isomorphic to the identity functor on A.

Coarse Philosophies of Coarse Embeddabilities: Metric Space Conjectures Act Algorithmically On Manifolds – Thought of the Day 145.0

A coarse structure on a set X is defined to be a collection of subsets of X × X, called the controlled sets or entourages for the coarse structure, which satisfy some simple axioms. The most important of these states that if E and F are controlled then so is

E ◦ F := {(x, z) : ∃y, (x, y) ∈ E, (y, z) ∈ F}

Consider the metric spaces Zⁿ and Rⁿ. Their small-scale structure, their topology is entirely different, but on the large scale they resemble each other closely: any geometric configuration in Rⁿ can be approximated by one in Zⁿ, to within a uniformly bounded error. We think of such spaces as “coarsely equivalent”. The other axioms require that the diagonal should be a controlled set, and that subsets, transposes, and (finite) unions of controlled sets should be controlled. It is accurate to say that a coarse structure is the large-scale counterpart of a uniformity than of a topology.

Coarse structures and coarse spaces enjoy a philosophical advantage over coarse metric spaces, in that, all left invariant bounded geometry metrics on a countable group induce the same metric coarse structure which is therefore transparently uniquely determined by the group. On the other hand, the absence of a natural gauge complicates the notion of a coarse family, while it is natural to speak of sets of uniform size in different metric spaces it is not possible to do so in different coarse spaces without imposing additional structure.

Mikhail Leonidovich Gromov introduced the notion of coarse embedding for metric spaces. Let X and Y be metric spaces.

A map f : X → Y is said to be a coarse embedding if ∃ nondecreasing functions ρ₁ and ρ₂ from R₊ = [0, ∞) to R such that

• ρ₁(d(x,y)) ≤ d(f(x),f(y)) ≤ ρ₂(d(x,y)) ∀ x, y ∈ X.
• lim_r→∞ ρ_i(r) = +∞ (i=1, 2).

Intuitively, coarse embeddability of a metric space X into Y means that we can draw a picture of X in Y which reflects the large scale geometry of X. In early 90’s, Gromov suggested that coarse embeddability of a discrete group into Hilbert space or some Banach spaces should be relevant to solving the Novikov conjecture. The connection between large scale geometry and differential topology and differential geometry, such as the Novikov conjecture, is built by index theory. Recall that an elliptic differential operator D on a compact manifold M is Fredholm in the sense that the kernel and cokernel of D are finite dimensional. The Fredholm index of D, which is defined by

index(D) = dim(kerD) − dim(cokerD),

has the following fundamental properties:

(1) it is an obstruction to invertibility of D;

(2) it is invariant under homotopy equivalence.

The celebrated Atiyah-Singer index theorem computes the Fredholm index of elliptic differential operators on compact manifolds and has important applications. However, an elliptic differential operator on a noncompact manifold is in general not Fredholm in the usual sense, but Fredholm in a generalized sense. The generalized Fredholm index for such an operator is called the higher index. In particular, on a general noncompact complete Riemannian manifold M, John Roe (Coarse Cohomology and Index Theory on Complete Riemannian Manifolds) introduced a higher index theory for elliptic differential operators on M.

The coarse Baum-Connes conjecture is an algorithm to compute the higher index of an elliptic differential operator on noncompact complete Riemannian manifolds. By the descent principal, the coarse Baum-Connes conjecture implies the Novikov higher signature conjecture. Guoliang Yu has proved the coarse Baum-Connes conjecture for bounded geometry metric spaces which are coarsely embeddable into Hilbert space. The metric spaces which admit coarse embeddings into Hilbert space are a large class, including e.g. all amenable groups and hyperbolic groups. In general, however, there are counterexamples to the coarse Baum-Connes conjecture. A notorious one is expander graphs. On the other hand, the coarse Novikov conjecture (i.e. the injectivity part of the coarse Baum-Connes conjecture) is an algorithm of determining non-vanishing of the higher index. Kasparov-Yu have proved the coarse Novikov conjecture for spaces which admit coarse embeddings into a uniformly convex Banach space.