Ringed Spaces (2)

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Let |M| be a topological space. A presheaf of commutative algebras F on X is an assignment

U ↦ F(U), U open in |M|, F(U) is a commutative algebra, such that the following holds,

(1) If U ⊂ V are two open sets in |M|, ∃ a morphism rV, U: F(V) → F(U), called the restriction morphism and often denoted by rV, U(ƒ) = ƒ|U, such that

(i) rU, U = id,

(ii) rW, U = rV, U ○ rW, V

A presheaf ƒ is called a sheaf if the following holds:

(2) Given an open covering {Ui}i∈I of U and a family {ƒi}i∈I, ƒi ∈ F(Ui) such that ƒi|Ui ∩ Uj = ƒj|Ui ∩ Uj ∀ i, j ∈ I, ∃ a unique ƒ ∈ F(U) with ƒ|Ui = ƒi

The elements in F(U) are called sections over U, and with U = |M|, these are termed global sections.

The assignments U ↦ C(U), U open in the differentiable manifold M and U ↦ OX(U), U open in algebraic variety X are examples of sheaves of functions on the topological spaces |M| and |X| underlying the differentiable manifold M and the algebraic variety X respectively.

In the language of categories, the above definition says that we have defined a functor, F, from top(M) to (alg), where top(M) is the category of the open sets in the topological space |M|, the arrows given by the inclusions of open sets while (alg) is the category of commutative algebras. In fact, the assignment U ↦ F(U) defines F on the objects while the assignment

U ⊂ V ↦ rV, U: F(V) → F(U)

defines F on the arrows.

Let |M| be a topological space. We define a presheaf of algebras on |M| to be a functor

F: top(M)op → (alg)

The suffix “op” denotes as usual the opposite category; in other words, F is a contravariant functor from top(M) to (alg). A presheaf is a sheaf if it satisfies the property (2) of the above definition.

If F is a (pre)sheaf on |M| and U is open in |M|, we define F|U, the (pre)sheaf F restricted to U, as the functor F restricted to the category of open sets in U (viewed as a topological space itself).

Let F be a presheaf on the topological space |M| and let x be a point in |M|. We define the stalk Fx of F, at the point x, as the direct limit

lim F(U)

where the direct limit is taken ∀ the U open neighbourhoods of x in |M|. Fx consists of the disjoint union of all pairs (U, s) with U open in |M|, x ∈ U, and s ∈ F(U), modulo the equivalence relation: (U, s) ≅ (V, t) iff ∃ a neighbourhood W of x, W ⊂ U ∩ V, such that s|W = t|W.

The elements in Fx are called germs of sections.

Let F and G be presheaves on |M|. A morphism of presheaves φ: F → G, for each open set U in |M|, such that ∀ V ⊂ U, the following diagram commutes

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Equivalently and more elegantly, one can also say that a morphism of presheaves is a natural transformation between the two presheaves F and G viewed as functors.

A morphism of sheaves is just a morphism of the underlying presheaves.

Clearly any morphism of presheaves induces a morphism on the stalks: φx: Fx → Gx. The sheaf property, i.e., property (2) in the above definition, ensures that if we have two morphisms of sheaves φ and ψ, such that φx = ψx ∀ x, then φ = ψ.

We say that the morphism of sheaves is injective (resp. surjective) if x is injective (resp. surjective).

On the notion of surjectivity, however, one should exert some care, since we can have a surjective sheaf morphism φ: F → G such that φU: F(U) → G(U) is not surjective for some open sets U. This strange phenomenon is a consequence of the following fact. While the assignment U ↦ ker(φ(U)) always defines a sheaf, the assignment

U ↦ im( φ(U)) = F(U)/G(U)

defines in general only a presheaf and not all the presheaves are sheaves. A simple example is given by the assignment associating to an open set U in R, the algebra of constant real functions on U. Clearly this is a presheaf, but not a sheaf.

We can always associate, in a natural way, to any presheaf a sheaf called its sheafification. Intuitively, one may think of the sheafification as the sheaf that best “approximates” the given presheaf. For example, the sheafification of the presheaf of constant functions on open sets in R is the sheaf of locally constant functions on open sets in R. We construct the sheafification of a presheaf using the étalé space, which we also need in the sequel, since it gives an equivalent approach to sheaf theory.

Let F be a presheaf on |M|. We define the étalé space of F to be the disjoint union ⊔x∈|M| Fx. Let each open U ∈ |M| and each s ∈ F(U) define the map šU: U ⊔x∈|U| Fx, šU(x) = sx. We give to the étalé space the finest topology that makes the maps š continuous, ∀ open U ⊂ |M| and all sections s ∈ F(U). We define Fet to be the presheaf on |M|:

U ↦ Fet(U) = {šU: U → ⊔x∈|U| Fx, šU(x) = sx ∈ Fx}

Let F be a presheaf on |M|. A sheafification of F is a sheaf F~, together with a presheaf morphism α: F → Fsuch that

(1) any presheaf morphism ψ: F → G, G a sheaf factors via α, i.e. ψ: F →α F~ → G,

(2) F and Fare locally isomorphic, i.e., ∃ an open cover {Ui}i∈I of |M| such that F(Ui) ≅ F~(Ui) via α.

Let F and G be sheaves of rings on some topological space |M|. Assume that we have an injective morphism of sheaves G → F such that G(U) ⊂ F(U) ∀ U open in |M|. We define the quotient F/G to be the sheafification of the image presheaf: U ↦ F(U)/G(U). In general F/G (U) ≠ F(U)/G(U), however they are locally isomorphic.

Ringed space is a pair M = (|M|, F) consisting of a topological space |M| and a sheaf of commutative rings F on |M|. This is a locally ringed space, if the stalk Fx is a local ring ∀ x ∈ |M|. A morphism of ringed spaces φ: M = (|M|, F) → N = (|N|, G) consists of a morphism |φ|: |M| → |N| of the topological spaces and a sheaf morphism φ*: ON → φ*OM, where φ*OM is a sheaf on |N| and defined as follows:

*OM)(U) = OM-1(U)) ∀ U open in |N|

Morphism of ringed spaces induces a morphism on the stalks for each

x ∈ |M|: φx: ON,|φ|(x) → OM,x

If M and N are locally ringed spaces, we say that the morphism of ringed spaces φ is a morphism of locally ringed spaces if φx is local, i.e. φ-1x(mM,x) = mN,|φ|(x), where mN,|φ|(x) and mM,x are the maximal ideals in the local rings ON,|φ|(x) and OM,x respectively.

Austrian School of Economics: The Praxeological Synthetic. Thought of the Day 135.0

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Within the Austrian economics (here, here, here and here), the a priori stance has dominated a tradition running from Carl Menger to Murray Rothbard. The idea here is that the basic structures of economy is entrenched in the more basic structures of human action as such. Nowhere is this more evident than in the work of Ludwig von Mises – his so-called ‘praxeology’, which rests on the fundamental axiom that individual human beings act on the primordial fact that individuals engage in conscious actions toward chosen goals, is built from the idea that all basic laws of economy can be derived apriorically from one premiss: the concept of human action. Of course, this concept is no simple concept, containing within itself purpose, product, time, scarcity of resources, etc. – so it would be more fair to say that economics lies as the implication of the basic schema of human action as such.

Even if the Austrian economists’ conception of the a priori is decidedly objectivist and anti-subjectivist, it is important to remark their insistence on subjectivity within their ontological domain. The Austrian economics tradition is famous exactly for their emphasis on the role of subjectivity in economy. From Carl Menger onwards, they protest against the mainstream economical assumption that the economic agent in the market is fully rational, knows his own preferences in detail, has constant preferences over time, has access to all prices for a given commodity at a given moment, etc. Thus, von Mises’ famous criticism of socialist planned economy is built on this idea: the system of ever-changing prices in the market constitutes a dispersed knowledge about the conditions of resource allocation which is a priori impossible for any single agent – let alone, any central planner’s office – to possess. Thus, their conception of the objective a priori laws of the economic domain perhaps surprisingly had the implication that they warned against a too objectivist conception of economy not taking into account the limits of economic rationality stemming from the general limitations of the capacities of real subjects. Their ensuing liberalism is thus built on a priori conclusions about the relative unpredictability of economics founded on the role played by subjective intentionality. For the same reason, Hayek ended up with a distinction between simple and complex processes, respectively, cutting across all empirical disciplines, where only the former permit precise, predictive, quantitative calculi based on mathemathical modeling while the latter permit only recognition of patterns (which may also be mathematically modeled, to be sure, but without quantitative predictability). It is of paramount importance, though, to distinguish this emphasis on the ineradicable role of subjectivity in certain regional domains from Kantian-like ideas about the foundational role of subjectivity in the construction of knowledge as such. The Austrians are as much subjectivists in the former respect as they are objectivists in the latter. In the history of economics, the Austrians occupy a middle position, being against historicism on the one hand as well as against positivism on the other. Against the former, they insist that a priori structures of economy transgress history which does not possess the power to form institutions at random but only as constrained by a priori structures. And against the latter, they insist that the mere accumulation of empirical data subject to induction will never in itself give rise to the formation of theoretical insights. Structures of intelligible concepts are in all cases necessary for any understanding of empirical regularities – in so far, the Austrian a priori approach is tantamount to a non-skepticist version of the doctrine of ‘theory-ladenness’ of observations.

A late descendant of the Austrian tradition after its emigration to the Anglo-Saxon world (von Mises, Hayek, and Schumpeter were such emigrés) was the anarcho-liberal economist Murray Rothbard, and it is the inspiration from him which allows Barry Smith to articulate the principles underlying the Austrians as ‘fallibilistic apriorism’. Rothbard characterizes in a brief paper what he calls ‘Extreme Apriorism’ as follows:

there are two basic differences between the positivists’ model science of physics on the one hand, and sciences dealing with human actions on the other: the former permits experimental verification of consequences of hypotheses, which the latter do not (or, only to a limited degree, we may add); the former admits of no possibility of testing the premisses of hypotheses (like: what is gravity?), while the latter permits a rational investigation of the premisses of hypotheses (like: what is human action?). This state of affairs makes it possible for economics to derive its basic laws with absolute – a priori – certainty: in addition to the fundamental axiom – the existence of human action – only two empirical postulates are needed: ‘(1) the most fundamental variety of resources, both natural and human. From this follows directly the division of labor, the market, etc.; (2) less important, that leisure is a consumer good’. On this basis, it may e.g. be inferred, ‘that every firm aims always at maximizing its psychic profit’.

Rothbard draws forth this example so as to counterargue traditional economists who will claim that the following proposition could be added as a corollary: ‘that every firm aims always at maximizing its money profit’. This cannot be inferred and is, according to Rothbard, an economical prejudice – the manager may, e.g. prefer for nepotistic reasons to employ his stupid brother even if that decreases the firm’s financial profit possibilities. This is an example of how the Austrians refute the basic premiss of absolute rationality in terms of maximal profit seeking. Given this basis, other immediate implications are:

the means-ends relationship, the time-structure of production, time-preference, the law of diminishing marginal utility, the law of optimum returns, etc.

Rothbard quotes Mises for seeing the fundamental Axiom as a ‘Law of Thought’ – while he himself sees this as a much too Kantian way of expressing it, he prefers instead the simple Aristotelian/Thomist idea of a ‘Law of Reality’. Rothbard furthermore insists that this doctrine is not inherently political – in order to attain the Austrians’ average liberalist political orientation, the preference for certain types of ends must be added to the a priori theory (such as the preference for life over death, abundance over poverty, etc.). This also displays the radicality of the Austrian approach: nothing is assumed about the content of human ends – this is why they will never subscribe to theories about Man as economically rational agent or Man as necessarily economical egotist. All different ends meet and compete on the market – including both desire for profit in one end and idealist, utopian, or altruist goals in the other. The principal interest, in these features of economical theory is the high degree of awareness of the difference between the – extreme – synthetic a priori theory developed, on the one hand, and its incarnation in concrete empirical cases and their limiting conditions on the other.

 

Revisiting Catastrophes. Thought of the Day 134.0

The most explicit influence from mathematics in semiotics is probably René Thom’s controversial theory of catastrophes (here and here), with philosophical and semiotic support from Jean Petitot. Catastrophe theory is but one of several formalisms in the broad field of qualitative dynamics (comprising also chaos theory, complexity theory, self-organized criticality, etc.). In all these cases, the theories in question are in a certain sense phenomenological because the focus is different types of qualitative behavior of dynamic systems grasped on a purely formal level bracketing their causal determination on the deeper level. A widespread tool in these disciplines is phase space – a space defined by the variables governing the development of the system so that this development may be mapped as a trajectory through phase space, each point on the trajectory mapping one global state of the system. This space may be inhabited by different types of attractors (attracting trajectories), repellors (repelling them), attractor basins around attractors, and borders between such basins characterized by different types of topological saddles which may have a complicated topology.

Catastrophe theory has its basis in differential topology, that is, the branch of topology keeping various differential properties in a function invariant under transformation. It is, more specifically, the so-called Whitney topology whose invariants are points where the nth derivative of a function takes the value 0, graphically corresponding to minima, maxima, turning tangents, and, in higher dimensions, different complicated saddles. Catastrophe theory takes its point of departure in singularity theory whose object is the shift between types of such functions. It thus erects a distinction between an inner space – where the function varies – and an outer space of control variables charting the variation of that function including where it changes type – where, e.g. it goes from having one minimum to having two minima, via a singular case with turning tangent. The continuous variation of control parameters thus corresponds to a continuous variation within one subtype of the function, until it reaches a singular point where it discontinuously, ‘catastrophically’, changes subtype. The philosophy-of-science interpretation of this formalism now conceives the stable subtype of function as representing the stable state of a system, and the passage of the critical point as the sudden shift to a new stable state. The configuration of control parameters thus provides a sort of map of the shift between continuous development and discontinuous ‘jump’. Thom’s semiotic interpretation of this formalism entails that typical catastrophic trajectories of this kind may be interpreted as stable process types phenomenologically salient for perception and giving rise to basic verbal categories.

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One of the simpler catastrophes is the so-called cusp (a). It constitutes a meta-diagram, namely a diagram of the possible type-shifts of a simpler diagram (b), that of the equation ax4 + bx2 + cx = 0. The upper part of (a) shows the so-called fold, charting the manifold of solutions to the equation in the three dimensions a, b and c. By the projection of the fold on the a, b-plane, the pointed figure of the cusp (lower a) is obtained. The cusp now charts the type-shift of the function: Inside the cusp, the function has two minima, outside it only one minimum. Different paths through the cusp thus corresponds to different variations of the equation by the variation of the external variables a and b. One such typical path is the path indicated by the left-right arrow on all four diagrams which crosses the cusp from inside out, giving rise to a diagram of the further level (c) – depending on the interpretation of the minima as simultaneous states. Here, thus, we find diagram transformations on three different, nested levels.

The concept of transformation plays several roles in this formalism. The most spectacular one refers, of course, to the change in external control variables, determining a trajectory through phase space where the function controlled changes type. This transformation thus searches the possibility for a change of the subtypes of the function in question, that is, it plays the role of eidetic variation mapping how the function is ‘unfolded’ (the basic theorem of catastrophe theory refers to such unfolding of simple functions). Another transformation finds stable classes of such local trajectory pieces including such shifts – making possible the recognition of such types of shifts in different empirical phenomena. On the most empirical level, finally, one running of such a trajectory piece provides, in itself, a transformation of one state into another, whereby the two states are rationally interconnected. Generally, it is possible to make a given transformation the object of a higher order transformation which by abstraction may investigate aspects of the lower one’s type and conditions. Thus, the central unfolding of a function germ in Catastrophe Theory constitutes a transformation having the character of an eidetic variation making clear which possibilities lie in the function germ in question. As an abstract formalism, the higher of these transformations may determine the lower one as invariant in a series of empirical cases.

Complexity theory is a broader and more inclusive term covering the general study of the macro-behavior of composite systems, also using phase space representation. The theoretical biologist Stuart Kauffman (intro) argues that in a phase space of all possible genotypes, biological evolution must unfold in a rather small and specifically qualified sub-space characterized by many, closely located and stable states (corresponding to the possibility of a species to ‘jump’ to another and better genotype in the face of environmental change) – as opposed to phase space areas with few, very stable states (which will only be optimal in certain, very stable environments and thus fragile when exposed to change), and also opposed, on the other hand, to sub-spaces with a high plurality of only metastable states (here, the species will tend to merge into neighboring species and hence never stabilize). On the base of this argument, only a small subset of the set of virtual genotypes possesses ‘evolvability’ as this special combination between plasticity and stability. The overall argument thus goes that order in biology is not a pure product of evolution; the possibility of order must be present in certain types of organized matter before selection begins – conversely, selection requires already organized material on which to work. The identification of a species with a co-localized group of stable states in genome space thus provides a (local) invariance for the transformation taking a trajectory through space, and larger groups of neighboring stabilities – lineages – again provide invariants defined by various more or less general transformations. Species, in this view, are in a certain limited sense ‘natural kinds’ and thus naturally signifying entities. Kauffman’s speculations over genotypical phase space have a crucial bearing on a transformation concept central to biology, namely mutation. On this basis far from all virtual mutations are really possible – even apart from their degree of environmental relevance. A mutation into a stable but remotely placed species in phase space will be impossible (evolution cannot cross the distance in phase space), just like a mutation in an area with many, unstable proto-species will not allow for any stabilization of species at all and will thus fall prey to arbitrary small environment variations. Kauffman takes a spontaneous and non-formalized transformation concept (mutation) and attempts a formalization by investigating its condition of possibility as movement between stable genomes in genotype phase space. A series of constraints turn out to determine type formation on a higher level (the three different types of local geography in phase space). If the trajectory of mutations must obey the possibility of walking between stable species, then the space of possibility of trajectories is highly limited. Self-organized criticality as developed by Per Bak (How Nature Works the science of self-organized criticality) belongs to the same type of theories. Criticality is here defined as that state of a complicated system where sudden developments in all sizes spontaneously occur.

Ringed Spaces (1)

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A ringed space is a broad concept in which we can fit most of the interesting geometrical objects. It consists of a topological space together with a sheaf of functions on it.

Let M be a differentiable manifold, whose topological space is Hausdorff and second countable. For each open set U ⊂ M , let C(U) be the R-algebra of smooth functions on U .

The assignment

U ↦ C(U)

satisfies the following two properties:

(1) If U ⊂ V are two open sets in M, we can define the restriction map, which is an algebra morphism:

rV, U : C(V) → C(U), ƒ ↦ ƒ|U

which is such that

i) rU, U = id

ii) rW, U = rV, U ○ rW, V

(2) Let {Ui}i∈I be an open covering of U and let {ƒi}i∈I, ƒi ∈ C(Ui) be a family such that ƒi|Ui ∩ Uj = ƒj| Ui ∩ Uj ∀ i, j ∈ I. In other words the elements of the family {ƒi}i∈I agree on the intersection of any two open sets Ui ∩ Uj. Then there exists a unique ƒ ∈ C(U) such that ƒ|Ui = ƒi.

Such an assignment is called a sheaf. The pair (M, C), consisting of the topological space M, underlying the differentiable manifold, and the sheaf of the C functions on M is an example of locally ringed space (the word “locally” refers to a local property of the sheaf of C functions.

Given two manifolds M and N, and the respective sheaves of smooth functions CM and CN, a morphism ƒ from M to N, viewed as ringed spaces, is a morphism |ƒ|: M → N of the underlying topological spaces together with a morphism of algebras,

ƒ*: CN(V) →  CM-1(V)), ƒ*(φ)(x) = φ(|ƒ|(x))

compatible with the restriction morphisms.

Notice that, as soon as we give the continuous map |ƒ| between the topological spaces, the morphism ƒ* is automatically assigned. This is a peculiarity of the sheaf of smooth functions on a manifold. Such a property is no longer true for a generic ringed space and, in particular, it is not true for supermanifolds.

A morphism of differentiable manifolds gives rise to a unique (locally) ringed space morphism and vice versa.

Moreover, given two manifolds, they are isomorphic as manifolds iff they are isomorphic as (locally) ringed spaces. In the language of categories, we say we have a fully faithful functor from the category of manifolds to the category of locally ringed spaces.

The generalization of algebraic geometry to the super-setting comes somehow more naturally than the similar generalization of differentiable geometry. This is because the machinery of algebraic geometry was developed to take already into account the presence of (even) nilpotents and consequently, the language is more suitable to supergeometry.

Let X be an affine algebraic variety in the affine space An over an algebraically closed field k and let O(X) = k[x1,…., xn]/I be its coordinate ring, where the ideal I is prime. This corresponds topologically to the irreducibility of the variety X. We can think of the points of X as the zeros of the polynomials in the ideal I in An. X is a topological space with respect to the Zariski topology, whose closed sets are the zeros of the polynomials in the ideals of O(X). For each open U in X, consider the assignment

U ↦ OX(U)

where OX(U) is the k-algebra of regular functions on U. By definition, these are the functions ƒ X → k that can be expressed as a quotient of two polynomials at each point of U ⊂ X. The assignment U ↦ OX(U) is another example of a sheaf is called the structure sheaf of the variety X or the sheaf of regular functions. (X, OX) is another example of a (locally) ringed space.

Defaultable Bonds. Thought of the Day 133.0

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Defaultable bonds are bonds that have a positive possibility of default.  Most corporate bonds and some government bonds are defaultable.  When a bond defaults, its coupon and principal payments will be altered.  Most of the time, only a portion of the principal, and sometimes, also a portion of the coupon, will be paid. A defaultable (T, x) – bond with maturity T > 0 and credit rating x ∈ I ⊆ [0, 1], is a financial contract which pays to its holder 1 unit of currency at time T provided that the writer of the bond hasn’t bankrupted till time T. The set I stands for all possible credit ratings. The bankruptcy is modeled with the use of a so called loss process {L(t), t ≥ 0} which starts from zero, increases and takes values in the interval [0, 1]. The bond is worthless if the loss process exceeds its credit rating. Thus the payoff profile of the (T, x) – bond is of the form

1{LT ≤ x}

The price P(t, T, x) of the (T, x) – bond is a stochastic process defined by

P(t, T, x) = 1{LT ≤ x}e−∫tT f(t, u, x)du, t ∈ [0, T] —– (1)

where f (·, ·, x) stands for an x-forward rate. The value x = 1 corresponds to the risk-free bond and f(t, T, 1) determines the short rate process via f(t, t, 1), t ≥ 0.

The (T, x) – bond market is thus fully determined by the family of x-forward rates and the loss process L. This is an extension of the classical non-defaultable bond market which can be identified with the case when I is a singleton, that is, when I = {1}.

The model of (T, x) – bonds does not correspond to defaultable bonds which are directly traded on a real market. For instance, in this setting the bankruptcy of the (T, x2) – bond automatically implies the bankruptcy of the (T, x1) – bond if x1 < x2. In reality, a bond with a higher credit rating may, however, default earlier than that with a lower one. The (T, x) – bonds are basic instruments related to the pool of defaultable assets called Collateralized Debt Obligations (CDOs), which are actually widely traded on the market. In the CDO market model, the loss process L(t) describes the part of the pool which has defaulted up to time t > 0 and F(LT), where F as some function, specifies the CDO payoff at time T > 0. In particular, (T, x) – bonds may be identified with the digital-type CDO payoffs with the function F of the form

F(z) = Fx(z) := 1[0,x](z), x ∈ I, z ∈ [0,1]

Then the price of that payoff pt(Fx(LT)) at time t ≤ T equals P(t, T, x). Moreover, each regular CDO claim can be replicated, and thus also priced, with a portfolio consisting of a certain combination of (T, x) – bonds. Thus it follows that the model of (T, x) – bonds determines the structure of the CDO payoffs. The induced family of prices

P(t, T, x), T ≥ 0, x ∈ I

will be a CDO term structure. On real markets the price of a claim which pays more is always higher. This implies

P(t, T, x1) = pt(Fx1(LT)) ≤ pt(Fx2(LT)) = P(t, T, x2), t ∈ [0, T], x1 < x2, x1, x2 ∈ I —– (2)

which means that the prices of (T, x) – bonds are increasing in x. Similarly, if the claim is paid earlier, then it has a higher value and hence

P(t, T1, x) = pt(Fx(LT1)) ≥ pt(Fx(LT2)) = P(t, T2, x), t ∈ [0, T1], T1 < T2, x ∈ I —– (3)

which means that the (T, x) – bond prices are decreasing in T. The CDO term structure is monotone if both (2) and (3) are satisfied. Surprisingly, monotonicity of the (T, x) – bond prices is not always preserved in mathematical models even if they satisfy severe no-arbitrage conditions.

Breakdown of Lorentz Invariance: The Order of Quantum Gravity Phenomenology. Thought of the Day 132.0

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The purpose of quantum gravity phenomenology is to analyze the physical consequences arising from various models of quantum gravity. One hope for obtaining an experimental grasp on quantum gravity is the generic prediction arising in many (but not all) quantum gravity models that ultraviolet physics at or near the Planck scale, MPlanck = 1.2 × 1019 GeV/c2, (or in some models the string scale), typically induces violations of Lorentz invariance at lower scales. Interestingly most investigations, even if they arise from quite different fundamental physics, seem to converge on the prediction that the breakdown of Lorentz invariance can generically become manifest in the form of modified dispersion relations

ω2 = ω02 + (1 + η2) c2k2 + η4(ħ/MLorentz violation)2 + k4 + ….

where the coefficients ηn are dimensionless (and possibly dependent on the particle species under consideration). The particular inertial frame for these dispersion relations is generally specified to be the frame set by cosmological microwave background, and MLorentz violation is the scale of Lorentz symmetry breaking which furthermore is generally assumed to be of the order of MPlanck.

Although several alternative scenarios have been considered to justify the modified kinematics,the most commonly explored avenue is an effective field theory (EFT) approach. Here, the focus is explicitly on the class of non-renormalizable EFTs with Lorentz violations associated to dispersion relations. Even if this framework as a “test theory” is successful, it is interesting to note that there are still significant open issues concerning its theoretical foundations. Perhaps the most pressing one is the so called naturalness problem which can be expressed in the following way: The lowest-order correction, proportional to η2, is not explicitly Planck suppressed. This implies that such a term would always be dominant with respect to the higher-order ones and grossly incompatible with observations (given that we have very good constraints on the universality of the speed of light for different elementary particles). If one were to take cues from observational leads, it is assumed either that some symmetry (other than Lorentz invariance) enforces the η2 coefficients to be exactly zero, or that the presence of some other characteristic EFT mass scale μ ≪ MPlanck (e.g., some particle physics mass scale) associated with the Lorentz symmetry breaking might enter in the lowest order dimensionless coefficient η2, which will be then generically suppressed by appropriate ratios of this characteristic mass to the Planck mass: η2 ∝ (μ/MPlanck)σ where σ ≥ 1 is some positive power (often taken as one or two). If this is the case then one has two distinct regimes: For low momenta p/(MPlanckc) ≪ (μ/MPlanck)σ the lower-order (quadratic in the momentum) deviations will dominate over the higher-order ones, while at high energies p/(MPlanckc) ≫ (μ/MPlanck)σ the higher order terms will be dominant.

The naturalness problem arises because such a scenario is not well justified within an EFT framework; in other words there is no natural suppression of the low-order modifications. EFT cannot justify why only the dimensionless coefficients of the n ≤ 2 terms should be suppressed by powers of the small ratio μ/MPlanck. Even worse, renormalization group arguments seem to imply that a similar mass ratio, μ/MPlanck would implicitly be present also in all the dimensionless n > 2 coefficients, hence suppressing them even further, to the point of complete undetectability. Furthermore, without some protecting symmetry, it is generic that radiative corrections due to particle interactions in an EFT with only Lorentz violations of order n > 2 for the free particles, will generate n = 2 Lorentz violating terms in the dispersion relation, which will then be dominant. Naturalness in EFT would then imply that the higher order terms are at least as suppressed as this, and hence beyond observational reach.

A second issue is that of universality, which is not so much a problem, as an issue of debate as to the best strategy to adopt. In dealing with situations with multiple particles one has to choose between the case of universal (particle-independent) Lorentz violating coefficients ηn, or instead go for a more general ansatz and allow for particle-dependent coefficients; hence allowing different magnitudes of Lorentz symmetry violation for different particles even when considering the same order terms (same n) in regards to momentum. Any violation of Lorentz invariance should be due to the microscopic structure of the effective space-time. This implies that one has to tune the system in order to cancel exactly all those violations of Lorentz invariance which are solely due to mode-mixing interactions in the hydrodynamic limit.

Tantric Initiation. Thought of the Day 131.0

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Man, universe, gods and ritual are not considered separate entities but rather different manifestations of the same Śakti. Therefore, during a particular ritual every element of it is symbolic of something else. The flowers are representative of something else, the incense is representative of something else and so on. This viewpoint is based upon the crucial teaching that “worldly and spiritual” are the two faces of a same coin. One often thinks that “spirituality” is associated with something which is “within”, while “worldliness” is associated with something which is “without”. So, if you see a light “within”, that is a “spiritual” experience, while if you see a light “without”, that is a “worldly” experience. Besides, the worldliness is based on “day-to-day experiences”. It is approximately so. Tantricism considers all to be the manifestation of Śakti, the Divine Mother. So, an external light is as spiritual as an internal one and vice versa. In fact, there is neither spirituality nor worldliness because only one Supreme Consciousness is permeating everything and everyone.

Śakti or the Divine Mother is the core of all tantric practices. She is known as Kuṇḍalinī when residing in a living being. She is the bestower of the Supreme Bliss for all those followers that worship Her according to the sacred rituals and meditations contained in the Tantra-s. Her importance has been emphasized in Niruttaratantra:

बहूनां जन्मनामन्ते शक्तिज्ञानं प्रजायते।
शक्तिज्ञानं विना देवि निर्वाणं नैव जायते॥

Bahūnaṁ janmanāmante śaktijñānaṁ prajāyate|
Śaktijñānaṁ vinā devi nirvāṇaṁ naiva jāyate||

After (ante) many (bahūnām) births (janmanām), the knowledge (jñānam) of Śakti (śakti) is born (in oneself) (prajāyate). Oh goddess (devi)!, without (vinā) the knowledge (jñānam) of Śakti (śakti), Nirvāṇa — final Liberation — (nirvāṇam) does not (na eva) spring up (jāyate).

However, Tantricism should not be “strictly” equated to Shaktism, because there are groups of Śākta-s (followers of Śakti) which are not “tantric” at all. In turn, there are tantric groups that worship Śiva, Viṣṇu, etc. as well as Śakti.

Consequently, one may use a set of elements as representative of other realities. For example: a man represents Śiva and a woman represents Śakti. Then, their union is representative of that of Śiva and Śakti. Microcosm and macrocosm are closely allied to each other, because the two are the manifestation of only one Power. The following fragment extracted from the ancient Tantra-s clearly shows the aforesaid correlation between man, universe, gods and ritual. The sādhaka or practitioner is meditating on the Divine Mother (Śakti) in his heart lotus. He forms a mental image of Śakti there, and begins worshipping Her this way:

हृत्पद्मासनं दद्यात् सहस्रारच्युतामृतैः।
पाद्यं चरणयोर्दद्यान्मनसार्घ्यं निवेदयेत्॥

तेनामृतेनाचमनं स्नानीयमपि कल्पयेत्।
आकाशतत्त्वं वसनं गन्धं तु गन्धतत्त्वकम्॥

चित्तं प्रकल्पयेत् पुष्पं धूपं प्राणान् प्रकल्पयेत्।
तेजस्तत्त्वं च दीपार्थे नैवेद्यं च सुधाम्बुधिम्॥

अनाहतध्वनिं घण्टां वायुतत्त्वं च चामरम्।
नृत्यमिन्द्रियकर्माणि चाञ्चल्यं मनसस्तथा॥

पुष्पं नानाविधं दद्यादात्मनो भावसिद्धये।
अमायामनहङ्कारमरागममदं तथा॥

अमोहकमदम्भं च अद्वेषाक्षोभके तथा।
अमात्सर्यमलोभं च दशपुष्पं प्रकीर्तितम्॥

अहिंसा परमं पुष्पं पुष्पमिन्द्रियनिग्रहम्।
दयाक्षमाज्ञानपुष्पं पञ्चपुष्पं ततः परम्॥

इति पञ्चदशैर्पुष्पैर्भावपुष्पैः प्रपूजयेत्॥

Hṛtpadmāsanaṁ dadyāt sahasrāracyutāmṛtaiḥ|
Pādyaṁ caraṇayordadyānmanasārghyaṁ nivedayet||

Tenāmṛtenācamanaṁ snānīyamapi kalpayet|
Ākāśatattvaṁ vasanaṁ gandhaṁ tu gandhatattvakam||

Cittaṁ prakalpayet puṣpaṁ dhūpaṁ prāṇān prakalpayet|
Tejastattvaṁ ca dīpārthe naivedyaṁ ca sudhāmbudhim||

Anāhatadhvaniṁ ghaṇṭāṁ vāyutattvaṁ ca cāmaram|
Nṛtyamindriyakarmāṇi cāñcalyaṁ manasastathā||

Puṣpaṁ nānāvidhaṁ dadyādātmano bhāvasiddhaye|
Amāyāmanahaṅkāramarāgamamadaṁ tathā||

Amohakamadambham ca adveṣākṣobhake tathā|
Amātsaryamalobhaṁ ca daśapuṣpaṁ prakīrtitam||

Ahiṁsā paramaṁ puṣpamindriyanigraham|
Dayākṣamājñānapuṣpaṁ pañcapuṣpaṁ tataḥ param||

Iti pañcadaśairpuṣpairbhāvapuṣpaiḥ prapūjayet||

He gives (dadyāt… dadyāt) (his) heart (hṛt) lotus (padma) as the seat (āsanam), and the water for washing (pādyam) the feet (caraṇayoḥ) in the form of the nectars (amṛtaiḥ) flowing (cyuta) from Sahasrāra — the supreme Cakra placed at the crown of the head– (sahasrāra). He presents (nivedayet) the offering — lit. water offered to a guest — (arghyam) in the form of (his) mind (manasā).

He also (api) prepares (kalpayet) the water to be sipped from the palm of the hand — a purificatory ceremony that is performed before any ritual or meal — (ācamanam) (as well as) the water to be used in ablutions (snānīyam) by means of that very (tena) nectar (amṛtena). (He gives) the principle (tattvam) of Ākāśa — ether or space– (ākāśa) as the dress (vasanam), and the power of smelling (gandhatattvakam) as the odor (gandham).

He prepares (prakalpayet) (his) mind (manas) as the flower (vai) (and) arranges (prakalpayet) (his) vital energies (prāṇān) as incense (dhūpam). (He) also (ca) (arranges) the principle (tattvam) of Tejas — fire — (tejas) for it to act as (arthe) the lamp (dīpa), and (ca) the ocean (ambudhim) of nectar (sudhā) as the offering of food (naivedyam).

(He prepares) the Anāhata (anāhata) sound — which keeps sounding constantly in the heart lotus — (dhvanim) as the bell (ghaṇṭām), and (ca) the principle (tattvam) of Vāyu –air– (vāyu) as the fly-whisk made of tail of Yak (cāmaram). (He offers) the actions (karmāṇi) of the senses (indriya) as well as (tathā) the unsteadiness (cāñcalyam) of mind (manasaḥ) as dance (nṛtyam).

For realizing (siddhaye) the state (bhāva) of the Self (ātmanaḥ), he gives (dadyāt) flower(s) (puṣpam) of various sorts (nānāvidham): absence of delusion (amāyām), nonegotism (anahaṅkāram), dispassion and detachment (arāgam) as well as (tathā) absence of arrogance (amadam);…… absence of both bewilderment (amohakam) and (ca) deceit (adambham), as well as (tathā) nonmalevolence (adveṣa) and freedom from agitation (akṣobhake); absence of envy (amātsaryam) and (ca) liberty from covetousness (alobham)” — (these virtues) are named (prakīrtitam) the ten (daśa) flower(s) (puṣpam) –.

The supreme (paramam) flower(s) (puṣpam) (known as) Áhiṁsā — nonviolence and harmlessness — (ahiṁsā) and subjugation (nigraham) of the senses (indriya) (along with) the flower(s) (puṣpam) (known as) compassion (dayā), patience (kṣamā) and knowledge (jñāna), (are) therefore (tatas) the highest (param) five (pañca) flowers (puspam). Thus (iti), through (these) fifteen (pañcadaśaiḥ) flowers (puṣpaiḥ), (which are actually fifteen) flowers (puṣpaiḥ) formed from feelings (bhāva), he performs the worship (prapūjayet).

The sādhaka or practitioner uses every object in the ritual as representative of a virtue, state and so on. Therefore, one “must” be initiated in order to understand the Truth according to the Tantra-s, since only then the well-known vedic spirit of renunciation could be replaced for “a reintegration of the worldly life to the purposes of Enlightenment”. The “desire” and all values associated with it are then employed to achieve final Liberation. The tantric practitioner is both a master in spiritual matters and a master in worldly matters, because, in fact, there is no difference between “spiritual” and “worldly”. They are the two aspects in which the Divine Mother (Śakti) is manifested. So, a freed person is one who has transcended all pains and Saṁsāra (transmigration of the souls, that is, to be born and then to die, and to die and then to be born), and one who has acquired astonishing skills to lead a mundane life which is full of fulfillments.

मद्यपानेन मनुजो यदि सिद्धिं लभेत वै।
मद्यपानरताः सर्वे सिद्धिं गच्छन्तु पामराः॥११७॥

मांसभक्षणमात्रेण यदि पुण्यगतिर्भवेत्।
लोके मांसाशिनः सर्वे पुण्यभाजो भवन्त्विह॥११८॥

स्त्रीसम्भोगेन देवेशि यदि मोक्षं व्रजन्ति वै।
सर्वेऽपि जन्तवो लोके मुक्ताः स्युः स्त्रीनिषेवणात्॥११९॥

Madyapānena manujo yadi siddhiṁ labheta vai|
Madyapānaratāḥ sarve siddhiṁ gacchantu pāmarāḥ||117||

Māṁsabhakṣaṇamātreṇa yadi puṇyagatirbhavet|
Loke māṁsāśinaḥ sarve puṇyabhājo bhavantviha||118||

Strīsambhogena deveśi yadi mokṣaṁ vrajanti vai|
Sarve’pi jantavo loke muktāḥ syuḥ strīniṣevaṇāt||119||

If (yadi) a man (manujaḥ) really (vai) could attain (labheta) to Perfection (siddhim) by drinking (pānena) wine (madya), (then) may all (sarve) (those) vile (pāmarāḥ) people who are addicted to drinking (pānaratāḥ) wine (madya) achieve (gacchantu) Perfection (siddhim)!||117||

If (yadi) the achievement (gatiḥ) of Virtue (puṇya) would result (bhavet) from merely (mātreṇa) eating (bhakṣaṇa) meat (māṁsa), (then) may all (sarve) carnivorous beings (māṁsāśinaḥ) in this world (loke… iha) be (bhavantu) virtuous (puṇyabhājaḥ)!||118||

Oh goddess (deveśi)!, if (yadi) (the beings) indeed (vai) attain (vrajanti) to Liberation (mokṣam) through the enjoyment (sambhogena) of women (strī), (then) all (sarve) creatures (jantavaḥ) in this world (loke) would become (syuḥ) liberated (muktāḥ) by frequenting (niṣevaṇāt) women (strī)||119||

Alt-Right Politics: Year-End Awards. Biggest Winner: Deep State?

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https://altright.com/2018/01/04/alt-right-politics-year-end-awards/

Richard Spencer, Eli Mosley, Don Camillo and Gregory Ritter present Alt-Right Politics’ year-end awards: Biggest Winner, Biggest Loser, Best Politician, Most Defining Political Moment, Most Boring, Best Comeback, Best Photo Op, Worst Lie and many more.

How Black Holes Emitting Hawking Radiation At Best Give Non-Trivial Information About Planckian Physics: Towards Entanglement Entropy.

The analogy between quantised sound waves in fluids and quantum fields in curved space-times facilitates an interdisciplinary knowhow transfer in both directions. On the one hand, one may use the microscopic structure of the fluid as a toy model for unknown high-energy (Planckian) effects in quantum gravity, for example, and investigate the influence of the corresponding cut-off. Examining the derivation of the Hawking effect for various dispersion relations, one reproduces Hawking radiation for a rather large class of scenarios, but there are also counter-examples, which do not appear to be unphysical or artificial, displaying strong deviations from Hawkings result. Therefore, whether real black holes emit Hawking radiation remains an open question and could give non-trivial information about Planckian physics.

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On the other hand, the emergence of an effective geometry/metric allows us to apply the vast amount of universal tools and concepts developed for general relativity (such as horizons), which provide a unified description and better understanding of (classical and quantum) non-equilibrium phenomena (e.g., freezing and amplification of quantum fluctuations) in condensed matter systems. As an example for such a universal mechanism, the Kibble-Zurek effect describes the generation of topological effects due to the amplification of classical/thermal fluctuations in non-equilibrium thermal phase transitions. The loss of causal connection underlying the Kibble-Zurek mechanism can be understood in terms of an effective horizon – which clearly indicates the departure from equilibrium. The associated breakdown of adiabaticity leads to an amplification of thermal fluctuations (as in the Kibble-Zurek mechanism) as well as quantum fluctuations (at zero temperature). The zero-temperature version of this amplification mechanism is completely analogous to the early universe and becomes particularly important for the new and rapidly developing field of quantum phase transitions.

Furthermore, these analogue models might provide the exciting opportunity of measuring the analogues of these exotic effects – such as Hawking radiation or the generation of the seeds for structure formation during inflation – in actual laboratory experiments, i.e., experimental quantum simulations of black hole physics or the early universe. Even though the detection of these exotic quantum effects is partially very hard and requires ultra-low temperatures etc., there is no (known) principal objection against it. The analogue models range from black and/or white hole event horizons in flowing fluids and other laboratory systems over apparent horizons in expanding Bose–Einstein condensates, for example, to particle horizons in quantum phase transitions etc.

However, one should stress that the analogy reproduces the kinematics (quantum fields in curved space-times with horizons etc.) but not the dynamics, i.e., the effective geometry/metric is not described by the Einstein equations in general. An important and strongly related problem is the correct description of the back-reaction of the quantum fluctuations (e.g., phonons) onto the background (e.g., fluid flow). In gravity, the impact of the (classical or quantum) matter is usually incorporated by the (expectation value of) energy-momentum tensor. Since this quantity can be introduced at a purely kinematic level, one may use the same construction for phonons in flowing fluids, for example, the pseudo energy-momentum tensor. The relevant component of this tensor describing the energy density (which is conserved for stationary flows) may become negative as soon as the flow velocity exceeds the sound speed. These negative contributions explain the energy balance of the Hawking radiation in black hole analogues as well as super-radiant scattering. However, the (expectation value of the) pseudo energy-momentum tensor does not determine the quantum back-reaction correctly.

One should not neglect to mention the occurrence of a horizon in the laboratory – the Unruh effect. A uniformly accelerated observer cannot see half of the (1+1- dimensional) space-time, the two Rindler wedges are completely causally disconnected by the horizon(s). In each wedge, one may introduce a set of observables corresponding to the measurements made by the observers confined to this wedge – thereby obtaining two equivalent copies of observables in one wedge. In terms of these two copies, the Minkowski vacuum is an entangled state which yields the usual phenomena (thermo-field formalism) including the Unruh effect – i.e., the uniformly accelerated observer experiences the Minkowski vacuum as a thermal bath: For rather general quantum fields (Bisognano-Wichmann theorem), the quantum state ρ obtained by restricting the Minkowski vacuum to one of the Rindler wedges behaves as a mixed state ρ = exp{−2πHˆτ/κ}/Z, where Hˆτ corresponds to the Hamiltonian generating the proper (co-moving wristwatch) time τ measured by the accelerated observer and κ is the analogue to the surface gravity and determines the acceleration.

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Space-time diagram with a trajectory of a uniformly accelerated observer and the resulting particle horizons. The observer is confined to the right Rindler wedge (region x > |ct| between the two horizons) and cannot influence or be influenced by all events in the left Rindler wedge (x < |ct|), which is completely causally disconnected.

The thermal character of this restricted state ρ arises from the quantum correlations of the Minkowski vacuum in the two Rindler wedges, i.e., the Minkowski vacuum is a multi-mode squeezed state with respect the two equivalent copies of observables in each wedge. This is a quite general phenomenon associated with doubling the degrees of freedom and describes the underlying idea of the thermo-field formalism, for example. The entropy of the thermal radiation in the Unruh and the Hawking effect can be understood as an entanglement entropy: For the Unruh effect, it is caused by averaging over the quantum correlations between the two Rindler wedges. In the black hole case, each particle of the outgoing Hawking radiation has its infalling partner particle (with a negative energy with respect to spatial infinity) and the entanglement between the two generates the entropy flux of the Hawking radiation. Instead of accelerating a detector and measuring its excitations, one could replace the accelerated observer by an accelerated scatterer. This device would scatter (virtual) particles from the thermal bath and thereby create real particles – which can be interpreted as a signature of Unruh effect.