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Réunion d'hiver SMC 2014

Université McMaster, 5 - 8 décembre 2014

Les EPD nonlinéaires venant de la physique mathématique
Org: Stan Alama et Lia Bronsard (McMaster)
[PDF]

GINO BIONDINI, State University of New York at Buffalo
The integrable nature of modulational instability  [PDF]

We investigate the nonlinear stage of the modulational instability (MI) by characterizing the IVP for the focusing NLS equation with non-zero boundary conditions (NZBC) at infinity, using the recently formulated inverse scattering transform (IST) for this problem. While the linearization of NLS ceases to be valid when perturbations have grown sufficiently large, the IST holds at all times, and therefore provides the best way to study the nonlinear stage of MI. First we study the scattering problem with piecewise constant ICs which are generalizations to NZBC of a potential well and barrier, and we obtain several results. We prove that there are arbitrarily small perturbations of the constant background for which there are discrete eigenvalues, which shows that no area theorem is possible for the NLS equation with NZBC. We prove that there there is a large class of perturbations for which no discrete eigenvalues are present, which shows that solitons cannot be the primary vehicle for the manifestation of MI, contrary to a recent conjecture. Finally, we compute the small-deviation limit of IST and we compare it with the linearization of NLS. This allows us to identify the precise nonlinear analogue of the unstable Fourier modes within the IST. These are the Jost eigenfunctions for values of the scattering parameter in a finite interval of the imaginary axis around the origin. Importantly, this shows that the IST contains an automatic mechanism for the saturation of the MI.

STEFANELLA BOATTO, Universidade Federal do Rio de Janeiro

ALMUT BURCHARD, University of Toronto
Geometric stability of the Coulomb energy  [PDF]

The Coulomb energy of a positive charge distribution increases under symmetrization. In particular, if the charge is uniformly distributed on some set, then the energy is maximized (among sets of given volume) by balls. I will present recent work with Greg Chambers where we show that near-maximizers are close to balls.

YURI CHER, University Of Toronto
On a class of Generalized Derivative Nonlinear Schr\"odinger Equations  [PDF]

We study a class of generalized Derivative Nonlinear Schr\"odinger (gDNLS) equations of the form $i \psi_t + \psi_{xx} + i |\psi|^{2\sigma} \psi_x = 0$ with $\sigma > 1$. When $\sigma = 1$, this equation reduces to the canonical DNLS equation that arises from magnetohydrodynamics as well as in studies of ultrashort optical pulses. The DNLS shares the same scaling properties ($L^2$ critical) as the quintic Nonlinear Schr\"odinger equation which is known to have finite time blow-up solutions. However, the long time existence/possible occurrence of singularities for the DNLS equation remains an open problem. Recent numerical studies of the gDNLS for a range of values of $\sigma \in (1, 2]$ indicate that finite time blow-up may occur and give precisely the local structure of the blow-up solutions in terms of the blow-up rate and a universal profile $Q$ depending only on the strength of the nonlinearity $\sigma$. This complex valued profile is a solution of a nonlinear elliptic ODE with 2 real valued parameters $a$ and $b$ with an integral constraint.

Using methods of asymptotic analysis, we study the deformation of this profile and the parameters as the nonlinearity $\sigma$ tends to $1$. We find that Q tends to the lump soliton of DNLS while the parameter $a$ tends to $0$ like a power law in $(\sigma-1)$. We compare our results to a numerical integration of the nonlinear elliptic ODE using continuation methods. This is a joint work with G. Simpson and C. Sulem.

Nonlinear Equations for Finite-Amplitude Wave Propagation in Fiber-Reinforced Hyperelastic Media  [PDF]

Various composite materials, including biological tissues, are modeled as nonlinear elastic materials reinforced with elastic fibers. We consider the full set of dynamic equations for finite deformations of incompressible hyperelastic solids reinforced by a single fiber family. Using finite-amplitude wave propagation ans\"{a}tze compatible with the incompressibility condition, we derive the corresponding nonlinear and linear wave equations. Properties of these equations, including point symmetries, are studied. Exact time-dependent solutions are derived in several physical situations, including the evolution of pre-strained configurations and traveling waves.

Nonnegative weak solutions for a degenerate system modeling the spreading of surfactant on thin films  [PDF]

Depending on the parameter range, we prove local and global in time existence of non- negative weak solutions to a coupled system of two degenerate parabolic equations. This system models the spreading of an insoluble surfactant on a thin liquid film. The model includes gravity, surface tension, capillarity effects, and van der Waals forces. The surface diffusion coefficient is not assumed constant and depends on the surfactant concentration.

Joint work with: Roman M. Taranets, UCLA

ANDRES CONTRERAS, Fields

WALTER CRAIG, McMaster University and The Fields Institute
Birkhoff normal form for wave equation null forms  [PDF]

Theorems on global existence of solutions of nonlinear wave equations in ${\mathbb R}^n$ depend upon a competition between the time decay of solutions and the degree of the nonlinearity. Decay estimates are more effective when inessential nonlinear terms are able to be removed through a well-chosen transformation. In this talk, we construct Birkhoff normal forms transformations for the class of wave equations which are Hamiltonian PDEs and null forms, giving a new proof via canonical transformations of the global (and almost global') existence theorems for null form wave equations of S. Klainerman and J. Shatah. I will also describe the potential further applications to PDEs of this technique of Hamiltonian dynamics. These results are work-in-progress with A. French and C.-R. Yang.

BERNARDO GALVAO-SOUSA, University of Toronto
Accelerating Fronts in Semilinear Wave Equations  [PDF]

I will study dynamics of interfaces in solutions of the equation $\varepsilon\, \square u + \frac{1}{\varepsilon} f_{\varepsilon}(u) = 0$, for $f_{\varepsilon}$ of the form $f_{\varepsilon}(u) = (u^2-1)(2u-\varepsilon \kappa)$, for $\kappa \in \mathbb{R}$, as well as more general, but qualitatively similar, nonlinearities. I will show that for suitable initial data, solutions exhibit interfaces that sweep out timelike hypersurfaces of mean curvature proportional to $\kappa$.

This is a joint work with Robert Jerrard (University of Toronto).

QI GAO, NCTS, National Taiwan University
Symmetric vortices of two-component Ginzburg-Landau system  [PDF]

We consider symmetric vortex solutions in the plane $\mathbb{R}^2$, $\Psi=(\psi_+ (x),\psi_-(x))=(f_+ (r)e^{in_+\theta},f_- (r)e^{in_-\theta} )$, with given degrees $n_\pm\in\mathbb Z$, and prove existence, uniqueness, and asymptotic behavior of solutions as $r\to\infty$. We also consider the monotonicity properties of solutions, and exhibit parameter ranges in which both vortex profiles $f_+,f_-$ are monotone, as well as parameter regimes where one component is non-monotone. The qualitative results are obtained by means of a sub- and supersolution construction and a comparison theorem for elliptic systems. This is joint work with S. Alama.

TIZIANA GIORGI, New Mexico State University
Field-induced smectic phases in liquid crystals  [PDF]

We will discuss a phenomenological Landau energy introduced by Vaupotič and Čopič to study the onset of layer-undulated structures, induced by an applied electric field, in bent-core molecules liquid crystals. We will also present an analysis of the chevron structure in Smectic A liquid crystals due to magnetic fields, as modelled by a Chen-Lubensky energy. We will refer to joint work with Carlos García-Cervera and Sookyung Joo.

DMITRY GOLOVATY, The University of Akron
Solutions of Ginzburg-Landau equations in the presence of weak electric currents  [PDF]

I will discuss a Ginzburg-Landau model that allows for weak electric currents and in which the magnetic field is neglected. I will demonstrate the existence of a steady-state solution in a vicinity of the purely superconducting state and show that this solution is linearly stable.

ROBERT L JERRARD, University of Toronto
Existence and uniqueness of minimizers of general least gradient problems  [PDF]

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems - under certain sharp conditions - for minimizers of the general least gradient problem $\inf_{u\in BV_f(\Omega)} \int_{\Omega}\varphi(x,Du),$ where $f:\partial \Omega\to {\mathbb R}$ is continuous, $BV_f(\Omega):=\{v\in BV(\Omega): \ \ \forall x\in \partial \Omega, \ \lim_{r\to 0} \ \mboxess sup_{y\in \Omega, |x-y|<r} |f(x) - v(y)| = 0 \ \}$ and $\varphi(x,\xi)$ is a function that, among other properties, is convex and homogeneous of degree $1$ with respect to the $\xi$ variable. In particular we prove that if $a\in C^{1,1}(\Omega)$ is bounded away from zero, then minimizers of the weighted least gradient problem $\inf_{u \in BV_f}\int_{\Omega} a|Du|$ are unique in $BV_f(\Omega)$. We construct counterexamples to show that the regularity assumption $a\in C^{1,1}$ is sharp, in the sense that it can not be replaced by $a\in C^{1,\alpha}(\Omega)$ with any $\alpha<1$. This is joint work with Amir Moradifam and Adrian Nachman.

LYUDMILA KOROBENKO, McMaster University
Infinitely degenerate elliptic equations and non doubling metrics  [PDF]

The talk is about regularity of weak solutions to second order infinitely degenerate elliptic equations. It is known that regularity of weak solutions can be studied by studying properties of certain metric spaces associated to the operator, namely subunit metric spaces. The problem arising in the infinitely degenerate case is that the metric is such that the Lebesgue measure of subunit balls is non doubling. This makes a well developed theory of homogeneous metric spaces not applicable. However, with some additional assumptions continuity of weak solutions can be obtained even in the non doubling case. The question that remains open is to prove certain assumptions (such as Sobolev inequality) for at least some types of non doubling metric spaces associated to infinitely degenerate operators.

XAVIER LAMY, McMaster and Université de Lyon
Uniaxial symmetry and biaxial escape' in nematic liquid crystals  [PDF]

In nematics , molecules tend to align in some preferred direction. The local alignment may be biaxial, or present uniaxial symmetry. It has long been predicted that biaxial escape' should occur in some particular regimes. We will present some new insight on the uniaxiality constraint, independently of the regime. We also present a rigorous proof of biaxial escape' in the low temperature regime (joint work with A.Contreras).

ROBERT MCCANN, University of Toronto
The spectrum for a family of fourth order diffusions near the self-similar attractor  [PDF]

The thin-film and quantum drift-diffusion equations belong to a fourth-order family of evolution equations proposed by Denzler and myself as analogous to the (second-order) porous medium family. They are 2-Wasserstein gradient flows of the generalized Fisher information (just as Otto showed the porous medium to be the 2-Wasserstein gradient flow of the Reyni entropy). In this talk we describe the linearization of the fourth-order dynamics around the self-similar solution. We diagonalize this linearization by relating it to analogous problem for the porous medium equation. This yields information about the leading- and higher-order asymptotics of the fourth-order flows on $R^n$ which — outside of special cases — were inaccessible previously. These results were obtained jointly with Christian Seis.

DMITRY PELINOVSKY, McMaster University
Orbital stability of periodic waves and black solitons in the cubic defocusing NLS equation  [PDF]

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown to be linearly stable and their Floquet-Bloch spectrum has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that small amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. We also show that the black soliton is an unconditional minimizer of a fourth-order energy functional, which gives a simple proof of orbital stability with respect to perturbations in $H^2(\mathbb{R})$.

MARY PUGH, University of Toronto
Special Solutions in Smectic Electroconvection  [PDF]

We discuss electroconvection in a free submicron-thick liquid crystal film in an annular geometry. The film is flat in the xy plane; seen from above it looks like a DVD. (Seen from above, it has two boundaries: concentric circles.) A voltage is applied across the film, from the inner boundary to the outer boundary; this voltage provides a convective forcing. Because of the annular geometry, the dynamics are periodic in the azimuthal direction and the only boundaries are those at which the convective forcing is applied. The liquid crystal is in smectic A phase, forming a nearly-perfect two-dimensional fluid because the film does not change thickness, even while flowing. Also, the inner electrode can be rotated and so the experiment can be used to study the interplay between a stabilizing force applied via the boundary (Couette shear) and convection. We present numerical simulations of special solutions such as convection cells, oscillatory convection cells, undulating convection cells, and localized vortex solutions. This is joint work with Stephen Morris (Physics, University of Toronto).

IHSAN TOPALOGLU, McMaster University
Nonlocal attractive-repulsive interaction energies of binary densities  [PDF]

In this talk I will consider the minimization of nonlocal interaction energies of the form $$E[\rho]=\int_{\mathbb{R}^n}\!\int_{\mathbb{R}^n} K(x-y)\rho(x)\rho(y)\,dxdy$$ over the set of admissible functions $\{\rho\in L^1(\mathbb{R}^n,\{0,1\}) \colon \|\rho\|_{L^1}=1\}$ where the interaction potential $K$ is of the form of a power-law with attractive and repulsive components. This type of energies arise naturally in descriptions of systems of interacting particles, as well as continuum descriptions of systems with long-range interactions and are used in modelling collective behavior of many-agent systems, granular media and self-assembly of nanoparticles. The additional nonconvex constraint $\rho\in\{0,1\}$, on one hand, poses new challenges such as the existence of minimizers. On the other hand, with this constraint the first and second variations of the energy $E$ can simply be expressed explicitly as conditions on the boundary of the set $\{\rho=1\}$. After establishing the existence of minimizers, I will characterize the ground state when the attraction is given by a quadratic interaction and the repulsion by Newtonian potential, and comment on qualitative properties of local minimizers. This is a joint work with R. Choksi.

VITALI VOUGALTER, University of Cape Town
Existence of stationary solutions for some nonlocal reaction-diffusion equations  [PDF]

The paper is devoted to the existence of solutions of a nonlocal reaction-diffusion equation arising in population dynamics. The proof is based on a fixed point technique. We use solvability conditions for elliptic operators in unbounded domains which do not satisfy the Fredholm property.


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# What are the truth-values of intuitionistic logic?

Classical propositional logic is bivalent, that is its set of truth-values has cardinality 2 (True & False). Intuitionistic logic drops the law of the excluded middle; does it have the same set of truth-values? If not - then what is its set of truth-values and its cardinality?

EDIT

As Rostomyan points out below, Godel discovered that intuitionistic logic is not a n-valent logic, for any value of n; this prompted Gentzen to write to Godel to say, somewhat sardonically:

“It is as if you had a malicious pleasure in showing the purposelessness of others’ investigations …

In the sense of economy of thought this work is certainly useful, and in addition to that comes the particular beauty of your short proof”

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Intuitionistic logic has no finite valuation system (Godel 1932). That means it's neither bivalent nor n-valent for any finite n. –  Hunan Rostomyan Mar 3 at 17:03
Complementing @HunanRostomyan's response: understood as characterised by a consequence relation, the propositional fragment of intuitionistic logic is indeed not finite-valued, but is still truth-functional (thus, it has in principle an infinite-valued semantics with non-denumerably many truth-values, which may not be very informative). The same is no longer true if you restrict to the Minimalkalkül (Shoesmith & Smiley 1971): in that case, if you're looking for a characterisation in terms of a single matrix, not even an infinite-valued one is available. –  J Marcos Mar 4 at 2:39
and further to @Rostomyan: according to this note n-valent truth-value semantics are sound but not complete. Presumably we want a model that is both sound & complete. –  Mozibur Ullah Mar 4 at 3:53

The truth values of classical propositional logic form a Boolean algebra. The only subdirectly irreducible Boolean algebra has cardinality 2 (True & False). Hence the equational theory of classical propositional logic is completely determined by the two element Boolean algebra.

The truth values of intuitionistic propositional logic form a Heyting algebra. Surprisingly, the class of subdirectly irreducible Heyting algebras is not really smaller than the class of Heyting algebras itself:

Every Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top. It follows that even among the finite Heyting algebras there exist infinitely many that are subdirectly irreducible, no two of which have the same equational theory. Hence no finite set of finite Heyting algebras can supply all the counterexamples to non-laws of Heyting algebra. This is in sharp contrast to Boolean algebras, whose only SI is the two-element one, which on its own therefore suffices for all counterexamples to non-laws of Boolean algebra, the basis for the simple truth table decision method. Nevertheless it is decidable whether an equation holds of all Heyting algebras.

This quote from Wikipedia makes it pretty obvious that studying the subdirectly irreducible Heyting algebras won't help much for the understanding of intuitionistic logic.

Every field of sets is a Boolean algebra, and every Boolean algebra can represented as a field of sets. As a consequence, classical propositional logic can also be regarded as the logic of subsets. In David Ellerman's paper about the dual of the logic of subsets, I read that (it is supposedly well known that) intuitionistic logic can also be interpreted as the logic of open subsets. There is also a later simplified (32 pages instead of 64) introduction to partition logic, which I haven't read yet. Every Heyting algebra can be represented as the open elements of an interior algebra. Every interior algebra can be represented as a topological field of sets with its interior and closure operators corresponding to those of the topological space. Another aspect is that every "n-ary operator between open sets" composed of pointwise operators and interior operators can be defined by an intuitionistic formula. But note that in the same way as a field of sets is not necessarily isomorphic to the set of all subsets of a set, a topological field of sets is not necessarily isomorphic to the set of all open subsets of a topological space.

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Should that be a complete Heyting algebra? I think that the locale of open sets for a topological space is alternatively, and equivalently thought of as a complete Heyting algebra, which matches what you about it being the logic of open sets of a topological space. –  Mozibur Ullah Mar 22 at 14:09
@MoziburUllah It's just a Heyting algebra, no completeness is needed. I think I did David (and me) a disfavor by trying to wiggle out of the explicit statement that intuitionistic propositional logic can be interpreted as the logic of the open subsets of a topological space. This statement is simply true, and I should better try to understand it properly, instead of claiming "it might be a more complicated correspondence than I expected upon learning this". I will edit my post after I have done this. –  Thomas Klimpel Mar 22 at 21:44
@Klimpel: ok; I think it is well-known that intuitionistic logic can be interpreted as the open-set logic, in the sense that well-known is understood in mathematics (or logic). I ask because given that there are arbitrary unions for open sets, one can interpret that as completeness. In the nomenclature of category theory, the logic of open sets, is a frame, and the dual is called a locale. Davids paper shows the 'dual' of subsets on a set X are partitions on X, –  Mozibur Ullah Mar 23 at 3:20
or equivalently equivalence relations on X. I'm not familiar with the notation or language that he uses, so I can't quite see how this dual is constructed. It isn't the categorical dual where one just reverses the arrows. Wikipedia confirms that intuitionistic logic can be interpreted topologically, and in fact the topology of the real line suffices, and they further claim though there are finite Heyting algebras, no finite Heyting algebra will interpret the logic. –  Mozibur Ullah Mar 23 at 3:24
@MoziburUllah I fixed my answer now. Most importantly, I replaced the statement "intuitionistic propositional logic can be interpreted as the logic of the open subsets of a topological space" by the statement "intuitionistic propositional logic can be interpreted as the logic of open subsets". Note also the subtle difference between "the open elements of an interior algebra" meaning "all open elements", and "topological field of sets" meaning just some open subsets of a topological space, but necessarily all. –  Thomas Klimpel Mar 24 at 0:01

See in SEP the entry on Intuitionistic Logic

The rejection of the Law of Excluded Middle does not means that intuitionsits adopt a three (or more) -valued logic.

According to Brouwer :

LEM was abstracted from finite situations, then extended without justification to statements about infinite collections. For example, if x, y range over the natural numbers 0, 1, 2, … and B(x) abbreviates the property (there is a y > x such that both y and y+2 are prime numbers), then we have no general method for deciding whether B(x) is true or false for arbitrary x, so ∀x(B(x) ∨ ¬B(x)) cannot be asserted in the present state of our knowledge. [...]

But to Brouwer the general LEM was equivalent to the a priori assumption that every mathematical problem has a solution — an assumption he rejected, anticipating Gödel's incompleteness theorem by a quarter of a century.

To see this, we need only reflect on the following Goldbach conjecture (GC):

Every even integer > 2 can be written as a sum of two primes,

which remains neither proved nor disproved despite the best efforts of many of the leading mathematicians since it was first raised in a letter from Goldbach to Euler in 1742.

Formalized intuitionistic logic is naturally motivated by the informal Brouwer-Heyting-Kolmogorov explication of intuitionistic truth.[...] From the B-H-K viewpoint, a sentence of the form (A ∨ B) asserts that either a proof of A, or a proof of B, has been constructed.

So, the rejection of LEM is motivated by the fact that we are not able to assert, for every mathematical problem A, that we posses a proof of A, or a proof of ¬A .

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Looking at Wittgensteins Tractatus, he identifies the truth values of elementary propositions with whether a certain states of affairs exists or does not exist. Here, is Brouwer saying that truth of A is a proof of A, and the falseness of A is a proof of (not A)? The model theory of classical propositional logic assigns truth values to variables, does this mean that the model theory of intuitionistic logic assigns proofs in some kind of sense? –  Mozibur Ullah Mar 3 at 12:23
Things are complicated; basically, for Brouwer mathematics is "basic" and logic is not good : see SEP on Brouwer. His followers, starting with Heyting, developed Brouwer's "alternative" mathematics, but abandoned Brouwer's rejection of logic (alas ! a double negation ...). The way logician "interpret" intuitionsitic logic is thorugh the B-H-K interpretation, or operational or computational meaning of the logical connectives and quantifiers. Roughly, I may assert a proposition when I have a proof of it : proof in place of truth. –  Mauro ALLEGRANZA Mar 3 at 13:46
@MoziburUllah - later W's phil of math (post-Tractatus) is deeply influenced by Brouwer; W adopted a sort of finitistic point of view, with a radical rejection of infinite (see M.Marion or P.Frascolla books on W's math phil). But it seems that W was scarcely interested and poor informed about contemporaries (post-Tractatus) developments in math log (see his underestimation of the significance of Godel's Incompleteness Th). –  Mauro ALLEGRANZA Mar 5 at 20:51

With intuition there is no right or wrong or more. It just is. There is a pull in one direction. There is one answer. It is correct as long as it is until it no longer is. Existent that is meaning the moment has passed.

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We're looking for long answers that provide some explanation and context. Don't just give a one-line answer; explain why your answer is right, ideally with citations. Answers that don't include explanations may be removed.

Unfortunately, both in classical and in intuitionistic propositional logics, once a formula is proved/correct, it will remain proved. –  Hunan Rostomyan Mar 3 at 17:49

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# Yes, ideas are harder to find. Don't panic yet.

Nick Bloom, Chad Jones, John van Reenen, and Michael Webb have a new paper on how productive we are at producing “ideas”, which one can argue are the ultimate source of all economic growth.

The splash graphic from the paper, which I believe made a brief tour of the Twitter-verse a month or so ago, is shown below. It shows that the TFP growth rate (in blue) peaked around the 1940’s, and has declined since then. The axis measurement here is a big funky, as you could see more of the details of this line if they had made the left axis go up to say 5%, rather than 25%. Regardless, the TFP growth rate is smaller than it was in the past, following Robert Gordon’s numbers.

In green is the relative number of researchers compared to the number in 1930. This isn’t the number of researchers, its a measure of effective researchers, found by dividing total R and D spending by the nominal wage of high-skilled workers. Relative to 1930, we have almost 25 times as many (effective) researchers at work today. This leads to the author’s title question - “Are ideas getting harder to find?”. And the author’s answer is “Yes”.

### Measuring Idea TFP

BJvRW measure how “hard” it is to find ideas by looking at what they call “idea TFP”. This is easy to back out from some data. Let’s write the growth rate of ideas (I) as

where the left side is the growth rate of ideas at time t, and on the right-hand side we’ve got the determinants of that growth. $S_t$ is research effort, and here think of the measure of effective researchers. How much work do we put into coming up with new ideas? The second term is $\alpha_t$, which is the “idea TFP”. It tells you how much idea growth you get out of those researchers.

If you tell me the growth rate of ideas (the left) and the number of effective researchers (S), then I (well, the authors) can back out the measure of $\alpha_t$ year by year. And this is what BJvRW do, in a variety of contexts.

What they do first is to equate the growth rate of ideas with the growth rate of TFP (actual aggregate TFP in producing GDP). That doesn’t mean that each idea is equal to one “unit” of TFP, it just means that if the number of ideas grows by 10%, then TFP grows by 10%. This is a big leap - but let’s not get hung up on this. It isn’t important in this first context. With the data from the first graph on the TFP growth rate ($\dot{I}/I$), and the number of effective researchers ($S_t$), we can back out $\alpha_t$. This “idea TFP” is, awkwardly, the “TFP TFP”, or our productivity at producing TFP given a set of researchers.

This second graph plots the $\alpha_t$ they back out, relative to 1930. You can see that it falls to about 1/64th of the level in 1930. Because we are 1/64th as effective at turning researchers into TFP, but we only have 32 times as many researchers, our growth rate of TFP is about 1/2 as high as it was in the 1930’s/1940’s. Hang on to the “why” question for a moment. For now, we’re just looking at the pure accounting.

This first exercise using (actual) TFP as the measure of ideas is a bit crude, and what BJvRW go on to do is examine individual industries or types of products to establish that the “idea TFP” is in fact falling within each of these as well. This is relevant because the first result using TFP does not tell us what is happening at the “micro” level. We could have constant (or even rising) “idea TFP” within individual industries or lines of research, and yet in aggregate falling “idea TFP”. It may be that ideas are not perfectly translated to TFP, or it may be that the we are spreading our research efforts over an ever-expanding list of industries or lines of research. In the latter case, idea growth in individual industries/lines may be falling because of the decline in numbers of researchers, not because of declining idea TFP. But in aggregate, this would look like declining idea TFP.

Thus looking at individual industries/lines will help us sort out what is happening. The first example is transistors, and the count of transistors per square inch on a chip. Famously, this appears to obey Moore’s Law, which is that the number of transistors per square inch doubles every 18 months. If we count the number of transistors per square inch as “ideas”, then this translates to a growth rate of 35% per year. At the same time, the effective number of researchers working at chip companies (Intel, AMD, etc..) has grown by a factor of about 80 since 1970. Plotting this together yields their next figure.

Relative to the first graph, there is no decline in the growth rate of transistors. But there also isn’t an increase, despite the increased number of researchers. What this means is that “transistor idea TFP” is falling over time, at about 10% per year. As they point out in the paper, Moore’s Law still holds, but it takes 78 times more researchers to figure out how to double transistor density than it did in 1970.

From here, the paper works through several other examples. I’ll leave it to you to look at the specifics. But overall, the message becomes clear. For crop yields, idea TFP is falling at between 3% and 9% per year, except for cotton, where it rose slightly over time. For “new molecular entities” (e.g. drugs) idea TFP has fallen to about 1/4 of its value in 1970, or about 3% per year. BJvRW also look at firm size, measured in different ways, which should be related to their productivity (and ideas). In each case, their size is not growing as fast as their RD spending, implying their idea TFP is falling at about 8-15% per year.

### Implications for Economic Growth

The looming question here is whether this implies economic growth is due to slow down to zero. And the answer is “No”. It is possible for ideas to have a constant growth rate (and GDP per capita to have a constant growth rate as well) even if idea TFP is falling over time. For “semi-endogenous growth theory”, a term often associated with Chad Jones, this falling idea TFP is consistent with constant, positive growth in GDP per capita.

This is what is in the Jones/Vollrath textbook (buy 1, buy 10, buy 1000!) when we teach how ideas accumulate. Modify that first equation I showed you to look like the following

where idea TFP is now the ratio of a constant, $\alpha$, and the number of ideas, $I_t$, raised to the power $\beta$. The value of $\beta$ is quite important. If $\beta=0$, then the level of ideas has no effect on idea TFP, and idea TFP should be constant. This is what was in the original Romer model, by the way. What Jones did was to say that this implies a very knife-edge condition, and there was no reason to assume that $\beta=0$. When $\beta>0$, as we accumulate more ideas, the growth rate of ideas falls - idea TFP goes down.

But just because idea TFP falls as we get more ideas doesn’t mean the growth rate of ideas (the left hand side) is falling. Why? Because $S_t$ is rising over time due to population growth. If the share of people working as researchers is constant, then their absolute number is rising over time. In that case, there is a growth rate of ideas such that the growth rate of ideas stays constant over time, $\dot{I}/I = n/\beta$, where $n$ is the growth rate of population (and hence of researchers). Idea growth need not slow down just because idea TFP falls over time.

Hence the “don’t panic yet” comment in the title. Falling idea TFP may well be the natural course of things, without resulting in lower actual TFP growth. The observed fall in actual TFP growth would have to come from some other forces involved in generating research effort. Start with an explanation that might imply a permanent fall in the TFP growth rate:

1. The growth rate of the effective population has fallen. If the population that provides researchers (and it is best here to think of the total population of advanced countries, not just the US) starts to grow more slowly, then S grows more slowly, and the long-run growth rate of ideas is going to be lower. Maybe the slowdown in TFP growth is due to slower population growth in developed countries. A counterargument would be that this is offset by the increasing shares of Chinese and Indian populations that are in the pool of possible researchers. A counter-counterargument would be that age distributions matter too, and we’ve got too many stodgy old people, and not enough inventive young whipper-snappers.

Other explanations might imply that the drop in the TFP growth rate will be temporary:

1. A downward shift in the share of that population that is doing research. This would generate a lower measured short-run growth rate of ideas because S would have a lower short-run growth rate for a while as we moved to a lower share of people doing research. This doesn’t appear to be true in the data, where an increasing share of workers are reported as doing RD work over time - and that should be pushing up the short-run growth rate of ideas.
2. An exogenous drop in the level of $\alpha$. In this case, the growth rate of ideas would eventually rise back up to what it was before, after we had lower measured TFP growth for a while on our way to a new equilibrium. If we’re going to argue about the value of $\alpha$, though, then we should start over with our theory about how ideas get generated, as saying “it was a random exogenous shock” ends up being kind of useless.

And of course, maybe semi-endogenous growth theory is wrong. Maybe this just isn’t the way to look at things. Alternative theories might say idea growth could be affected by policies (subsidies for basic research, tax policy regarding treatment of RD spending, etc..) and what we’ve seen is a continuous deterioration of the policies supporting idea generation. This one is a little hard to square, though, just because it seems odd that there would be such growth in the number of researchers if the returns to that research at the firm or individual level were going down so much. I can see how taxes on RD would lower the resources devoted to it, but why would that affect how effective RD efforts are? You’d have to argue that policies were incenting companies to work on useless ideas.


text:
Confidence intervals with tiers: functions for between-subjects (independent measures) ANOVA

June 21, 2012
By

(This article was first published on Serious Stats » R code, and kindly contributed to R-bloggers)

In a previous post I showed how to plot difference-adjusted CIs for between-subjects (independent measures) ANOVA designs (see here). The rationale behind this kind of graphical display is introduced in Chapter 3 of Serious stats (and summarized in my earlier blog post).

In a between-subjects – or in indeed in a within-subjects (repeated measures) – design you or your audience will not always be interested only in the differences between the means. Rarely, the main focus may even be on the individual estimates themselves. A CI for each of the individual means might be informative for several reasons.

First, it may be important to know that the interval excludes an important parameter value (e.g., zero). The example in Chapter 3 of  Serious Stats involved a task in which participants had to decide which of two diagrams matched a description they had just read. Chance performance is 50% matching accuracy, so a graphical display that showed that the 95% CI for each mean excludes 50% suggests that participants in each group were performing above chance.

Second the CI for an individual mean gives you an idea of the relative precision with which that quantity is measured. This may be particularly important in an applied domain. For example, you may want to be fairly sure that performance on a task is high in some conditions as well as being sure that there are differences between conditions.

Third, the CIs for the individual means are revealing about changes in the precision between conditions. If the sample sizes are equal (or nearly equal) they are also revealing about patterns in the variances. This is because the precision of the individual means is a function of the standard error and n. This may be obscured when difference-adjusted CIs are plotted – though mainly for within-subjects (repeated measures) designs which have to allow for the correlation between the samples.

In any case, it may be desirable to display CIs for individual means and difference-adjusted means on the same plot. This could be accomplished in several ways but I have proposed using a two-tiered CI plot (see here for a brief summary of my BRM paper on this or see Chapter 16 of Serious stats).

A common approach (for either individual means or difference-adjusted CIs) is to  adopt a pooled error term. This results in a more accurate CI if the homogeneity of variance assumption is met. For the purposes of a graphical display I would generally avoid pooled error terms (even if you use a pooled error term in your ANOVA). A graphical display of means is useful as an exploratory aid and supports informal inference. You want to be able to see any patterns in the precision (or variances) of the means. Sometimes these patterns are clear enough to be convincing without further (formal) inference or modeling. If they aren’t completely convincing it usually better to show the noisy graphic and supplement it with formal inference if necessary.

Experienced researchers understand that real data are noisy and may (indeed should!) get suspicious if data are too clean. (I’m perhaps being optimistic here – but we really ought to have more tolerance for noisy data, as this should reduce the pressure on honest researchers to ‘optimize’ their analyses – e.g., see here).

My earlier post on this blog provided functions for the single tier difference-adjusted CIs. Here is the two-tiered function (for a oneway design):

plot.bsci.tiered <- function(data.frame, group.var=1, dv.var=2,  var.equal=FALSE, conf.level = 0.95, xlab = NULL, ylab = NULL, level.labels = NULL, main = NULL, pch = 19, pch.cex = 1.3, text.cex = 1.2, ylim = c(min.y, max.y), line.width= c(1.5, 1.5), tier.width=0, grid=TRUE) {
data <- subset(data.frame, select=c(group.var, dv.var))
fact <- factor(data[[1]])
dv <- data[[2]]
J <- nlevels(fact)
ci.outer <- bsci(data.frame=data.frame , group.var=group.var, dv.var=dv.var,  difference=FALSE, var.equal=var.equal, conf.level =conf.level)
ci.inner <- bsci(data.frame=data.frame , group.var=group.var, dv.var=dv.var,  difference=TRUE, var.equal=var.equal, conf.level =conf.level)
moe.y <- max(ci.outer) - min(ci.outer)
min.y <- min(ci.outer) - moe.y/3
max.y <- max(ci.outer) + moe.y/3
if (missing(xlab))
xlab <- "Groups"
if (missing(ylab))
ylab <- "Confidence interval for mean"
plot(0, 0, ylim = ylim, xaxt = "n", xlim = c(0.7, J + 0.3), xlab = xlab,
ylab = ylab, main = main, cex.lab = text.cex)
if (grid == TRUE) grid()
points(ci.outer[,2], pch = pch, bg = "black", cex = pch.cex)
index <- 1:J
segments(index, ci.outer[, 1], index, ci.outer[, 3], lwd = line.width[1])
axis(1, index, labels = level.labels)
if(tier.width==0) {
segments(index - 0.025, ci.inner[, 1], index + 0.025, ci.inner[, 1], lwd = line.width[2])
segments(index - 0.025, ci.inner[, 3], index + 0.025, ci.inner[, 3], lwd = line.width[2])
}
else segments(index, ci.inner[, 1], index, ci.inner[, 3], lwd = line.width[1]*(1 + abs(tier.width)))
}

The following example uses the diagram data from the book:

source('http://www2.ntupsychology.net/seriousstats/SeriousStatsAllfunctions.txt')

plot.bsci.tiered(diag.dat, group.var=2, dv.var=4, ylab='Mean description quality', main = 'Two-tiered CIs for the Diagram data', tier.width=1)

The result is a plot that looks something like this (though I should probably have reordered the groups and labeled them):

For these data the group sizes are equal and thus the width of the outer tier reflect differences in variances between the groups. The variances are not very unequal, but neither are they particularly homogenous. The inner tier suggests group three is different from groups 2 and 4 (but not from group 1). This is a pretty decent summary of what’s going on and could be supplemented by formal inference (see Chapter 13 for a comparison of several formal approaches also using this data set).

N.B. R code formatted via Pretty R at inside-R.org

Footnote: The aesthetics of error bar plots

A major difference between the plot shown here and that in my BRM paper or in the book is that I have changed the method of plotting the tiers. The change is mainly aesthetic, but also reflects the desire not to emphasize the extremes of the error bar. The most plausible values of the parameter (e.g., mean) are towards the center of the interval – not at the extremes. I have discussed the reasons for my change of heart in a bit more detail elsewhere.

To this end I have also updated all my plotting functions. They still use the crossbar style from the book by default but this is controlled by a tier width argument. If tier.width=0 the crossbar style is used otherwise it used the tier.width to control the additional thickness of the difference-adjusted lines. In general, tier.width=1 seems to work well (but the crossbar style may be necessary for some unusual within-subject CIs where the difference-adjusted CI is wider than the CI for the individual means).

Filed under: R code, serious stats Tagged: ANOVA, confidence intervals, exploratory data analysis, R, R functions, repeated measures ANOVA, statistics


text:
# Quasi-fermi levels in a solar cell?

I was wondering, if my fermi levels splits up due to n and p type doping into two fermi levels, one for the p type one for the n type, and now due to light radiation my fermi levels split up into 2 quasi fermi levels for the n and p carriers, do i have 4 quasi levels in total then? If i recall correctly the electrochemical potential is defined by the difference of quasi fermi levels, so which quasi fermi levels do i take the difference of, if there exist 4 levels?

sincerely

• Level not niveaus – boyfarrell Sep 3 '15 at 13:56

In the dark you have a single Fermi level which is spatially dependent on the net free charges $E_f(z)$.
When the semiconductor is perturbed by light the Fermi level splits into two quasi fermi levels for electrons $\mu_e(z)$ holes holes $\mu_h(z)$. The quasi fermi level separation (or electrochemical potential) is, $\Delta E_f(z) = \mu_e(z) - \mu_h(z)$.
• No, because of the law of mass action $n_i(x)^2 = n(x) p(x)$. In the dark you always have one Fermi-level, because of the law of mass action that is all that is needed to get the correct electron and hole populations in the conduction and valence bands. When illuminated the Fermi-level can split and to get the correct number of electron and holes we need to use a quasi Fermi-level description. – boyfarrell Sep 11 '15 at 7:28


text:
# Thread: Euclidean norm of U*x when norm of x is 1!

1. ## Euclidean norm of U*x when norm of x is 1!

Let x be a nx1 vector in Cn and y = U*x is another vector in Cn, where U is a unitary matrix and * represents the conjugate transpose.

The Euclidean norm of y equals 1 if and only if the Euclidean norm of x equals 1:

||y|| = 1 if and only if ||x|| = 1.

I can't prove this statement and don't know where to start!

Thanks so much!

2. Originally Posted by Damien
Let x be a nx1 vector in Cn and y = U*x is another vector in Cn, where U is a unitary matrix and * represents the conjugate transpose.

The Euclidean norm of y equals 1 if and only if the Euclidean norm of x equals 1:

||y|| = 1 if and only if ||x|| = 1.

I can't prove this statement and don't know where to start!

Thanks so much!
The unitary matrix has at least one complex eigenvalue; what do you know about the eigenvalues of a unitary matrix? U is unitary if and only if U* is unitary right? Then since U*x = ax for some eigenvalue a, take the norm on both sides and use some properties of the norm to get what you want.

3. Originally Posted by Damien
Let x be a nx1 vector in Cn and y = U*x is another vector in Cn, where U is a unitary matrix and * represents the conjugate transpose.

The Euclidean norm of y equals 1 if and only if the Euclidean norm of x equals 1:

||y|| = 1 if and only if ||x|| = 1.

I can't prove this statement and don't know where to start!

Thanks so much!
let $e_1, \cdots , e_n$ be the columns of $U.$ since $U$ is unitary, we have $e_i \cdot \bar{e_j}=\delta_{ij},$ where $\delta_{ij}$ is the Kronecker's delta. let $x_i, y_i, \ \ 1 \leq i \leq n,$ be the entries of the vectors $x,y$ respectively.

now we have $\sum_{i=1}^n y_ie_i = Uy=x$ and thus $||x||^2=||Uy||^2=\left(\sum_{i=1}^ny_ie_i \right) \cdot \left(\sum_{j=1}^n \bar{y_j} \bar{e_j} \right)=\sum_{1 \leq i,j \leq n}y_i \bar{y_j} \delta_{ij}=\sum_{i=1}^n |y_i|^2=||y||^2$ and thus $||x||=||y||.$

4. ## Thank you!

Thanks a lot guys. 2 different proofs, but both very nice.

1.
Every matrix has at least one eigenvalue (fundamental theorem of algebra)
If U is unitary then U* is unitary too
Every eigenvalue of a unitary matrix has absolute value of 1
So: ||y|| = ||U*x|| = ||lambda x|| = |lambda| · ||x|| = ||x||

2.
If y = U*x then x = Uy
So: ||x||^2 = x*x = y*U*Uy = y*y = ||y||^2 <=> ||x|| = ||y||

Thanks a lot to both of you, really appreciate it!


text:
# wx.PickerBase¶

Base abstract class for all pickers which support an auxiliary text control.

This class handles all positioning and sizing of the text control like a an horizontal wx.BoxSizer would do, with the text control on the left of the picker button.

The proportion (see wx.Sizer documentation for more info about proportion values) of the picker control defaults to 1 when there isn’t a text control associated (see PB_USE_TEXTCTRL style) and to 0 otherwise.

## Window Styles¶

This class supports the following styles:

• wx.PB_USE_TEXTCTRL: Creates a text control to the left of the picker which is completely managed by this wx.PickerBase class.

## Class Hierarchy¶

Inheritance diagram for class PickerBase:

## Methods Summary¶

 __init__ CreateBase GetInternalMargin Returns the margin (in pixel) between the picker and the text control. GetPickerCtrl Returns the native implementation of the real picker control. GetPickerCtrlProportion Returns the proportion value of the picker. GetPickerStyle GetTextCtrl Returns a pointer to the text control handled by this window or None if the PB_USE_TEXTCTRL style was not specified when this control was created. GetTextCtrlProportion Returns the proportion value of the text control. GetTextCtrlStyle HasTextCtrl Returns True if this window has a valid text control (i.e. if the PB_USE_TEXTCTRL style was given when creating this control). IsPickerCtrlGrowable Returns True if the picker control is growable. IsTextCtrlGrowable Returns True if the text control is growable. PostCreation SetInternalMargin Sets the margin (in pixel) between the picker and the text control. SetPickerCtrl SetPickerCtrlGrowable Sets the picker control as growable when grow is True. SetPickerCtrlProportion Sets the proportion value of the picker. SetTextCtrl SetTextCtrlGrowable Sets the text control as growable when grow is True. SetTextCtrlProportion Sets the proportion value of the text control. UpdatePickerFromTextCtrl UpdateTextCtrlFromPicker

## Class API¶

class wx.PickerBase(Control)

Possible constructors:

PickerBase()

Base abstract class for all pickers which support an auxiliary text control.

### Methods¶

__init__(self)

CreateBase(self, parent, id=ID_ANY, text="", pos=DefaultPosition, size=DefaultSize, style=0, validator=DefaultValidator, name=ButtonNameStr)
Parameters
Return type

bool

GetInternalMargin(self)

Returns the margin (in pixel) between the picker and the text control.

This function can be used only when HasTextCtrl returns True.

Return type

int

GetPickerCtrl(self)

Returns the native implementation of the real picker control.

Return type

wx.Control

Note

The returned control in the generic implementation of wx.FilePickerCtrl, wx.DirPickerCtrl, wx.FontPickerCtrl and wx.ColourPickerCtrl is a specialized wx.Button class so that you can change its label doing, e.g.:

if wx.Platform == '__WXMSW__':
# wxMSW is one of the platforms where the generic implementation
# of wx.FilePickerCtrl is used...

pButt = myFilePickerCtrl.GetPickerCtrl()

if pButt is not None:
pButt.SetLabel('Custom browse string')

GetPickerCtrlProportion(self)

Returns the proportion value of the picker.

Return type

int

GetPickerStyle(self, style)
Parameters

style (long) –

Return type

long

GetTextCtrl(self)

Returns a pointer to the text control handled by this window or None if the PB_USE_TEXTCTRL style was not specified when this control was created.

Return type

wx.TextCtrl

Note

The contents of the text control could be an invalid representation of the entity which can be chosen through the picker (e.g. when the user enters an invalid colour syntax because of a typo). Thus you should never parse the content of the textctrl to get the user’s input; rather use the derived-class getter (e.g. wx.ColourPickerCtrl.GetColour , wx.FilePickerCtrl.GetPath , etc).

GetTextCtrlProportion(self)

Returns the proportion value of the text control.

This function can be used only when HasTextCtrl returns True.

Return type

int

GetTextCtrlStyle(self, style)
Parameters

style (long) –

Return type

long

HasTextCtrl(self)

Returns True if this window has a valid text control (i.e. if the PB_USE_TEXTCTRL style was given when creating this control).

Return type

bool

IsPickerCtrlGrowable(self)

Returns True if the picker control is growable.

Return type

bool

IsTextCtrlGrowable(self)

Returns True if the text control is growable.

This function can be used only when HasTextCtrl returns True.

Return type

bool

PostCreation(self)

SetInternalMargin(self, margin)

Sets the margin (in pixel) between the picker and the text control.

This function can be used only when HasTextCtrl returns True.

Parameters

margin (int) –

SetPickerCtrl(self, picker)
Parameters

picker (wx.Control) –

SetPickerCtrlGrowable(self, grow=True)

Sets the picker control as growable when grow is True.

Parameters

grow (bool) –

SetPickerCtrlProportion(self, prop)

Sets the proportion value of the picker.

Parameters

prop (int) –

SetTextCtrl(self, text)
Parameters

text (wx.TextCtrl) –

SetTextCtrlGrowable(self, grow=True)

Sets the text control as growable when grow is True.

This function can be used only when HasTextCtrl returns True.

Parameters

grow (bool) –

SetTextCtrlProportion(self, prop)

Sets the proportion value of the text control.

This function can be used only when HasTextCtrl returns True.

Parameters

prop (int) –

UpdatePickerFromTextCtrl(self)

UpdateTextCtrlFromPicker(self)

### Properties¶

InternalMargin

PickerCtrl

PickerCtrlProportion

TextCtrl

TextCtrlProportion


text:
# Taxes

Go back to  'Commercial Math'

Taxes and discounts are a natural extension of the application of percentages. These are probably the most frequent terms that come up during a conversation about shopping since they directly affect the amount of money involved. To understand tax and discount better, it is important to know the related terms.

## Marked price

Almost every item comes with a price tag attached to it (usually, by the manufacturer of the item). This price is called the marked price, abbreviated as MP.

Let’s try and understand this better using an example.

The price tag on a camera reads $$₹3999$$ but it is being sold for $$₹3900.$$

The marked price of the camera is $$₹3999.$$ The selling price is $$₹3900.$$

## Tax

You must have heard words like income tax, GST, service tax etc in daily conversations. In recent years, GST was introduced, while other taxes like excise duty, VAT, municipal tax, entertainment tax etc. have existed for a while now. The government charges a fee on every item that is sold. This fee is meant to be spent on public welfare and is known as tax. The tax charged is always a percentage of the SP.

For example, let’s say an item cost $$₹500$$ and a sales tax of $$5 \%$$ was charged. What would be the bill amount?

Let’s try and find how much is $$5 \%$$ of $$500.$$

\begin{align}{5 \over 100} \times 500 = 25 \end{align}

Now let’s add this amount to the cost price.

\begin{align}₹500 + ₹25 = ₹525 \end{align}

The total bill amount including taxes would be $$₹525.$$

Let’s now try and understand how taxes are levied on a discounted product.

-  Discount is calculated on selling price excluding taxes.
-  Tax is applied on the amount arrived at after subtracting all the discount from selling price.

Example: Suppose you are ordering a Pizza at a Pizza parlour and it costs $$₹400.$$ The restaurant is offering a discount of $$30 \%$$ on all dine in orders, so the final amount that it will come to, assuming the restaurant only charges $$10 \%$$ GST as tax is:

Step 1 - Calculate the discounted price

\begin{align} \text{Discount} & = {30 \over 100} \times 400 \\ & = ₹120 \end{align}

\begin{align} \text{Final Bill Amount} & = ₹400 - ₹120 \\ & = ₹280 \end{align}

Remember the tax is levied on the discounted price. $$10 \%$$ GST was levied on the final amount

\begin{align} {10 \over 100} \times ₹280 = ₹28 \end{align}

\begin{align} \text{Final Amount} &= ₹280 + ₹28 \\ & = ₹308 \end{align}

## Tips and Tricks:

Taxes are calculated on the final discounted price of the item and not on the marked price.

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# Can you tell me which test case it is failing?

 0 I am writing solution for http://codeforces.com/contest/768/problem/A but it is giving WA on 10th test case. Here is my code: #include #include #define ll long long int using namespace std; int main() { // your code goes here int n; cin>>n; int a[n]; for(int i=0;i>a[i]; cout<

 1 You have overlooked two important cases, I think. Firstly if N=1, your program will print N-2, which is equal to -1, while the answer should be 0. Another case that you haven't considered is that two or more stewards can have the same value. For example, if N=3, and the stewards have values: 4,4,4; your program will print 1, while the correct answer is 0. Hope this was helpful. answered 10 Jul '18, 01:11 681●12 accept rate: 14% 1 thanks for your time.. (10 Jul '18, 11:42) montycs0★
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text:
experimental

Because jupyter-book is built on top of Jekyll, we can use the excellent jekyll-scholar book to include citations and a bibliography with your book.

Note that this only works if you're building your book HTML locally and hosting the HTML files online somewhere. This can still use GitHub pages, but not the auto-generation of a cite from markdown files feature of GitHub pages. This is because GitHub pages doesn't include the jekyll-scholar plugin.

## How to use citations

Including citations with your markdown files or notebooks is done in the following way.

1. Modify the file in _bibliography/references.bib. This has a few sample citations in bibtex form. Update as you wish!
2. In your content, add the following text to include a citation

{% cite bibtex_shortname %}

For example, this text: {% cite holdgraf_evidence_2014 %} generates this citation: (Holdgraf, de Heer, Pasley, & Knight, 2014)

You can also include multiple citations in one go, like so:

{% cite holdgraf_evidence_2014 holdgraf_portable_2017 ruby %} becomes (Holdgraf, de Heer, Pasley, & Knight, 2014; Holdgraf et al., 2017; Flanagan & Matsumoto, 2008)

3. Generate a bibliography on your page by using the following text:

{% bibliography %}

This will generate a bibliography for your entire bibtex file. If you'd like to restrict the bibliography to only the citations you've used on a page, use the following syntax:

{% bibliography --cited %}

When your book is built, the bibliography and citations will now be included. For example, see the Bibliography generated below!

## References

1. Holdgraf, C. R., de Heer, W., Pasley, B. N., & Knight, R. T. (2014). Evidence for Predictive Coding in Human Auditory Cortex. In International Conference on Cognitive Neuroscience. Brisbane, Australia, Australia: Frontiers in Neuroscience.
2. Holdgraf, C. R., Culich, A., Rokem, A., Deniz, F., Alegro, M., & Ushizima, D. (2017). Portable learning environments for hands-on computational instruction using container-and cloud-based technology to teach data science. In ACM International Conference Proceeding Series (Vol. Part F1287). https://doi.org/10.1145/3093338.3093370
3. Flanagan, D., & Matsumoto, Y. (2008). The Ruby Programming Language. O’Reilly Media.


text:
## Nuclear Energy

Read this page, which explains how nuclear decay is used to produce electricity and presents real and imagined dangers of atomic energy. A coal power plant releases 100 times as much radiation as a nuclear power plant of the same wattage. The major concern with nuclear plants is high-level radioactive waste that should be safely stored for more than 10,000 years.


text:
# Parallel Model Calibration using MPI and Boost.MPI

The message passing standard MPI is a language-independent communication protocol. MPI supports the parallelization of numerical algorithms on both massive parallel computers and on symmetric multi processor systems. MPI is standardized, highly portable and the de facto standard on massive parallel supercomputers. Even though MPI can be used in a multi-threading environment it is normally used in a multi-process environment. Therefore MPI is tailor-made to parallelize algorithms based on the non thread-safe QuantLib.

The roots of the MPI specification are going back to the early 90’s and you will feel the age if you use the C-API, which is designed to achieve maximum performance. The Boost.MPI library – quoting from the web page – “is a C++ friendly interface to the standard Message Passing Interface… Boost.MPI can build MPI data types for user-defined types using the Boost.Serialization library”.

Model calibration can be a very time-consuming task, e.g. the calibration of a Heston or a Heston-Hull-White model using American puts with discrete dividends. The class MPICalibrationHelper acts as a MPI wrapper for a given CalibrationHelper and allows to parallelize an existing model calibration routine (hopefully with minimal impact/effort). The source code is available here. It contains the MPICalibrationHelper class and as an example the parallel version of the DAXCalibration test case (part of test-suite/hestonmodel.cpp). The code depends on QuantLib 1.0 or higher, Boost.Thread and Boost.MPI.

The diagram above shows the speed-up of a Heston-Hull-White calibration with discrete dividends on an eight core machine using a finite difference pricing engine. The main reason for the sub-linear scaling is the limited memory bandwidth between the CPUs and the main memory and not the MPI communication overhead.

# Pricing of a Virtual Power Plant on a GPU

Even the pricing of  a simple virtual power plant (VPP) is challenging. Main reasons are the high number of possible states of the VPP and the large number of possible exercise dates because often a VPP is priced as a bermudan-style option with hourly exercise rights. The implementation effort for an exact pricing engine based on finite difference methods (see e.g.[1]) or based on least squares Monte-Carlo is comparable large. As shown in [1] Monte-Carlo combined with  perfect foresight optimization can result in a very good approximation. The algorithm consists of a Monte-Carlo path generator and a dynamic programming optimization part, which calculates the optimal load schedule plan for each path separately. The stochastic processes involved are outlined in [1].

The CUDA based GPU implementation is available here.  It depends on the latest QuantLib version from the SVN trunk or the upcoming QuantLib 1.2 release and CUDA 4.0. The corresponding C++ implementation is a speed-optimized version of the test-case VPPTest::testVPPPricing. This version also supports multi-threading. The following hardware was used to compare both implementations:

• CPU: Core i5@3.0 GHz, quad-core
• GPU: GTX560@810/1620MHz, 336 cores

As can be seen in the diagram below the GPU outperforms the CPU roughly by a factor 100 for single precision and a factor of 50 if the GPU is using double precision.

The CUDA implementation consists of the following files:

gpuvpppricingengine.hpp / gpuvpppricingengine.cpp
A QuantLib pricing engine for a simple VPP based on a Monte-Carlo simulation and perfect foresight optimization via dynamic programming. The physical size of the Monte-Carlo simulation is controlled by the following parameters of the constructor

1. Size nSimulations: number of Monte-Carlo simulations carried out.
2. bool antithetic: enables/disables antithetic sampling
3. Size blockSize: number of threads in a CUDA block.
4. Size gridSize: number of CUDA blocks that are grouped together in a simulation kernel.

gpuvpppricingengine_kernel.hpp
gpuvpppricingengine_kernel.cu/ gpuvpppricingengine_kernel.def
The CUDA implementation consists of two kernels. The first kernel is the Monte-Carlo path generator, which calculates the paths on hourly granularity and stores them in the global memory of the graphic card.. The technics used are outlined e.g. in [2], [3] and [4]. The second kernel performs the optimization of the load schedule based on dynamic programming. The memory layout of this step depends on the number of possible states of the VPP because every possible state is stored in the shared memory of the GPU. The number of states is given by $N_{states} = 2t_{up}+t_{down}$. CUDA does not support efficient dynamic shared memory allocation. Therefore the sizes of all shared memory arrays must be given at compile time. To allow an optimal use of the limited shared memory capacity different kernels with different $N_{states}$ values are generated using X-macros and the appropriate kernel is chosen at runtime.

cudatype.hpp
defines basic CUDA types, especially the typedef for the type “real” can be used to compile the code either for single or double precision.

gpurand.hpp
C++ interface for a GPU random number generator

gpucurand.hpp / gpucurand.cpp / gpucurand_kernel.hpp / gpucurand_kernel.cu
implementation of the GPURand interface based on the CURAND library, which is part of CUDA 4.0.

[2] L. Howes, D. Thomas, Efficient Random Number Generation and Application Using CUDA.

[3] A. Bernemann, R. Schreyer, K. Spanderen, Accelerating Exotic Option Pricing and Model Calibration Using GPUs

[4] M. Joshi, Graphical Asian Options

# VPP Pricing III: Exact Pricing based on Finite Difference Methods

The total value of a virtual power pant (VPP) can be decomposed in an intrinsic part plus an extrinsic part. The intrinsic value is given by the cash-flows that the VPP would generate based on the current power and gas forward curve. Therefore the intrinsic value can be calculated without defining a stochastic model for the power and gas prices using either linear or dynamic optimization methods. Calculating the extrinsic value implies pricing the VPP exactly to calculate the total value for a given stochastic process. The model in use here is outlined in the article VPP Pricing I: Stochastic Processes & Partial Integro Differential Equation. Exact pricing can  be done using least square Monte-Carlo or finite difference methods using dynamic programming for the local optimization [1].

The focus is on the finite difference method, which involves solving the three-dimensional partial integro differential equation

$\begin{array}{rcl} rV&=&\frac{\partial V}{\partial t}+\frac{\sigma_x^2}{2}\frac{\partial^2 V}{\partial x^2}-\alpha x\frac{\partial V}{\partial x}-\beta y \frac{\partial V}{\partial y} \\[6pt]&+&\frac{\sigma_u^2}{2}\frac{\partial^2 V}{\partial u^2}- \kappa u\frac{\partial V}{\partial u} +\rho\sigma_x\sigma_u\frac{\partial^2 V}{\partial x\partial u}\\[6pt] &+&\lambda\int_\mathbb{R}\left(V(x,y+z,u,t)-V(x,y,u,t) \right )\omega(z)dz \\ \end{array}$

An additional fourth dimension is needed to keep track of the different states of the VPP. Based on the characteristics of the VPP type outlined in the article VPP Pricing II: Mixed Integer Linear Programming a VPP with three possible load levels $P \in \{0, P_{min}, P_{max} \}$ has $2 t_{up} + t_{down}$ different states.

Pricing via Monte-Carlo and perfect foresight involves simulating the stochastic process

$\begin{array}{rcl} P_t &=& \exp(p_t + X_t + Y_t) \\ dX_t &=& -\alpha X_tdt + \sigma_x dW_t^x \\ dY_t &=& -\beta Y_{t-}dt+J_tdN_t \\ \omega(J) &=& \eta e^{-\eta J} \\ G_t &=& \exp(g_t + U_t) \\ dU_t &=& -\kappa U_tdt+\sigma_udW_t^u \\ \rho &=& \mathrm{corr} (dW_t^x, dW_t^u) \end{array}$

and optimize the power plant load schedule for each path separately in the same way the intrinsic value is calculated. This procedure will result in an upper bound for the exact price of the VPP. The prices for a 4 weeks VPP contract based on this two methods and with the parameters outlined in VPP Pricing II show almost no differences between the “exact” finite difference method and the perfect foresight  upper bound value beside the Monte-Carlo error (s. diagram below, compare with [1]).

The size of the lattice for the finite difference method was

$(t, nPower, nJump, nGas, nStates) = (672, 101, 51, 20, 14)$.

The code is available here. It depends on the latest QuantLib version from the SVN trunk or the upcoming QuantLib 1.2 release. If you want to generate the plot you’ll also need R

[1] H. van Dijken,  D. van Abbena, H.S. Los, C. de Jong, The value of starting up the power plant.

# VPP Pricing II: Mixed Integer Linear Programming

The next two steps are defining a simple VPP contract (or a simplified gas-run power plant) and setting up a mixed integer linear programming optimization (MIP) to calculated the intrinsic value and an upper bound for the extrinsic value based on a Monte-Carlo simulation and assuming perfect foresight. The third step outlined in the next part will then be the “exact” pricing of the extrinsic value using dynamic programming and finite difference methods.

The set-up of the simplified gas-run power plant is similar to the one explained in chapter 4.2.3 of the text-book [1]. In general the power plant has three power output level:

• Plant is off, $P=0$
• Generation at minimum load $P=P_{min}$
• Generation at maximum load $P=P_{max}$

The power plant has a fixed efficiency rate

$\zeta=\frac{MWh Power Output}{MWh Heat}$.

Ramp rates will be neglected, but the power plant has a minimum uptime $t_{up}$ and a minimum downtime $t_{down}$. The start-up costs are given by a fixed start-up cost $\eta$ (in €) and the price of the gas needed to produce the start-up heat $\theta$ (in MWh).

The mixed integer linear optimization is running in one hour blocks and is using three decision variables per hour $i$. The binary decision variable $\beta_i$ is true if the power plant is running at minimum load $P_{min}$ or at maximum load $P_{max}$ and $\beta_i$ is false if the plant is off. The real decision variable $0\le s_i \le 1$ is equal to one if the plant is started in hour $i$, which is implied by the following constraint

$\beta_i - \beta_{i-1} \le s_i$.

The minimum up-time $t_{up}$ and the minimum down-time $t_{down}$ is a consequence of the constraints

The real decision variable $0\le \gamma \le 1$ is equal to one if the power plant is running at maximum load $P_{max}$ and zero if the power plant is either running at minimum load $P_{min}$ or if the plant is off, that means

$\beta_i \ge \gamma_i$

Let $P_i$ be the power price, $G_i$ be the gas price and $CO_2^i$ be the carbon dioxide price at hour $i$. The objective function is then given by

For a one year span the problem consists of $3\cdot 365 \cdot 24 = 26280$ decision variables $\{\beta_i, \gamma_i, s_i\}$ and $4 \cdot 365\cdot 24 = 35040$ constraints. This comparable small problem can be solved using e.g. the Gnu Linear Programming Kit (GLPK). For an overview on open source linear/mixed integer programming solver see [2].

The model parameters and the example forward curves are outlined in the previous entry VPP Pricing I. The diagram below shows the intrinsic value and the upper bound for the total value (intrinsic plus extrinsic value) based on Monte-Carlo, perfect foresight and MIP for different  power plant efficiencies $\zeta$. The parameters of the VPP contract are given by

$t_{up}=t_{down}=2h, P_{min}=8MW, P_{max}=40MW, \eta=300 EUR, \theta=20MWh$,

the (fixed) carbon dioxide price is 3.0€ per MWh heat.

The source code is available here. It depends on GLPK and the latest QuantLib version from the SVN trunk or the next QuantLib 1.2 release.

It is now quite easy to add and price time-integral constraints, e.g. the following constraint restricts the number of starts within a year to be less than or equal to a given number

$\sum_{t=1}^N s_i \le \#Starts$.

The following diagram shows the results for $\#Starts \le 25$ and a minimum load $P_{min}=25MW$.

The source code is available here. It depends on QuantLib 1.1 and if  you want to generate the plot directly from the C++ program you’ll also need R, RCPP and RInside.

[1] M. Burger, B. Graeber, G. Schindlmayr, Managing Energy Risk, ISDN 978-0-470-ß2962-6

# Swing Option: Linear vs. Dynamic Programming II

In order to get an improved impression on the differences between Monte-Carlo/linear programming and  finite difference methods/dynamic programming  a more realistic swing option with hourly payoff profile is evaluated based on a German hourly forward curve. The forward curve is taken from the Kyos example download page. The parameterization of the Kluge model is outlined in [1].

With maturity of twelve weeks the example swing option provides $12*7*24=2016$ exercise opportunities. The number of overall exercised swing opportunities is constraint by

$100 \le \sum_{i=1}^{2016} \beta_i \le 500$.

The size of two dimensions of the finite difference method (dynamic programming) is therefore already been given. 2016 steps are needed in time direction and 500 steps are needed in the “consumed exercises” direction. Together with the two other directions – one for the power price and one dimension for the jump process – and without further simplidsfications this forms a pretty large finite difference problem. The Monte-Carlo based linear programming approach reduces the computational burden but will lead to an upper bound of the swing option price. The diagram below shows the corresponding results.

The code is available here. It depends on the GNU Linear Programming Kit, the Boost Thread library for parallelization and at the time of writing on the latest QuantLib version from the SVN trunk. If you want to generate the plot directly out of the C++ program you also need R, RCPP and RInside. To utilize all CPU cores please use the -DNTHREADS=(number of CPU cores)  compiler switch. In addition to run the program you have to download the forward curve German power from the Kyos web page and convert it into a text file EEX_2010-2013.txt of the folliwing format

4-Oct-2010 36.73 32.09 27.32 23.22 25.71 35.60 47.32 …

5-Oct-2010 38.12 31.12 22.76 25.65 27.87 34.60 50.01 …

# Swing Option: Linear vs. Dynamic Programming I

Finding the optimal exercise strategy for a swing option is a challenging task, even for simple payoff structures. In the absence of complicated time-integral constraints dynamic programming can be used to calculate the optimal exercise strategy. Linear programming applied ex post over the whole spot price path is capable to deal with more complicated time-integral constraints but the algorithm leads only to an upper bound of the real option price [1].

A basic payoff structure is e.g.: On the exercise dates $t_i, i\in\{1..N\}$ the decision variables $\beta_i\in\{0,1\}$ indicate whether to exercise $(\beta_i=1)$ or not to exercise $(\beta_i=0)$ the option at a given point $t_i$ in time. Payoff is either $\beta_i (P(t_i)-K)$ for a call or $\beta_i(K-P(t_i))$ for a put option. Here $K$ is the strike price and $P(t_i)$ is the spot price . The number of exercise opportunities is constraint by

$E_{min} \le \sum_{i=1}^N \beta_i \le E_{max}$.

The price of a swing call option is then given by

$\text{npv} = \max_{\beta}\mathbb{E}^\mathbb{Q} \left[ \sum_{i=1}^N \beta_i \left( P(t_i) - K\right)e^{rt_i}\right]$

Let the dynamics of the spot price in the risk neutral measure $\mathbb{Q}$ be given by the Kluge model [2]:

$\begin{array}{rcl} S &=& exp(X_t + Y_t) \\ dX_t &=& \alpha(\mu(t)-X_t)dt + \sigma dW_t \\ dY_t &=& -\beta Y_{t-}dt + J_tdN_t \\ \omega(J)&=& \eta_u e^{-\eta_u J} \end{array}$

This model is composed of an Ornstein-Uhlenbeck (OU) process $X_t$, a deterministic period function $\mu(t)$ characterizing the seasonality and a mean reverting process $Y_t$ with jumps to incorporate spikes. $N_t$ is a Poisson  process with jump intensity $\lambda$ and the jump size itself is exponentially distributed. The random variables $W_t, N_t$ and $J_t$ are independent. The Monte-Carlo path simulation will be built on top of standard components for an OU process and a Poisson jump-diffusion process [3]. The dynamic programming approach can either be carried out using least squares Monte-Carlo ansatz (Longstaff Schwartz algorithm) or using finite difference methods. We prefer finite difference methods to avoid e.g. the problem of choosing an appropriate basis function set. The Feynman-Kac theorem leads to the corresponding partial integro differential equation (PIDE):

$\begin{array}{rcl} rV =&&\frac{\partial V}{\partial t}+\frac{\sigma^2}{2}\frac{\partial^2 V}{\partial x^2} + \alpha(\mu(t)-x)\frac{\partial V}{\partial x}-\beta y\frac{\partial V}{\partial y}\\&+&\lambda \int_\mathbb{R} \left( V(x,y+z,t)-V(x,y,t)\right )\omega(z) dz \end{array}$

A Gauss-Laguerre quadrature is appropriate to calculate the integral part of the PIDE. Beside this two-dimensional PIDE an additional dimension is needed to keep track of the already consumed exercise rights. The three-dimensional formulation together with Bellman’s principle of optimality transforms the global optimization problem into a local optimization problem.

Target of the linear programming approach is the upper bound

$\text{npv} \le \mathbb{E}^\mathbb{Q}\left[ \max_{\beta} \sum_{i=1}^N \beta_i\left(P(t_i) - K \right) e^{rt_i}\right]$

The linear programming algorithm will calculate the optimal exercise strategy $\left\{ \beta \right\}$ on each Monte-Carlo path separately (perfect foresight) with respect to the given constraints. For this basic type of swing option linear programming is sort of over-engineering because a simple sort algorithm will also reproduce the optimal exercise strategy. But introducing linear programming and mixed integer programming now will enable us later on to deal with more complicated time-integral constraints. Libraries for this task are freely available, e.g. the GNU Linear Programming Kit.

The test parameterization of the model is

$\alpha=1.0, \sigma=200\%, \mu(t)=3.0,\beta=5.0, \eta=2.0, \lambda=1.0, X_0=3.0, Y_0=0.0$,

the example swing call option has maturity of one year, strike price is equal 40, one exercise opportunity per month and the minimum number of exercise rights is equal to the maximum number of exercise rights, $E_{min} = E_{max}$. Interest rates are at $r=10\%$.

As the diagram above shows for this set-up the upper bound for the swing option price calculated using linear programming differs significantly from the correct price calculated based on dynamic programming on a lattice. For twelve exercise rights both algorithms have to provide the same results because the constraint forces to always exercise on each exercise date. Next to come is to rerun the simulation with a more realistic forward curve and process parameters.

The code is available here. It depends on the GNU Linear Programming Kit, the Boost Thread library for parallelization and at the time of writing on the latest QuantLib version from the SVN trunk. If you want to generate the plot directly out of the C++ program you also need R, RCPP and RInside.

[1] M. Burger, B. Graeber, G. Schindlmayr, Managing Energy Risk, ISDN 978-0-470-ß2962-6

[3] P. Glasserman, Monte Carlo Methods in Financial Engineering.  ISBN-0387004513

# Calibration: Comparison of deterministic Optimizers

The gaol is to calibrate a time dependent Heston model defined by the following SDE

The parameter set $\{\kappa_t, \theta_t, ,\sigma_t, \rho_t\}$ is supposed to be piecewise constant in time. This model has a semi-closed solution for plain vanilla European put/call options based on the characteristic function method [1].

The DAX implied volatility surface based on July 5, 2002 data taken from [2] defines the “benchmark” calibration problem. The benchmark model parameters for the optimization problem are given by

$\Theta = \{\nu_0, \kappa_{0\leq t < 0.25}, \kappa_{0.25 \leq t},\theta, \sigma, \rho\}$.

The non-linear least square optimization problem is defined by the goodness of fit measure

$\zeta=\min_{\Theta} \sum_{i=1}^N \left( \sigma_i^{market}(K,T) - \sigma_i^{model}(K,T, \Theta) \right)^2$

where $\sigma_i^{market}(K,T)$ is the market implied volatility for strike K and maturity T and $\sigma_i^{model}(K,T, \Theta)$ is the corresponding Black-Scholes volatility implied from the model price. The optimal solution for this problem is

$\Theta_{min} = \{ 0.2231, 39.651, 7.546, 0.0954, 5.1865, -0.5004 \}$

leading to a goodness of fit measure of $\zeta_{min}=74.4731$ (Please keep in mind that this result is the outcome of a naive calibration procedure. Due to the large $\kappa$ and $\sigma$ values I’d not use these parameters to price a derivative.).

The diagram above shows the “goodness of fit”-surface for the parameter sets in

$\Theta = \{ 0.2231, 39.651, \kappa_{0.25 \leq t} \in [0, 16], 0.0954, \sigma \in [0, 16], -0.5004 \}$

To be able to compare a larger number of deterministic optimizers the model calibration will be carried out using R and with help of the additional packages minpack.lm and minqa.

Non-linear Least Square Optimization:

1. nls.lm: Levenberg-Marquardt algorithm(based on MINPACK, also available in QuantLib)
2. nls: Gauss-Newton algorithm
3. nl2sol:based on the PORT library.

Non-linear (trusted region) Minimization:

1. nlm:Newton style minizer


text:
TITLE

Electromagnetic multipole moments of spin 3/2 particles in NKR formalism

AUTHOR(S)
E. Germán Delgado A.; Napsuciale, Mauro
PUB. DATE
April 2009
SOURCE
AIP Conference Proceedings;4/20/2009, Vol. 1116 Issue 1, p415
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this work we present results for Compton scattering off spin 3/2 particles in NKR formalism and relate the behavior of the cross section to the electromagnetic multipole moments of the particle included by this formalism. We obtain expressions for such multipole moments using model independent definitions.
ACCESSION #
37831726

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#### Vol. 2, No. 2, 2009

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
Bounds for Fibonacci period growth

### Chuya Guo and Alan Koch

Vol. 2 (2009), No. 2, 195–210
##### Abstract

We study the Fibonacci sequence mod $n$ for some positive integer $n$. Such a sequence is necessarily periodic; we introduce a function $Q\left(n\right)$ which gives the ratio of the length of this period to $n$ itself. We compute $Q\left(n\right)$ in certain cases and provide bounds for it which depend on the nature of the prime divisors of $n$.

##### Keywords
Fibonacci sequence, Fibonacci periods, growth of Fibonacci periods, Fibonacci period mod $n$
Primary: 11B39
Secondary: 11B50
##### Milestones
Accepted: 5 December 2008
Published: 7 May 2009

Communicated by Arthur T. Benjamin
##### Authors
 Chuya Guo Agnes Scott College Department of Mathematics 141 E. College Ave. Decatur, GA 30030 United States Alan Koch Agnes Scott College Department of Mathematics 141 E. College Ave. Decatur, GA 30030 United States


text:
#### Conformal Bayesian Computation

We develop scalable methods for producing conformal Bayesian predictive intervals with finite sample calibration guarantees. Bayesian posterior predictive distributions, $p(y \mid x)$, characterize subjective beliefs on outcomes of interest, $y$, conditional on predictors, $x$. Bayesian prediction is well-calibrated when the model is true, but the predictive intervals may exhibit poor empirical coverage when the model is misspecified, under the so called ${\cal{M}}$-open perspective. In contrast, conformal inference provides finite sample frequentist guarantees on predictive confidence intervals without the requirement of model fidelity. Using 'add-one-in' importance sampling, we show that conformal Bayesian predictive intervals are efficiently obtained from re-weighted posterior samples of model parameters. Our approach contrasts with existing conformal methods that require expensive refitting of models or data-splitting to achieve computational efficiency. We demonstrate the utility on a range of examples including extensions to partially exchangeable settings such as hierarchical models.