Relational Graph Models at Work

We study the relational graph models that constitute a natural subclass of relational models of lambda-calculus. We prove that among the lambda-theories induced by such models there exists a minimal one, and that the corresponding relational graph model is very natural and easy to construct. We then study relational graph models that are fully abstract, in the sense that they capture some observational equivalence between lambda-terms. We focus on the two main observational equivalences in the lambda-calculus, the theory H+ generated by taking as observables the beta-normal forms, and H* generated by considering as observables the head normal forms. On the one hand we introduce a notion of lambda-K\"onig model and prove that a relational graph model is fully abstract for H+ if and only if it is extensional and lambda-K\"onig. On the other hand we show that the dual notion of hyperimmune model, together with extensionality, captures the full abstraction for H*

Glueability of Resource Proof-Structures: Inverting the Taylor Expansion

A Multiplicative-Exponential Linear Logic (MELL) proof-structure can be expanded into a set of resource proof-structures: its Taylor expansion. We introduce a new criterion characterizing those sets of resource proof-structures that are part of the Taylor expansion of some MELL proof-structure, through a rewriting system acting both on resource and MELL proof-structures

Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion

We study the semantics of a resource-sensitive extension of the lambda calculus in a canonical reflexive object of a category of sets and relations, a relational version of Scott's original model of the pure lambda calculus. This calculus is related to Boudol's resource calculus and is derived from Ehrhard and Regnier's differential extension of Linear Logic and of the lambda calculus. We extend it with new constructions, to be understood as implementing a very simple exception mechanism, and with a "must" parallel composition. These new operations allow to associate a context of this calculus with any point of the model and to prove full abstraction for the finite sub-calculus where ordinary lambda calculus application is not allowed. The result is then extended to the full calculus by means of a Taylor Expansion formula. As an intermediate result we prove that the exception mechanism is not essential in the finite sub-calculus

Opportunistic CP Violation

In the electroweak sector of the Standard Model, CP violation arises through a very particular interplay between the three quark generations, as described by the Cabibbo--Kobayashi--Maskawa (CKM) mechanism and the single Jarlskog invariant $J_4$. Once generalized to the Standard Model Effective Field Theory (SMEFT), this peculiar pattern gets modified by higher-dimensional operators, whose associated Wilson coefficients are usually split into CP-even and odd parts. However, CP violation at dimension four, i.e., at the lowest order in the EFT expansion, blurs this distinction: any Wilson coefficient can interfere with $J_4$ and mediate CP violation. In this paper, we study such interferences at first order in the SMEFT expansion, \order{1/\Lambda^2}, and we capture their associated parameter space via a set of 1551 linear CP-odd flavor invariants. This construction describes both new, genuinely CP-violating quantities as well as the interference between $J_4$ and CP-conserving ones. We call this latter possibility \textit{opportunistic CP violation}. Relying on an appropriate extension of the matrix rank to Taylor expansions, which we dub \emph{Taylor rank}, we define a procedure to organize the invariants in terms of their magnitude, so as to retain only the relevant ones at a given precision. We explore how this characterization changes when different assumptions are made on the flavor structure of the SMEFT coefficients. Interestingly, some of the CP-odd invariants turn out to be less suppressed than $J_4$, even when they capture opportunistic CPV, demonstrating that CP-violation in the SM, at dimension 4, is \textit{accidentally small.}Comment: 34 p. + 29 p. of appendices, 6 figure

Dynamical Emergence of FRW Cosmological Models

Recent astronomical observations strongly indicate that the current Universe is undergoing an accelerated phase of expansion. The discovery of this fact was unexpected and resulted in the comeback of cosmological constant. The conception of standard cosmological model has its roots in this context. The paper relates to the methodological status of effective theories in the context of cosmological investigations. We argue that the standard cosmological model (LCDM model) as well as the CDM have a status of effective theories only, similarly to the standard model of particle physics. The LCDM model is studied from the point of view of the methodological debate on reductionism and epistemological emergence in the science. It is shown in the paper that bifurcation as well as structural instability notion can be useful in the detection of emergence the LCDM model from the CDM model. We demonstrate that the structural stability of the LCDM model can explain the flexibility of the model to accommodation of the observational data. We show that LCDM model can be derived from CDM as the bifurcation. It is an example of acausal derivation of Lambda term. The case study of emergence of LCDM model suggests that it can be understood in terms of bifurcation and structural stability issue. The reduction from the upper models represented in terms of dynamical system to low-level ones can be realized in any case by application of a mathematical limit (boundary crossing) with respect to the model parameter. It is a simple consequence of mathematical theorem about smooth dependence solutions with respect to time, initial condition and the parameters.Comment: 8ssmmp.cls, 14 pages, 3 figures; rev. 2: added section on emergence from bifurcation; rev. 3: reorganized and shortened text; transition from the LCDM to CDM model was explicitly shown, for this aim it was used the theorem on smooth dependence of solution on initial conditions and change of parameters. In: B. Dragovich, I. Salom (eds) Proceedings of the 8th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics. August 24-31, 2014, Belgrade, Serbia. Institute of Physics, Belgrade, 201

Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors

It has been known since Ehrhard and Regnier's seminal work on the Taylor expansion of $\lambda$-terms that this operation commutes with normalization: the expansion of a $\lambda$-term is always normalizable and its normal form is the expansion of the B\"ohm tree of the term. We generalize this result to the non-uniform setting of the algebraic $\lambda$-calculus, i.e. $\lambda$-calculus extended with linear combinations of terms. This requires us to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's techniques rely heavily on the uniform, deterministic nature of the ordinary $\lambda$-calculus, and thus cannot be adapted; second is the absence of any satisfactory generic extension of the notion of B\"ohm tree in presence of quantitative non-determinism, which is reflected by the fact that the Taylor expansion of an algebraic $\lambda$-term is not always normalizable. Our solution is to provide a fine grained study of the dynamics of $\beta$-reduction under Taylor expansion, by introducing a notion of reduction on resource vectors, i.e. infinite linear combinations of resource $\lambda$-terms. The latter form the multilinear fragment of the differential $\lambda$-calculus, and resource vectors are the target of the Taylor expansion of $\lambda$-terms. We show the reduction of resource vectors contains the image of any $\beta$-reduction step, from which we deduce that Taylor expansion and normalization commute on the nose. We moreover identify a class of algebraic $\lambda$-terms, encompassing both normalizable algebraic $\lambda$-terms and arbitrary ordinary $\lambda$-terms: the expansion of these is always normalizable, which guides the definition of a generalization of B\"ohm trees to this setting

Existence and Uniqueness of Perturbation Solutions to DSGE Models

We prove that standard regularity and saddle stability assumptions for linear approximations are sufficient to guarantee the existence of a unique solution for all undetermined coefficients of nonlinear perturbations of arbitrary order to discrete time DSGE models. We derive the perturbation using a matrix calculus that preserves linear algebraic structures to arbitrary orders of derivatives, enabling the direct application of theorems from matrix analysis to prove our main result. As a consequence, we provide insight into several invertibility assumptions from linear solution methods, prove that the local solution is independent of terms first order in the perturbation parameter, and relax the assumptions needed for the local existence theorem of perturbation solutions.Perturbation, matrix calculus, DSGE, solution methods, Bézout theorem; Sylvester equations

Parametric arbitrage-free models for implied smile dynamics

Based on the theory of Tangent Levy model [1] developed by R. Carmona and S. Nadtochiy, this thesis gives a paramatrized realization of dynamic implied smile.\ud \ud After specifying a Dirac style Levy measure, we give argument about the consistency issue of our model with the Tangent Levy Model. A corresponding no arbitrage drift condition is derived for the parameters. Numerical setup under our model for option pricing and parameter estimation for calibration is given. Implementation results are illustrated in detail and in the end we provide with simulation results of one day ahead implied smile