Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling

The objective of this paper is to complete certain issues from our recent contribution [J. Calatayud, J.-C. Cort\'es, M. Jornet, L. Villafuerte, Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties, Advances in Difference Equations, 2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via an habitual Lipschitz condition that extends the classical Picard Theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, being the latter a reformulation of our random Fr\"obenius method.Comment: 15 pages, 0 figures, 2 table

Topological Excitations in the Thirring model

The quantization of the massless Thirring model in the light-cone using functional methods is considered. The need to compactify the coordinate $x^-$ in the light-cone spacetime implies that the quantum effective action for left-handed fermions contains excitations similar to abelian instantons produced by composite of left-handed fermions. Right-handed fermions don't have a similar effective action. Thus, quantum mechanically, chiral symmetry must be broken as a result of the topological excitations. The conserved charge associated to the topological states is quantized. Different cases with only fermionic excitations or bosonic excitations or both can occur depending on the boundary conditions and the value of the coupling.Comment: title changed, one reference added, accepted in Phys. Lett.

Relativistic kinematics beyond Special Relativity

In the context of departures from Special Relativity written as a momentum power expansion in the inverse of an ultraviolet energy scale M, we derive the constraints that the relativity principle imposes between coefficients of a deformed composition law, dispersion relation, and transformation laws, at first order in the power expansion. In particular, we find that, at that order, the consistency of a modification of the energy-momentum composition law fixes the modification in the dispersion relation. We therefore obtain the most generic modification of Special Relativity that preserves the relativity principle at leading order in 1/M.Comment: Version with minor corrections, to appear in Phys. Rev.

Enlarging instruction streams

The stream fetch engine is a high-performance fetch architecture based on the concept of an instruction stream. We call a sequence of instructions from the target of a taken branch to the next taken branch, potentially containing multiple basic blocks, a stream. The long length of instruction streams makes it possible for the stream fetch engine to provide a high fetch bandwidth and to hide the branch predictor access latency, leading to performance results close to a trace cache at a lower implementation cost and complexity. Therefore, enlarging instruction streams is an excellent way to improve the stream fetch engine. In this paper, we present several hardware and software mechanisms focused on enlarging those streams that finalize at particular branch types. However, our results point out that focusing on particular branch types is not a good strategy due to Amdahl's law. Consequently, we propose the multiple-stream predictor, a novel mechanism that deals with all branch types by combining single streams into long virtual streams. This proposal tolerates the prediction table access latency without requiring the complexity caused by additional hardware mechanisms like prediction overriding. Moreover, it provides high-performance results which are comparable to state-of-the-art fetch architectures but with a simpler design that consumes less energy.Peer ReviewedPostprint (published version

Numerical solution of random differential models

This paper deals with the construction of a numerical solution of random initial value problems by means of a random improved Euler method. Conditions for the mean square convergence of the proposed method are established. Finally, an illustrative example is included in which the main statistics properties such as the mean and the variance of the stochastic approximation solution process are given. © 2011 Elsevier Ltd.This work has been partially supported by the Spanish M.C.Y.T. grants MTM2009-08587, DPI2010-20891-C02-01, Universidad Politecnica de Valencia grant PAID06-09-2588 and Mexican Conacyt.Cortés López, JC.; Jódar Sánchez, LA.; Villafuerte Altuzar, L.; Company Rossi, R. (2011). Numerical solution of random differential models. Mathematical and Computer Modelling. 54(7):1846-1851. https://doi.org/10.1016/j.mcm.2010.12.037S1846185154

Static locality analysis for cache management

Most memory references in numerical codes correspond to array references whose indices are affine functions of surrounding loop indices. These array references follow a regular predictable memory pattern that can be analysed at compile time. This analysis can provide valuable information like the locality exhibited by the program, which can be used to implement more intelligent caching strategy. In this paper we propose a static locality analysis oriented to the management of data caches. We show that previous proposals on locality analysis are not appropriate when the proposals have a high conflict miss ratio. This paper examines those proposals by introducing a compile-time interference analysis that significantly improve the performance of them. We first show how this analysis can be used to characterize the dynamic locality properties of numerical codes. This evaluation show for instance that a large percentage of references exhibit any type of locality. This motivates the use of a dual data cache, which has a module specialized to exploit temporal locality, and a selective cache respectively. Then, the performance provided by these two cache organizations is evaluated. In both organizations, the static locality analysis is responsible for tagging each memory instruction accordingly to the particular type(s) of locality that it exhibits.Peer ReviewedPostprint (published version