Derivations on Pseudoquotients

A space of pseudoquotients denoted by B(X, S) is defined as equivalence classes of pairs (x, f); where x is an element of a nonempty set X, f is an element of S; a commutative semigroup of injective maps from X to X; and (x, f) ~ (y, g) for gx = fy: If X is a ring and elements of S are ring homomorphisms, then B(X, S) is a ring. We show that, under natural conditions, a derivation on X has a unique extension to a derivation on B(X, S): We also consider (α, β) -Jordan derivations, inner derivations, and generalized derivations.Введено означення простору псевдочасток B(X, S) як класів еквiвалентностi пар (x, f), де x — елемент непорожньої множини X, f — елемент комутативної напівгрупи S ін'єктивних відображень із X у X; та (x, f) ~ (y, g), якщо gx = fy. Якщо X — кільце та елементи S є гомоморфізмами кільця, то B(X, S) є кільцем. Показано, що за природних умов похідна на X має єдине розширення до похідної на B(X, S). Також розглянуто (α, β)-жорданові похідні, внутрішні похідні та узагальнені похідні

On De Graaf spaces of pseudoquotients

A space of pseudoquotients, $\mathcal{B}(X,S)$, is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) \sim (y,g)$ if $gx=fy$. In this note, we consider a generalization of this construction where the assumption of commutativity of $S$ is replaced by Ore type conditions. As in the commutative case, $X$ can be identified with a subset of $\mathcal{B}(X,S)$, and $S$ can be extended to a group, $G$, of bijections on $\mathcal{B}(X,S)$. We introduce a natural topology on $\mathcal{B}(X,S)$ and show that all elements of $G$ are homeomorphisms on $\mathcal{B}(X,S)$

Exponential sums for eighth degree polynomial

Let p > 7 be a prime, the exponential sums of any polynomial f(x, y) is given by S(f; p α ) = ∑x,y mod p e 2πif(x,y)/ pα, where the sum is taken over a complete set of residue modulo p. Firstly, Newton Polyhedron technique was used to determine the estimation for the p-adic sizes of common zeros of the partial derivative polynomials fx, fy which derive from f(x, y). We continue by estimating the cardinality N(g, h; p α ) as well as the exponential sums of polynomial f(x, y). Throught out this paper, we consider the polynomial of eighth degree with two variables in the form f(x, y) = ax8 +bx7 y+cx6 y 2 +dx5 y 3 +ex4 y 4 +kx3 y 5 +mx2 y 6 + nxy7 + uy8 + rx + sy + t

High-contrast 40 Gb/s operation of a 500 um long silicon carrier-depletion slow wave modulator

This paper was published in OPTICS LETTERS and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://dx.doi.org/10.1364/OL.37.003504. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law[EN] In this Letter, we demonstrate a highly efficient, compact, high-contrast and low-loss silicon slow wave modulator based on a traveling-wave Mach¿Zehnder interferometer with two 500 μm long slow wave phase shifters. 40 Gb ∕ s operation with 6.6 dB extinction ratio at quadrature and with an on-chip insertion loss of only 6 dB is shown. These results confirm the benefits of slow light as a means to enhance the performance of silicon modulators based on the plasma dispersion effect.Funding by the European Commission (EC) under project Photonics Electronics Functional Integration on CMOS (HELIOS) (FP7224312) and PROMETEO-2010- 087 R&D Excellency Program are acknowledged. F.Y.G, D.J.T. and G.T.R. acknowledge funding support from the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) under the grant “UK Silicon Photonics”.Brimont, ACJ.; Thomson, DJ.; Gardes, FY.; Fedeli, JM.; Reed, GT.; Martí Sendra, J.; Sanchis Kilders, P. (2012). High-contrast 40 Gb/s operation of a 500 um long silicon carrier-depletion slow wave modulator. Optics Letters. 37(17):3504-3506. https://doi.org/10.1364/OL.37.003504S350435063717Liao, L., Liu, A., Rubin, D., Basak, J., Chetrit, Y., Nguyen, H., … Paniccia, M. (2007). 40 Gbit/s silicon optical modulator for high-speed applications. Electronics Letters, 43(22), 1196. doi:10.1049/el:20072253Gardes, F. Y., Thomson, D. J., Emerson, N. G., & Reed, G. T. (2011). 40 Gb/s silicon photonics modulator for TE and TM polarisations. Optics Express, 19(12), 11804. doi:10.1364/oe.19.011804Thomson, D. J., Gardes, F. Y., Hu, Y., Mashanovich, G., Fournier, M., Grosse, P., … Reed, G. T. (2011). High contrast 40Gbit/s optical modulation in silicon. Optics Express, 19(12), 11507. doi:10.1364/oe.19.011507Brimont, A., Thomson, D. J., Sanchis, P., Herrera, J., Gardes, F. Y., Fedeli, J. M., … Martí, J. (2011). High speed silicon electro-optical modulators enhanced via slow light propagation. Optics Express, 19(21), 20876. doi:10.1364/oe.19.020876Ziebell, M., Marris-Morini, D., Rasigade, G., Fédéli, J.-M., Crozat, P., Cassan, E., … Vivien, L. (2012). 40 Gbit/s low-loss silicon optical modulator based on a pipin diode. Optics Express, 20(10), 10591. doi:10.1364/oe.20.010591Dong, P., Chen, L., & Chen, Y. (2012). High-speed low-voltage single-drive push-pull silicon Mach-Zehnder modulators. Optics Express, 20(6), 6163. doi:10.1364/oe.20.006163Taylor, H. F. (1999). Enhanced electrooptic modulation efficiency utilizing slow-wave optical propagation. Journal of Lightwave Technology, 17(10), 1875-1883. doi:10.1109/50.793770O’Faolain, L., Beggs, D. M., White, T. P., Kampfrath, T., Kuipers, K., & Krauss, T. F. (2010). Compact Optical Switches and Modulators Based on Dispersion Engineered Photonic Crystals. IEEE Photonics Journal, 2(3), 404-414. doi:10.1109/jphot.2010.2047918Brimont, A., Vicente Galán, J., Maria Escalante, J., Martí, J., & Sanchis, P. (2010). Group-index engineering in silicon corrugated waveguides. Optics Letters, 35(16), 2708. doi:10.1364/ol.35.002708Soref, R., & Bennett, B. (1987). Electrooptical effects in silicon. IEEE Journal of Quantum Electronics, 23(1), 123-129. doi:10.1109/jqe.1987.1073206Nguyen, H. C., Sakai, Y., Shinkawa, M., Ishikura, N., & Baba, T. (2011). 10 Gb/s operation of photonic crystal silicon optical modulators. Optics Express, 19(14), 13000. doi:10.1364/oe.19.013000Dong, P., Liao, S., Liang, H., Qian, W., Wang, X., Shafiiha, R., … Asghari, M. (2010). High-speed and compact silicon modulator based on a racetrack resonator with a 1 V drive voltage. Optics Letters, 35(19), 3246. doi:10.1364/ol.35.00324

Ecuaciones diferenciales estocásticas conservativas sobre grafos: Distribución invariante y relación con la topología del grafo

En este trabajo abordaremos las condiciones de necesidad y su ciencia para la existencia de una medida invariante para el proceso estocástico que da solución a la ecuación diferencial dX = LXdt + dY; X 2 Rn; donde Y = fY (t) : t 0g es un proceso de Lévy n-dimensional y L 2 Rn n es una variación de la matriz Laplaciana de un grafo G de n vértices. L es una matriz que induce un sistema conservativo, es decir d dt Xn i=1 Xi(t) = 0; t 0: El objetivo es calcular la distribución invariante y analizar la relación con la topología del grafo G. (Tomado de la fuente)MaestríaMagister en Ciencias - Matemática

A Unifying Variational Perspective on Some Fundamental Information Theoretic Inequalities

This paper proposes a unifying variational approach for proving and extending some fundamental information theoretic inequalities. Fundamental information theory results such as maximization of differential entropy, minimization of Fisher information (Cram\'er-Rao inequality), worst additive noise lemma, entropy power inequality (EPI), and extremal entropy inequality (EEI) are interpreted as functional problems and proved within the framework of calculus of variations. Several applications and possible extensions of the proposed results are briefly mentioned

Pointwise asymptotic behavior of modulated periodic reaction-diffusion waves

By working with the periodic resolvent kernel and Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion equations.With our linearized estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, we obtain $L^p$- behavior($p \geq 1$) of a nonlinear solution to a perturbation equation of a reaction-diffusion equation with respect to initial data in $L^1 \cap H^1$ recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations $|u_0|\leq E_0e^{-|x|^2/M}$ and $|u_0| \leq E_0(1+|x|)^{-3/2}$, respectively, $E_0>0$ sufficiently small and $M>1$ sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques

How to read probability distributions as statements about process

Probability distributions can be read as simple expressions of information. Each continuous probability distribution describes how information changes with magnitude. Once one learns to read a probability distribution as a measurement scale of information, opportunities arise to understand the processes that generate the commonly observed patterns. Probability expressions may be parsed into four components: the dissipation of all information, except the preservation of average values, taken over the measurement scale that relates changes in observed values to changes in information, and the transformation from the underlying scale on which information dissipates to alternative scales on which probability pattern may be expressed. Information invariances set the commonly observed measurement scales and the relations between them. In particular, a measurement scale for information is defined by its invariance to specific transformations of underlying values into measurable outputs. Essentially all common distributions can be understood within this simple framework of information invariance and measurement scale.Comment: v2: added table of contents, adjusted section numbers v3: minor editing, updated referenc