Supercritical biharmonic equations with power-type nonlinearity

The biharmonic supercritical equation $\Delta^2u=|u|^{p-1}u$, where $n>4$ and $p>(n+4)/(n-4)$, is studied in the whole space $\mathbb{R}^n$ as well as in a modified form with $\lambda(1+u)^p$ as right-hand-side with an additional eigenvalue parameter $\lambda>0$ in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the explicitly known radial {\it singular} solution, provided $p\in((n+4)/(n-4),p_c)$, where $p_c\in ((n+4)/(n-4),\infty]$ is a further critical exponent, which was introduced in a recent work by Gazzola and the second author. The third author proved already that these oscillations do not occur in the complementing case, where $p\ge p_c$. Concerning the Dirichlet problem we prove existence of at least one singular solution with corresponding eigenvalue parameter. Moreover, for the extremal solution in the bifurcation diagram for this nonlinear biharmonic eigenvalue problem, we prove smoothness as long as $p\in((n+4)/(n-4),p_c)$

Short-time dynamics of finite-size mean-field systems

We study the short-time dynamics of a mean-field model with non-conserved order parameter (Curie-Weiss with Glauber dynamics) by solving the associated Fokker-Planck equation. We obtain closed-form expressions for the first moments of the order parameter, near to both the critical and spinodal points, starting from different initial conditions. This allows us to confirm the validity of the short-time dynamical scaling hypothesis in both cases. Although the procedure is illustrated for a particular mean-field model, our results can be straightforwardly extended to generic models with a single order parameter.Comment: accepted for publication in JSTA

100 Gbps/λ PON downstream O- And C-band alternatives using direct-detection and linear-impairment equalization [Invited]

The future-generation passive optical network (PON) physical layer, targeting 100 Gbps/wavelength, will have to deal with severe optoelectronics bandwidth and chromatic dispersion limitations. In this paper, largely extending our Optical Fiber Communication Conference (OFC) 2020 invited paper, we review 100 Gbps/wavelength PON downstream alternatives over standard single-mode fiber in the O- and C-bands, analyzing three modulation formats (PAM-4, partial-response PAM-4, and PAM-8), two types of direct-detection receivers (APD- and SOA $+$+ PIN-based), and three digital reception strategies (unequalized, feed-forward equalized, and decision-feedback equalized). We evaluate by means of simulations the performance of these alternatives under different optoelectronics bandwidth and dispersion scenarios, identifying O-band feasible solutions able to reach 20 km of fiber and an optical path loss of at least 29 dB over a wide wavelength range of operation. Finally, we compare two digitally precompensated modulation schemes that are highly tolerant of chromatic dispersion, showing a possible extension to C-band operation, preserving direct-detection and linear-impairment equalization at the optical network unit side

Experimental Demonstration of 100 Gbps/λ C-Band Direct-Detection Downstream PON Using Non-Linear and CD Compensation with 29 dB+ OPL over 0 Km-100 Km

Passive Optical Networks (PON), able to operate at 50 Gbps per wavelength (λ), are under development and standardization, based on intensity-modulation (IM) and direct-detection (DD) systems. The next step in PON evolution will be driven by 5G/6G fronthauling capacity demands, and will require the development of 100 Gbps/λ (and beyond) systems, which poses big challenges if maintaining the DD-format. In this contribution, we analyze a 100 Gbps/λ PON architecture able to preserve the IM-DD approach at the Optical Network Unit (ONU), placing the complexity at the Optical Line Terminal (OLT), thanks to Digital Signal Processing (DSP). We experimentally demonstrate a 100 Gbps/λ transmission using this architecture in the downstream (DS) direction. Chromatic dispersion digital pre-compensation (CD-DPC) in combination with an IQ Mach-Zehnder Modulator (IQ-MZM) is used at the transmitter (TX). Keeping the ONU DSP as simple as possible, as compared with current DSP proposals for 50 Gbps/λ PON, is another main goal of this work. Adaptive equalization (AEQ) is used to correct for linear impairments, in addition to digital non-linear correction (NLC) at the receiver (RX). We compare two NLC approaches: a full Volterra Non-Linear Equalizer (VNLE) and a simpler NLC technique based on a square-root like function (SQRT). Operation over standard single-mode fiber (SMF) in C-band, achieving reaches from 0 km to 100 km and Optical Path Loss (OPL) values higher than 29 dB, are shown. The analyzed proposal is directly applicable to Terabit-capable wavelength division multiplexing (WDM)-PON, and can be extended to very high-speed Time Division Multiplexing (TDM)-PON and TWDM-PON, with some modifications discussed here

Efectos agudos de las vibraciones de cuerpo completo sobre variables funcionales en niños con parálisis cerebral

Accésit Congreso SIBB 2015El objetivo del presente estudio fue analizar los efectos agudos de las vibraciones de cuerpo completo (VCC) sobre el equilibrio y la movilidad en niños con parálisis cerebral, y comparar los efectos producidos por el estímulo vibratorio vertical y oscilante sobre las variables mencionadas. Se utilizó un diseño cruzado aleatorio en el que participaron 6 niños y adolescentes con parálisis cerebral. Se llevaron a cabo 5 sesiones de tratamiento, utilizando las dos primeras como familiarización. En cada una de las tres sesiones restantes, el paciente recibía de forma aleatoria un tratamiento de VCC diferente consistentes en 5 series de 1 minuto de vibración seguidas de 1 minuto de descanso. Antes y después de cada intervención se realizaron valoraciones de esfuerzo percibido, equilibrio y movilidad a través de una escala visual analógica y los test de Romberg y timed up and go, respectivamente. La escala visual analógica mostró un incremento del esfuerzo percibido tras la aplicación del tratamiento (+75%, p=0,007). En el equilibrio se observó un aumento en la velocidad de desplazamiento del centro de presiones tras la intervención (+11,3%, p=0,024).Todos los protocolos aplicados en niños y adolescentes con parálisis cerebral, modificaron de manera aguda el equilibrio, no mostrándose cambios respecto a la movilidad. Los resultados del presente estudio demuestran que, el estímulo vibratorio no parece inducir mayores modificaciones sobre el equilibrio y la movilidad que el propio trabajo isométrico sobre la plataforma.Peer ReviewedAward-winnin