On the stability of a modified Nyström method for Mellin convolution equations in weighted spaces

This paper deals with the numerical solution of second kind integral equations with fixed singularities of Mellin convolution type. The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral operator the chief theoretical barrier. Here, we propose a Nyström method suitably modified in order to achieve the theoretical stability under proper assumptions on the Mellin kernel. We also provide an error estimate in weighted uniform norm and prove the well-conditioning of the involved linear systems. Some numerical tests which confirm the efficiency of the method are shown

Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis

This paper deals with a quadrature rule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis. The rule is of product type and consists in approximating the density function f by a truncated interpolation process based on the zeros of generalized Laguerre polynomials and an additional point. We prove the stability and the convergence of the rule, giving error estimates for functions belonging to weighted Sobolev spaces equipped with uniform norm. We also show how the proposed rule can be used for the numerical solution of hypersingular integral equations. Numerical tests which confirm the theoretical estimates and comparisons with other existing quadrature rules are presented

Data-driven model reduction-based nonlinear MPC for large-scale distributed parameter systems

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordModel predictive control (MPC) has been effectively applied in process industries since the 1990s. Models in the form of closed equation sets are normally needed for MPC, but it is often difficult to obtain such formulations for large nonlinear systems. To extend nonlinear MPC (NMPC) application to nonlinear distributed parameter systems (DPS) with unknown dynamics, a data-driven model reduction-based approach is followed. The proper orthogonal decomposition (POD) method is first applied off-line to compute a set of basis functions. Then a series of artificial neural networks (ANNs) are trained to effectively compute POD time coefficients. NMPC, using sequential quadratic programming is then applied. The novelty of our methodology lies in the application of POD's highly efficient linear decomposition for the consequent conversion of any distributed multi-dimensional space-state model to a reduced 1-dimensional model, dependent only on time, which can be handled effectively as a black-box through ANNs. Hence we construct a paradigm, which allows the application of NMPC to complex nonlinear high-dimensional systems, even input/output systems, handled by black-box solvers, with significant computational efficiency. This paradigm combines elements of gain scheduling, NMPC, model reduction and ANN for effective control of nonlinear DPS. The stabilization/destabilization of a tubular reactor with recycle is used as an illustrative example to demonstrate the efficiency of our methodology. Case studies with inequality constraints are also presented.The authors would like to acknowledge the financial support of the EC FP6 Project: CONNECT [COOP-2006-31638] and the EC FP7 project CAFE [KBBE-212754]

Ultrasensitive Displacement Noise Measurement of Carbon Nanotube Mechanical Resonators

Mechanical resonators based on a single carbon nanotube are exceptional sensors of mass and force. The force sensitivity in these ultra-light resonators is often limited by the noise in the detection of the vibrations. Here, we report on an ultra-sensitive scheme based on a RLC resonator and a low-temperature amplifier to detect nanotube vibrations. We also show a new fabrication process of electromechanical nanotube resonators to reduce the separation between the suspended nanotube and the gate electrode down to $\sim 150$~nm. These advances in detection and fabrication allow us to reach $0.5~\mathrm{pm}/\sqrt{\mathrm{Hz}}$ displacement sensitivity. Thermal vibrations cooled cryogenically at 300~mK are detected with a signal-to-noise ratio as high as 17~dB. We demonstrate $4.3~\mathrm{zN}/\sqrt{\mathrm{Hz}}$ force sensitivity, which is the best force sensitivity achieved thus far with a mechanical resonator. Our work is an important step towards imaging individual nuclear spins and studying the coupling between mechanical vibrations and electrons in different quantum electron transport regimes.Comment: 9 pages, 5 figure

Q2Q_2-free families in the Boolean lattice

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N),$ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.Comment: 18 pages, 2 figure

A NUMERICAL METHOD FOR SOLVING SYSTEMS OF HYPERSINGULAR INTEGRO-DIFFERENTIAL EQUATIONS

This paper is concerned with a collocation-quadrature method for solving systems of Prandtl's integro-differential equations based on de la Vallee Poussin filtered interpolation at Chebyshev nodes. We prove stability and convergence in Holder-Zygmund spaces of locally continuous functions. Some numerical tests are presented to examine the method's efficacy

MatLab Toolbox for the numerical solution of linear Volterra integral equations arising in metastatic tumor growth models

This paper introduces VIE Toolbox composed by fourteen MatLab functions used for the numerical resolution of Volterra Integral Equations (VIEs) of the second kind on infinite intervals. An application to metastatic tumor growth models is also considered, assuming five different tumor growth laws, e.g. exponential, power-law, Gompertz, generalized logistic and von Bertalanffy-West laws, for lung and breast tumors data

The Cerebellar Fastigial Nucleus Contributes to CO\u3csub\u3e2\u3c/sub\u3e-H\u3csup\u3e+\u3c/sup\u3e Ventilatory Sensitivity in Awake Goats

The purpose of this study was to test the hypothesis that an intact cerebellar fastigial nucleus (CFN) is an important determinant of CO2-H+ sensitivity during wakefulness. Bilateral, stainless steel microtubules were implanted into the CFN (N = 9) for injection (0.5–10 μl) of the neurotoxin ibotenic acid. Two or more weeks after implantation of the microtubules, eupneic breathing and CO2-H+ sensitivity did not differ significantly (P \u3e 0.10) from pre-implantation conditions. Injection of ibotenic acid (50 mM) did not significantly alter eupneic PaCO2 (P \u3e 0.10). The coefficient of variation of eupneic PaCO2 was 4.0 ± 0.6 and 3.7 ± 0.4% over the 2 weeks before and after the lesion, respectively. CO2-H+ sensitivity expressed as inspired ventilation/PaCO2 decreased from 2.15 ± 0.17 pre-lesion to 1.58 ± 0.26 l/(min mmHg) 3–6 days post-lesion (P \u3c 0.02, −27%). There was no significant (P \u3e 0.10) recovery of sensitivity between 7 and 10 days post-lesion. The lesion also increased (P \u3c 0.05) the day-to-day variability of this index by nearly 100%. When CO2 sensitivity was expressed as elevated inspired CO2/room air VI, values at 7%, but not 3 and 5% inspired CO2, were reduced and more variable (P \u3c 0.05) after the ibotenic acid injections. We conclude that during wakefulness, the CFN contributes relatively more to overall ventilatory drive at high relative to low levels of hypercapnia

High performance bilayer-graphene Terahertz detectors

We report bilayer-graphene field effect transistors operating as THz broadband photodetectors based on plasma-waves excitation. By employing wide-gate geometries or buried gate configurations, we achieve a responsivity $\sim 1.2V/W (1.3 mA/W)$ and a noise equivalent power $\sim 2\times 10^{-9} W/Hz^{-1/2}$ in the 0.29-0.38 THz range, in photovoltage and photocurrent mode. The potential of this technology for scalability to higher frequencies and the development of flexible devices makes our approach competitive for a future generation of THz detection systems.Comment: 8 pages, 5 figures. Submitted to Applied Physics Letter