Nonlinear Schroedinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the original nonlinear problem

The importance of circulating tumor products as „liquid biopsies” in colorectal cancer

Liquid biopsies represent an array of plasma analysis tests that are studied to evaluate and identify circulating tumor products, especially circulating tumor cells (CTCs) and circulating tumor DNA (ctDNA). Examining such biomarkers in the plasma of colorectal cancer patients has attracted attention due to its clinical significance in the treatment of malignant diseases. Given that tissue samples are sometimes challenging to procure or unsatisfactory for genomic profiling from patients with colorectal cancer, trustworthy biomarkers are mandatory for guiding treatment, monitoring therapeutic response, and detecting recurrence. This review considers the relevance of flowing tumor products like circulating tumor cells (CTCs), circulating tumor DNA (ctDNA), circulating messenger RNA (mRNA), circulating micro RNA (miRNA), circulating exosomes, and tumor educated platelets (TEPs) for patients with colorectal cancer

DC magnetron discharge fluid model using a new numerical scheme

Numerical modelling of an electrical discharge by fluid model can be accomplished through different procedures and approaches. A 2D time-dependent one was applied in order to describe a cylindrical symmetry Argon DC planar magnetron discharge. All transport equations, which means continuity and momentum transfer equations for electrons and ions and electron mean energy transport equation are solved in the same manner, using corrected classical drift-diffusion expressions for fluxes. For the validity of this last approach, the presence of magnetic field has been introduced as a correction in the electronic flux expression, while for ions an effective electric field has been considered. Plasma potential is given by Poisson equation

2D fluid approaches of DC magnetron discharge

A two dimensional (r,z) time-dependent fluid model was developed and used to describe a DC planar magnetron discharge with cylindrical symmetry. The transport description of the charged species uses the corresponding first three moments of Boltzmann equation: continuity, momentum transfer and mean energy transfer (the latter one only for electrons), coupled with Poisson equation. An original way is proposed to treat the transport equations. Electron and ion momentum transport equations are reduced to the classical drift-diffusion expression of the fluxes since the presence of the magnetic field is introduced as an additional part in the electron flux, while for ions an effective electric field was considered. Thus, both continuity and mean energy transfer equations are solved in a classical manner. Numerical simulations were performed considering Argon as buffer gas, with a neutral pressure varying between 5 and 30 mtorr, a gas temperature from 300 to 350 K and cathode voltages lying from -200 up to -600 V. Results obtained for densities of the charged particle, fluxes and plasma potential are in good agreement with previous works.Laboratoire de Physique des Gaz el des Plasmas (LPGP).Consiliul National al Cercetarii Stiintifice din Invatamantul Superior (CNCSIS) - A/1344/40213/2003

Clinical-evolutional particularities of the cryoglobulinemic vasculitis in the case of a patient diagnosed with hepatitis C virus in the predialitic phase

Hepatitis C virus (HCV) represents a fundamental issue for public health, with long term evolution and the gradual appearance of several complications and associated pathologies. One of these pathologies is represented by cryoglobulinemic vasculitis, a disorder characterized by the appearance in the patient’s serum of the cryoglobulins, which typically precipitate at temperatures below normal body temperature (37°C) and dissolve again if the serum is heated. Here, we describe the case of a patient diagnosed with HCV that, during the evolution of the hepatic disease, developed a form of cryoglobulinemic vasculitis. The connection between the vasculitis and the hepatic disorder was revealed following treatment with interferon, with the temporary remission of both pathologies and subsequent relapse at the end of the 12 months of treatment, the patient becoming a non-responder. The particularity of the case is represented by both the severity of the vasculitic disease from its onset and the deterioration of renal function up to the predialitic phase, a situation not typical of the evolution of cryoglobulinemia. Taking into account the hepatic disorder, the inevitable evolution towards cirrhosis, and the risk of developing the hepatocellular carcinoma, close monitoring is necessary

Decay versus survival of a localized state subjected to harmonic forcing: exact results

We investigate the survival probability of a localized 1-d quantum particle subjected to a time dependent potential of the form $rU(x)\sin{\omega t}$ with $U(x)=2\delta (x-a)$ or $U(x)= 2\delta(x-a)-2\delta (x+a)$. The particle is initially in a bound state produced by the binding potential $-2\delta (x)$. We prove that this probability goes to zero as $t\to\infty$ for almost all values of $r$, $\omega$, and $a$. The decay is initially exponential followed by a $t^{-3}$ law if $\omega$ is not close to resonances and $r$ is small; otherwise the exponential disappears and Fermi's golden rule fails. For exceptional sets of parameters $r,\omega$ and $a$ the survival probability never decays to zero, corresponding to the Floquet operator having a bound state. We show similar behavior even in the absence of a binding potential: permitting a free particle to be trapped by harmonically oscillating delta function potential

Cohomological BRST aspects of the massless tensor field with the mixed symmetry (k,k)

The main BRST cohomological properties of a free, massless tensor field that transforms in an irreducible representation of GL(D,R), corresponding to a rectangular, two-column Young diagram with k>2 rows are studied in detail. In particular, it is shown that any non-trivial co-cycle from the local BRST cohomology group H(s|d) can be taken to stop either at antighost number (k+1) or k, its last component belonging to the cohomology of the exterior longitudinal derivative H(gamma) and containing non-trivial elements from the (invariant) characteristic cohomology H^{inv}(delta|d).Comment: Latex, 50 pages, uses amssym

Hierarchical pinning models, quadratic maps and quenched disorder

We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in 1992, which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {R_n}_{n=1,2,...}, which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known Logistic Map. The large-n limit of the sequence of random variables 2^{-n} log R_n, a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter alpha>0, related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R_0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured by Derrida et al. (1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2<alpha<1 (respectively, alpha<1/2 or alpha=1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., non-disordered) model if alpha is larger or equal to 1/2, but not if alpha is smaller than 1/2. Our main result is a proof of these conjectures for the case alpha different from 1/2. We emphasize that for alpha>1/2 we find the correct scaling form (for weak disorder) of the critical point shift.Comment: 26 pages, 2 figures. v3: Theorem 1.6 improved. To appear on Probab. Theory Rel. Field

A quantitative central limit theorem for linear statistics of random matrix eigenvalues

It is known that the fluctuations of suitable linear statistics of Haar distributed elements of the compact classical groups satisfy a central limit theorem. We show that if the corresponding test functions are sufficiently smooth, a rate of convergence of order almost $1/n$ can be obtained using a quantitative multivariate CLT for traces of powers that was recently proven using Stein's method of exchangeable pairs.Comment: Title modified; main result stated under slightly weaker conditions; accepted for publication in the Journal of Theoretical Probabilit