Osculating spaces to secant varieties

We generalize the classical Terracini's Lemma to higher order osculating spaces to secant varieties. As an application, we address with the so-called Horace method the case of the $d$-Veronese embedding of the projective 3-space

Time-optimal Unitary Operations in Ising Chains II: Unequal Couplings and Fixed Fidelity

We analytically determine the minimal time and the optimal control laws required for the realization, up to an assigned fidelity and with a fixed energy available, of entangling quantum gates ($\mathrm{CNOT}$) between indirectly coupled qubits of a trilinear Ising chain. The control is coherent and open loop, and it is represented by a local and continuous magnetic field acting on the intermediate qubit. The time cost of this local quantum operation is not restricted to be zero. When the matching with the target gate is perfect (fidelity equal to one) we provide exact solutions for the case of equal Ising coupling. For the more general case when some error is tolerated (fidelity smaller than one) we give perturbative solutions for unequal couplings. Comparison with previous numerical solutions for the minimal time to generate the same gates with the same Ising Hamiltonian but with instantaneous local controls shows that the latter are not time-optimal.Comment: 11 pages, no figure

Brachistochrone of Entanglement for Spin Chains

We analytically investigate the role of entanglement in time-optimal state evolution as an appli- cation of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly cou- pled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entan- glement plays a role for W and GHZ initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.Comment: 9 pages, 4 figure

A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied recently by the authors for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games

On the dimensions of secant varieties of Segre-Veronese varieties

This paper explores the dimensions of higher secant varieties to Segre-Veronese varieties. The main goal of this paper is to introduce two different inductive techniques. These techniques enable one to reduce the computation of the dimension of the secant variety in a high dimensional case to the computation of the dimensions of secant varieties in low dimensional cases. As an application of these inductive approaches, we will prove non-defectivity of secant varieties of certain two-factor Segre-Veronese varieties. We also use these methods to give a complete classification of defective s-th Segre-Veronese varieties for small s. In the final section, we propose a conjecture about defective two-factor Segre-Veronese varieties.Comment: Revised version. To appear in Annali di Matematica Pura e Applicat

Time-optimal CNOT between indirectly coupled qubits in a linear Ising chain

We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3 is $T=\sqrt{3/2} J^{-1}$, i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space of qubit 3 shows that the time-optimal synthesis of the $\mathrm{CNOT}^{\pm}(1, 3)$ (which acts as the identity when the control qubit 1 is in the state $\ket{0}$, while if the control qubit is in the state $\ket{1}$ the target qubit 3 is flipped as $\ket{\pm}\rightarrow \ket{\mp}$) also requires the same time $T$.Comment: 9 pages; minor modification

Immature platelet fraction as predictive index of sepsis

Introduction The incidence of sepsis is reported around 37% in European ICUs [1]. The mortality rate depends on the severity of organ failure, up to 65% if four or more organs are involved. Multiple organ failure (MOF) is due to microcirculatory dysfunction with microthrombosis resulting from coagulation disorders including platelets’ activation. An early diagnosis should identify the microcirculatory dysfunction before MOF became clinically evident. The diagnosis of sepsis is commonly based on clinical criteria, pathogen identifi cation and use of markers like procalcitonin (PCT) and C-reactive protein (PCR) associated with infection. The aim of our study is to evaluate whether the routine measurement of immature platelet fraction (IPF), considered a precocious marker of platelet production, is associated with sepsis and its severity and/or whether it could be used as a predicting marker of sepsis. Methods We enrolled 66 consecutive patients admitted to the ICU, dividing them into two groups: septic (n = 44) and no septic (n = 22). The severity of sepsis was evaluated. The exclusion criterion was a platelet count <150,000/mm3. Blood count, coagulation, PCR, PCT, and IPF were collected every day. Results The IPF values between septic (4.6 ± 3.1) and no septic patients (3.3 ± 1.5) did not diff er (P = 0.16). No correlation was found between IPF values and the severity of septic condition (no sepsis 11.7 ± 10.1; sepsis 14.3 ± 10.5; severe sepsis 10.5 ± 9.1; septic shock 19.5 ± 12.4; P = 0.3). When we considered only subjects who did not have sepsis at the ICU admission we found that patients who developed sepsis during the recovery had IPF values higher than patients who did not develop sepsis (Table 1). Conclusions From our results IPF cannot be considered a marker of sepsis. Conversely it could be used as predictive index of sepsis because it can identify patients who will develop sepsis. References 1. Vincent et al.: Sepsis in European intensive care units: results of the SOAP study. Intensive Care Med 2006, 34:344-353

Cladribine Tablets Mode of Action, Learning from the Pandemic: A Narrative Review

Multiple sclerosis (MS) is an immune-mediated disease of the central nervous system, characterized by chronic, inflammatory, demyelinating, and neurodegenerative processes. MS management relies on disease-modifying drugs that suppress/modulate the immune system. Cladribine tablets (CladT) have been approved by different health authorities for patients with various forms of relapsing MS. The drug has been demonstrated to deplete CD4+ and CD8+ T-cells, with a higher effect described in the former, and to decrease total CD19+, CD20+, and naive B-cell counts. COVID-19 is expected to become endemic, suggesting its potential infection risk for immuno-compromised patients, including MS patients treated with disease-modifying drugs. We report here the available data on disease-modifying drug-treated-MS patients and COVID-19 infection and vaccination, with a focus on CladT. MS patients treated with CladT are not at higher risk of developing severe COVID-19. While anti-SARS-CoV-2 vaccination is recommended in all MS patients with guidelines addressing vaccination timing according to the different disease-modifying drugs, no vaccination timing restrictions seem to be necessary for cladribine, based on its mechanism of action and available evidence. Published data suggest that CladT treatment does not impact the production of anti-SARS-CoV-2 antibodies after COVID-19 vaccination, possibly due to its relative sparing effect on naïve B-cells and the rapid B-cell reconstitution following treatment. Slightly lower specific T-cell responses are likely not impacting the risk of breakthrough COVID-19. It could be stated that cladribine’s transient effect on innate immune cells likely contributes to maintaining an adequate first line of defense against the SARS-CoV-2 virus