Optimal Interpolating Spaces Preserving Shape

In this paper we study the existence and characterization of spaces which are images of minimal-norm projections that are required to interpolate at given functionals and satisfy additional shape-preserving requirements. We will call such spaces optimal interpolating spaces preserving shape. This investigation leads to concrete solutions in classical settings and, as examples, Π_n will be determined to be such spaces with regard to certain interpolation and shape-preserving requirements on the projections. Restated, the theory of this paper gives rist to an n-dimensional Hahn–Banach extension theorem, where the minimal-norm extension is required to keep invariant a fixed cone.     

Nearly monotone spline approximation in

It is shown that the rate of -approximation of a non-decreasing function in , , by ``nearly non-decreasing" splines can be estimated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. It is known that these estimates are no longer true for ``purely" monotone spline approximation, and properties of intervals where the monotonicity restriction can be relaxed in order to achieve better approximation rate are investigated.

     

Free-knot Splines Approximation of s-monotone Functions

Let I be a finite interval and r,sN. Given a set M, of functions defined on I, denote by ₊ ^s M the subset of all functions yM such that the s-difference ^s y() is nonnegative on I, >0. Further, denote by ₊ ^s W _p ^r , the class of functions x on I with the seminorm x ^(r)L _p 1, such that ^s x0, >0. Let M _n (h _k ):={ _i=1 ⁿ c _i h _k (w _i t– _i )c _i ,w _i , _i R, be a single hidden layer perceptron univariate model with n units in the hidden layer, and activation functions h _k (t)=t ₊ ^k , tR, kN ₀. We give two-sided estimates both of the best unconstrained approximation E( ₊ ^s W _p ^r ,M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1, and of the best s-monotonicity preserving approximation E( ₊ ^s W _p ^r , ₊ ^s M _n (h _k ))L _q , k=r–1,r, s=0,1,...,r+1. The most significant results are contained in theorem 2.2.     

Statistical Physics of Fracture Surfaces Morphology

Experiments on fracture surface morphologies offer increasing amounts of data that can be analyzed using methods of statistical physics. One finds scaling exponents associated with correlation and structure functions, indicating a rich phenomenology of anomalous scaling. We argue that traditional models of fracture fail to reproduce this rich phenomenology and new ideas and concepts are called for. We present some recent models that introduce the effects of deviations from homogeneous linear elasticity theory on the morphology of fracture surfaces, successfully reproducing the multiscaling phenomenology at least in 1+1 dimensions. For surfaces in 2+1 dimensions we introduce novel methods of analysis based on projecting the data on the irreducible representations of the SO(2) symmetry group. It appears that this approach organizes effectively the rich scaling properties. We end up proposing new experiments in which the rotational symmetry is not broken, such that the scaling properties should be particularly simple.     

Development and Validation of an Innovative Analytical Approach for the Quantitation of Tris(Hydroxymethyl)Aminomethane (TRIS) in Pharmaceutical Formulations by Liquid Chromatography Tandem Mass Spectrometry

A novel COVID-19 vaccine (BriLife^®) has been developed by the Israel Institute for Biological Research (IIBR) to prevent the spread of the SARS-CoV-2 virus throughout the population in Israel. One of the components in the vaccine formulation is tris(hydroxymethyl)aminomethane (tromethamine, TRIS), a buffering agent. TRIS is a commonly used excipient in various approved parenteral medicinal products, including the mRNA COVID-19 vaccines produced by Pfizer/BioNtech and Moderna. TRIS is a hydrophilic basic compound that does not contain any chromophores/fluorophores and hence cannot be retained and detected by reverse-phase liquid chromatography (RPLC)-ultraviolet (UV)/fluorescence methods. Among the few extant methods for TRIS determination, all exhibit a lack of selectivity and/or sensitivity and require laborious sample treatment. In this study, LC–mass spectrometry (MS) with its inherent selectivity and sensitivity in the multiple reaction monitoring (MRM) mode was utilized, for the first time, as an alternative method for TRIS quantitation. Extensive validation of the developed method demonstrated suitable specificity, linearity, precision, accuracy and robustness over the investigated concentration range (1.2–4.8 mg/mL). Specifically, the R² of the standard curve was >0.999, the recovery was >92%, and the coefficient of variance (%CV) was <12% and <6% for repeatability and intermediate precision, respectively. Moreover, the method was validated in accordance with strict Good Manufacturing Practice (GMP) guidelines. The developed method provides valuable tools that pharmaceutical companies can use for TRIS quantitation in vaccines and other pharmaceutical products.     

Non-degenerate single-particle energies in the Ginocchio model

A one-body operator expressing the breaking of the degeneracy of the single-nucleon energies is added to the pairing interaction of the Ginocchio model. This operator couples states inside the model's SD space to states outside it. The influence of this coupling on the effective interaction in the SD space and the possibility of expressing the results in terms of renormalization of parameters in the fermion hamiltonian or the IBM are investigated. The effective interaction is found to be almost diagonal in seniority, while splitting the previously-degenerate seniority multiplets. Appropriately renormalized Ginocchio and IBM hamiltonians can approximately reproduce the results, but fermion-number dependence of the hamiltonian parameters and explicit three-body interactions are needed to reproduce the computed effects exactly.     

Kolmogorov and linear widths of weighted Sobolev-type classes on a finite interval

Let I be a finite interval, r N and p(t)=dist{t,I}, tI. Denote by W ^r _{p ,}, 0<<, the class of functions x on I with the seminorm x ^(r) p _Lp1. We obtain two-sided estimates of the Kolmogorov widths d _n(W^r _p, )_Lq and of the linear widths d _n(W^r _p,)_Lq ^lin