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Our work goes in two directions. At first we want to transfer definitions, concepts and results of the theory of hyperidentities and solid varieties from the total to the partial case. (1) We prove that the operators chi^A_RNF and chi^E_RNF are only monotone and additive and we show that the sets of all fixed points of these operators are characterized only by three instead of four equivalent conditions for the case of closure operators. (2) We prove that V is n − SF-solid iff clone^SF V is free with respect to itself, freely generated by the independent set {[fi(x_1, . . . , x_n)]Id^SF_n V | i \in I}. (3) We prove that if V is n-fluid and ~V |P(V ) =~V −iso |P(V ) then V is kunsolid for k >= n (where P(V ) is the set of all V -proper hypersubstitutions of type \tau ). (4) We prove that a strong M-hyperquasi-equational theory is characterized by four equivalent conditions. The second direction of our work is to follow ideas which are typical for the partial case. (1) We characterize all minimal partial clones which are strongly solidifyable. (2)We define the operator Chi^A_Ph where Ph is a monoid of regular partial hypersubstitutions.Using this concept, we define the concept of a Phyp_R(\tau )-solid strong regular variety of partial algebras and we prove that a PHyp_R(\tau )-solid strong regular variety satisfies four equivalent conditions.
The objectives of this study were the identification in (morbidly) obese and nonobese patients of (i) the most appropriate body size descriptor for fosfomycin dose adjustments and (ii) adequacy of the currently employed dosing regimens. Plasma and target site (interstitial fluid of subcutaneous adipose tissue) concentrations after fosfomycin administration (8 g) to 30 surgery patients (15 obese/15 nonobese) were obtained from a prospective clinical trial. After characterization of plasma and microdialysis-derived target site pharmacokinetics via population analysis, short-term infusions of fosfomycin 3 to 4 times daily were simulated. The adequacy of therapy was assessed by probability of pharmacokinetic/pharmacodynamic target attainment (PTA) analysis based on the unbound drug-related targets of an %fT(>= MIC) (the fraction of time that unbound fosfomycin concentrations exceed the MIC during 24 h) of 70 and an fAUC(0-24h)/MIC (the area under the concentration-time curve from 0 to 24 h for the unbound fraction of fosfomycin relative to the MIC) of 40.8 to 83.3. Lean body weight, fat mass, and creatinine clearance calculated via adjusted body weight (ABW) (CLCRCG_ABW) of all patients (body mass index [BMI] = 20.1 to 52.0 kg/m(2)) explained a considerable proportion of between-patient pharmacokinetic variability (up to 31.0% relative reduction). The steady-state unbound target site/plasma concentration ratio was 26.3% lower in (morbidly) obese than nonobese patients. For infections with fosfomycin-susceptible pathogens (MIC <= 16 mg/L), intermittent "high-dosage" intravenous (i.v.) fosfomycin (8 g, three times daily) was sufficient to treat patients with a CLCRCG_ABW of,130 mL/min, irrespective of the pharmacokinetic/pharmacodynamic indices considered. For infections by Pseudomonas aeruginosa with a MIC of 32 mg/L, when the index fAUC0-24h/MIC is applied, fosfomycin might represent a promising treatment option in obese and nonobese patients, especially in combination therapy to complement beta-lactams, in which carbapenem-resistant P. aeruginosa is critical. In conclusion, fosfomycin showed excellent target site penetration in obese and nonobese patients. Dosing should be guided by renal function rather than obesity status.
For a singularly perturbed parabolic - ODE system we construct the asymptotic expansion in the small parameter in the case, when the degenerate equation has a double root. Such systems, which are called partly dissipative reaction-diffusion systems, are used to model various natural processes, including the signal transmission along axons, solid combustion and the kinetics of some chemical reactions. It turns out that the algorithm of the construction of the boundary layer functions and the behavior of the solution in the boundary layers essentially differ from that ones in case of a simple root. The multizonal initial and boundary layers behaviour was stated.
We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.
We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions, its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres, and 3-dimensional spherical space forms. Published under license by AIP Publishing.
A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S-4 or diffeomorphic to CP2.
This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not too nonconstant. In particular, the Hopf conjecture on S-2 x S-2 holds in this class of manifolds.
Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green's operators. The most prominent examples are wave operators and Dirac-type operators. This paper is devoted to a systematic study of this class of differential operators. For instance, we show that this class is closed under taking restrictions to suitable subregions of the manifold, under composition, under taking "square roots", and under the direct sum construction. Symmetric hyperbolic systems are studied in detail.
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.