Sufficient conditions for the convergence of the Magnus expansion

Two different sufficient conditions are given for the convergence of the Magnus expansion arising in the study of the linear differential equation $Y' = A(t) Y$. The first one provides a bound on the convergence domain based on the norm of the operator $A(t)$. The second condition links the convergence of the expansion with the structure of the spectrum of $Y(t)$, thus yielding a more precise characterization. Several examples are proposed to illustrate the main issues involved and the information on the convergence domain provided by both conditions.Comment: 20 page

On an asymptotic method for computing the modified energy for symplectic methods

We revisit an algorithm by Skeel et al. [5,16] for computing the modified, or shadow, energy associated with symplectic discretizations of Hamiltonian systems. We amend the algorithm to use Richardson extrapolation in order to obtain arbitrarily high order of accuracy. Error estimates show that the new method captures the exponentially small drift associated with such discretizations. Several numerical examples illustrate the theory

Convergence of the Magnus series

The Magnus series is an infinite series which arises in the study of linear ordinary differential equations. If the series converges, then the matrix exponential of the sum equals the fundamental solution of the differential equation. The question considered in this paper is: When does the series converge? The main result establishes a sufficient condition for convergence, which improves on several earlier results.Comment: 11 pages; v2: added justification for conjecture, minor clarifications and correction

Transformation kinetics of alloys under non-isothermal conditions

The overall solid-to-solid phase transformation kinetics under non-isothermal conditions has been modeled by means of a differential equation method. The method requires provisions for expressions of the fraction of the transformed phase in equilibrium condition and the relaxation time for transition as functions of temperature. The thermal history is an input to the model. We have used the method to calculate the time/temperature variation of the volume fraction of the favored phase in the alpha-to-beta transition in a zirconium alloy under heating and cooling, in agreement with experimental results. We also present a formulation that accounts for both additive and non-additive phase transformation processes. Moreover, a method based on the concept of path integral, which considers all the possible paths in thermal histories to reach the final state, is suggested.Comment: 16 pages, 7 figures. To appear in Modelling Simul. Mater. Sci. En

Topical application of ALA and ALA hexyl ester on a subcutaneous murine mammary adenocarcinoma: tissue distribution

Although 5-aminolevulinic acid (ALA)-based photodynamic therapy (PDT) has proven to be clinically beneficial for the treatment of certain cancers, including a variety of skin cancers, optimal tissue localisation still remains a problem. An approach to improve the bioavailability of protoporphyrin IX (PpIX) is the use of ALA derivatives instead of ALA. In this work, we employed a subcutaneous murine mammary adenocarcinoma to study the tissue distribution pattern of the ALA hexyl ester (He-ALA) in comparison with ALA after their topical application in different vehicles. He-ALA induced porphyrin synthesis in the skin overlying the tumour (SOT), but it did not reach the tumour tissue as efficiently. Only 5 h after He-ALA lotion application, tumour porphyrin levels surpassed control values. He-ALA delivered in cream induced a substantially lower porphyrin synthesis in SOT, reinforcing the importance of the vehicle in the use of topical PDT. Porphyrin levels in internal organs remained almost within control values when He-ALA was employed. The addition of DMSO to ALA formulation slightly increased tumour and SOT porphyrin biosynthesis, but it did not when added to He-ALA lotion

Photostability of commercial sunscreens upon sun exposure and irradiation by ultraviolet lamps

BACKGROUND: Sunscreens are being widely used to reduce exposure to harmful ultraviolet (UV) radiation. The fact that some sunscreens are photounstable has been known for many years. Since the UV-absorbing ingredients of sunscreens may be photounstable, especially in the long wavelength region, it is of great interest to determine their degradation during exposure to UV radiation. Our aim was to investigate the photostability of seven commercial sunscreen products after natural UV exposure (UVnat) and artificial UV exposure (UVart). METHODS: Seven commercial sunscreens were studied with absorption spectroscopy. Sunscreen product, 0.5 mg/cm(2), was placed between plates of silica. The area under the curve (AUC) in the spectrum was calculated for UVA (320–400 nm), UVA1 (340–400 nm), UVA2 (320–340 nm) and UVB (290–320 nm) before (AUC(before)) and after (AUC(after)) UVart (980 kJ/m(2 )UVA and 12 kJ/m(2 )of UVB) and before and after UVnat. If theAUC Index (AUCI), defined as AUCI = AUC(after)/AUC(before), was > 0.80, the sunscreen was considered photostable. RESULTS: Three sunscreens were unstable after 90 min of UVnat; in the UVA range the AUCI was between 0.41 and 0.76. In the UVB range one of these sunscreens was unstable with an AUCI of 0.75 after 90 min. Three sunscreens were photostable after 120 min of UVnat; in the UVA range the AUCI was between 0.85 and 0.99 and in the UVB range between 0.92 and 1.0. One sunscreen showed in the UVA range an AUCI of 0.87 after UVnat but an AUCI of 0.72 after UVart. Five of the sunscreens were stable in the UVB region. CONCLUSION: The present study shows that several sunscreens are photounstable in the UVA range after UVnat and UVart. There is a need for a standardized method to measure photostability, and the photostability should be marked on the sunscreen product

ALA and ALA hexyl ester induction of porphyrins after their systemic administration to tumour bearing mice

The use of synthetic lipophilic molecules derived from 5-aminolevulinic acid (ALA) is currently under investigation to enhance cellular ALA penetration. In this work we studied the effect of systemic administration to mice of the hexyl ester of ALA (He-ALA) on porphyrin tissue synthesis as compared to ALA. In most normal tissues as well as in tumour, He-ALA induced less porphyrin synthesis than ALA after its systemic administration either intravenous or intraperitoneal, although explant organ cultures exposed to either ALA or He-ALA revealed equally active esterases. The only tissue that accumulated higher porphyrin levels from He-ALA (seven times more than ALA) was the brain, and this correlated well with a rapid increase in ALA/He-ALA content in brain after administration of He-ALA. This may be ascribed to a differential permeability to lipophilic substances controlled by the blood–brain barrier, a feature which could be further exploited to treat brain tumours

Enhanced photodynamic destruction of a transplantable fibrosarcoma using photochemical internalisation of gelonin

Photochemical internalisation (PCI) is a technique for releasing biologically active macromolecules from endocytic vesicles by light activation of a photosensitiser localised in the same vesicles of targeted cells. This study investigated the PCI of the toxin gelonin as a way of enhancing the effect of photodynamic therapy (PDT) on a human malignant fibrous histiocytoma transplanted into nude mice using the photosensitiser disulphonated aluminium phthalocyanine (AlPcS2a). Pharmacokinetic studies after intraperitoneal administration showed that the serum level of AlPcS2a fitted a biexponential model (half-lives of 1.8 and 26.7 h). The tumour concentration was roughly constant up to 48 h, although fluorescence microscopy showed that the drug location was initially mainly vascular, but became intracellular by 48 h. To compare PDT with PCI, 48 h after intraperitoneal injection of 10 mg kg−1 AlPcS2a, and 6 h after direct intratumour injection of 50 μg gelonin (PCI) or a similar volume of phosphate-buffered saline (PDT controls), tumour-bearing animals were exposed to red light (150 J cm−2). Complete response was observed for more than 100 days in 50% of the PCI tumours but only 10% of the PDT tumours (P<0.01). In tumours examined histologically 4 days after light delivery, the depth of necrosis was 3–4 mm after PDT, but 7 mm after PCI. The deeper effect after PCI demonstrates that the light fluence needed to kill tumour is less than with PDT. We conclude that PCI with gelonin can markedly enhance the effect of PDT on this type of tumour and may have a role clinically as an adjunct to surgery to control localised disease