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  • 1
    ISSN: 1432-0940
    Keywords: Primary ; 41A65 ; Secondary ; 41A29 ; Best constrained approximation ; Shape-preserving interpolation ; Cones ; Dual cones ; Duality ; n-Convex functions ; Hilbert space
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper continues the study of best approximation in a Hilbert spaceX from a subsetK which is the intersection of a closed convex coneC and a closed linear variety, with special emphasis on application to then-convex functions. A subtle separation theorem is utilized to significantly extend the results in [4] and to obtain new results even for the “classical” cone of nonnegative functions. It was shown in [4] that finding best approximations inK to anyf inX can be reduced to the (generally much simpler) problem of finding best approximations to a certain perturbation off from either the coneC or a certain subconeC F. We will show how to determine this subconeC F, give the precise condition characterizing whenC F=C, and apply and strengthen these general results in the practically important case whenC is the cone ofn-convex functions inL 2 (a,b),
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  • 2
    ISSN: 1432-0940
    Keywords: 41A05 ; 41A63 ; Conditionally negative definite ; Completely monotone ; Radial ; Interpolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Motivated by the problem of multivariate scattered data interpolation, much interest has centered on interpolation by functions of the form $$f(x) = \sum\limits_{j = 1}^N {a_j g(\parallel x - x_j \parallel ),x \in R^s }$$ whereg:R + →R is some prescribed function. For a wide range of functionsg, it is known that the interpolation matricesA=g(∥x i −x j ∥) i,j=1 N are invertible for given distinct data pointsx 1,x 2,...,x N. More recently, progress has been made in quantifying these interpolation methods, in the sense of estimating the (l 2) norms of the inverses of these interpolation matrices as well as their condition numbers. In particular, given a suitable functiong:R + →R, and data inR s having minimal separationq, there exists a functionh s:R + →R +, which depends only ong ands, and a constantC s , which depends only ons, such that the inverse of the associated interpolation matrixA satisfies the estimate ‖A −1‖≤C s h s (q). The present paper seeks “converse” results to the inequality given above. That is, given a suitable functiong, a spatial dimensions, and a parameterq〉0 (which is usually assumed to be small), it is shown that there exists a data set inR s having minimal separationq, a constant $$\tilde C_s$$ depending only ons, and a functionk s (q), such that the inverse of the interpolation matrixA associated with this data set satisfies $$\parallel A^{ - 1} \parallel \geqslant \tilde C_s k_s (q)$$ . In some cases, it is seen thath s(q)=k s (q), so the bounds are optimal up to constants. In certain others,k s (q) is less thanh s (q), but nevertheless exhibits a behavior comparable to that ofh s (q). That is, even in these cases, the bounds are close to being optimal.
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  • 3
    ISSN: 1432-0533
    Keywords: Diabetic neuropathy ; Axonal regeneration ; Nerve growth factor receptors ; Schwann cells ; Basal lamina
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Abstract Observations were made on myelinated fibre regeneration in diabetic sensory polyneuropathy assessed in sural nerve biopsy specimens. These confirmed that regenerative clusters initially develop within abnormally persistent Schwann cell basal laminal tubes. The number of regenerating fibres, identified by light microscopy, was found to decline in proportion to the reduction in total myelinated fibre density. The relative number of regenerating fibres was significantly greater in patients with insulin-dependent as compared with those with non-insulin-dependent diabetes after correction for age. There was a slight negative correlation between the relative proportion of regenerating fibres and age, but this was not statistically significant. The progressive reduction in the number of regenerating fibres with declining total fibre density indicates that axonal regeneration fails with advancing neuropathy. The production of nerve growth factor (NGF) and NGF receptors by denervated Schwann cells is likely to be important for axonal regeneration. To investigate whether the failure of axonal regeneration could be related to a lack of NGF receptor production by Schwann cells, we examined the expression of p75 NGF receptors by Büngner bands immunocytochemically. In comparison with other types of peripheral neuropathy, p75 NGF receptor expression appeared to take place normally. It is concluded that failure of axonal regeneration constitutes an important component in diabetic neuropathy. Its explanation requires further investigation.
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  • 4
    ISSN: 1432-0533
    Keywords: Key words Diabetic polyneuropathy ; Nodes of ; Ranvier ; Paranodes ; Axoglial dysjunction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Abstract Observations have been made on the structure of the paranodal region at nodes of Ranvier in the sural nerve of patients with diabetic sensory polyneuropathy. The structure of the paranodes was examined with particular attention to the definition and assessment of axoglial dysjunction, which has been claimed to be a characteristic feature of both human and experimental diabetic neuropathy and which has been related to paranodal swelling. In the present series of cases it was not possible to confirm that axoglial dysjunction is a distinctive feature of diabetic polyneuropathy in fibres not undergoing active demyelination or wallerian-type degeneration, neither was excessive paranodal enlargement found.
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  • 5
    ISSN: 1432-0533
    Keywords: Key words Diabetic neuropathy ; Axonal regeneration ; Nerve growth factor receptors ; Schwann cells ; Basal lamina
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Abstract Observations were made on myelinated fibre regeneration in diabetic sensory polyneuropathy assessed in sural nerve biopsy specimens. These confirmed that regenerative clusters initially develop within abnormally persistent Schwann cell basal laminal tubes. The number of regenerating fibres, identified by light microscopy, was found to decline in proportion to the reduction in total myelinated fibre density. The relative number of regenerating fibres was significantly greater in patients with insulin-dependent as compared with those with non-insulin-dependent diabetes after correction for age. There was a slight negative correlation between the relative proportion of regenerating fibres and age, but this was not statistically significant. The progressive reduction in the number of regenerating fibres with declining total fibre density indicates that axonal regeneration fails with advancing neuropathy. The production of nerve growth factor (NGF) and NGF receptors by denervated Schwann cells is likely to be important for axonal regeneration. To investigate whether the failure of axonal regeneration could be related to a lack of NGF receptor production by Schwann cells, we examined the expression of p75 NGF receptors by Büngner bands immunocytochemically. In comparison with other types of peripheral neuropathy, p75 NGF receptor expression appeared to take place normally. It is concluded that failure of axonal regeneration constitutes an important component in diabetic neuropathy. Its explanation requires further investigation.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Diabetologia 22 (1982), S. 141-147 
    ISSN: 1432-0428
    Keywords: Arteriovenous anastamoses ; blood flow velocity ; diabeticneuropathies ; electrophysiology ; ultrasonics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Summary Many clinical problems arise as a result of tissue pathology in the diabetic leg. Neuropathic or vascular ischaemic syndromes are readily identified but on occasions differentiation of the two may be difficult. This survey reviews the aetiological background of neuropathy and ischaemia, examines the relationship of physiological blood flow abnormalities to both areas and comments on management of the clinical states encountered, including the common problem of the ulcerated diabetic foot.
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  • 7
    ISSN: 1432-0428
    Keywords: Diabetic neuropathy ; nerve blood flow ; sural nerve ; sural sensory conduction velocity ; temperature ; exercise
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Summary Severe microvascular disease exists at the stage of clinical diabetic neuropathy. A non-invasive test that will identify those diabetic subjects who will eventually develop neuropathy is essential for early intervention. Sural sensory conduction velocity was recorded (x 3) in 12 non-neuropathic diabetic subjects, 15 diabetic subjects with established neuropathy and 16 age-matched normal control subjects, before and after exercise to 80% age/sex predicted maximum heart rate. Fixed sural electrodes were used. Subcutaneous temperature was recorded by a needle thermocouple placed near the sural nerve. Sural sensory conduction velocity increased significantly after exercise in normal subjects (p〈0.01, mean increase 5.07 m/s) and non-neuropathic diabetic subjects (p〈0.02, mean increase 3.99 m/s) but not in neuropathic subjects (mean increase 0.99 m/s). Subcutaneous temperature rose significantly in normal subjects (p〈0.01, mean increase 2.07°C) and non-neuropathic diabetic subjects (p〈0.001, mean increase 2.52 °C) but not in neuropathic subjects (mean increase 0.15 °C). However, sural sensory conduction velocity increased by 1.2 m · s−1. °C−1 following direct warming of the limb in six neuropathic subjects which was comparable to that of normal and non-neuropathic subjects (1.49 and 1.48 m · s−1. °C−1). The impairment of exercise conduction increment in diabetic neuropathy suggests impaired nerve blood flow in diabetic neuropathy.
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  • 8
    ISSN: 1432-0428
    Keywords: Diabetic neuropathy ; sural nerve ; nerve blood flow ; epineurial vessel photography ; fluorescein angiography ; arterio-venous shunting ; vasa nervorum
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Summary New techniques of sural nerve photography and fluorescein angiography which are able to provide an index of nerve blood flow have been developed. Under local anaesthetic, 3 cm of sural nerve was exposed at the ankle using an operating microscope. Without disturbing the epineurium, vessels were identified and photographed at a standard magnification (× 30). These were independently graded by an ophthalmologist not otherwise involved with the study. Fluorescein angiography was then carried out on the exposed nerve. The fluorescein appearance time and intensity of fluorescence were quantified, using computer analysis of digitised images. Thirteen subjects with chronic sensory motor neuropathy, five non-neuropathic diabetic and nine normal control subjects were studied. The mean epineurial vessel pathology score of the neuropathic group was significantly higher than the combined normal control and non-neuropathic diabetic groups (p 〈0.01). Direct epineurial arteriovenous shunting was observed in six neuropathic and one non-neuropathic diabetic patients and not in any of the normal control subjects. The nerve fluorescein appearance time was significantly delayed in subjects with chronic sensory motor neuropathy (51.5 ± 12 s) compared to both normal (34.7 ± 9 s, p 〈0.01) and non-neuropathic diabetic subjects (33.4 ± 11 s, p 〈0.025). The mean intensity of fluorescence at 96, 252 and 576 s, was significantly lower in subjects with chronic sensory motor neuropathy compared with both of the other groups (p 〈0.05). The epineurial vessel pathology score was significantly related to reduced sural (p 〈0.01) and peroneal (p 〈0.001) nerve conduction velocities, elevated vibration (p 〈0.01) and thermal (p 〈0.001) perception and the severity of retinopathy (p 〈0.002). The fluorescein appearance time was significantly related to reduced sural sensory (p 〈0.02) conduction velocity, elevated vibration (p 〈0.01) perception and epineurial vessel (p 〈0.002) pathology score, but it failed to relate to peroneal motor (p = 0.06) conduction velocity, thermal (p = 0.1) perception and the severity of retinopathy (p = 0.3). Intensity of fluorescence was significantly related to fluorescein appearance time (at 96 s, p 〈0.001; at 576 s, p 〈0.05) but did not relate to measures of neuropathic severity. These techniques have enabled us to observe that epineurial vessel anatomy is abnormal and that nerve blood flow is impaired in subjects with chronic sensory motor neuropathy. In addition epineurial arterio-venous shunting may be a feature of diabetic neuropathy. These techniques may further be applied to study nerve blood flow in early diabetic neuropathy.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Diabetologia 37 (1994), S. 847-854 
    ISSN: 1432-0428
    Keywords: Key words Diabetic neuropathy ; microangiopathy ; nerve ; hypoxia ; blood flow.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Summary Despite considerable research we still do not have a comprehensive explanation for the pathogenesis of diabetic neuropathy. Although chronic hyperglycaemia is almost certainly involved, it is not known whether the primary pathology is metabolic, microvascular, or an interaction between the two. Hyperglycaemia-induced polyol pathway hyperactivity associated with nerve sorbitol accumulation and myo-inositol depletion may play a part in the genesis of diabetic neuropathy. The case for microvascular disease in diabetic neuropathy is now strong. Fibre loss in human sural nerve is multifocal, suggesting ischaemia. The degree of vessel disease has been related to the severity of neuropathy. People with chronic obstructive pulmonary disease develop the so called “hypoxic neuropathy” in which similar microvascular changes occur as in diabetic neuropathy. In rats with experimental diabetic neuropathy nerve blood flow is reduced and oxygen supplementation or vasodilator treatment improved the deterioration in conduction velocity and nerve blood flow. Similarly, in human diabetic neuropathy, there is impaired nerve blood flow, epineurial arterio-venous shunting and a reduction in sural nerve oxygen tension. At what stage during the development of nerve damage these changes occur is yet to be determined. [Diabetologia (1994) 37: 847–854]
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  • 10
    ISSN: 1432-0428
    Keywords: Keywords Insulin-dependent diabetes mellitus ; diabetic neuropathy ; prevalence ; glycaemic control ; microalbuminuria ; impotence ; epidemiology.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Medicine
    Notes: Summary The EURODIAB IDDM Complications Study involved the examination of 3250 randomly selected insulin-dependent diabetic patients, from 31 centres in 16 European countries. Part of the examination included an assessment of neurological function including neuropathic symptoms and physical signs, vibration perception threshold, tests of autonomic function and the prevalence of impotence. The prevalence of diabetic neuropathy across Europe was 28 % with no significant geographical differences. Significant correlations were observed between the presence of diabetic peripheral neuropathy with age (p 〈 0.05), duration of diabetes (p 〈 0.001), quality of metabolic control (p 〈 0.001), height (p 〈 0.01), the presence of background or proliferative diabetic retinopathy (p 〈 0.01), cigarette smoking (p 〈 0.001), high-density lipoprotein cholesterol (p 〈 0.001) and the presence of cardiovascular disease (p 〈 0.05), thus confirming previous associations. New associations have been identified from this study – namely with elevated diastolic blood pressure (p 〈 0.05), the presence of severe ketoacidosis (p 〈 0.001), an increase in the levels of fasting triglyceride (p 〈 0.001), and the presence of microalbuminuria (p 〈 0.01). All the data were adjusted for age, duration of diabetes and HbA1c. Although alcohol intake correlated with absence of leg reflexes and autonomic dysfunction, there was no overall association of alcohol consumption and neuropathy. The reported problems of impotence were extremely variable between centres, suggesting many cultural and attitudinal differences in the collection of such information in different European countries. In conclusion, this study has identified previously known and new potential risk factors for the development of diabetic peripheral neuropathy. [Diabetologia (1996) 39: 1377–1384]
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