Lagrange multiplier method for convex programs

The problem of minimizing a convex function that is subject to the constraint that a number of other convex functions be nonpositive can be treated by the Lagrange multiplier method. Such a treatment was revived by Kuhn and Tucker and further studied by many other scientists. These studies led to an associated maximizing problem on the Lagrange function. The aim of this note is to give a short elementary proof that the infimum of the first problem is equal to the supremum of the second problem. To carry this out it is necessary to relax the constraints of the first (or the second) problem so that the constraints are enforced only in the limit. This relaxation of constraints is not necessary should prescribing upper bounds to all the convex functions define a bounded set of points in the domain of the functions. The domain of the functions can be n-dimensional space or a reflexive Banach space.     

Convex programs having some linear constraints

The problem of concern is the minimization of a convex function over a normed space (such as a Hilbert space) subject to the constraints that a number of other convex functions are not positive. As is well known, there is a dual maximization problem involving Lagrange multipliers. Some of the constraint functions are linear, and so the Uzawa, Stoer, and Witzgall form of the Slater constraint qualifications is appropriate. A short elementary proof is given that the infimum of the first problem is equal to the supremum of the second problem.     

Use of linear and convex ultrasound transducers for evaluation of retrosternal area in patients after cardiac surgery

The use of linear and convex transducers is usually reserved for ultrasound imaging of vessels, abdominal organs, and lungs. However, the use of these transducers may be extended for the evaluation of thoracic structures. We describe our experience with the use of linear and convex transducers in the evaluation of the retrosternal area, which is usually difficult to visualize by conventional echocardiography and so could provide improved diagnostic information in patients after cardiac surgery.     

Higher dimensional mappings for which the area formula holds

For each continuous mapping of 2 space into n space, n >/= 2, the Lebesgue area is given by the classical formula provided that the partial derivatives exist almost everywhere and belong to the class L(2). The analogous question for mappings of m space into n space, 2 < m 相似文献   

The inverse problem for convex bodies

A completely rigorous study of the high frequency asymptotics for the scattering matrix in the exterior of convex bodies is developed. These results are then applied to a variety of inverse problems for convex bodies.     

Inflammatory bowel disease: Traditional knowledge holds the seeds for the future

Despite the level of sophistication they have reached nowadays, the available tools for treatment of inflammatory bowel disease(IBD) can at best chronicize the disease but not cure it. Chances to make leap forward from this hold-back may include designs to reach personalized treatment strategies taking advantage of modern genome associated studies, and shift resources towards unfolding inciting pathogenetic steps rather than continuing to develop drugs that address down-stream phenomena. We have arbitrarily chosen to scrutinize a few projects that may make their way in 2015 and mark the history of IBD research. The list includes: the role of appendix as a regulating factor in pathogenesis of ulcerative colitis/proctitis; the reappraisal of(auto)immune phenomena in the era of microbiome; projects to treat IBD by stem cell infusion; recognition of the crucial pathogenetic role of gut microbiome, and attempts to modify it to treat enteric diseases, from clostridium difficile infection to IBD.