STRONG EQUIVALENCES OF APPROXIMATION NUMBERS AND TRACTABILITY OF WEIGHTED ANISOTROPIC SOBOLEV EMBEDDINGS

In this article,we study multivariate approximation defined over weighted anisotropic Sobolev spaces which depend on two sequences a={a j}j≥1 and b={b j}j≥1 of positive numbers.We obtain strong equivalences of the approximation numbers,and necessary and sufficient conditions on a,b to achieve various notions of tractability of the weighted anisotropic Sobolev embeddings.

An AW-HARIS Based Automated Segmentation of Human Liver Using CT Images

In the digestion of amino acids,carbohydrates,and lipids,as well as protein synthesis from the consumed food,the liver has many diverse responsibilities and functions that are to be performed.Liver disease may impact the hormonal and nutritional balance in the human body.The earlier diagnosis of such critical conditions may help to treat the patient effectively.A computationally efficient AW-HARIS algorithm is used in this paper to perform automated segmentation of CT scan images to identify abnormalities in the human liver.The proposed approach can recognize the abnormalities with better accuracy without training,unlike in supervisory procedures requiring considerable computational efforts for training.In the earlier stages,the CT images are pre-processed through an Adaptive Multiscale Data Condensation Kernel to normalize the underlying noise and enhance the image’s contrast for better segmentation.Then,the preliminary phase’s outcome is being fed as the input for the Anisotropic Weighted—Heuristic Algorithm for Real-time Image Segmentation algorithm that uses texture-related information,which has resulted in precise outcome with acceptable computational latency when compared to that of its counterparts.It is observed that the proposed approach has outperformed in the majority of the cases with an accuracy of 78%.The smart diagnosis approach would help the medical staff accurately predict the abnormality and disease progression in earlier ailment stages.

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R^n×R^m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R^n× R^m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R^n× R^m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R^n× R^m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R^n× R^m) to L~φ(R^n× R^m)and from H~φ_A(R^n×R^m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R^n× R^m and are new even for classical product Orlicz-Hardy spaces.