The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

Ductal carcinoma in situ (DCIS) of the breast represents a heterogeneous group of neoplastic lesions in the breast ducts. The goal for management of DCIS is to prevent the development of invasive breast cancer. This manuscript focuses on the NCCN Guidelines Panel recommendations for the workup, primary treatment, risk reduction strategies, and surveillance specific to DCIS.

https://www.unitypoint.org/desmoines/filesimages/Cancer%20Education/Oncology%20Nursing%20Conference/2015/2014%20NCCN%20Clin%20Pract%20Guidelines%20Male%20Sex%20Dysf.pdf

Clinical practice guidelines in oncology

Exploring and identifying structure is even more important for multivariate data than univariate data, given the difficulties in graphically presenting multivariate data and the comparative lack of parametric models to represent it. Unfortunately, such exploration is also inherently more difficult.

Multivariate Density Estimation

Studies in Advanced Mathematics

The NCCN Guidelines for Kidney Cancer provide multidisciplinary recommendations for the clinical management of patients with clear cell and non-clear cell renal cell carcinoma, and are intended to assist with clinical decision-making. These NCCN Guidelines Insights summarize the NCCN Kidney Cancer Panel discussions for the 2020 update to the guidelines regarding initial management and first-line systemic therapy options for patients with advanced clear cell renal cell carcinoma.

/pdf/featured-updates-to-the-nccn-guidelines-e6q3nzm0p4.pdf

Featured Updates to the NCCN Guidelines

The major prognostic indicator in patients with breast cancer is the presence of metastases in axillary lymph nodes. The authors developed a mathematical model, based on 1446 complete axillary dissections performed in Milan between 1983 and 1986, and determined the following: (1) the sample size from Level I necessary for a 90% certainty degree of N0 axillary status; (2) the probability of residual tumor in the axilla after axillary sampling from Level I; and (3) the maximum number of involved axillary nodes in Levels I, II, and III to be expected (90% certainty) after sampling from Level I. Thus, this model permitted the determination of the cutoff level for a true N0 axillary status when only a few nodes are sampled from Level I. The cutoff level for a T1 primary tumor is ten axillary nodes removed and found uninvolved. Also, this model provides guidance in managing possible residual tumor after an incomplete axillary dissection. This information is important in indicating adjuvant axillary radiation therapy and chemotherapy or hormone therapy.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/1097-0142%2819920515%2969%3A10%3C2496%3A%3AAID-CNCR2820691018%3E3.0.CO%3B2-T

A mathematical model of axillary lymph node involvement based on 1446 complete axillary dissections in patients with breast carcinoma.

A fast method for solving the heat equation by layer potentials

A multilevel transform is introduced to represent discretizations of integral operators from potential theory by nearly sparse matrices. The new feature presented here is to construct the basis in a hierarchical decomposition of the three-space and not, as in previous approaches, in a parameter space of the boundary manifold. This construction leads to sparse representations of the operator even for geometrically complicated, multiply connected domains. We will demonstrate that the numerical cost to apply a vector to the operator using the nonstandard form is essentially equal to performing the same operation with the fast multipole method. With a second compression scheme the multiscale approach can be further optimized. The diagonal blocks of the transformed matrix can be used as an inexpensive preconditioner which is empirically shown to reduce the condition number of discretizations of the single layer operator so as to be independent of mesh size.

Multiscale Bases for the Sparse Representation of Boundary Integral Operators on Complex Geometry

The present paper is concerned with the numerical solution of a shape identification problem for the heat equation. The goal is to determine of the shape of a void or an inclusion of zero temperature from measurements of the temperature and the heat flux at the exterior boundary. This nonlinear and ill-posed shape identification problem is reformulated in terms of three different shape optimization problems: (a) minimization of a least-squares energy variational functional, (b) tracking of the Dirichlet data, and (c) tracking of the Neumann data. The states and their adjoint equations are expressed as parabolic boundary integral equations and solved using a Nystrom discretization and a space-time fast multipole method for the rapid evaluation of thermal potentials. Special quadrature rules are derived to handle singularities of the kernel and the solution. Numerical experiments are carried out to demonstrate and compare the different formulations.

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

The many levels of metal used in aggressive deep submicron process technologies has made fast and accurate capacitance extraction of complicated 3D geometries of conductors essential, and many novel approaches have been recently developed. In this paper we present an accelerated boundary-element method, like the well-known FASTCAP program, but instead of using an adaptive fast multipole algorithm we use a numerically generated multiscale basis for constructing a sparse representation of the dense boundary-element matrix. Results are presented to demonstrate that the multiscale method can be applied to complicated geometries, generates a sparser boundary-element matrix than the adaptive fast multipole method and provides an inexpensive but effective preconditioner. Examples are used to show that the better sparsification and the effective preconditioner yield a method that can be 25 times faster than FASTCAP while still maintain accuracy in the smallest coupling capacitances.

/pdf/a-multiscale-method-for-fast-capacitance-extraction-38bme1bwu0.pdf

Johannes Tausch

Papers

A mathematical model of axillary lymph node involvement based on 1446 complete axillary dissections in patients with breast carcinoma.

A fast method for solving the heat equation by layer potentials

Multiscale Bases for the Sparse Representation of Boundary Integral Operators on Complex Geometry

On the Numerical Solution of a Shape Optimization Problem for the Heat Equation

A multiscale method for fast capacitance extraction