Selected Open Problems in Discrete Geometry and Optimization

A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers

Fluorescence Microscopy Visualization of Contacts Between Objects

The area of contact between two objects was detected by using the strong enhancement of the fluorescence of rigidochromic probe molecules attached to one of the surfaces. Confinement of the molecules suppresses nonradiative decay and turns on the fluorescence. The approach is demonstrated by imaging of the contact area of a plastic sphere in contact with a flat glass surface. Our results agree excellently with the prediction of Hertz's classical theory based on elastic deformation

Fluorescence Microscopy Visualization of Contacts Between Objects

The area of contact between two objects was detected by using the strong enhancement of the fluorescence of rigidochromic probe molecules attached to one of the surfaces. Confinement of the molecules suppresses nonradiative decay and turns on the fluorescence. The approach is demonstrated by imaging of the contact area of a plastic sphere in contact with a flat glass surface. Our results agree excellently with the prediction of Hertz's classical theory based on elastic deformation

Fluorescence Microscopy Visualization of Contacts Between Objects

The area of contact between two objects was detected by using the strong enhancement of the fluorescence of rigidochromic probe molecules attached to one of the surfaces. Confinement of the molecules suppresses nonradiative decay and turns on the fluorescence. The approach is demonstrated by imaging of the contact area of a plastic sphere in contact with a flat glass surface. Our results agree excellently with the prediction of Hertz's classical theory based on elastic deformation

Chemokine receptors in advanced breast cancer: differential expression in metastatic disease sites with diagnostic and therapeutic implications

Background: We investigated the expression of CXCR4, CCR7, estrogen receptor (ER), progesterone receptor (PR) and HER2-neu in human metastatic breast cancers to determine whether these biological biomarkers were preferentially expressed in any organ-specific metastases. Materials and methods: CXCR4, CCR7, ER, PR and HER2-neu expression levels were evaluated by immunohistochemical staining using paraffin-embedded tissue sections of metastatic breast cancers (n = 41) obtained by either diagnostic biopsy or surgical resection. Results: The metastatic sites included the following: bone (n = 15), brain (n = 14), lung (n = 6), liver (n = 2), and omental metastases (n = 2). CXCR4 was expressed in 41% of cases, CCR7 expression was demonstrated in 10%, and HER2-neu overexpression was present in 27%. CXCR4 was more likely to be expressed in bone metastases than visceral metastases (67% versus 26%, P = 0.020). Visceral sites demonstrated a lower rate of CXCR4 positivity (33% and 23%, respectively, for lung and brain metastases). Similarly, CCR7 was more likely to be found in bone metastases than visceral sites (27% versus 0%, P = 0.037). Conclusions: These results indicate that CXCR4 can contribute to the homing of breast cancer cells to the bone. This finding might have important clinical implications since patients with metastatic bone disease may achieve the highest benefit from a CXCR4-targeted therapy