The Lavrentiev gap phenomenon for harmonic maps into spheres holds on a dense set of zero degree boundary data

We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to \mathbb{S}^2$ which agrees with $\psi$ on the boundary of the unit ball; (2) this map $u$ has at least $N$ singular points in $\mathbb{B}^3$; (3) the Lavrentiev gap phenomenon holds for $\psi$, i.e., the infimum of the Dirichlet energies $E(w)$ of all smooth extensions $w\colon \mathbb{B}^3\to\mathbb{S}^2$ of $\psi$ is strictly larger than the Dirichlet energy $\int_{\mathbb{B}^3} |\nabla u|^2$ of the (irregular) minimizer $u$, is dense in the set of all smooth zero degree maps $\phi\colon \mathbb{S}^2\to\mathbb{S}^2$ endowed with the $W^{1,p}$-topology, where $1\le p < 2$. This result is sharp: it fails in the $W^{1,2}$ topology on the set of all smooth boundary data.Comment: 14 pages, 3 figures; minor typos etc. correcte

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is semi-conjugate to $z^d\colon S^1\to S^1$, where $d$ is equal to the degree of $F$. More precisely, for such an $F$ we construct a Peano curve $\gamma\colon S^1\to S^2$ (onto), such that $F\circ \gamma(z) = \gamma(z^d)$ (for all $z\in S^1$).Comment: 63 pages, 12 figure

WISP genes are members of the connective tissue growth factor family that are up-regulated in Wnt-1-transformed cells and aberrantly expressed in human colon tumors

Wnt family members are critical to many developmental processes, and components of the Wnt signaling pathway have been linked to tumorigenesis in familial and sporadic colon carcinomas. Here we report the identification of two genes, WISP-1 and WISP-2, that are up-regulated in the mouse mammary epithelial cell line C57MG transformed by Wnt-1, but not by Wnt-4. Together with a third related gene, WISP-3, these proteins define a subfamily of the connective tissue growth factor family. Two distinct systems demonstrated WISP induction to be associated with the expression of Wnt-1. These included (i) C57MG cells infected with a Wnt-1 retroviral vector or expressing Wnt-1 under the control of a tetracyline repressible promoter, and (ii) Wnt-1 transgenic mice. The WISP-1 gene was localized to human chromosome 8q24.1-8q24.3. WISP-1 genomic DNA was amplified in colon cancer cell lines and in human colon tumors and its RNA overexpressed (2- to >30-fold) in 84% of the tumors examined compared with patient-matched normal mucosa. WISP-3 mapped to chromosome 6q22-6q23 and also was overexpressed (4- to >40-fold) in 63% of the colon tumors analyzed. In contrast, WISP-2 mapped to human chromosome 20q12-20q13 and its DNA was amplified, but RNA expression was reduced (2- to >30-fold) in 79% of the tumors. These results suggest that the WISP genes may be downstream of Wnt-1 signaling and that aberrant levels of WISP expression in colon cancer may play a role in colon tumorigenesis

Multi-peak solutions for magnetic NLS equations without non--degeneracy conditions

In the work we consider the magnetic NLS equation (\frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u = 0 \quad {in} \R^N where $N \geq 3$, $A \colon \R^N \to \R^N$ is a magnetic potential, possibly unbounded, $V \colon \R^N \to \R$ is a multi-well electric potential, which can vanish somewhere, $f$ is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u\colon \R^N \to \C, under conditions on the nonlinearity which are nearly optimal.Comment: Important modification in the last part of the pape

Induction of Colonic Aberrant Crypts in Mice by Feeding Apparent N-Nitroso Compounds Derived From Hot Dogs

Nitrite-preserved meats (e.g., hot dogs) may help cause colon cancer because they contain N-nitroso compounds. We tested whether purified hot-dog-derived total apparent N-nitroso compounds (ANC) could induce colonic aberrant crypts, which are putative precursors of colon cancer. We purified ANC precursors in hot dogs and nitrosated them to produce ANC. In preliminary tests, CF1 mice received 1 or 3 i.p. injections of 5mg azoxymethane (AOM)/kg. In Experiments 1 and 2, female A/J mice received ANC in diet. In Experiment 1, ANC dose initially dropped sharply because the ANC precursors had mostly decomposed but, later in Experiment 1 and throughout Experiment 2, ANC remained at 85 nmol/g diet. Mice were killed after 8 (AOM tests) or 17–34 (ANC tests) wk.Median numbers of aberrant crypts in the distal 2 cm of the colon for 1 and 3 AOMinjections, CF1 controls, ANC (Experiment 1), ANC (Experiment 2),and untreated A/J mice were 31, 74, 12, 20, 12, and 5–6, with P < 0.01 for both ANC tests. Experiment 2 showed somewhat increased numbers of colonic mucin-depleted foci in the ANC-treated group. We conclude that hot-dog-derived ANC induced significant numbers of aberrant crypts in the mouse colon

Fixed points of n-valued maps on surfaces and the Wecken property -- a configuration space approach

In this paper, we explore the fixed point theory of $n$-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp.\ the $2$-sphere ${\mathbb S}^{2}$) has the Wecken property for $n$-valued maps for all $n\in {\mathbb N}$ (resp.\ all $n\geq 3$). In the case $n=2$ and ${\mathbb S}^{2}$, we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split $n$-valued map $\phi\colon\thinspace X \multimap X$ of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering $q\colon\thinspace \widehat{X} \to X$ with a subset of the coordinate maps of a lift of the $n$-valued split map $\phi\circ q\colon\thinspace \widehat{X} \multimap X$.Comment: To appear in Science China Mat

Restrictions of Brownian motion

Let $\{ B(t) \colon 0\leq t\leq 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $\alpha>\frac12$ and $1\leq \beta \leq 2$. We prove that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac12$ and $B\colon A\to\mathbb{R}$ is $\alpha$-H\"older continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac{\beta}{2}$ and $B\colon A\to\mathbb{R}$ has finite $\beta$-variation. The zero set of $B$ and a deterministic construction witness that the above theorems give the optimal dimensions.Comment: 6 page

Diabetes alone should not be a reason for withholding adjuvant chemotherapy for stage III colon cancer

Background: With increasing prevalence of diabetes mellitus and colon cancer, the number of patients suffering from both diseases is growing, and physicians are being faced with complicated treatment decisions. Objective: To investigate the association between diabetes and treatment/course of stage III colon cancer and the association between colon cancer and course of diabetes. Materials and Methods: Additional information was collected from the medical records of all patients with both stage III colon cancer and diabetes (n=201) and a random sample of stage III colon cancer patients without diabetes (n=206) in the area of the population-based Eindhoven Cancer Registry (1998–2007). Results: Colon cancer patients without diabetes were more likely to receive adjuvant chemotherapy compared with diabetic colon cancer patients (OR 1.8; 95% CI 1.2–2.7). After adjustment for age, this difference was borderline significant (OR 1.6; 95% CI 1.0–2.6). Diabetic patients did not have: significantly more side-effects from surgery or adjuvant chemotherapy; more recurrence from colon cancer; significantly shorter time interval until recurrence; or a poorer disease-free survival or overall survival. Age and withholding of adjuvant chemotherapy were most predictive of all-cause mortality. After colon cancer diagnosis, the dose of antiglycaemic medications was increased in 22% of diabetic patients, resulting in significantly lower glycaemic indexes than before colon cancer diagnosis. Conclusions: Since diabetic patients did not have more side-effects of adjuvant chemotherapy, and adjuvant chemotherapy had a positive effect on survival for both patients with and without diabetes, diabetes alone should not be a reason for withholding adjuvant chemotherapy.Journal of Comorbidity 2011;1(1):19–2