Testing Linear-Invariant Non-Linear Properties

We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for "triangle freeness": a function f:\cube^{n}\to\cube satisfies this property if $f(x),f(y),f(x+y)$ do not all equal 1, for any pair x,y\in\cube^{n}. Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by $k$ points v_{1},...,v_{k}\in\cube^{k} and f:\cube^{n}\to\cube satisfies the property that if for all linear maps L:\cube^{k}\to\cube^{n} it is the case that $f(L(v_{1})),...,f(L(v_{k}))$ do not all equal 1. We show that this property is testable if the underlying matroid specified by $v_{1},...,v_{k}$ is a graphic matroid. This extends Green's result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of "1-complexity linear systems" of Green and Tao, and graphic matroids, to derive the results.Comment: This is the full version; conference version appeared in the proceedings of STACS 200

A study of patent thickets

Report analysing whether entry of UK enterprises into patenting in a technology area is affected by patent thickets in the technology area

Sublinear Algorithms for Approximating String Compressibility

We raise the question of approximating the compressibility of a string with respect to a fixed compression scheme, in sublinear time. We study this question in detail for two popular lossless compression schemes: run-length encoding (RLE) and a variant of Lempel-Ziv (LZ77), and present sublinear algorithms for approximating compressibility with respect to both schemes. We also give several lower bounds that show that our algorithms for both schemes cannot be improved significantly. Our investigation of LZ77 yields results whose interest goes beyond the initial questions we set out to study. In particular, we prove combinatorial structural lemmas that relate the compressibility of a string with respect to LZ77 to the number of distinct short substrings contained in it (its ℓth subword complexity , for small ℓ). In addition, we show that approximating the compressibility with respect to LZ77 is related to approximating the support size of a distribution.National Science Foundation (U.S.) (Award CCF-1065125)National Science Foundation (U.S.) (Award CCF-0728645)Marie Curie International Reintegration Grant PIRG03-GA-2008-231077Israel Science Foundation (Grant 1147/09)Israel Science Foundation (Grant 1675/09

The Leukemia-Associated Mllt10/Af10-Dot1l Are Tcf4/β-Catenin Coactivators Essential for Intestinal Homeostasis

The leukemia-associated Mllt10/Af10 and its partner the histone methyltransferase Dot1l are identified as Tcf4/β-catenin co-activators and shown to be essential for Wnt-driven endogenous gene expression, intestinal development and homeostasis

Role of Wnt canonical pathway in hematological malignancies

Wnt canonical signaling pathway plays a diverse role in embryonic development and maintenance of organs and tissues in adults. It has been observed that Wnt/β-catenin signaling pathway is involved in the pathogenesis of many carcinomas. Moreover, Wnt/β-catenin pathway has been revealed to be associated with angiogenesis. Wnt canonical pathway signaling has great potential as a therapeutic target. It has been disclosed that some hematological malignancies, such as chronic lymphocytic leukemia, mantle cell lymphoma, may occur partly due to the constitutive activation of Wnt canonical signaling pathway. This review will summarize the latest development in Wnt canonical signaling pathway and its roles in tumorigenesis and angiogenesis

Genetic Mechanisms in Apc-Mediated Mammary Tumorigenesis

Many components of Wnt/β-catenin signaling pathway also play critical roles in mammary tumor development, yet the role of the tumor suppressor gene APC (adenomatous polyposis coli) in breast oncongenesis is unclear. To better understand the role of Apc in mammary tumorigenesis, we introduced conditional Apc mutations specifically into two different mammary epithelial populations using K14-cre and WAP-cre transgenic mice that express Cre-recombinase in mammary progenitor cells and lactating luminal cells, respectively. Only the K14-cre–mediated Apc heterozygosity developed mammary adenocarcinomas demonstrating histological heterogeneity, suggesting the multilineage progenitor cell origin of these tumors. These tumors harbored truncation mutation in a defined region in the remaining wild-type allele of Apc that would retain some down-regulating activity of β-catenin signaling. Activating mutations at codons 12 and 61 of either H-Ras or K-Ras were also found in a subset of these tumors. Expression profiles of acinar-type mammary tumors from K14-cre; ApcCKO/+ mice showed luminal epithelial gene expression pattern, and clustering analysis demonstrated more correlation to MMTV-neu model than to MMTV-Wnt1. In contrast, neither WAP-cre–induced Apc heterozygous nor homozygous mutations resulted in predisposition to mammary tumorigenesis, although WAP-cre–mediated Apc deficiency resulted in severe squamous metaplasia of mammary glands. Collectively, our results suggest that not only the epithelial origin but also a certain Apc mutations are selected to achieve a specific level of β-catenin signaling optimal for mammary tumor development and explain partially the colon- but not mammary-specific tumor development in patients that carry germline mutations in APC