Representing Symmetric Rank Two Updates

Various quasi-Newton methods periodically add a symmetric "correction" matrix of rank at most 2 to a matrix approximating some quantity A of interest (such as the Hessian of an objective function). In this paper we examine several ways to express a symmetric rank 2 matrix [delta] as the sum of rank 1 matrices. We show that it is easy to compute rank 1 matrices [delta1] and [delta2] such that [delta] = [delta1] + [delta2] and [the norm of delta1]+ [the norm of delta2] is minimized, where ||.|| is any inner product norm. Such a representation recommends itself for use in those computer programs that maintain A explicitly, since it should reduce cancellation errors and/or improve efficiency over other representations. In the common case where [delta] is indefinite, a choice of the form [delta1] = [delta2 to the power of T] = [xy to the power of T] appears best. This case occurs for rank 2 quasi- Newton updates [delta] exactly when [delta] may be obtained by symmetrizing some rank 1 update; such popular updates as the DFP, BFGS, PSB, and Davidon's new optimally conditioned update fall into this category.

On Modifying Singular Values to Solve Possible Singular Systems of Non-Linear Equations

We show that if a certain nondegeneracy assumption holds, it is possible to guarantee the existence of a solution to a system of nonlinear equations f(x) = 0 whose Jacobian matrix J(x) exists but maybe singular. The main idea is to modify small singular values of J(x) in such away that the modified Jacobian matrix J^(x) has a continuous pseudoinverse J^+(x)and that a solution x* of f(x) = 0 may be found by determining an asymptote of the solution to the initial value problem x(0) = x[sub0}, x’(t) = -J^+(x)f(x). We briefly discuss practical (algorithmic) implications of this result. Although the nondegeneracy assumption may fail for many systems of interest (indeed, if the assumption holds and J(x*) is non-singular, then x is unique), algorithms using(x) may enjoy a larger region of convergence than those that require(an approximation to) J[to the -1 power[(x).

Some Convergence Properties of Broyden's Method

In 1965 Broyden introduced a family of algorithms called(rank-one) quasi—New-ton methods for iteratively solving systems of nonlinear equations. We show that when any member of this family is applied to an n x n nonsingular system of linear equations and direct-prediction steps are taken every second iteration, then the solution is found in at most 2n steps. Specializing to the particular family member known as Broyden’s (good) method, we use this result to show that Broyden's method enjoys local 2n-step Q-quadratic convergence on nonlinear problems.

Solving Systems of Non-Linear Equations by Broyden's Method with Projected Updates

We introduce a modification of Broyden's method for finding a zero of n nonlinear equations in n unknowns when analytic derivatives are not available. The method retains the local Q-superlinear convergence of Broyden's method and has the additional property that if any or all of the equations are linear, it locates a zero of these equations in n+1 or fewer iterations. Limited computational experience suggests that our modification often improves upon Eroyden's method.

The expansion of thymopoiesis in neonatal mice is dependent on expression of high mobility group a 2 protein (Hmga2).

Cell number in the mouse thymus increases steadily during the first two weeks after birth. It then plateaus and begins to decline by seven weeks after birth. The factors governing these dramatic changes in cell production are not well understood. The data herein correlate levels of High mobility group A 2 protein (Hmga2) expression with these temporal changes in thymopoiesis. Hmga2 is expressed at high levels in murine fetal and neonatal early T cell progenitors (ETP), which are the most immature intrathymic precursors, and becomes almost undetectable in these progenitors after 5 weeks of age. Hmga2 expression is critical for the initial, exponential expansion of thymopoiesis, as Hmga2 deficient mice have a deficit of ETPs within days after birth, and total thymocyte number is repressed compared to wild type littermates. Finally, our data raise the possibility that similar events occur in humans, because Hmga2 expression is high in human fetal thymic progenitors and falls in these cells during early infancy

This is the Piece that Everyone Here Has Come to Experience: The Challenges to Copyright of John Cage’s 4’33”

Framed within the broader context of law's engagement with modernism, this paper offers an argument in defence of copyright protection of John Cage's 4′33″ as a ‘musical work’ under the Copyright, Designs and Patents Act 1988. This argument approaches the issues involved analytically and contextually. In doing so, it draws on both legal and non-legal sources. Throughout the paper, the underlying question remains as to whether Cage's 4′33″ really is – or is not – a challenge to law (and to music)

Long non-coding RNA profiling of human lymphoid progenitor cells reveals transcriptional divergence of B cell and T cell lineages.

To elucidate the transcriptional 'landscape' that regulates human lymphoid commitment during postnatal life, we used RNA sequencing to assemble the long non-coding transcriptome across human bone marrow and thymic progenitor cells spanning the earliest stages of B lymphoid and T lymphoid specification. Over 3,000 genes encoding previously unknown long non-coding RNAs (lncRNAs) were revealed through the analysis of these rare populations. Lymphoid commitment was characterized by lncRNA expression patterns that were highly stage specific and were more lineage specific than those of protein-coding genes. Protein-coding genes co-expressed with neighboring lncRNA genes showed enrichment for ontologies related to lymphoid differentiation. The exquisite cell-type specificity of global lncRNA expression patterns independently revealed new developmental relationships among the earliest progenitor cells in the human bone marrow and thymus

Subjective Patriotism: A Cross-sectional Comparison of the Millennial, Generation X, Baby Boom, and Silent Generation Birth Cohorts

With the power to unite or divide a nation, patriotism is an integral component of the many attitudes, beliefs, and behaviors which collectively influence the nature and direction of society. In its extreme forms, patriotism can lead to systematic social disorganization, foster intolerance and fear, or give rise to fascist or neo-fascist movements. Given the diverse and potentially problematic roles patriotism can play, identifying determinants for patriotism represents an important line of inquiry. Although several studies have explored the origins of patriotism in society, prior research has yet to examine the influence of birth cohort effects on patriotic self-identification the United States. This study examines generational birth cohort effects on a subjective measure of American patriotism. Data for this research were collected from the Pew Research Center 2014 Political Survey. Results from our analysis suggest that younger cohorts are significantly less patriotic than preceding generations. Cohort effects on patriotism were significant with and without controls. To our knowledge, this is the first study to identify a systematic link between birth cohort and patriotism within the context of a multivariate examination. We also found evidence of a significant interaction between religiosity and birth cohort. Specifically, the positive effect of religiosity on patriotism is considerably more pronounced among Millennials than Baby Boomers. In addition to identifying a new source of variation in American patriotism, this study demonstrates the continued relevance and utility of birth cohort research for identifying and describing broad patterns of variation in society

mTORC2 signaling drives the development and progression of pancreatic cancer

mTOR signaling controls several critical cellular functions and is deregulated in many cancers, including pancreatic cancer. To date, most efforts have focused on inhibiting the mTORC1 complex. However, clinical trials of mTORC1 inhibitors in pancreatic cancer have failed, raising questions about this therapeutic approach. We employed a genetic approach to delete the obligate mTORC2 subunit Rictor and identified the critical times during which tumorigenesis requires mTORC2 signaling. Rictor deletion resulted in profoundly delayed tumorigenesis. Whereas previous studies showed most pancreatic tumors were insensitive to rapamycin, treatment with a dual mTORC1/2 inhibitor strongly suppressed tumorigenesis. In late-stage tumor-bearing mice, combined mTORC1/2 and PI3K inhibition significantly increased survival. Thus, targeting mTOR may be a potential therapeutic strategy in pancreatic cancer