We have quantified the galaxy environments around a sample of 0.5 ≤ z ≤ 0.8 radioquiet quasars using the amplitude of the spatial galaxy–quasar correlation function, Bgq. The quasars exist in a wide variety of environments, some sources are located in clusters as rich as Abell class 1–2 clusters, whereas others exist in environments comparable to the field. We find that on average, the quasars ...

Locally-biased graph algorithms are algorithms that attempt to find local or small-scale structure in a typically large data graph. In some cases, this can be accomplished by adding some sort of locality constraint and calling a traditional graph algorithm; but more interesting are locally-biased graph algorithms that compute answers by running a procedure that does not even look at most of the...

A biased graph consists of a graph G together with a collection of distinguished cycles of G, called balanced, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on G arise from orienting G and then labelling the edges of G with elements of a group Γ. In this case, we may define a biased graph by declaring a cycle to be balanced...

It is a challenging topic to perform pattern reconstruction from a unit-width skeleton, which is obtained by a parallel thinning algorithm. The bias skeleton yielded by a fully-parallel thinning algorithm, which usually results from the so-called hidden deletable points, will result in the difficulty of pattern reconstruction. In order to make a fully-parallel thinning algorithm pattern reconst...

Let M be a frame matroid or a lifted-graphic matroid and let (G,B) be a biased graph representing M . Given a field F, a canonical F-representation of M particular to (G,B) is a matrix A arising from a gain function over the multiplicative or additive group of F that realizes (G,B). First, for a biased graph (G,B) that is properly unbalanced, loopless, and vertically 2-connected, we show that t...

A gain graph is a graph whose oriented edges are labelled invertibly from a group G, the gain group. A gain graph determines a biased graph and therefore has three natural matroids (as shown in Parts I–II): the bias matroid G has connected circuits; the complete lift matroid L0 and its restriction to the edge set, the lift matroid L, have circuits not necessarily connected. We investigate repre...

A multiary (polyadic, n-ary) quasigroup is an n-ary operation which is invertible with respect to each of its variables. A biased expansion of a graph is a kind of branched covering graph with additional structure similar to combinatorial homotopy of circles. A biased expansion of a circle with chords encodes a multiary quasigroup, the chords corresponding to factorizations, i.e., associative s...

Contrastive Divergence (CD) learning is frequently applied to Restricted Boltzmann Machines (RBMs), the building blocks of deep believe networks. It relies on biased approximations of the log-likelihood gradient. This bias can deteriorate the learning process. It was claimed that the signs of most components of the CD update are equal to the corresponding signs of the log-likelihood gradient. T...