Convex optimization for the planted k-disjoint-clique problem

We consider the $k$ -disjoint-clique problem. The input is an undirected graph $G$ in which the nodes represent data items, and edges indicate a similarity between the corresponding items. The problem is to find within the graph $k$ disjoint cliques that cover the maximum number of nodes of $G$ . This problem may be understood as a general way to pose the classical ‘clustering’ problem. In clustering, one is given data items and a distance function, and one wishes to partition the data into disjoint clusters of data items, such that the items in each cluster are close to each other. Our formulation additionally allows ‘noise’ nodes to be present in the input data that are not part of any of the cliques. The $k$ -disjoint-clique problem is NP-hard, but we show that a convex relaxation can solve it in polynomial time for input instances constructed in a certain way. The input instances for which our algorithm finds the optimal solution consist of $k$ disjoint large cliques (called ‘planted cliques’) that are then obscured by noise edges inserted either at random or by an adversary, as well as additional nodes not belonging to any of the $k$ planted cliques.     

Guaranteed clustering and biclustering via semidefinite programming

Identifying clusters of similar objects in data plays a significant role in a wide range of applications. As a model problem for clustering, we consider the densest \(k\) -disjoint-clique problem, whose goal is to identify the collection of \(k\) disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques. In this paper, we establish conditions ensuring exact recovery of the densest \(k\) cliques of a given graph from the optimal solution of a particular semidefinite program. In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of \(k\) large, distinct clusters and a smaller number of outliers. This approach also yields a semidefinite relaxation with similar recovery guarantees for the biclustering problem. Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels. This problem may be posed as that of partitioning the nodes of a weighted bipartite complete graph such that the sum of the densities of the resulting bipartite complete subgraphs is maximized. As in our analysis of the densest \(k\) -disjoint-clique problem, we show that the correct partition of the objects and features can be recovered from the optimal solution of a semidefinite program in the case that the given data consists of several disjoint sets of objects exhibiting similar features. Empirical evidence from numerical experiments supporting these theoretical guarantees is also provided.     

Resonance ionization mass spectroscopy for trace analysis of neptunium

Resonance ionization mass spectroscopic (RIMS) measurements for trace analysis and spectroscopy of ²³⁷Np, the ecologically most important isotope of neptunium, are described. The chemical procedure for the separation of neptunium from aqueous samples as well as the preparation of filaments for RIMS are outlined. Several two- and three-step excitation schemes have been investigated in order to find suitable conditions for the sensitive detection of ²³⁷Np. Using a three-step, three-color excitation and ionization scheme an overall detection efficiency of 3×10^–8 was obtained, resulting in a detection limit of 4×10⁸ atoms (160 fg) of ²³⁷Np. The hyperfine structure splittings of the levels under investigation, which influence the detection limit, were measured. A new method to determine the first ionization potential (IP) was applied to neptunium yielding a value of IP=6.2655(2) eV.     

Boundedness and Global Nonexistence for a Nonlinear Convective Parabolic-System

A two-component parabolic system with convective terms is studied. It is shown that if the forces on the right hand side are no worse than quadratic in the variables then the solution remains bounded for all time. It is also proved that if the forces are allowed to grow faster than quadratic then finite time blow-up is to be expected.     

An Empirical Investigation of the Adaptive-Filtering Learning Factor

Acceptance of the adaptive-filtering parameter-estimation process has been mixed. Much of the variability between studies may result from the particular learning factor selected and may be further affected by the type of standardization used. The author presents a modification of the Makridakis and Wheelwright adaptive-filtering algorithm. This modified algorithm provides superior performance and simplifies the selection of the learning factor when used with simple autoregressive processes.     

Measurement of the sign of the spectroscopic quadrupole moment for the 2(1)+ state in 70Se: no evidence for oblate shape

Using a method whereby molecular and atomic ions are independently selected, an isobarically pure beam of 70Se ions was postaccelerated to an energy of 206 MeV using REX-ISOLDE. Coulomb-excitation yields for states in the beam and target nuclei were deduced by recording deexcitation gamma rays in the highly segmented MINIBALL gamma-ray spectrometer in coincidence with scattered particles in a silicon detector. At these energies, the Coulomb-excitation yield for the first 2+ state is expected to be strongly sensitive to the sign of the spectroscopic quadrupole moment through the nuclear reorientation effect. Experimental evidence is presented here for a prolate shape for the first 2+ state in 70Se, reopening the question over whether there are, as reported earlier, deformed oblate shapes near to the ground state in the light selenium isotopes.     

0(gs)+ -->2(1)+ transition strengths in 106Sn and 108Sn

The reduced transition probabilities, B(E2; 0(gs)+ -->2(1)+), have been measured in the radioactive isotopes (108,106)Sn using subbarrier Coulomb excitation at the REX-ISOLDE facility at CERN. Deexcitation gamma rays were detected by the highly segmented MINIBALL Ge-detector array. The results, B(E2;0(gs)+ -->2(1)+)=0.222(19)e2b2 for 108Sn and B(E2; 0(gs)+-->2(1)+)=0.195(39)e2b2 for 106Sn were determined relative to a stable 58Ni target. The resulting B(E2) values are approximately 30% larger than shell-model predictions and deviate from the generalized seniority model. This experimental result may point towards a weakening of the N=Z=50 shell closure.