Sparse nonnegative solution of underdetermined linear equations by linear programming

Consider an underdetermined system of linear equations y = Ax with known y and d x n matrix A. We seek the nonnegative x with the fewest nonzeros satisfying y = Ax. In general, this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. We explain this by the theory of convex polytopes. Let a(j) denote the jth column of A, 1 < or = j < or = n, let a0 = 0 and P denote the convex hull of the a(j). We say the polytope P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. We also consider weak neighborliness, where the overwhelming majority of k-sets of a(j)s not containing 0 span a face of P. This implies that most nonnegative vectors x with k nonzeros are uniquely recoverable from y = Ax by linear programming. Numerous corollaries follow by invoking neighborliness results. For example, for most large n by 2n underdetermined systems having a solution with fewer nonzeros than roughly half the number of equations, the sparsest solution can be found by linear programming.     

Prolonged low-dose dexamethasone, in early gestation, has no long-term deleterious effect on normal ovine fetuses

Low-dose dexamethasone (D) treatment is used in pregnancies where the fetus is suspected to be at risk of congenital/virilizing adrenal hyperplasia. To study if this treatment had any immediate or long-term effects in normal fetuses, pregnant ewes were treated with D (20 microg/kg maternal body weight x d) or saline (S), from d 25-45 of gestation. Tissue was collected from fetuses killed at 45 d (S = 6; D = 8), 130 d (S = 8; D = 8), or lambs at 2 months of age (S = 6; D = 6) and mRNA levels measured using real-time PCR. D treatment reduced adrenal wt at 45 d (S, 12.2 +/- 0.7 mg; D, 6.3 +/- 0.4 mg) and significantly decreased adrenal mRNA for P(450scc). At 130 d, fetuses from the D treatment were growth retarded (S, 3.2 +/- 0.1 kg; D, 2.5 +/- 0.1 g), but the adrenals were appropriate for the body weight. mRNA levels of angiotensinogen, the AT(1) receptor and mineralocorticoid receptor (MR) and GR were similar in kidney and brain (hypothalamus, hippocampus, medulla oblongata) except for hippocampal expression of MR and GR, which was significantly decreased by D treatment. By 2 months, BW and hippocampal MR and GR mRNA levels were similar, and lambs were normotensive (S, 83 +/- 3 mm Hg; D, 78 +/- 3 mm Hg). Thus, there were no persistent, long-term effects of prolonged low-dose D treatment in normal ovine fetuses.     

Neighborliness of randomly projected simplices in high dimensions

Let A be a d x n matrix and T = T(n-1) be the standard simplex in Rn. Suppose that d and n are both large and comparable: d approximately deltan, delta in (0, 1). We count the faces of the projected simplex AT when the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rn. We derive rhoN(delta) > 0 with the property that, for any rho < rhoN(delta), with overwhelming probability for large d, the number of k-dimensional faces of P = AT is exactly the same as for T, for 0 < or = k < or = rhod. This implies that P is left floor rhod right floor-neighborly, and its skeleton Skel(left floor rhod right floor)(P) is combinatorially equivalent to Skel( left floor rhod right floor)(T). We also study a weaker notion of neighborliness where the numbers of k-dimensional faces f(k)(P) > or = f(k)(T)(1-epsilon). Vershik and Sporyshev previously showed existence of a threshold rhoVS(delta) > 0 at which phase transition occurs in k/d. We compute and display rhoVS and compare with rhoN. Corollaries are as follows. (1) The convex hull of n Gaussian samples in Rd, with n large and proportional to d, has the same k-skeleton as the (n-1) simplex, for k < rhoN (d/n)d(1 + oP(1)). (2) There is a "phase transition" in the ability of linear programming to find the sparsest nonnegative solution to systems of underdetermined linear equations. For most systems having a solution with fewer than rhoVS(d/n)d(1 + o(1)) nonzeros, linear programming will find that solution.     

Dynamic pattern evolution on scale-free networks

A general class of dynamic models on scale-free networks is studied by analytical methods and computer simulations. Each network consists of N vertices and is characterized by its degree distribution, P(k), which represents the probability that a randomly chosen vertex is connected to k nearest neighbors. Each vertex can attain two internal states described by binary variables or Ising-like spins that evolve in time according to local majority rules. Scale-free networks, for which the degree distribution has a power law tail P(k) approximately k(-gamma), are shown to exhibit qualitatively different dynamic behavior for gamma < 5/2 and gamma > 5/2, shedding light on the empirical observation that many real-world networks are scale-free with 2 < gamma < 5/2. For 2 < gamma < 5/2, strongly disordered patterns decay within a finite decay time even in the limit of infinite networks. For gamma > 5/2, on the other hand, this decay time diverges as ln(N) with the network size N. An analogous distinction is found for a variety of more complex models including Hopfield models for associative memory networks. In the latter case, the storage capacity is found, within mean field theory, to be independent of N in the limit of large N for gamma > 5/2 but to grow as N(alpha) with alpha = (5 - 2gamma)/(gamma - 1) for 2 < gamma < 5/2.     

Noninvasive determination of stiffness of the left ventricle by combined M-mode echo- and apexcardiography. An experimental method

In order to compare an invasive with a noninvasive index of left ventricular stiffness during late diastole, M-mode echocardiogram, left ventricular pressure (LVP) and apexcardiogram (QLAC) were recorded simultaneously during acute dog experiments. The slope of the log pressure (P)-log volume (V) relationship is expressed by the ratio VdP/PdV. This index has been proven to be valid for the evaluation of left ventricular stiffness during late diastole. Diameter (D) changes were assumed to be proportional to volume changes. Continuous data from the ascending part of the A wave (due to atrial contraction) and the corresponding diameter change were used to calculate the slope (k1) of the log LVP--log D relation. During the same period the slope (k2) of the log QLAC--log D relation was also calculated. A significant linear correlation was found between values obtained semi-invasively from LVP-D (k1) and noninvasively from QLAC-D (k2) data: n = 21, r = 0.93, p less than 0.001. This shows the usefulness of VdP/PdV derived noninvasively from QLAC and M-mode echo for the assessment of stiffness of the left ventricle during late diastole.     

Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(phi) of a two-dimensional modular Galois representation phi. We start with the p-adic Galois representation phi0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(phi0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(phi0)) from the proof of Wiles of the Shimura-Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let phi denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting phi0, and the characteristic power series of the Selmer group Sel(ad(phi)) is given by a p-adic L-function interpolating L(1, ad(phik)) for weight k + 2 specialization phik of phi. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(phi) [symbol, see text] nu-1), where nu is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.     

Unwinding of double-stranded DNA helix by dehydration.

Conformation changes of the double-stranded DNA helix in response to dehydration were investigated by monitoring, by agarose gel electrophoresis, the linking number of covalently closed circular DNA generated by ligation of linear DNA in the presence of different organic solvents or different temperatures. It was found that: (i) The DNA helix unwinds upon addition of certain organic solvents or elevation of temperature. (ii) The conformational change observed under the experimental conditions is a continuous process in response to the organic solvent concentration. (iii) The delta H of unwinding one linking of the DNA helix is constant at approximately 12.2 +/- 0.4 kcal/mol (1 kcal = 4.184 kJ); the corresponding delta S and d(delta S)/dn are 2nkR and 2kR, in which n is the relative equivalent linking number (referred to the state of delta S = 0 for unwinding) of the DNA, R is the gas constant, and k is equal to 1117/number of base pairs. The delta H, delta S, and d(delta S)/dn for unwinding i linkings are i X 12.2 kcal/mol, 2inkR, and 2ikR, respectively. (iv) d(delta S)/dn, like k, is inversely proportional to the number of base pairs in DNA. (v) Double-stranded DNAs of different chain lengths have average delta S = 35 cal/mol.K for unwinding one linking under the experimental conditions; this corresponds to 127 +/- 14 base pairs per "relative linking."     

Reactivations of Latent Viral Infections Are Associated with an Increased Thr389 p70S6k Phosphorylation in Peripheral Lymphocytes of Renal Transplant Recipients

Reactivations of BK polyoma virus (BKPyV) and human cytomegalovirus (HCMV) frequently cause life- and graft-threatening complications after renal transplantation. Both viruses are dependent on the mTOR pathway for replication. In this study we investigated the association of viral replication with mTOR activity in peripheral lymphocytes of renal transplant recipients. A flow-cytometry based assay for the measurement of Thr389 p70S6k phosphorylation, a surrogate marker of the mTOR pathway was established. Forty-eight adult renal transplant recipients were recruited to measure p70S6k activity in their peripheral blood mononuclear cells. This data set in conjunction with information concerning previous replication of BKPyV and HCMV was examined for correlations. Episodes of BKPyV replication were significantly associated with increased p70S6k phosphorylation in CD4⁺ T lymphocytes (p = 0.0002) and CD19⁺ B lymphocytes (p = 0.0073). HCMV infection of patients with a high-risk HCMV constellation of donor and recipient (D+/R−) was associated with increased p70S6k phosphorylation in CD19⁺ B lymphocytes (p = 0.0325). These associations were found to be independent of the trough levels of the immunosuppressive drugs. Conclusion: P70S6k phosphorylation in peripheral lymphocytes is associated with BKPyV reactivations and to a lesser extent with HCMV infections in renal transplant recipients.     

Diffusion of isolated DNA molecules: dependence on length and topology

The conformation and dynamics of circular polymers is a subject of considerable theoretical and experimental interest. DNA is an important example because it occurs naturally in different topological states, including linear, relaxed circular, and supercoiled circular forms. A fundamental question is how the diffusion coefficients of isolated polymers scale with molecular length and how they vary for different topologies. Here, diffusion coefficients D for relaxed circular, supercoiled, and linear DNA molecules of length L ranging from approximately 6 to 290 kbp were measured by tracking the Brownian motion of single molecules. A topology-independent scaling law D approximately L(-nu) was observed with nu(L) = 0.571 +/- 0.014, nu(C) = 0.589 +/- 0.018, and nu(S) = 0.571 +/- 0.057 for linear, relaxed circular, and supercoiled DNA, respectively, in good agreement with the scaling exponent of nu congruent with 0.588 predicted by renormalization group theory for polymers with significant excluded volume interactions. Our findings thus provide evidence in support of several theories that predict an effective diameter of DNA much greater than the Debye screening length. In addition, the measured ratio D(Circular)/D(Linear) = 1.32 +/- 0.014 was closer to the value of 1.45 predicted by using renormalization group theory than the value of 1.18 predicted by classical Kirkwood hydrodynamic theory and agreed well with a value of 1.31 predicted when incorporating a recently proposed expression for the radius of gyration of circular polymers into the Zimm model.     

Homologous pairing in genetic recombination: complexes of recA protein and DNA.

recA protein, which is essential for general genetic recombination in Escherichia coli, promotes the homologous pairing of single-stranded DNA with double-stranded DNA to form a D loop. The amount of recA protein required for the reaction was directly proportional to the amount of single stranded DNA and was unaffected by similar variations in the amount of double-stranded DNA. The ATP analog, adenosine 5'-O-(3-thiotriphosphate) (ATP gamma S), which was not rapidly hydrolyzed by recA protein, blocked the formation of D loops but promoted the formation of stable complexes of recA protein and single-stranded DNA. These complexes, in turn, bound homologous or heterologous double-stranded DNA and partially unwound it. Because ATP gamma S competitively inhibited the ATPase activity of recA protein (Km/Ki approximately 300), we infer that ATP gamma S binds at a site that overlaps the site for ATP and that the functional complexes formed in the presence of the analog probably represent partial steps in the overall reaction. If the complexes formed in the presence of ATP gamma S reflect natural intermediates in the formation of D loops, recA protein must promote homologous pairing either by moving juxtaposed single-stranded and double-stranded DNA relative to one another or by forming and dissociating complexes reiteratively until a homologous match occurs.     

多普勒超声估测右心室dp/dt max的研究

应用脉冲多普勒超声测定２５例先天性心脏病患儿的肺动脉血流频谱参数并与心导管测定的右心室ｄｐ／ｄｔｍａｘ进行对比分析。结果显示:肺动脉血流频谱参数（动能最大变化率,ｄｋ／ｄｔｍａｘ）与右心室ｄｐ／ｄｔｍａｘ呈良好相关（ｒ＝０．９２６,Ｐ＜０．０１）,直线回归方程为:ｄｐ／ｄｔｍａｘ＝０．５９ｄｋ／ｄｔｍａｘ＋１１．２（ｋＰａ／ｓ）;平均加速度ｍＡ及Ｖｐ２／ＡＴ与ｄｐ／ｄｔｍａｘ亦有较好的相关性。表明由多普勒超声获得的肺动脉血流频谱参数ｄｋ／ｄｔｍａｘ能较准确地估测右室ｄｐ／ｄｔｍａｘ,因而可作为评价心肌收缩功能的指标。     

Analysis of a break in chromosome 14 mapping to the region of the immunoglobulin heavy chain locus.

We have detected restriction fragment length polymorphisms associated with the immunoglobulin heavy chain C gamma genes. DNA from both parents of an individual having an unbalanced rearrangement of the long arm of chromosome 14, region q32 [Cox, D. W., Markovic, V. D. & Teshima, I. E. (1982) Nature (London) 297, 428-430], revealed distinctive patterns of BamHI fragments which hybridized with cloned probes from the C gamma 2-C gamma 4 gene cluster. The number of hybridizing fragments in both cases (five) equaled the number of known C gamma genes. Pedigree and densitometric analyses indicated that the proband did not have any maternal complement of C gamma gene-hybridizing fragments. Included on the deleted chromosomal segment was a C gamma gene having properties of the previously reported C gamma pseudogene. We also examined DNA from this family with a probe for the highly polymorphic locus D14S1, which recently was demonstrated to be tightly linked to the C gamma 1 gene locus [Balazs, I., Purrello, M., Rubinstein, P., Alhadeff, B. & Siniscalco, M. (1982) Proc. Natl. Acad. Sci. USA 79, 7395-7399]. EcoRI and EcoRI-BamHI fragments from both parents hybridized with a probe for this locus in DNA from the proband, indicating that, unlike the C gamma gene family, D14S1 was not deleted from the abnormal chromosome. Thus, the chromosomal breakpoint in the proband lies within region 14q32 between the two tightly linked markers, D14S1 and the C gamma 1 heavy chain gene locus. The D14S1 locus must lie proximal to the centromere relative to the C gamma gene family. The genetic variability detected with C gamma gene probes may prove useful for genetic analysis of structural rearrangements involving this region of chromosome 14.     

Results from a genome-wide search for predisposing genes in sarcoidosis.

Sarcoidosis is a systemic disease of granulomatous inflammation and unknown etiology. An inherited predisposition is involved, and many candidate susceptibility genes have been tested in association studies. We have applied the more general strategy of genome-wide microsatellite linkage analysis to identify chromosomal regions that contribute to the risk of sarcoidosis. On the basis of 225 microsatellite markers tested in 63 families with affected siblings (138 patients) and multipoint nonparametric linkage (NPL) analysis, we found the most prominent peak (six adjacent markers including D6S1666; NPL score = 2.99; p = 0.001) at the major histocompatibility complex (MHC). Six minor peaks (p < 0.05) were found on chromosomes 1 (D1S1665 ), 3 (D3S1766 ), 7 (D7S821 and D7S3070), 9 (D9S934), and the X chromosome (DXS6789). A subset of nine families with more than two affected siblings (30 patients) contributed little to the peak at the MHC (D6S1666; NPL score = 0.79; p = 0.21). Our results point to locus heterogeneity of susceptibility to sarcoidosis, with a major effect of the MHC.     

DNA rearrangements in human follicular lymphoma can involve the 5'' or the 3'' region of the bcl-2 gene.

In most human follicular lymphomas, the chromosome translocation t(14;18) occurs within two breakpoint clustering regions on chromosome 18, the major one at the 3' untranslated region of the bcl-2 gene and the minor one at 3' of the gene. Analysis of a panel of follicular lymphoma DNAs using probes for the first exon of the bcl-2 gene indicates that DNA rearrangements may also occur 5' to the involved bcl-2 gene. In this case the IgH locus and the bcl-2 gene are found in the order 3' C gamma S gamma/mu JH 5'::5' bcl-2 3' (where C = constant, S = switch, and JH = joining segment of the heavy chain locus), suggesting that an inversion also occurred during the translocation process. The coding regions of the bcl-2 gene, however, are left intact in all cases of follicular lymphoma studied to date.     

Fibrinogen Kyoto III: a congenital dysfibrinogen with a gamma aspartic acid-330 to tyrosine substitution manifesting impaired fibrin monomer polymerization

A congenital dysfibrinogen characterized by impaired fibrin monomer polymerization was found in an asymptomatic 50-year-old woman and her two sons. On sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) run according to the method of Laemmli, we noticed two gamma chain species in fibrinogen and its plasmic fragments D1 and D2, consisting of a normal species and an apparently lower molecular weight (mol wt) variant in respective fractions. However, in fragment D3 only a single gamma chain remnant was observed. By chromatofocusing of the plasmic-CaCl2 digests of the abnormal fibrinogen, we separately isolated the normal and abnormal D1 species, the latter having been eluted in a slightly higher pH range. As expected, the abnormal D1 species failed to interfere with thrombin clotting of normal fibrinogen and normal fibrin monomer polymerization, whereas the normal D1 species inhibited them markedly. By analyzing the lysyl endopeptidase digests of the isolated gamma chain, we identified a replacement of aspartic acid by tyrosine at position 330 of the mutant gamma chain. The point mutation from an aspartic acid (pK for the beta-carboxyl = 3.86) to a tyrosine (pK for the aromatic hydroxyl = 10.07) may have perturbed the folding gamma chain structure in the D domain of fibrinogen specifically required for polymerization.     

Genome-wide search for susceptibility genes to type 2 diabetes in West Africans: potential role of C-peptide

C-peptide is a substance that the pancreas releases into the circulation in equimolar amounts to insulin and has demonstrated important physiological effects which relate to the vascular field, in particular the microcirculation. For this analysis, we included 321 full and 36 half sibling pairs affected with type 2 diabetes (T2D) from West Africa. A genome-wide panel of 390 tri-nucleotide and tetra-nucleotide repeats with an average distance of 8.9 cM was performed on a total of 691 persons. Variance components based on multipoint linkage approach as implemented in SOLAR were performed for log C-peptide. Significant linkage evidences were observed on 10q23 at D10S2327 with a LOD score of 4.04 (nominal p-value=0.000008, empirical p-value=0.0004); and on 4p15 at D4S2632 with a LOD score of 3.48 (nominal p-value=0.000031, empirical p-value=0.0013). Other suggestive evidence of linkage were observed on 15q14 at D15S659 with a LOD score 2.41 (nominal p-value=0.000435, empirical p-value=0.0068), and on 18p11 near D18S976 with a LOD score 2.18 (nominal p-value=0.000771 and empirical p-value=0.0094). Interestingly, five positional candidate genes for diabetes and related complications are located in our linkage region (the pituitary adenylate cyclase activating polypeptide (PACAP in 18p11); the peroxisome proliferator-activated receptor gamma coactivator 1 (PPARGC1 in 4p15); PTEN, PPP1R5, and IDE located in 10q23. In conclusion, we identified four major genetic loci (10q23, 4p15, 15q14, and 18p11) influencing C-peptide concentration in West Africans with T2D.     

Higher-Order Curvature and Local Solvability of D(theta)

Let E and F be vector bundles and D:E --> F an operator of order k. We associate a sequence of invariants ((l))(D), l >/= 0 with D which generalize the concept of curvature in a natural way. In the case where D = D(theta) is the differential operator of a connection theta on a vector bundle E, ((1))(D(theta)) is the classical curvature. Furthermore, we find an interesting geometric interpretation for ((2))(D(theta)). Finally, given regularity assumptions, we find, with the aid of these invariants, necessary and sufficient conditions for local solvability of D(theta).     

A combinatorial model for the Macdonald polynomials

We introduce a polynomial C(mu)[Z; q, t], depending on a set of variables Z = z(1), z(2),..., a partition mu, and two extra parameters q, t. The definition of C(mu) involves a pair of statistics (maj(sigma, mu), inv(sigma, mu)) on words sigma of positive integers, and the coefficients of the z(i) are manifestly in N[q,t]. We conjecture that C(mu)[Z; q, t] is none other than the modified Macdonald polynomial H(mu)[Z; q, t]. We further introduce a general family of polynomials F(T)[Z; q, S], where T is an arbitrary set of squares in the first quadrant of the xy plane, and S is an arbitrary subset of T. The coefficients of the F(T)[Z; q, S] are in N[q], and C(mu)[Z; q, t] is a sum of certain F(T)[Z; q, S] times nonnegative powers of t. We prove F(T)[Z; q, S] is symmetric in the z(i) and satisfies other properties consistent with the conjecture. We also show how the coefficient of a monomial in F(T)[Z; q, S] can be expressed recursively. maple calculations indicate the F(T)[Z; q, S] are Schur-positive, and we present a combinatorial conjecture for their Schur coefficients when the set T is a partition with at most three columns.     

Human U1 RNA genes contain an unusually sensitive nuclease S1 cleavage site within the conserved 3'' flanking region.

We find that the cloned DNAs of human U1 small nuclear RNA genes contain two nuclease S1-sensitive sites, one about 1.8 kilobases downstream of the U1 RNA coding region and the other around 0.3 kilobase upstream. The downstream site is unusually sensitive to the nuclease, being cleaved in both linear and negatively supercoiled DNAs. The extent of cleavage at this site is enhanced at lower pH and reduced concentrations of NaCl; the effects of salt are more apparent on linear than supercoiled DNAs. The nuclease S1 sensitivity of this downstream site is dependent on the presence of the sequence (dC-dT)n X (dA-dG)n, where n = 15-25. (One gene with n = 5 is resistant to nuclease S1 cleavage in this region.) In contrast, the nuclease S1 site upstream of the coding region is cleaved only when the DNA is supercoiled. This site also has a homopyrimidine X homopurine bias in the DNA strands, but the sequence is less regular. In the course of these studies, we detected several discrepancies between our restriction maps of some U1 RNA genes and those published by others. Our maps demonstrate that all seven cloned human U1 RNA genes are very similar in sequence for as much as 2.3 kilobases downstream of the U1 RNA coding region.