Lower central series and free resolutions of hyperplane arrangements

If $M$ is the complement of a hyperplane arrangement, and A=H^*(M,\k) is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the Betti numbers, b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i, of the linear strand in a (minimal) free resolution of \k over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j, of a (minimal) resolution of $A$ over the exterior algebra $E$. From this analysis, we recover a formula of Falk for $\phi_3$, and obtain a new formula for $\phi_4$. The exact sequence of low degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra $A$ is Koszul iff the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b'_{i,i+1}$, of the linear strand of the free resolution of $A$ over $E$; if the lower bound is attained for $i = 2$, then it is attained for all $i \ge 2$. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of $A$ are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid sub-arrangements), we show that $b'_{i,i+1}$ is determined by the number of triangles and $K_4$ subgraphs in the graph.Comment: 25 pages, to appear in Trans. Amer. Math. So

Chen ranks and resonance

The Chen groups of a group $G$ are the lower central series quotients of the maximal metabelian quotient of $G$. Under certain conditions, we relate the ranks of the Chen groups to the first resonance variety of $G$, a jump locus for the cohomology of $G$. In the case where $G$ is the fundamental group of the complement of a complex hyperplane arrangement, our results positively resolve Suciu's Chen ranks conjecture. We obtain explicit formulas for the Chen ranks of a number of groups of broad interest, including pure Artin groups associated to Coxeter groups, and the group of basis-conjugating automorphisms of a finitely generated free group.Comment: final version, to appear in Advances in Mathematic

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a module over the exterior algebra E on \A: \theta_k(G) = \dim_\k Tor^E_{k-1}(A,\k)_k, where \k is a field of characteristic 0, and k\ge 2. The Chen ranks conjecture asserts that, for k sufficiently large, \theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}, where h_r is the number of r-dimensional components of the projective resonance variety R^1(\A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R^1(\A) and a localization argument, we establish the conjectured lower bound for the Chen ranks of an arbitrary arrangement \A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R^1(\A), such that \theta_k(G) = P(k), for k sufficiently large.Comment: 21 pages; final versio

Fat Points, Inverse Systems, and Piecewise Polynomial Functions

AbstractWe explore the connection between ideals of fat points (which correspond to subschemes of Pnobtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on ad-dimensional simplicial complex Δ embedded inRd. Using the inverse system approach introduced by Macaulay [11], we give a complete characterization of the free resolutions possible for ideals ink[x,y] generated by powers of homogeneous linear forms (we allow the powers to differ). We show how ideals generated by powers of homogeneous linear forms are related to the question of determining, for some fixed Δ, the dimension of the vector space of splines on Δ of degree less than or equal tok. We use this relationship and the results above to derive a formula which gives the number of planar (mixed) splines in sufficiently high degree

Importance of Cement Market Characteristics to the Industrial Geologist

Author Institution: Mineral Economics Department, College of Mineral Industries, The Pennsylvania State University, University Park, PennsylvaniaDecisions to develop deposits of calcareous materials for cement production are based largely on anticipated markets for heavy construction which utilizes concrete. The geologist should understand the fundamental factors that underlie such management decisions, so that his field work and reports will more effectively relate to the information requirements of managers in the cement industry. The paper examines the relationships that exist between exploration and development of calcareous raw materials for cement manufacture and changing factors in the marketing of cement. Continued overcapacity, improved technology, changing distribution patterns,ease of entry, economics of scale, and the nature of product demand are some of thecharacteristics of the cement industry that are basic to decisions regarding explorationprograms and property development

Possible Magnetic Chirality in Optically Chiral Magnet [Cr(CN)6_6][Mn(SS)-pnH(H2_2O)](H2_2O) Probed by Muon Spin Rotation and Relaxation

Local magnetic fields in a molecule-based optically chiral magnet [Cr(CN)$_6$][Mn($S$)-pnH(H$_2$O)](H$_2$O) (GN-S) and its enantiomer (GN-R) are studied by means of muon spin rotation and relaxation (muSR). Detailed analysis of muon precession signals under zero field observed below T_c supports the average magnetic structure suggested by neutron powder diffraction. Moreover, comparison of muSR spectra between GN-S and GN-R suggests that they are a pair of complete optical isomers in terms of both crystallographic and magnetic structure. Possibility of magnetic chirality in such a pair is discussed.Comment: 5 pages, 5 figures, submitted to J. Phys. Soc. Jp

Muon-spin-rotation measurements of the penetration depth in Li_2Pd_3B

Measurements of the magnetic field penetration depth $\lambda$ in the ternary boride superconductor Li$_2$Pd$_3$B ($T_c\simeq7.3$ K) have been carried out by means of muon-spin rotation ($\mu$SR). The absolute values of $\lambda$, the Ginzburg-Landau parameter $\kappa$, and the first $H_{c1}$ and the second $H_{c2}$ critical fields at T=0 obtained from $\mu$SR were found to be $\lambda(0)=252(2)$ nm, $\kappa(0)=27(1)$, $\mu_0H_{c1}(0)=9.5(1)$ mT, and $\mu_0H_{c2}(0)=3.66(8)$ T, respectively. The zero-temperature value of the superconducting gap $\Delta_0=$1.31(3) meV was found, corresponding to the ratio $2\Delta_0/k_BT_c=4.0(1)$. At low temperatures $\lambda(T)$ saturates and becomes constant below $T\simeq 0.2T_c$, in agreement with what is expected for s-wave BCS superconductors. Our results suggest that Li$_2$Pd$_3$B is a s-wave BCS superconductor with the only one isotropic energy gap.Comment: 6 pages, 7 figure

Pressure Induced Static Magnetic Order in Superconducting FeSe_1-x

We report on a detailed investigation of the electronic phase diagram of FeSe_1-x under pressures up to 1.4GPa by means of AC magnetization and muon-spin rotation. At a pressure \simeq0.8GPa the non-magnetic and superconducting FeSe_1-x enters a region where long range static magnetic order is realized above T_c and bulk superconductivity coexists and competes on short length scales with the magnetic order below T_c. For even higher pressures an enhancement of both the magnetic and the superconducting transition temperatures as well as of the corresponding order parameters is observed. These exceptional properties make FeSe1-x to be one of the most interesting superconducting systems investigated extensively at present.Comment: 5 pages, 3 figure

Food Habits of Deer in the Black Hills

This study was conducted in two parts for Master of Science theses by the senior authors of Parts I and II. The study in the Northem Black Hills (Part I) was completed in 1968. Principal and preferred foods were determined for the winter and summer and a pasture study was conducted to measure production and utilization of foods in a typical aspen stand during the summer months. The study in the Southern Black Hills (Part II) was made in 1968 and 1969. Objectives were to determine the principal plants used by mule and white-tailed deer in fall, winter and summer. The utility of the point-analysis technique for measuring rumen contents was evaluated and the technique was applied to rumen contents examined. The studies were supported by the South Dakota Department of Game, Fish and Parks under Federal Aid Project W-75-R through the South Dakota Cooperative Wildlife Research Unit (South Dakota State University, the Bureau of Sport Fisheries and Wildlife, the South Dakota Department of Game, Fish and Parks and the Wildlife Management Institute, cooperating). Special acknowledgement is extended to the personnel of the Wildlife Habitat Project, Rocky Mountain Forest and Range Experiment Station Rapid City, for their assistance in both studies. Dr. Donald Dietz, Project Leader, and Harold E. Messner, Range Technician, were particularly helpful in the development of techniques for analysis of rumen content and the equipment used for the point-analysis method described in Part II. The assistance of William Hepworth, Director of Technical Research, Wyoming Game and Fish Department, in securing the two deer used in the pasture study described in Part I is gratefully acknowledged