Zero excess and minimal length in finite coxeter groups

Let \mathcal{W} be the set of strongly real elements of W, a Coxeter group. Then for $w \in \mathcal{W}$, $e(w)$, the excess of w, is defined by e(w) = \min min \{l(x)+l(y) - l(w)| w = xy; x^2 = y^2 =1}. When $W$ is finite we may also define E(w), the reflection excess of $w$. The main result established here is that if $W$ is finite and $X$ is a $W$-conjugacy class, then there exists $w \in X$ such that $w$ has minimal length in $X$ and $e(w) = 0 = E(w)$

Involution products in Coxeter groups

For W a Coxeter group, let = {w ∈ W | w = xy where x, y ∈ W and x 2 = 1 = y 2}. It is well known that if W is finite then W = . Suppose that w ∈ . Then the minimum value of ℓ(x) + ℓ(y) – ℓ(w), where x, y ∈ W with w = xy and x 2 = 1 = y 2, is called the excess of w (ℓ is the length function of W). The main result established here is that w is always W-conjugate to an element with excess equal to zero

Complex Structure and Stratigraphy of Lower Slices of the Taconic Allochthon Near Middle Granville, New York

The Precambrian (?) to medial Ordovician rocks of the Taconic Allochthon are characterized by argillaceous and arenaceous sediments with lesser associated carbonates, carbonate breccias, and cherts of predominantly deep-water aspect. These allochthonous rocks tectonically overlie an autochthonous to parautochthonous coeval sequence of dominantly shallow marine clastics and carbonates of the Champlain and Vermont Valley sequences. Facies, thickness, sedimentologic, and paleontologic considerations suggest that these coeval sequences represent a carbonate shelf continental rise pair of the east-facing early Paleozoic Atlantic-type margin of North America. This margin formed by the opening of an ocean in latest Precambrian time. The stratigraphy of the shelf suggests that it experienced a complex transgressive-regressive history which is recorded on the rise by marked changes in type of sediment and mode of sedimentation. This Atlantic-type margin was destroyed in the medial Ordovician by eastward subduction and consequent collision beneath the Ammonoosuc volcanic arc. This resulted in the progressive east to west stacking of the rise sequence and subsequent obduction onto the shelf. Obduction involved an exceedingly complex deformation history of folding and imbrication of the shelf, Allochthon and Grenville basement. The stratigraphy of the study area varies considerably across strike. Regions of different, though comparable stratigraphy occur in thrust bonded slices. In the west a stratigraphy closely similar to that defined by Jacobi (1977) is observed. All units, including Bomoseen, Truthville, Browns Pond, Mettawee, Hatch Hill-West Castleton, Poultney, Indian River, Mount Merino, and Pawlet are present. A central region with a similar stratigraphy is recognized, but characterized by less carbonate, thinner and commonly more fine-grained quartzites, which among other aspects suggests that it represents a somewhat more distal (easterly) facies. To the east, the sequence is Bullfrog Hollow Lithozone, Poultney, Indian River (?), Mount Merino (?), and Pawlet. The name Bullfrog Hollow Lithozone is introduced for the basal, apparently thick sequence of purple, green and gray slates and argillites, with associated minor thin quartzites. A thin gray slate with interbedded quartzite and black calcareous quartz wacke lies within the Bullfrog Hollow and is tentatively correlated with the Browns Pond. A new name is used because direct correlations with the Truthville and Mettawee slates of western regions was not possible and other names, such as Bull, St. Catherine, or Mettawee were considered inappropriate because of misuse, poor definition, or the inclusion of untis not observed in this area. Pawlet and Poultney are usually in stratigraphic contact, but locally Indian River and/or Mount Merino are also observed. The Poultney-Pawlet contact appears to be a disconformity. Pawlet and Bullfrog Hollow are locally juxtaposed, but their contact is everywhere interpreted to be structural. Structurally, the study area is quite complex. Four phases of tectonic deformation associated with at least three generations of thrust faults are recognized. Earlier, pre-tectonic, syndepositional deformation features (DO) are also recognized. The earliest tectonic deformation (Dl) is only locally recognizable. It involves macroscopic isoclinal and initially recumbent folds (Fl) and axial surface-parallel thrusts (T1). F1 folds and T1 thrusts are refolded by prominent west-verging, asymmetric, overturned folds (F2) with an axial surface slaty cleavage (S2). Thrusting (T2) parallel or somewhat less steep than F2 axial surfaces imbricates and dismembers the F2 folds. These structures pre-date the Giddings Brook Thrust. Mesoscopic refolding of D2 and earlier structures by F3 folds which are associated with an axial surface crenulation cleavage (S3) is observed, but is not macroscopically significant. A third generation of thrusts (T3) that dip significantly less steeply east than F2 axial surfaces are prominent in this area and may be temporally associated with F3 folds, but this cannot be proven. T3 thrusts may be of the same age as the Giddings Brook Thrust. Rare vertical kink bands (F4) represent the fourth tectonic deformation and are not considered to be significant to the regional structure

Fusion at deep subbarrier energies: potential inversion revisited

For a single potential barrier, the barrier penetrability can be inverted based on the WKB approximation to yield the barrier thickness. We apply this method to heavy-ion fusion reactions at energies well below the Coulomb barrier and directly determine the inter-nucleus potential between the colliding nuclei. To this end, we assume that fusion cross sections at deep subbarrier energies are governed by the lowest barrier in the barrier distribution. The inverted inter-nucleus potentials for the $^{16}$O +$^{144}$Sm and $^{16}$O +$^{208}$Pb reactions show that they are much thicker than phenomenological potentials. We discuss a consequence of such thick potential by fitting the inverted potentials with the Bass function.Comment: 8 pages, 5 figures. Uses aipxfm.sty. A talk given at the FUSION08: New Aspects of Heavy Ion Collisions Near the Coulomb Barrier, September 22-26, 2008, Chicago, US

Application of the Van\u27t Hoff Equation to Adsorption Equilibria

Isothermal adsorption data for many gases and vapors on charcoal and other adsorbents have been shown by various investigators (1), (2), (3), (4), to agree satisfactorily with the Langmuir adsorption isotherm, except for deviations, possibly due to multilayer adsorption, as the pressure of the saturated vapor is approached. The Langmuir equation is derived on the hypothesis of a unimolecular adsorbed layer. The rate of adsorption, assumed proportional to the pressure p and the fraction of the surface unoccupied, (1 - ϑ), is equated to the rate of desorption which is assumed proportional to the fraction of the surface covered, ϑ, giving the equation below

Measurement of Adsorption Isotherms for Mixed Vapors

A great deal of work has been done in the measurement of adsorption isotherms of pure gases and vapors, but very little of the measurement of adsorption isotherms of mixed gases or vapors and much of this has been unintentional, being due to impurities in the adsorbate. Papers dealing with measurements of this type include those of Richardson and Woodhouse (2) and Bakr and King. (3) The method of Bakr and King has the objection that each experiment yields an isolated value. Richardson\u27s and Woodhouse’s method requires extensive gas analysis and yields an isotherm in which the composition of the adsorbate changes