Periodic solutions of arbitrary length in a simple integer iteration

We prove that all solutions to the nonlinear second-order difference equation in integers yn+1 = ⌈ay n ⌉-yn-1, {a ∈ ℝ:|a|a≠0,±1}, y0, y1 ∈ ℤ, are periodic. The first-order system representation of this equation is shown to have self-similar and chaotic solutions in the integer plane

Endocrine and Physiological Responses of the Female Goat During Three Reproductive Phases

Female Spanish x Dairy cross goats were divided into three reproductive phases--breeding season or cyclic (20), seasonal anestrus (20), and early postpartum (20). Each group of animals was further divided into three treatments--control (10), Gonadotropin Releasing Hormone (GnRH) treated (5), and pituitary Luteinizing Hormone (LH) (5). During each of the above mentioned reproductive phases, the following were measured: progesterone and LH profiles, serum LH levels following GnRH injection, pituitary LH concentration and ovarian responses following GnRH treatment. Progesterone levels indicative of luteal development were present only in the breeding season. Progesterone levels during the seasonal anestrous and postpartum periods were 0.65 ± 0.03 and 0.3 ± 0.02 ng/ml, respectively, and significantly different (P\u3c0.01) The LH surge mechanism was operational only in breeding season does,showing LH peaks of 187.86 ± 3.95 ng/ml an average of 13.2 ± 1.47 hours after the onset of estrus. LH levels returned to 2 ng/ml by 48 hours post estrus. There was an inverse relationship between plasma progesterone and LH with mean LH levels of 1.1 ± 0.13 ng/ml during the luteal phase of the cycle. Serum LH levels during the seasonal anestrous and postpartum period were 0.71 ± 0.02 and 0.32 ± 0.02 ng/ml, respectively. These differences in tonic LH secretion were significant. Tonic LH levels (0.5 ± 0.2 ng/ml) were recorded in all does, regardless of the reproductive phase, prior to the initial injection of GnRH. Serum LH increased to 182.96 ± 54.56, 209.38 ± 41.38, and 97.84 ± 55.84 ng/ml, these peak levels were recorded at 114 ± 4, 135 ± 7, and 135 ± 11 minutes post injection in cyclic, seasonal anestrous, and postpartum does, respectively. Response to the second injection was more rapid and heightened, peaks were achieved from 27 ± 3 to 66 ± 14 minutes post injection, LH peak levels were higher than those recorded following the first injection. Postpartum animals showed reduced responses in all cases. Pituitary LH concentrations were 1711 ± 378, 2069 ± 265, and 3542 ± 398 μg LH/g tissue in the postpartum, seasonal anestrous, and cyclic animals respectively. Because of high nonspecific binding, these concentrations are considered as estimates, nevertheless, trends observed are considered to be real

Stratigraphy, genesis, and economic potential of the southern part of the Florida land-pebble phosphate field

The important and highly productive Florida land-pebble phosphate field is located in west central peninsular Florida in portions of Hillsborough, Polk, Hardee, and Manatee counties. The Pine Level phosphate area, described in detail in this report, is south of the previously known and mined deposits and occurs in portions of Manatee, Sarasota, and De Soto counties. Results of the current geologic study of the Pine Level phosphate deposit and the evaluation of the overall potential of this southern part of the land-pebble field are presented. The entire southern part of the phosphate field is underlain by more than 15,000 feet of Cretaceous and Tertiary carbonate strata. Phosphate deposits are confined to a thin clastic veneer of sediments that overlie the carbonate strata, and include the upper clastic member of the Hawthorn Formation of Miocene age, the Bone Valley Formation of Pliocene age, and unnamed strata of Pleistocene age. The total thickness of the phosphatic veneer is somewhat more than 100 feet. The Pine Level phosphate deposit, characteristic of the heretofore undescribed phosphate deposits in the southern part of the Florida land-pebble phosphate field, is compared with the deposits of the main producing area in the northern part of the field. The Pine Level deposit differs markedly from the deposits in the main producing district. The differences include the more localized and erratic distribution of mineable phosphate concentrations, inclusion of portions of the upper clastic member of the Hawthorn Formation within the mineable unit, origin and age or the deposits, significant contrasts in pebble and concentrate quality and quantity related to the mode of origin, the lack of development of the aluminum phosphate zone, and the enrichment of the contained carbonate fluorapatite by replacement processes. Very gentle scarps, representing Pleistocene sea standstills, divide the land-pebble field into several physiographic subdivisions. The physiographic provinces of the land-pebble field and the origin of the Pine Level deposit are related to three, and possibly four, Pleistocene interglacial marine advances that have reworked and recycled the apatite particles of the Bone Valley and Hawthorn Formations into new lower-level Pleistocene deposits that surround and flank buried remnant paleo-islands of the Bone Valley Formation. Field relations, chemical analyses and petrographic studies of a series of apatite pebbles ranging from deeply buried, low-grade, black, impure apatite to shallow, high-grade, relatively pure, white apatite, indicates that high-grade white apatite in the Pine Level deposit is derived from initial low-grade black apatite. The alteration occurs by progressive replacement of mineral impurities within the black apatite as erosion continually reduces the depth of burial and the black apatite is subjected to increasingly acidic and oxidizing ground water activity. Simplified evaluation criteria that serve to identify economically valuable deposits of the Pine Level phosphate type are described. These criteria are easily and readily determinable by the exploration geologist or engineer in search or such deposits. The slimes (clay) content of the Pine Level phosphate deposit is much lower than in the deposits of the main producing area and provides the basis for a new method of land reclamation that may eliminate the expensive and difficult conventional method or slimes disposal in permanent storage reservoirs --Abstract, pages ii-iii

Circular avoiding sequences with prescribed sum

AbstractFor given positive integers x, n, and s an x-avoiding circular sequence (of positive integers) of length n and sum s has no set of consecutive terms summing to x, even if wraparound is allowed. A necessary and sufficient condition for the existence of such a sequence is obtained. An effective method to construct avoiding sequences is given. For the cases of most interest the number of avoiding sequences is found