Some Two Color, Four Variable Rado Numbers

There exists a minimum integer $N$ such that any 2-coloring of $\{1,2,...,N\}$ admits a monochromatic solution to $x+y+kz =\ell w$ for $k,\ell \in \mathbb{Z}^+$, where $N$ depends on $k$ and $\ell$. We determine $N$ when $\ell-k \in \{0,1,2,3,4,5\}$, for all $k,\ell$ for which ${1/2}((\ell-k)^2-2)(\ell-k+1)\leq k \leq \ell-4$, as well as for arbitrary $k$ when $\ell=2$.Comment: 13 page

Geographic variation and evolutionary history of Dipodomys nitratoides (Rodentia: Heteromyidae), a species in severe decline

We examined geographic patterns of diversification in the highly impacted San Joaquin kangaroo rat, Dipodomys nitratoides, throughout its range in the San Joaquin Valley and adjacent basins in central California. The currently recognized subspecies were distinct by the original set of mensural and color variables used in their formal diagnoses, although the Fresno kangaroo rat (D. n. exilis) is the most strongly differentiated with sharp steps in character clines relative to the adjacent Tipton (D. n. nitratoides) and short-nosed (D. n. brevinasus) races. The latter two grade more smoothly into one another but still exhibit independent, and different, character clines within themselves. At the molecular level, as delineated by mtDNA cytochrome b sequences, most population samples retain high levels of diversity despite significant retraction in the species range and severe fragmentation of local populations in recent decades due primarily to landscape conversion for agriculture and secondarily to increased urbanization. Haplotype apportionment bears no relationship to morphologically defined subspecies boundaries. Rather, a haplotype network is shallow, most haplotypes are single-step variants, and the time to coalescence is substantially more recent than the time of species split between D. nitratoides and its sister taxon, D. merriami. The biogeographic history of the species within the San Joaquin Valley appears tied to mid-late Pleistocene expansion following significant drying of the valley resulting from the rain shadow produced by uplift of the Central Coastal Ranges

Boolean algebras and Lubell functions

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection \B\subset 2^{[n]} forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps the exception of $X_0$, so that \B={X_0\cup \bigcup_{i\in I} X_i\colon I\subseteq [d]}. Let $b(n,d)$ be the maximum cardinality of a family \F\subset 2^X that does not contain a $d$-dimensional Boolean algebra. Gunderson, R\"odl, and Sidorenko proved that $b(n,d) \leq c_d n^{-1/2^d} \cdot 2^n$ where $c_d= 10^d 2^{-2^{1-d}}d^{d-2^{-d}}$. In this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets \F with \F\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}. We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains no $d$-dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for sufficiently large $n$. This results implies $b(n,d) \leq C n^{-1/2^d} \cdot 2^n$, where $C$ is an absolute constant independent of $n$ and $d$. As a consequence, we improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page

Coexistence of 'alpha+ 208Pb' cluster structures and single-particle excitations in 212Po

Excited states in 212Po have been populated by alpha transfer using the 208Pb(18O,14C) reaction at 85MeV beam energy and studied with the EUROBALL IV gamma multidetector array. The level scheme has been extended up to ~ 3.2 MeV excitation energy from the triple gamma coincidence data. Spin and parity values of most of the observed states have been assigned from the gamma angular distributions and gamma -gamma angular correlations. Several gamma lines with E(gamma) < 1 MeV have been found to be shifted by the Doppler effect, allowing for the measurements of the associated lifetimes by the DSAM method. The values, found in the range [0.1-0.6] ps, lead to very enhanced E1 transitions. All the emitting states, which have non-natural parity values, are discussed in terms of alpha-208Pb structure. They are in the same excitation-energy range as the states issued from shell-model configurations.Comment: 21 pages, 19 figures, corrected typos, revised arguments in Sect. III

On the degree of regularity of a certain quadratic Diophantine equation

We show that, for every positive integer r, there exists an integer b = b(r) such that the 4-variable quadratic Diophantine equation (x1 − y1)(x2 − y2) = b is r-regular. Our proof uses Szemerédi’s theorem on arithmetic progressions