A normed space $mathfrak{X}$ is said to have the fixed point property, if for each nonexpansive mapping $T : E longrightarrow E $ on a nonempty bounded closed convex subset $ E $ of $ mathfrak{X} $ has a fixed point. In this paper, we first show that if $ X $ is a locally compact Hausdorff space then the following are equivalent: (i) $X$ is infinite set, (ii) $C_0(X)$ is infinite dimensional, (...

We show that for every rooted, finitely branching, pruned tree T of height ω there exists a family F which consists of order isomorphic to T subtrees of the dyadic tree C = {0, 1}<N with the following properties: (i) the family F is a Gδ subset of 2C ; (ii) every perfect subtree of C contains a member of F ; (iii) if K is an analytic subset of F , then for every perfect subtree S of C there exi...

THEOREM 1.1 (U. Sattes). Let r > 2 and g E C[O, l]\B$,‘. Then f”EB$’ is a best approximation to g, in L” (such a best approximation necessari/J) exisrs) if and only if there exists a subinterual (a, /?) c IO. 1 I and a positilse integer M > r + 1 for which the following conditions hold (i) f”l,n.ll, is a Perfect spline of degree r with exactly) M ~ r -1 knots arzd I.f”““(s)l = I a. e. on [u,pI....

Archaeologists working on plantation sites have typically ignored the role that violence played in the lives of enslaved African-Americans. By drawing together examples from archaeological publications in which violence has been mentioned, the commentary illustrates not only both the types and scale of violence endured by African-Americans, but also how archaeologists have effectively sanitized...

A graph is h-perfect if its stable set polytope can be completely described by nonnegativity, clique and odd-hole constraints. It is t-perfect if it furthermore has no clique of size 4. For every graph G and every c ∈ Z (G) + , the weighted chromatic number of (G, c) is the minimum cardinality of a multi-set F of stable sets of G such that every v ∈ V (G) belongs to at least cv members of F . W...

Let P be a perfect subset of the real line, and let the «-element subsets of P be partitioned into finitely many classes, each open (or just Borel) in the natural topology on the collection of such subsets. Then P has a perfect subset whose «-element subsets he in at most (n — 1)! of the classes. Let C be the set of infinite sequences of zeros and ones, topologized as the product of countably m...

We consider the size of circuits that perfectly hash an arbitrary subset S⊂{0, 1} of cardinality 2 into {0, 1}. We observe that, in general, the size of such circuits is exponential in 2k−m, and provide a matching upper bound.