The problem of monotonicity testing of the boolean hypercube is a classic well-studied, yet unsolved question in property testing. We are given query access to f : {0, 1} 7→ R (for some ordered range R). The boolean hypercube B has a natural partial order, denoted by ≺ (defined by the product of coordinate-wise ordering). A function is monotone if all pairs x ≺ y in B, f(x) ≤ f(y). The distance...

1 Overview 1.1 Last time • Proper learning for P implies property testing of P (generic, but quite inefficient) • Testing linearity (over GF[2]), i.e. P = {all parities}: (optimal) O 1-query 1-sided non-adaptive tester. • Testing monotonicity (P = {all monotone functions}: an efficient O n-query 1-sided non-adaptive tester.

Given a monotone property P of graphs, write Pn for the set of graphs with vertex set [n] having property P. Building on recent results in the enumeration of graphical properties, we prove numerous results about the structure of graphs in Pn and the functions |Pn|. We also examine the measure eP (n), the maximum number of edges in a graph of Pn.

Absac A graph property is called elusive (or evasive) if every algorithm for testing this property has to read in the worst case Q) entries of the adjacency matrix of the given graph. Several graph properties have been shown to be elusive, e.g. planarity [2] or fc-colorability [4]. A famous conjecture of Karp [11] says that every non-trivial monotone graph property is elusive. We prove that a n...

Let R be an expansion of a dense linear order (R,<) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f : [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (R, <), as well as every o-minimal expansion of (R,<), has the intermediate value property. Conversely, some ni...

Given a set P of n planar points, two axes and a realvalued score function f() on subsets of P , the Optimal Planar Box problem consists in finding a box (i.e. an axis-aligned rectangle) H maximizing f(H ∩ P ). We consider the case where f() is monotone decomposable, i.e. there exists a composition function g() monotone in its two arguments such that f(A) = g(f(A1), f(A2)) for every subset A ⊆ ...

In this paper we study some monotone behavior of the residual (past) entropy of order . We prove that, under some relation between the hazard rates (reversed hazard rates) of two distributions functions F and G, when the residual (past) entropy of order of F is decreasing (increasing) then the residual (past) entropy of G is decreasing (increasing). Using this, several conclusions regarding mo...

This paper deals with a new family of (A, N)-implications that derived from fuzzy negation and continuous monotone function, and some properties, like the left neutrally, the exchange principle, the ordering property and the laws of contraposition, of the proposed family of (A, N)-implications are studied. Moreover, the T-conditionality and the Modus tollens for the new family of (A, N)-i...

Given a graph G, a k-dominating set of G is a subset S of its vertices with the property that every vertex of G is either in S or has at least k neighbors in S. We present a new incremental local algorithm to construct a k-dominating set. The algorithm constructs a monotone family of dominating sets D1 ⊆ D2 . . . ⊆ Di . . . ⊆ Dk such that each Di is an i-dominating set. For unit disk graphs, th...

in this paper we investigate the existence and uniqueness for volterra-fredholm type integral equations and extension of this type of integral equations. the result is obtained by using the  coupled fixed point theorems in the framework of banach space $ x=c([a,b],mathbb{r})$. finally, we  give an example to illustrate the applications of our results.