Terms Of Monotonicity Research Articles

Stochastic Monotonicity and Conditioning in the Limit

Suppose that {(Xn, Yn)} is a sequence of pairs of cector‐valued stochastic variables which converges weakly to (X, Y), and that {yn} converges to y. Sufficient conditions for the conditional distribution of Xn given Y = y are given in terms of stochastic monotonicity. Conditions, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.

Exact learning of linear combinations of monotone terms from function value queries

We investigate the problem of exactly identifying a real-valued function of {0, 1} n represented by a weighted sum of a number of monotone terms by querying for the values of the target function at assignments of the learner's choice. When all coefficients are nonnegative, we exhibit an efficient learning algorithm requiring at most ( n − ⌊ log s⌋ + 1) s queries, where n is the number of variables and s is the number of terms in the target formula. We prove a lower bound of Ω( ns log s ) on the number of queries necessary for learning this class, so no algorithm can reduce the number of queries dramatically. The algorithm runs in time O( ns 2) in the worst case. The same algorithm can be used to learn the ‘inductive-read- k’ subclass, a proper super class of the ‘read- k’ subclass, with a number of queries not exceeding 1 2 ((n − ⌊log k⌋)(n − ⌊log k⌋ + 1) + 2)k , which improves upon the bound achievable by a naive learning algorithm by a factor of two. In addition, the above method can be extended to handle the nonmonotone case in some restricted sense: A similar algorithm can learn the unate linear combinations of terms with a comparable number of queries. In the general case, namely, when the coefficients vary over the reals (or any arbitrary field), we show that the number of queries required for exact learning of the k-term subclass is upper bounded by q( n,⌊ log k⌋ + 1) and is lower bounded by q( n,⌊ log k⌋), where q(n, l) = ∑ l i = 0 ( n i) . These bounds are shown by generalizing Roth and Benedek's technique for analyzing the learning problem for k-sparse multivariate polynomials over GF(2) (Roth and Benedek, 1991) to those over an arbitrary field.

Subgradient Criteria for Monotonicity, The Lipschitz Condition, and Convexity

AbstractLet ƒ H → (—∞,∞] be lower semicontinuous, where H is a real Hilbert space. An approach based upon nonsmooth analysis and optimization is used in order to characterize monotonicity of ƒ with respect to a cone, as well as Lipschitz behavior and constancy. The results, which involve hypotheses on the proximal subgradient ∂ πƒ, specialize on the real line to yield classical characterizations of these properties in terms of the Dini derivate. They also give new extensions of these results to the multidimensional case. A new proof of a known characterization of convexity in terms of proximal subgradient monotonicity is also given.

Some results on quantifiers.

Introduction We shall study (generalized) quantifiers in the framework introduced by Barwise and Cooper in [ 1 ], the logical investigation of which has been continued in van Ben them [3] and Keenan and Stavi [2]. The present paper, although self-contained, is in the spirit of [3]. Its main characteristic is a systematic use of a method of proof introduced by van Benthem, which is based on a representation of quantifiers as relations on the natural numbers. With this method we give simplified proofs of a number of van Benthem's results and prove some new ones. In particular, the number-theoretic representation proves convenient for studying monotonicity behavior of quantifiers, and our main result is a characterization of the first-order definable quantifiers in terms of monotonicity.

Evidence for Logarithmic Singularities in the Specific Heats of the Ferroelectric Crystals KH2PO4and KH2AsO4

By drawing interpolation curves between the high- and low-temperature branches of the specific heats of K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$ and K${\mathrm{H}}_{2}$As${\mathrm{O}}_{4}$ (measured by Stephenson and co-workers) we have decomposed each specific-heat function into the sum of a monotonic term plus an anomolous term, peaked about the transition temperature. In the region of the transition temperature ${\ensuremath{\theta}}_{c}$ we find that the paraelectric and ferroelectric branches of the anomalous specific heat of each crystal have the limiting behavior ${C}^{a}=\ensuremath{-}A\mathrm{ln}|\ensuremath{\theta}\ensuremath{-}{\ensuremath{\theta}}_{c}|+B,$ where $A=1.2\ifmmode\pm\else\textpm\fi{}0.3$ cal/mole deg, $B=0.95\ifmmode\pm\else\textpm\fi{}0.07$ cal/mole deg for $\ensuremath{\theta}g{\ensuremath{\theta}}_{c}$, and $A=14.2\ifmmode\pm\else\textpm\fi{}1.4$ cal/mole deg, $B=14.4\ifmmode\pm\else\textpm\fi{}0.3$ cal/mole deg for $\ensuremath{\theta}l{\ensuremath{\theta}}_{c}$. These constants have the same values for both crystals.

Infinite Series with Multiply Monotone Terms

Infinite Series with Multiply Monotone Terms