We show that if a Boolean function f : {0, 1}n → {0, 1} can be computed by a monotone real circuit of size s using k-ary monotone gates then f can be computed by a monotone real circuit of size O(snk−2) which uses unary or binary monotone gates only. This partially solves an open problem presented in [2]. In fact, in size O(snk−1), the circuit uses only unary monotone gates and binary addition....

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed parabolic problem. A block monotone domain decomposition algorithm based on a Schwarz alternating method and on a block iterative scheme is constructed. This monotone algorithm solves only linear discrete systems at each time level and converges monotonically to the exact solution of the no...

We study monotonicity properties for minimizers of transport problems. In the one-dimensional case, we present an algorithm to construct minimizing monotone transport plans by “monotonizing” a given minimizing transport plan. This method applies in particular to the case of the L1-Wasserstein metric where we prove the existence of monotone minimizers for arbitrary marginals. We find that monoto...

We study the learnability of monotone term decision lists in the exact model of equivalence and membership queries. We show that, for any constant k 0, k-term monotone decision lists are exactly and properly learnable with n O(k) membership queries in O(n k 3) time. We also show n (k) membership queries are necessary for exact learning. In contrast, both k-term monotone decision lists (k 2) and...

The subject is the overview of the use of quasi-entropy in finite dimensional spaces. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the f -divergence, monotone metrics, covariance and the χ2-divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone m...

We establish a formal connection between algorithmic correspondence theory and certain dual characterization results for finite lattices, similar to Nation’s characterization of a hierarchy of pseudovarieties of finite lattices, progressively generalizing finite distributive lattices. This formal connection is mediated through monotone modal logic. Indeed, we adapt the correspondence algorithm ...

We show that the problem of guarding an x-monotone terrain from an altitude line and the problem of guarding a uni-monotone polygon are equivalent. We present a polynomial time algorithm for both problems, and show that the cardinality of a minimum guard set and the cardinality of a maximum witness set coincide. Thus, uni-monotone polygons are perfect; this result also extends to monotone mount...

Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone Lévy-Khintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal's " remarkable transformation. " A monotone analogue of the family of e...

A metric space (X, d) is called c-monotone if there is a linear order < on X and c > 0 such that d(x, y) 6 c d(x, z) for all x < y < z in X. A brief account of investigation of monotone spaces including applications is presented. 1 Monotone and σ-Monotone Spaces. In [6] I investigated existence of sets in Euclidean spaces that have large Hausdorff dimension and yet host no continuous finite Bor...

We give a general complexity classi cation scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes includingmAC , mNC , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We de ne a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing r...