We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

Reasoning with quantifier expressions in natural language combines logical and arithmetical features, transcending strict divides between qualitative and quantitative. Our topic is this cooperation of styles as it occurs in common linguistic usage and its extension into the broader practice of natural language plus ‘grassroots mathematics’.

We begin with a brief review of $\mathsf {FO}(\#)$, first-order logic with counting operators and cardinality comparisons. This system is known to be of very high complexity, and drowns out finer aspects of the combination of logic and counting. We therefore move to a small fragment that can represent numerical syllogisms and basic reasoning about comparative size: monadic first-order logic with counting, $\mathsf {MFO}(\#)$. We provide normal forms that allow for axiomatization, determine which arithmetical notions can be defined on finite and on infinite models, and conversely, we discuss which logical notions can be defined out of purely arithmetical ones, and what sort of (non-)classical logics can be induced.

Next, we investigate a series of strengthenings of $\mathsf {MFO}(\#)$, again using normal form methods. The monadic second-order version is close, in a precise sense, to additive Presburger Arithmetic, while versions with the natural device of tuple counting take us to Diophantine equations, making the logic undecidable. We also define a system $\mathsf {ML}(\#)$ that combines basic modal logic over binary accessibility relations with counting, needed to formulate ubiquitous reasoning patterns such as the Pigeonhole Principle. We prove decidability of $\mathsf {ML}(\#)$, and provide a new kind of bisimulation matching the expressive power of the language.

As a complement to the fragment approach pursued here, we also discuss two other ways of lowering the complexity of $\mathsf {FO}(\#)$ by changing the semantics of counting in natural ways. A first approach replaces cardinalities by abstract but well-motivated values of ‘mass’ or other mereological aggregating notions. A second approach keeps the cardinalities but generalizes the meaning of counting to work in models that allow dependencies between variables.

Finally, we return to our starting point in natural language, confronting the architecture of our formal systems with linguistic quantifier vocabulary and syntax, as well as with natural reasoning modules such as the monotonicity calculus. In addition to these encounters with formal semantics, we discuss the role of counting in semantic evaluation procedures for quantifier expressions and determine, for instance, which binary quantifiers are computable by finite ‘semantic automata’. We conclude with some general thoughts on yet further entanglements of logic and counting in formal systems, on rethinking the qualitative/quantitative divide, and on connecting our analysis to empirical findings in cognitive science.

Mathematics is a particularly important challenge for embodied approaches to cognition, as it is probably the most abstract domain of human knowledge. Humans use metaphors in all aspects of life. This paper studies the effects of human body parts on numerals, numeral systems, and mensural and sortal classifiers. The evidence for this paper comes from the Tāti language group, an endangered Iranian language of the Indo-European language family. The Tāti data shows these languages make use of base-10, base-20, and base-50 numeral systems, some of which are among the most common and earliest counting systems worldwide, while the last is unique and peculiar to the area. Body parts may also play an important role in forming mensural and sortal classifiers, as is the case in the Tāti language group.

Bounded treewidth is one of the most cited combinatorial invariants in the literature. It was also applied for solving several counting problems efficiently. A canonical counting problem is #Sat, which asks to count the satisfying assignments of a Boolean formula. Recent work shows that benchmarking instances for #Sat often have reasonably small treewidth. This paper deals with counting problems for instances of small treewidth. We introduce a general framework to solve counting questions based on state-of-the-art database management systems (DBMSs). Our framework takes explicitly advantage of small treewidth by solving instances using dynamic programming (DP) on tree decompositions (TD). Therefore, we implement the concept of DP into a DBMS (PostgreSQL), since DP algorithms are already often given in terms of table manipulations in theory. This allows for elegant specifications of DP algorithms and the use of SQL to manipulate records and tables, which gives us a natural approach to bring DP algorithms into practice. To the best of our knowledge, we present the first approach to employ a DBMS for algorithms on TDs. A key advantage of our approach is that DBMSs naturally allow for dealing with huge tables with a limited amount of main memory (RAM).

We present a computational method for pseudo-circular object detection and quantitative characterization in digital images, using the gradient accumulation matrix as a basic tool. This Gradient Accumulation Transform (GAT) was first introduced in 1992 by Kierkegaard and recently used by Kaytanli & Valentine. In the present article, we modify the approach by using the phase coding studied by Cicconet, and by adding a “local contributor list” (LCL) as well as a “used contributor matrix” (UCM), which allow for accurate peak detection and exploitation. These changes help make the GAT algorithm a robust and precise method to automatically detect pseudo-circular objects in a microscopic image. We then present an application of the method to cell counting in microbiological images.

We answer a question of Masser by showing that for the Weierstrass zeta function ζ corresponding to a given lattice Λ, the density of algebraic points of absolute multiplicative height bounded by T and degree bounded by k lying on the graph of ζ, restricted to an appropriate domain, does not exceed c(log T)15 for an effective constant c > 0 depending on k and on Λ. Using different methods, we also give two bounds of the same form for the density of algebraic points of bounded height in a fixed number field lying on the graph of ζ restricted to an appropriate subset of (0, 1). In one case the constant c can be shown not to depend on the choice of lattice; in the other, the exponent can be improved to 12.

We provide an algorithm for listing all minimal 2-dominating sets of a tree of ordern in time 𝒪(1.3248n). This implies that every tree has at most1.3248n minimal 2-dominating sets. We also show that thisbound is tight.

Mixed-level orthogonal arrays are basic structures in experimental design. We develop three algorithms that compute Rao- and Gilbert-Varshamov-type bounds for mixed-level orthogonal arrays. The computational complexity of the terms involved in the original combinatorial representations of these bounds can grow fast as the parameters of the arrays increase and this justifies the construction of these algorithms. The first is a recursive algorithm that computes the bounds exactly, the second is based on an asymptotic analysis, and the third is a simulation algorithm. They are all based on the representation of the combinatorial expressions that appear in the bounds as expectations involving a symmetric random walk. The Markov property of the underlying random walk gives the recursive formula to compute the expectations. A large deviation (LD) analysis of the expectations provides the asymptotic algorithm. The asymptotically optimal importance sampling (IS) of the same expectation provides the simulation algorithm. Both the LD analysis and the construction of the IS algorithm use a representation of these problems as a sequence of stochastic optimal control problems converging to a limit calculus of a variations problem. The construction of the IS algorithm uses a recently discovered method of using subsolutions to the Hamilton-Jacobi-Bellman equations associated with the limit problem.

Preschoolers with spina bifida (SB) were compared to typically developing (TD) children on tasks tapping mathematical knowledge at 36 months (n = 102) and 60 months of age (n = 98). The group with SB had difficulty compared to TD peers on all mathematical tasks except for transformation on quantities in the subitizable range. At 36 months, vocabulary knowledge, visual–spatial, and fine motor abilities predicted achievement on a measure of informal math knowledge in both groups. At 60 months of age, phonological awareness, visual–spatial ability, and fine motor skill were uniquely and differentially related to counting knowledge, oral counting, object-based arithmetic skills, and quantitative concepts. Importantly, the patterns of association between these predictors and mathematical performance were similar across the groups. A novel finding is that fine motor skill uniquely predicted object-based arithmetic abilities in both groups, suggesting developmental continuity in the neurocognitive correlates of early object-based and later symbolic arithmetic problem solving. Models combining 36-month mathematical ability and these language-based, visual–spatial, and fine motor abilities at 60 months accounted for considerable variance on 60-month informal mathematical outcomes. Results are discussed with reference to models of mathematical development and early identification of risk in preschoolers with neurodevelopmental disorder. (JINS, 2011, 17, 431–444)

Numerical competence is one of the many aspects of animal cognition that have enjoyed a resurgence of interest during the past decade. Evidence for numerical abilities in animals has followed a tortuous path to respectability, however, from Clever Hans, the counting horse, to modern experimental studies. Recent surveys of the literaturereveal theoretical as well as definitional confusion arising from inconsistent terminology for numerical processes and procedures. The term “counting” has been applied to situations having little to do with its meaning in the human literature. We propose a consistent vocabulary and theoretical framework for evaluating numerical competence. Relative numerousness judgments, subitizing, counting, and estimation may be the essential processes by which animals perform numerical discriminations. Ordinality, cardinality, and transitivity also play an important role in these processes. Our schema is applied to a variety of recent experimental situations. Some evidence of transfer is essential in demonstrating higher-order ability such as counting or “sense of number.” Those instances of numerical competence in which all viable alternatives to counting (e.g., subitizing) have been precluded, but no evidence of transfer has been demonstrated might be described as “protocounting.” To show that animals are capable of “true” counting future research will have to demonstrate generality across situations.

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV1 + dWPHP(PV).

Pollen counting is an important element of research in pollination, and the majority of current methods involve manual counting. An image-processing technique has been developed for accurate counting and sizing of small particles, and this was tested on almond pollen grains. Our automatic process was found to be several times faster than manual counting and more accurate, with an average error of only 3%. Because of its simplicity, it is an attractive tool for research on such topics as pollen viability or germination. It was found that in the methods using vials there were ‘residues’ of c. 17% on the vial walls and on the handling tools. In addition, the first drop from each vial contained, on average, 1·65 times more grains than the average number of grains in all drops, whereas the last drop from each vial contained fewer grains than the average. This phenomenon was correlated with the non-uniform pollen distribution in the suspension. An automatic counting system is therefore, recommended, to avoid the problem, otherwise a calibration procedure should be used that accounts for both the residues on the tools and the non-uniformity of contents of sample drops.