Much like other social and nonsocial evaluations, estimates of numerical quantities are susceptible to comparative influences. However, numerical representations can take either a nonsymbolic (e.g., a grouping of dots) or a symbolic numerical form (e.g., Hindu–Arabic numerals), which each produce comparative biases in opposite directions. The current work takes a fine-grained curve fitting approach across a wide range of values to the investigation of their potential nonlinear influence in the context of a numerical estimation task. A series of 3 experiments (N = 1,613) showed how nonsymbolic standards produce linear contrastive patterns (Study 1), whereas symbolic numerical anchors show a cubic assimilative effect with a leveling off in strength for more extreme standards (Study 2). Sequential contrast from the previous patterns and assimilation to the previous response were found to be present and additive in the presence of both representations but were larger in absence of the symbolic numerical anchors (Study 3).

We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

In this study, several joint axis orientations on equilateral platforms and the limbs of 3-UPU parallel manipulators (PMs) are examined. The generated joint layouts for the platforms were matched with each other to generate and enumerate manipulator architectures based on certain assumptions. The structures of thus obtained manipulators are examined and limb types were determined. These limb types were analyzed using screw theory. The instantaneous mobility of the manipulators and the motion characteristics of the moving platforms are tabulated. The finite mobility analysis of one of the manipulators is performed using a software package as an example. Among several different 3-UPU PM architectures, 118 novel 3-UPU PMs with non-parasitic 3-degrees-of-freedom are significantly important. The classified 3-UPU PMs with determined motion characteristics can be used by researchers as a design alternative for their specific design task.

We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous $\text{C}^{\ast }$-algebras with exactly $n$ Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.

A classical result of Honsberger states that the number of incongruent triangles with integer sides and perimeter $n$ is the nearest integer to $n^{2}/48$ ($n$ even) or $(n+3)^{2}/48$ ($n$ odd). We solve the analogous problem for $m$-gons (for arbitrary but fixed $m\geq 3$) and for polygons (with arbitrary number of sides).

For a lattice $L$ of ${{\mathbb{R}}^{n}}$, a sphere $S\left( c,r \right)$ of center $c$ and radius $r$ is called empty if for any $v\,\in \,L$ we have $\parallel v\,-\,c\parallel \,\,\ge \,r$. Then the set $S\left( c,r \right)\,\cap \,L$ is the vertex set of a $Delaunay\,polytope\,P\,=\,\text{conv}\left( S\left( c,r \right)\cap L \right)$. A Delaunay polytope is called perfect if any affine transformation $\phi $ such that $\phi \left( P \right)$ is a Delaunay polytope is necessarily an isometry of the space composed with an homothety.

Perfect Delaunay polytopes are remarkable structures that exist only if $n\,=\,1$ or $n\,\ge \,6$, and they have shown up recently in covering maxima studies. Here we give a general algorithm for their enumeration that relies on the Erdahl cone. We apply this algorithm in dimension seven, which allows us to find that there are only two perfect Delaunay polytopes: 321, which is a Delaunay polytope in the root lattice ${{\text{E}}_{7}}$, and the Erdahl Rybnikov polytope.

We then use this classification in order to get the list of all types of Delaunay simplices in dimension seven and found that there are eleven types.

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {ck}. Notoriously hard to compute, the exact fractions ck have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c4 and c5 as well; both are ratios of enormous integers, the denominator of c5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of ck is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.

Researchers have focused on identifying the mechanisms involved in subitizing and its differences with estimation. Some suggest that subitizing relies on a visual indexing system in charge of the simultaneous individuation of objects that is also used by visuospatial working memory (VSWM). In adults, studies found associations between subitizing and VSWM, in the absence of correlation between VSWM and estimation. The present study analyzed the performance of 120 4 and 6-year-old children in three tasks: dot enumeration to measure subitizing capacity, quantity discrimination for estimation, and Corsi Block-tapping task for VSWM. In the enumeration task RTs (F(9, 1062)=720.59, MSE=734394, p<.001, η2=.86) and errors (F(9, 1062)=42.15, MSE=.194, p<.001, η2=.26.) increased with the array, but this growth was statistically significant only from 4 dots onward. Each subject’s subitizing range was estimated by fitting RTs with a sigmoid function of number of dots and obtaining the bend point of the curve. Data fit (age 4: R2 = .88; SD = .08; age 6: R2 = .91, SD = .08) showed a mean subitizing range of 2.79 (SD = .66) for 4 year-olds and of 3.11 (SD = .64) for 6 year-olds. Subitizing ranges and average RTs showed low association with storage (r = .274; p < .05; r = –.398; p < .001) and average RTs with concurrent processing (r = –.412; p < .001) in VSWM. Subitizing range and speed showed no association with estimation speed and a poor association with accuracy (r = .234, p < .01; r = –.398, p < .001), which suggests independent systems for small and large quantities. Subitizing and estimation measures correlated with VSWM (p < .01), which suggests that both processes may require VSWM resources.

This study aims to describe in detail the temporal dynamics of E. coli O157 shedding and risk factors for shedding in a grass-fed beef herd. During a 9-month period, 23 beef cows were sampled twice a week (58 sampling points) and E. coli O157 was enumerated from faecal samples. Isolates were screened by PCR for presence of rfbE, stx1 and stx2. The prevalence per sampling day ranged from 0% to 57%. This study demonstrates that many members of the herd were concurrently shedding E. coli O157. Occurrence of rainfall (P < 0·01), feeding silage (P < 0·01) and lactating (P < 0·01) were found to be predictors of shedding. Moving cattle to a new paddock had a negative effect on shedding. This approach, based on short-interval sampling, confirms the known variability of shedding within a herd and highlights that high shedding events are rare.

Presently, there is no shortage of methods for collecting data on populations requiring assistance from humanitarian health interventions. However, utilizing a working group, the authors of this paper have looked at these methods through a critical lens and found that there is need for improvement upon existing systems of data collection and analysis. The authors concluded that efforts to standardize the methods of data collection are needed to achieve universal uniformity, and that more funding should be allocated to analyze the data collected.

Miller (1956) summarized evidence that people can remember about seven chunks in short-term memory (STM) tasks. However, that number was meant more as a rough estimate and a rhetorical device than as a real capacity limit. Others have since suggested that there is a more precise capacity limit, but that it is only three to five chunks. The present target article brings together a wide variety of data on capacity limits suggesting that the smaller capacity limit is real. Capacity limits will be useful in analyses of information processing only if the boundary conditions for observing them can be carefully described. Four basic conditions in which chunks can be identified and capacity limits can accordingly be observed are: (1) when information overload limits chunks to individual stimulus items, (2) when other steps are taken specifically to block the recoding of stimulus items into larger chunks, (3) in performance discontinuities caused by the capacity limit, and (4) in various indirect effects of the capacity limit. Under these conditions, rehearsal and long-term memory cannot be used to combine stimulus items into chunks of an unknown size; nor can storage mechanisms that are not capacity-limited, such as sensory memory, allow the capacity-limited storage mechanism to be refilled during recall. A single, central capacity limit averaging about four chunks is implicated along with other, noncapacity-limited sources. The pure STM capacity limit expressed in chunks is distinguished from compound STM limits obtained when the number of separately held chunks is unclear. Reasons why pure capacity estimates fall within a narrow range are discussed and a capacity limit for the focus of attention is proposed.

In this paper, we examine the class of "deco" polyominoes and the succession rule describing their construction. These polyominoes are enumerated according to their directed height by factorial numbers. By changing some aspects of the "factorial" rule, we obtain some succession rules that describe various "deco" polyomino subclasses. By enumerating the subclasses according to their height and width, we find the following well-known numbers: Stirling numbers of the first and second kind, Narayana and odd index Fibonacci numbers. We wish to point out how the changes made on the original succession rule yield some new succession rules that produce transcendental, algebraic and rational generating functions.

We present a new algorithm which uses a cohomological approach to determine the groups of order pn, where p is a prime. We develop two methods to enumerate p-groups using the Cauchy-Frobenius Lemma. As an application we show that there are 10 494213 groups of order 29.

1991 Mathematics subject classification (Amer. Math. Soc.): primary 20D15.

This paper discusses two-terminal series-parallel networks occurring in Applied Probability and other contexts where the mathematical theory of reliability is developed. Eighty years ago Mac Mahon successfully discussed the number of different structures built of n identical components [1]; later Knödel [3] and Carlitz and Riordan [4] in the 1950's investigated the number of different structures built of n different components. However, in many new applications, engineers and statisticians have to study structures having n components with a specification defining various types of components and stating how many identical components of each kind should be used. This general problem is investigated in Section 7 whilst in Sections 1–6 results obtained earlier are presented and it is shown there how modern approaches to combinatorics (particularly Pólya's Enumerative Combinatorial Analysis) can simplify some reasoning of previous authors.