A census of critical sets in the Latin squares of order at most six

A critical set in a Latin square of order n is a set of entries from the square which can be embedded in precisely one Latin square of order n, Such that if any element of the critical set. is deleted, the remaining set can be embedded, in more than one Latin square of order n.. In this paper we find all the critical sets of different sizes in the Latin squares of order at most six. We count the number of main and isotopy classes of these critical sets and classify critical sets from the main classes into various strengths. Some observations are made about the relationship between the numbers of classes, particularly in the 6 x 6 case. Finally some examples are given of each type of critical set.

An investigation of 2-critical sets in latin squares

In this paper we focus on the existence of 2-critical sets in the latin square corresponding to the elementary abelian 2-group of order 2(n). It has been shown by Stinson and van Rees that this latin square contains a 2-critical set of volume 4(n) - 3(n). We provide constructions for 2-critical sets containing 4(n) - 3(n) + 1 - (2(k-1) + 2(m-1) + 2(n-(k+m+1))) entries, where 1 less than or equal to k less than or equal to n and 1 less than or equal to m less than or equal to n - k. That is, we construct 2-critical sets for certain values less than 4(n) - 3(n) + 1 - 3 (.) 2([n /3]-1). The results raise the interesting question of whether, for the given latin square, it is possible to construct 2-critical sets of volume m, where 4(n) - 3(n) + 1 - 3 (.) 2([n/3]-1) < m < 4(n) - 3(n).

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Webers Law signature, and was independent of conventional number knowledge.

The choice of the model in inventory control and the cost of sophistication

This thesis is concerned with the inventory control of items that can be considered independent of one another. The decisions when to order and in what quantity, are the controllable or independent variables in cost expressions which are minimised. The four systems considered are referred to as (Q, R), (nQ,R,T), (M,T) and (M,R,T). Wiith ((Q,R) a fixed quantity Q is ordered each time the order cover (i.e. stock in hand plus on order ) equals or falls below R, the re-order level. With the other three systems reviews are made only at intervals of T. With (nQ,R,T) an order for nQ is placed if on review the inventory cover is less than or equal to R, where n, which is an integer, is chosen at the time so that the new order cover just exceeds R. In (M, T) each order increases the order cover to M. Fnally in (M, R, T) when on review, order cover does not exceed R, enough is ordered to increase it to M. The (Q, R) system is examined at several levels of complexity, so that the theoretical savings in inventory costs obtained with more exact models could be compared with the increases in computational costs. Since the exact model was preferable for the (Q,R) system only exact models were derived for theoretical systems for the other three. Several methods of optimization were tried, but most were found inappropriate for the exact models because of non-convergence. However one method did work for each of the exact models. Demand is considered continuous, and with one exception, the distribution assumed is the normal distribution truncated so that demand is never less than zero. Shortages are assumed to result in backorders, not lost sales. However, the shortage cost is a function of three items, one of which, the backorder cost, may be either a linear, quadratic or an exponential function of the length of time of a backorder, with or without period of grace. Lead times are assumed constant or gamma distributed. Lastly, the actual supply quantity is allowed to be distributed. All the sets of equations were programmed for a KDF 9 computer and the computed performances of the four inventory control procedures are compared under each assurnption.

On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials

This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.

A Geometrical Construction for the Polynomial Invariants of some Reflection Groups

2000 Mathematics Subject Classification: Primary 20F55, 13F20; Secondary 14L30.

Lp extremal polynomials. Results and perspectives

2000 Mathematics Subject Classification: 30C40, 30D50, 30E10, 30E15, 42C05.

External Characterization of I-Favorable Spaces

1991 AMS Math. Subj. Class.:Primary 54C10; Secondary 54F65

Monogenic polynomials of four variables with binomial expansion

In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.

On Ordinal VC-Dimension and Some Notions of Complexity

We generalize the classical notion of Vapnik–Chernovenkis (VC) dimension to ordinal VC-dimension, in the context of logical learning paradigms. Logical learning paradigms encompass the numerical learning paradigms commonly studied in Inductive Inference. A logical learning paradigm is defined as a set W of structures over some vocabulary, and a set D of first-order formulas that represent data. The sets of models of ϕ in W, where ϕ varies over D, generate a natural topology W over W. We show that if D is closed under boolean operators, then the notion of ordinal VC-dimension offers a perfect characterization for the problem of predicting the truth of the members of D in a member of W, with an ordinal bound on the number of mistakes. This shows that the notion of VC-dimension has a natural interpretation in Inductive Inference, when cast into a logical setting. We also study the relationships between predictive complexity, selective complexity—a variation on predictive complexity—and mind change complexity. The assumptions that D is closed under boolean operators and that W is compact often play a crucial role to establish connections between these concepts. We then consider a computable setting with effective versions of the complexity measures, and show that the equivalence between ordinal VC-dimension and predictive complexity fails. More precisely, we prove that the effective ordinal VC-dimension of a paradigm can be defined when all other effective notions of complexity are undefined. On a better note, when W is compact, all effective notions of complexity are defined, though they are not related as in the noncomputable version of the framework.

Cultural selection and its influence on perceptions of reputation within organisations

The capacity to identify, interpret, and prioritise environmental issues is critical in the management of corporate reputation. In spite of the significance of these abilities for corporate reputation management, there has been little effort to document and describe internal organizational influences on these capacities. Contrary to this state of affairs in the discipline of public relations, a long history of ethnographic research in cultural anthropology documents how sets of shared environmental perceptions can influence and moderate environmental factors in cultural populations (see for example, Durham, 1991 ). This study explores how cultural “frames of reference” derived from shared values and assumptions among organizational members influence organizational perceptions, and consequently, organizational actions. Specifically, this study explores how a central attribute of organizational culture--the property of cultural selection-- influences perceptions of organizational reputation held by organizational members. Perceptions of reputation among organizational members are obvious drivers to both the nature of and rationale for organizational communication strategies and responses. These perceptions are the result of collective processes that synthesise (with varying degrees of consensus) member conceptualisations, interpretations, and representations of the environmental realities in which their organization operate. To explore how cultural selection influences member perceptions of organizational reputation, this study employs ethnographic research including 20 depth interviews and six months of organizational observation in the focal organization. We argue that while external indicators of organizational reputation are acknowledged by members as significant, the internal action of cultural selection is a far stronger influence on organizational action.

A review of LCADesign

Objective The review addresses two distinct sets of issues: 1. specific functionality, interface, and calculation problems that presumably can be fixed or improved; and 2. the more fundamental question of whether the system is close to being ready for ‘commercial prime time’ in the North American market. Findings Many of our comments relate to the first set of issues, especially sections B and C. Sections D and E deal with the second set. Overall, we feel that LCADesign represents a very impressive step forward in the ongoing quest to link CAD with LCA tools and, more importantly, to link the world of architectural practice and that of environmental research. From that perspective, it deserves continued financial support as a research project. However, if the decision is whether or not to continue the development program from a purely commercial perspective, we are less bullish. In terms of the North American market, there are no regulatory or other drivers to press design teams to use a tool of this nature. There is certainly interest in this area, but the tools must be very easy to use with little or no training. Understanding the results is as important in this regard as knowing how to apply the tool. Our comments are fairly negative when it comes to that aspect. Our opinion might change to some degree when the ‘fixes’ are made and the functionality improved. However, as discussed in more detail in the following sections, we feel that the multi-step process — CAD to IFC to LCADesign — could pose a serious problem in terms of market acceptance. The CAD to IFC part is impossible for us to judge with the information provided, and we can’t even begin to answer the question about the ease of using the software to import designs, but it appears cumbersome from what we do know. There does appear to be a developing North American market for 3D CAD, with a recent survey indicating that about 50% of the firms use some form of 3D modeling for about 75% of their projects. However, this does not mean that full 3D CAD is always being used. Our information suggests that AutoDesk accounts for about 75 to 80% of the 3D CAD market, and they are very cautious about any links that do not serve a latent demand. Finally, other system that link CAD to energy simulation are using XML data transfer protocols rather than IFC files, and it is our understanding that the market served by AutoDesk tends in that direction right now. This is a subject that is outside our area of expertise, so please take these comments as suggestions for more intensive market research rather than as definitive findings.

Raman spectroscopic study of the magnesium carbonate minerals– artinite and dypingite

Magnesium minerals are important in the understanding of the concept of geosequestration. The two hydrated hydroxy magnesium carbonate minerals artinite and dypingite have been studied by Raman spectroscopy. Intense bands are observed at 1092 cm-1 for artinite and at 1120 cm-1 for dypingite attributed CO32- ν1 symmetric stretching mode. The CO32- ν3 antisymmetric stretching vibrations are extremely weak and are observed at1412 and 1465 cm-1 for artinite and at 1366, 1447 and 1524 cm-1 for dypingite. Very weak Raman bands at 790 cm-1 for artinite and 800 cm-1 for dypingite are assigned to the CO32- ν2 out-of-plane bend. The Raman band at 700 cm-1 of artinite and at 725 and 760 cm-1 of dypingite are ascribed to CO32- ν2 in-plane bending mode. The Raman spectrum of artinite in the OH stretching region is characterised by two sets of bands: (a) an intense band at 3593 cm-1 assigned to the MgOH stretching vibrations and (b) the broad profile of overlapping bands at 3030 and 3229 cm-1 attributed to water stretching vibrations. X-ray diffraction studies show the minerals are disordered. This is reflected in the difficulty of obtaining Raman spectra of reasonable quality and explains why the Raman spectra of these minerals have not been previously or sufficiently described.

A Raman spectroscopic study of the uranyl mineral rutherfordine – revisited

The molecular structure of the uranyl mineral rutherfordine has been investigated by the measurement of the NIR and Raman spectra and complemented with infrared spectra including their interpretation. The spectra of the rutherfordine show the presence of both water and hydroxyl units in the structure as evidenced by IR bands at 3562 and 3465 cm-1 (OH) and 3343, 3185 and 2980 cm-1 (H2O). Raman spectra show the presence of four sharp bands at 3511, 3460, 3329 and 3151 cm-1. Corresponding molecular water bending vibrations were only observed in both Raman and infrared spectra of one of two studied rutherfordine samples. The second rutherfordine sample studied contained only hydroxyl ions in the equatorial uranyl plane and did not contain any molecular water. The infrared spectra of the (CO3)2- units in the antisymmetric stretching region show complexity with three sets of carbonate bands observed. This combined with the observation of multiple bands in the (CO3)2- bending region in both the Raman and IR spectra suggests that both monodentate and bidentate (CO3)2- units may be present in the structure. This cannot be exactly proved and inferred from the spectra; however, it is in accordance with the X-ray crystallographic studies. Complexity is also observed in the IR spectra of (UO2)2+ antisymmetric stretching region and is attributed to non-identical UO bonds. U-O bond lengths were calculated using wavenumbers of the 3 and 1 (UO2)2+ and compared with data from X-ray single crystal structure analysis of rutherfordine. Existence of solid solution having a general formula (UO2)(CO3)1-x(OH)2x.yH2O ( x, y  0) is supported in the crystal structure of rutherfordine samples studied.

Authentic and conventional assessment in Singapore schools : an empirical study of teacher assignments and student work

This is an empirical examination of the quality of teacher assignments and student work in Singapore schools. Using a theoretical framework based on principles of authentic assessment and intellectual quality, two sets of criteria and scoring rubrics were developed for the training of expert teachers to judge the quality of assignments and student work. Following rigorous training, the inter-rater reliability of expert teacher scoring was high. Samples of teacher assignments and student work were collected in English, social studies, mathematics, and science subject areas from a random stratified sample of 30 elementary schools and 29 high schools. For both grade levels, there were significant differences for the authentic intellectual quality of teachers’ assignments by subject area. Likewise, the differences of authentic intellectual quality for student work were significant and varied by subject area. Subject area effect was large. The correlations between the quality of teachers’ assignment tasks and student work were strong and significant at both grade levels. Where teachers set more intellectually demanding tasks, students were more likely to generate work or artefacts judged to be of higher quality. The findings suggest that teacher professional development in authentic intellectual assessment task design can contribute to the improvement of student learning and performance. It is argued that this will be a key requisite of educational systems like Singapore that are seeking to expand pedagogy and student outcomes beyond a focus on factual and rote knowledge.