The partially alternating ternary sum in an associative dialgebra

The alternating ternary sum in an associative algebra, $abc - acb - bac + bca + cab - cba$, gives rise to the partially alternating ternary sum in an associative dialgebra with products $\dashv$ and $\vdash$ by making the argument $a$ the center of each term: $a \dashv b \dashv c - a \dashv c \dashv b - b \vdash a \dashv c + c \vdash a \dashv b + b \vdash c \vdash a - c \vdash b \vdash a$. We use computer algebra to determine the polynomial identities in degree $\le 9$ satisfied by this new trilinear operation. In degrees 3 and 5 we obtain $[a,b,c] + [a,c,b] \equiv 0$ and $[a,[b,c,d],e] + [a,[c,b,d],e] \equiv 0$; these identities define a new variety of partially alternating ternary algebras. We show that there is a 49-dimensional space of multilinear identities in degree 7, and we find equivalent nonlinear identities. We use the representation theory of the symmetric group to show that there are no new identities in degree 9.Comment: 14 page

A formula for the First Eigenvalue of the Dirac Operator on Compact Spin Symmetric Spaces

Let $G/K$ be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of $G$ sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of $G$. As an example of application, we give the list of the first eigenvalues for the spin compact irreducible symmetric spaces endowed with a quaternion-K\"{a}hler structure

School gardens and the school food plan: contributing to a culture of healthy living

Significantly revised and updated the second edition now includes 7 brand new chapters on: · Methods of Assessment and Evaluation · Global perspectives on Outdoor Learning · Developing whole school approaches to indoor and outdoor ..

Four types of special functions of G_2 and their discretization

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G2, are compared and described. Two of the four families (called here C- and S-functions) are well known, whereas the other two (S^L- and S^S-functions) are not found elsewhere in the literature. It is shown explicitly that all four families have similar properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space, and they are discretely orthogonal when their values, sampled at the lattice points F_M \subset F, are added up with a weight function appropriate for each family. Products of ten types among the four families of functions, namely CC, CS, SS, SS^L, CS^S, SS^L, SS^S, S^SS^S, S^LS^S and S^LS^L, are completely decomposable into the finite sum of the functions. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.Comment: 18 pages, 4 figures, 4 table

Overcoming the uncanny valley: Displays of emotions reduce the uncanniness of humanlike robots

In this paper we show empirically that highly humanlike robots make thoughts of death more accessible, leading to perceptions of uncanniness and eeriness of such robots. Rather than reducing the humanlikeness of robots, our research suggests the addition of emotion displays to decrease a sense of uncanniness. We show that a highly humanlike robot displaying emotions in a social context reduces death-thought accessibility (DTA), which in turn reduces uncanniness. In a pre-test with N = 95 participants, we established that not all humanoid robots elicit thoughts of death and that the extent to which a robot appears humanlike may be linked to DTA. In our Main Study, N = 44 participants briefly interacted with a highly humanlike robotic head that either showed appropriate basic emotions or reacted by blinking. The display of emotions significantly reduced perceptions of uncanniness, which was mediated by a corresponding reduction in DTA. Implications for the design of humanoid robots are proposed.EPSR

Non-adaptive Measurement-based Quantum Computation and Multi-party Bell Inequalities

Quantum correlations exhibit behaviour that cannot be resolved with a local hidden variable picture of the world. In quantum information, they are also used as resources for information processing tasks, such as Measurement-based Quantum Computation (MQC). In MQC, universal quantum computation can be achieved via adaptive measurements on a suitable entangled resource state. In this paper, we look at a version of MQC in which we remove the adaptivity of measurements and aim to understand what computational abilities still remain in the resource. We show that there are explicit connections between this model of computation and the question of non-classicality in quantum correlations. We demonstrate this by focussing on deterministic computation of Boolean functions, in which natural generalisations of the Greenberger-Horne-Zeilinger (GHZ) paradox emerge; we then explore probabilistic computation, via which multipartite Bell Inequalities can be defined. We use this correspondence to define families of multi-party Bell inequalities, which we show to have a number of interesting contrasting properties.Comment: 13 pages, 4 figures, final version accepted for publicatio

String Branchings on Complex Tori and Algebraic Representations of Generalized Krichever-Novikov Algebras

The propagation differential for bosonic strings on a complex torus with three symmetric punctures is investigated. We study deformation aspects between two point and three point differentials as well as the behaviour of the corresponding Krichever-Novikov algebras. The structure constants are calculated and from this we derive a central extension of the Krichever-Novikov algebras by means of b-c systems. The defining cocycle for this central extension deforms to the well known Virasoro cocycle for certain kinds of degenerations of the torus. AMS subject classification (1991): 17B66, 17B90, 14H52, 30F30, 81T40Comment: 11 pages, amste

Outcome Independence of Entanglement in One-Way Computation

We show that the various intermediate states appearing in the process of one-way computation at a given step of measurement are all equivalent modulo local unitary transformations. This implies, in particular, that all those intermediate states share the same entanglement irrespective of the measurement outcomes, indicating that the process of one-way computation is essentially unique with respect to local quantum operations.Comment: 6 pages, 4 figure

Centralizers of maximal regular subgroups in simple Lie groups and relative congruence classes of representations

In the paper we present a new, uniform and comprehensive description of centralizers of the maximal regular subgroups in compact simple Lie groups of all types and ranks. The centralizer is either a direct product of finite cyclic groups, a continuous group of rank 1, or a product, not necessarily direct, of a continuous group of rank 1 with a finite cyclic group. Explicit formulas for the action of such centralizers on irreducible representations of the simple Lie algebras are given.Comment: 27 page