K-Theory of Azumaya Algebras over Schemes

Let X be a connected, noetherian scheme and A{script} be a sheaf of Azumaya algebras on X, which is a locally free O{script}-module of rank a. We show that the kernel and cokernel of K(X) ? K(A{script}) are torsion groups with exponent a for some m and any i = 0, when X is regular or X is of dimension d with an ample sheaf (in this case m = d + 1). As a consequence, K(X, Z/m) ? K(A{script}, Z/m), for any m relatively prime to a. © 2013 Copyright Taylor and Francis Group, LLC.

The Engine of Thought Is a Hybrid: Roles of Associative and Structured Knowledge in Reasoning

Across a range of domains in psychology different theories assume different mental representations of knowledge. For example, in the literature on category-based inductive reasoning, certain theories (e.g., Rogers & McClelland, 2004; Sloutsky & Fisher, 2008) assume that the knowledge upon which inductive inferences are based is associative, whereas others (e.g., Heit & Rubinstein, 1994; Kemp & Tenenbaum, 2009; Osherson, Smith, Wilkie, López, & Shafir, 1990) assume that knowledge is structured. In this article we investigate whether associative and structured knowledge underlie inductive reasoning to different degrees under different processing conditions. We develop a measure of knowledge about the degree of association between categories and show that it dissociates from measures of structured knowledge. In Experiment 1 participants rated the strength of inductive arguments whose categories were either taxonomically or causally related. A measure of associative strength predicted reasoning when people had to respond fast, whereas causal and taxonomic knowledge explained inference strength when people responded slowly. In Experiment 2, we also manipulated whether the causal link between the categories was predictive or diagnostic. Participants preferred predictive to diagnostic arguments except when they responded under cognitive load. In Experiment 3, using an open-ended induction paradigm, people generated and evaluated their own conclusion categories. Inductive strength was predicted by associative strength under heavy cognitive load, whereas an index of structured knowledge was more predictive of inductive strength under minimal cognitive load. Together these results suggest that associative and structured models of reasoning apply best under different processing conditions and that the application of structured knowledge in reasoning is often effortful.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Ideals of A(G) and bimodules over maximal abelian selfadjoint algebras

This paper is concerned with weak⁎ closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of B(L2(G)) which are invariant under both Schur multipliers and a canonical action of M(G) on B(L2(G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.

A class of almost C_0(K)-C*-algebras

We consider in this paper the family of exponential Lie groups Gn,µ, whose Lie algebra is an extension of the Heisenberg Lie algebra by the reals and whose quotient group by the centre of the Heisenberg group is an ax + b-like group. The C*-algebras of the groups Gn,µ give new examples of almost C0(K)-C*-algebras.

Norms of basic elementary operators on algebras of regular operators

We show that if E is an atomic Banach lattice with an ordercontinuous norm, A, B ∈ L^r(E) and M_A,_B is the operator on L^r(E) defined by M_A,_B(T) = AT B then ||M_A,_B||r = ||A||_r||B||_r but that there is no real α > 0 such that ||MA,B || ≥ α ||A||r||B ||r.

Two dimensional unital Riesz algebras, their representations and norms

We describe all two dimensional unital Riesz algebras and study representations of them in Riesz algebras of regular operators. Although our results are not complete, we do demonstrate that very varied behaviour can occur even though all these algebras can be given a Banach lattice algebra norm.

Paired associative transcranial alternating current stimulation increases the excitability of corticospinal projections in humans

Many types of non-invasive brain stimulation alter corticospinal excitability (CSE). Paired associative stimulation (PAS) has attracted particular attention as its effects ostensibly adhere to Hebbian principles of neural plasticity. In prototypical form, a single electrical stimulus is directed to a peripheral nerve in close temporal contiguity with transcranial magnetic stimulation delivered to the contralateral primary motor cortex (M1). Repeated pairing of the two discrete stimulus events (i.e. association) over an extended period either increases or decreases the excitability of corticospinal projections from M1, contingent on the interstimulus interval. We studied a novel form of associative stimulation, consisting of brief trains of peripheral afferent stimulation paired with short bursts of high frequency (≥80 Hz) transcranial alternating current stimulation (tACS) over contralateral M1. Elevations in the excitability of corticospinal projections to the forearm were observed for a range of tACS frequency (80, 140 and 250 Hz), current (1, 2 and 3 mA) and duration (500 and 1000 ms) parameters. The effects were at least as reliable as those brought about by PAS or transcranial direct current stimulation. When paired with tACS, muscle tendon vibration also induced elevations of CSE. No such changes were brought about by the tACS or peripheral afferent stimulation alone. In demonstrating that associative effects are expressed when the timing of the peripheral and cortical events is not precisely circumscribed, these findings suggest that multiple cellular pathways may contribute to a long term potentiation-type response. Their relative contributions will differ depending on the nature of the induction protocol that is used.

Modulation of human corticospinal excitability by paired associative stimulation

Paired Associative Stimulation (PAS) has come to prominence as a potential therapeutic intervention for the treatment of brain injury/disease, and as an experimental method with which to investigate Hebbian principles of neural plasticity in humans. Prototypically, a single electrical stimulus is directed to a peripheral nerve in advance of transcranial magnetic stimulation (TMS) delivered to the contralateral primary motor cortex (M1). Repeated pairing of the stimuli (i.e., association) over an extended period may increase or decrease the excitability of corticospinal projections from M1, in manner that depends on the interstimulus interval (ISI). It has been suggested that these effects represent a form of associative long-term potentiation (LTP) and depression (LTD) that bears resemblance to spike-timing dependent plasticity (STDP) as it has been elaborated in animal models. With a large body of empirical evidence having emerged since the cardinal features of PAS were first described, and in light of the variations from the original protocols that have been implemented, it is opportune to consider whether the phenomenology of PAS remains consistent with the characteristic features that were initially disclosed. This assessment necessarily has bearing upon interpretation of the effects of PAS in relation to the specific cellular pathways that are putatively engaged, including those that adhere to the rules of STDP. The balance of evidence suggests that the mechanisms that contribute to the LTP- and LTD-type responses to PAS differ depending on the precise nature of the induction protocol that is used. In addition to emphasizing the requirement for additional explanatory models, in the present analysis we highlight the key features of the PAS phenomenology that require interpretation.

Using Graph Components Derived from an Associative Concept Dictionary to Predict fMRI Neural Activation Patterns that Represent the Meaning of Nouns

In this study, we introduce an original distance definition for graphs, called the Markov-inverse-F measure (MiF). This measure enables the integration of classical graph theory indices with new knowledge pertaining to structural feature extraction from semantic networks. MiF improves the conventional Jaccard and/or Simpson indices, and reconciles both the geodesic information (random walk) and co-occurrence adjustment (degree balance and distribution). We measure the effectiveness of graph-based coefficients through the application of linguistic graph information for a neural activity recorded during conceptual processing in the human brain. Specifically, the MiF distance is computed between each of the nouns used in a previous neural experiment and each of the in-between words in a subgraph derived from the Edinburgh Word Association Thesaurus of English. From the MiF-based information matrix, a machine learning model can accurately obtain a scalar parameter that specifies the degree to which each voxel in (the MRI image of) the brain is activated by each word or each principal component of the intermediate semantic features. Furthermore, correlating the voxel information with the MiF-based principal components, a new computational neurolinguistics model with a network connectivity paradigm is created. This allows two dimensions of context space to be incorporated with both semantic and neural distributional representations.

Spectral isometries into commutative Banach algebras

We determine the structure of spectral isometries between unital Banach algebras under the hypothesis that the codomain is commutative.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).