GraphAttack

Graph structures are a natural representation of important and pervasive data. While graph applications have significant parallelism, their characteristic pointer indirect loads to neighbor data hinder scalability to large datasets on multicore systems. A scalable and efficient system must tolerate latency while leveraging data parallelism across millions of vertices. Modern Out-of-Order (OoO) cores inherently tolerate a fraction of long latencies, but become clogged when running severely memory-bound applications. Combined with large power/area footprints, this limits their parallel scaling potential and, consequently, the gains that existing software frameworks can achieve. Conversely, accelerator and memory hierarchy designs provide performant hardware specializations, but cannot support diverse application demands. To address these shortcomings, we present GraphAttack, a hardware-software data supply approach that accelerates graph applications on in-order multicore architectures. GraphAttack proposes compiler passes to (1) identify idiomatic long-latency loads and (2) slice programs along these loads into data Producer/ Consumer threads to map onto pairs of parallel cores. Each pair shares a communication queue; the Producer asynchronously issues long-latency loads, whose results are buffered in the queue and used by the Consumer. This scheme drastically increases memory-level parallelism (MLP) to mitigate latency bottlenecks. In equal-area comparisons, GraphAttack outperforms OoO cores, do-all parallelism, prefetching, and prior decoupling approaches, achieving a 2.87× speedup and 8.61× gain in energy efficiency across a range of graph applications. These improvements scale; GraphAttack achieves a 3× speedup over 64 parallel cores. Lastly, it has pragmatic design principles; it enhances in-order architectures that are gaining increasing open-source support.

On the derivative formulas of the rotation minimizing frame in Lorentz–Minkowski 3-space

The main intention of this paper is to analyze integrability for the derivative formulas of the rotation minimizing frame in the Lorentz–Minkowski 3-space. As far as we know, no one has yet given a method to study their integrability in the Lorentz–Minkowski 3-space. So, we introduce the coordinate system in order to provide a tool for studying the integrability. As an application, the position vectors of some special curves having an important place in mathematical and physical research are obtained in the natural representation form. Finally, we support our work with examples.

On the realization of the Gelfand character of a finite group as a twisted trace

We show that the Gelfand character χ G \chi_{G} of a finite group 𝐺 (i.e. the sum of all irreducible complex characters of 𝐺) may be realized as a “twisted trace” g ↦ Tr ⁡ ( ρ g ∘ T ) g\mapsto\operatorname{Tr}(\rho_{g}\circ T) for a suitable involutive linear automorphism 𝑇 of L 2 ⁢ ( G ) L^{2}(G) , where ( L 2 ⁢ ( G ) , ρ ) (L^{2}(G),\rho) is the right regular representation of 𝐺. Moreover, we prove that, under certain hypotheses, we have T ⁢ ( f ) = f ∘ L T(f)=f\circ L ( f ∈ L 2 ⁢ ( G ) f\in L^{2}(G) ), where 𝐿 is an involutive anti-automorphism of 𝐺. The natural representation 𝜏 of 𝐺 associated to the natural 𝐿-conjugacy action of 𝐺 in the fixed point set Fix G ⁡ ( L ) \operatorname{Fix}_{G}(L) of 𝐿 turns out to be a Gelfand model for 𝐺 in some cases. We show that ( L 2 ⁢ ( Fix G ⁡ ( L ) ) , τ ) (L^{2}(\operatorname{Fix}_{G}(L)),\tau) fails to be a Gelfand model if 𝐺 admits non-trivial central involutions.

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

The Calculus of Natural Calculation

The calculus of Natural Calculation is introduced as an extension of Natural Deduction by proper term rules. Such term rules provide the capacity of dealing directly with terms in the calculus instead of the usual reasoning based on equations, and therefore the capacity of a natural representation of informal mathematical calculations. Basic proof theoretic results are communicated, in particular completeness and soundness of the calculus; normalisation is briefly investigated. The philosophical impact on a proof theoretic account of the notion of meaning is considered.

A Natural Representation of Functions for Exact Learning

We present a collection of mathematical tools and emphasise a fundamental representation of analytic functions. Connecting these concepts leads to a framework for `exact learning', where an unknown numeric distribution could in principle be assigned an exact mathematical description. This is a new perspective on machine learning with potential applications in all domains of the mathematical sciences and the generalised representations presented here have not yet been widely considered in the context of machine learning and data analysis. The moments of a multivariate function or distribution are extracted using a Mellin transform and the generalised form of the coefficients is trained assuming a highly generalised Mellin-Barnes integral representation. The functions use many fewer parameters than contemporary machine learning methods and any implementation that connects these concepts successfully will likely carry across to non-exact problems and provide approximate solutions. We compare the equations for the exact learning method with those for a neural network which leads to a new perspective on understanding what a neural network may be learning and how to interpret the parameters of those networks.

p-adic numbers encode complex networks

The Erdős-Rényi (ER) random graph G(n, p) analytically characterizes the behaviors in complex networks. However, attempts to fit real-world observations need more sophisticated structures (e.g., multilayer networks), rules (e.g., Achlioptas processes), and projections onto geometric, social, or geographic spaces. The p-adic number system offers a natural representation of hierarchical organization of complex networks. The p-adic random graph interprets n as the cardinality of a set of p-adic numbers. Constructing a vast space of hierarchical structures is equivalent for combining number sequences. Although the giant component is vital in dynamic evolution of networks, the structure of multiple big components is also essential. Fitting the sizes of the few largest components to empirical data was rarely demonstrated. The p-adic ultrametric enables the ER model to simulate multiple big components from the observations of genetic interaction networks, social networks, and epidemics. Community structures lead to multimodal distributions of the big component sizes in networks, which have important implications in intervention of spreading processes.

Nonnatural Mental Representation

This chapter distinguishes between two types of representation, natural and nonnatural. It argues that nonnatural representation is necessary to explain intentionality. It also argues that traditional accounts of the semantic content of mental representations are insufficient to explain nonnatural representation and, therefore, intentionality. To remedy this, the chapter sketches an account of nonnatural representation in terms of natural representation plus offline simulation of nonactual environments plus tracking the ways in which a simulation departs from the actual environment. To represent nonnaturally, a system must be able to decouple internal simulations from sensory information by activating representational resources offline. The system must be able to represent things that are not in the actual environment and to track that it’s doing so; i.e., there must be an internal signal or state that can indicate whether what is represented departs from the actual environment. In addition, the system must be able to manipulate a representation independently of what happens in the actual environment and keep track that it’s doing so. In short, nonnatural representations are offline simulations whose departure from the actual environment the system has the function to keep track of. This is a step toward a naturalistic, mechanistic, neurocomputational account of intentionality.

A Weighted-Logic Representation of C-Revising Ordinal Conditional Functions

The problem of belief change is considered as a major issue in managing the dynamics of an information system. It consists in modifying an uncertainty distribution, representing agents’ beliefs, in the light of a new information. In this paper, we focus on the so-called multiple iterated belief revision or C-revision, proposed for conditioning or revising uncertain distributions under uncertain inputs. Uncertainty distributions are represented in terms of ordinal conditional functions. We will use prioritized or weighted knowledge bases as a compact representation of uncertainty distributions. The input information leading to a revision of an uncertainty distribution is also represented by a set of consistent weighted formulas. This paper shows that C-revision, defined at a semantic level using ordinal conditional functions, has a very natural representation using weighted knowledge bases. We propose simple syntactic methods for revising weighted knowledge bases, that are semantically meaningful in the frameworks of possibility theory and ordinal conditional functions. In particular, we show that the space complexity of the proposed syntactic C-revision is linear with respect to the size of initial weighted knowledge bases.