Isometries between quantum convolution algebras

Given locally compact quantum groups \G_1 and \G_2, we show that if the convolution algebras L^1(\G_1) and L^1(\G_2) are isometrically isomorphic as algebras, then \G_1 is isomorphic either to \G_2 or the commutant \G_2'. Furthermore, given an isometric algebra isomorphism \theta:L^1(\G_2) \rightarrow L^1(\G_1), the adjoint is a *-isomorphism between L^\infty(\G_1) and either L^\infty(\G_2) or its commutant, composed with a twist given by a member of the intrinsic group of L^\infty(\G_2). This extends known results for Kac algebras (although our proofs are somewhat different) which in turn generalised classical results of Wendel and Walter. We show that the same result holds for isometric algebra homomorphisms between quantum measure algebras (either reduced or universal). We make some remarks about the intrinsic groups of the enveloping von Neumann algebras of C$^*$-algebraic quantum groups.Comment: 23 pages, typos corrected, references adde

A general theorem of existence of quasi absolutely minimal Lipschitz extensions

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier "quasi" to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.Comment: 33 pages. v3: Correction to Example 2.4.3. Specifically, alpha-H\"older continuous functions, for alpha strictly less than one, do not satisfy (P3). Thus one cannot conclude that quasi-AMLEs exist in this case. Please note that the error remains in the published version of the paper in Mathematische Annalen. v2: Several minor corrections and edits, a new appendix (Appendix A

Preduals of semigroup algebras

For a locally compact group $G$, the measure convolution algebra $M(G)$ carries a natural coproduct. In previous work, we showed that the canonical predual $C_0(G)$ of $M(G)$ is the unique predual which makes both the product and the coproduct on $M(G)$ weak$^*$-continuous. Given a discrete semigroup $S$, the convolution algebra $\ell^1(S)$ also carries a coproduct. In this paper we examine preduals for $\ell^1(S)$ making both the product and the coproduct weak$^*$-continuous. Under certain conditions on $S$, we show that $\ell^1(S)$ has a unique such predual. Such $S$ include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on $\ell^1(S)$ when $S$ is either $\mathbb Z_+\times\mathbb Z$ or $(\mathbb N,\cdot)$.Comment: 17 pages, LaTe

Adaptive Element-Free Galerkin method applied to the limit analysis of plates

The implementation of an h-adaptive Element-Free Galerkin (EFG) method in the framework of limit analysis is described. The naturally conforming property of mesh- free approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. Nodes may be moved, discarded or introduced without the need for complex manipulation of the data structures involved. With the use of the Taylor expansion technique, the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy. A stabilized conforming nodal integration scheme is extended to error estimators and results in an efficient and truly meshfree adaptive method. To demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions

Learning a local-variable model of aromatic and conjugated systems

A collection of new approaches to building and training neural networks, collectively referred to as deep learning, are attracting attention in theoretical chemistry. Several groups aim to replace computationally expensive <i>ab initio</i> quantum mechanics calculations with learned estimators. This raises questions about the representability of complex quantum chemical systems with neural networks. Can local-variable models efficiently approximate nonlocal quantum chemical features? Here, we find that convolutional architectures, those that only aggregate information locally, cannot efficiently represent aromaticity and conjugation in large systems. They cannot represent long-range nonlocality known to be important in quantum chemistry. This study uses aromatic and conjugated systems computed from molecule graphs, though reproducing quantum simulations is the ultimate goal. This task, by definition, is both computable and known to be important to chemistry. The failure of convolutional architectures on this focused task calls into question their use in modeling quantum mechanics. To remedy this heretofore unrecognized deficiency, we introduce a new architecture that propagates information back and forth in waves of nonlinear computation. This architecture is still a local-variable model, and it is both computationally and representationally efficient, processing molecules in sublinear time with far fewer parameters than convolutional networks. Wave-like propagation models aromatic and conjugated systems with high accuracy, and even models the impact of small structural changes on large molecules. This new architecture demonstrates that some nonlocal features of quantum chemistry can be efficiently represented in local variable models

Partial Aborts for Transactions via First Class Continuations

Software transactional memory (STM) has proven to be a useful abstraction for developing concurrent applications where programmers denote transactions with an atomic construct that delimits a collection of reads and writes to shared mutable references. The runtime system then guarantees that all transactions are observed to execute atomically with respect to each other. Traditionally, when the runtime system detects that one transaction conflicts with another, it aborts one of the transactions and restarts its execution from the beginning. This can lead to problems with both execution time and throughput. This thesis presents a novel approach that uses first-class continuations to restart a conflicting transaction at the point of a conflict, avoiding the re-execution of any work from the beginning of the transaction that has not been compromised. In practice, this allows transactions to complete more quickly, decreasing execution time and increasing throughput. The ideas presented in this thesis have been implemented in the context of the Manticore project, an ML-family language with support for parallelism and concurrency. Crucially, this work relies on constant-time continuation capturing via a continuation-passing-style (CPS) transformation and heap-allocated continuations. The partial abort scheme has been implemented as a part of three modern STM implementations: TL2, TinySTM, and NOrec. Experimental results show that, while no base STM implementation is universally best, each partial-abort implementation compares favorably to its full-abort counterpart. In addition to an implementation, this thesis presents a formal semantics for partial aborts. A proof of correctness is given by relating the partial-abort semantics to an analogous full-abort semantics via a simulation. All proofs have been formally verified using the Coq Theorem Prover