Isometries of the product of composition operators on weighted Bergman space

In this paper, the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on [Formula: see text] and [Formula: see text] the composition operator on [Formula: see text] induced by an analytic self-map on [Formula: see text] with fixed origin need not be of norm one. We have generalized the Schwartz’s [Composition operators on [Formula: see text], thesis, University of Toledo (1969)] well-known result on [Formula: see text] which characterizes the almost multiplicative operator on [Formula: see text]

Autotuning Parallel Programs by Model Checking

The paper presents a new approach to autotuning data-parallel programs. Autotuning is a search for optimal program settings which maximize its performance. The novelty of the approach lies in the use of the model checking method to find the optimal tuning parameters by the method of counterexamples. In our work, we abstract from specific programs and specific processors by defining their representative abstract patterns. Our method of counterexamples implements the following four steps. At the first step, an execution model of an abstract program on an abstract processor is described in the language of a model checking tool. At the second step, in the language of the model checking tool, we formulate the optimality property that depends on the constructed model. At the third step, we find the optimal values of the tuning parameters by using a counterexample constructed during the verification of the optimality property. In the fourth step, we extract the information about the tuning parameters from the counter-example for the optimal parameters. We apply this approach to autotuning parallel programs written in OpenCL, a popular modern language that extends the C language for programming both standard multi-core processors (CPUs) and massively parallel graphics processing units (GPUs). As a verification tool, we use the SPIN verifier and its model representation language Promela, whose formal semantics is good for modelling the execution of parallel programs on processors with different architectures.

Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on X. For X=c0, we introduce ε-slice G-analytic functions that have a behavior similar to G-analytic functions at points x∈c0 such that all coordinates of x are greater than ε, and we prove a theorem on approximations of uniformly continuous functions on c0 by ε-slice G-analytic functions.

On -genericity of distributional chaos

Let M be a compact smooth manifold without boundary. Based on results by Good and Meddaugh [Invent. Math.220 (2020), 715–736], we prove that a strong distributional chaos is $C^0$ -generic in the space of continuous self-maps (respectively, homeomorphisms) of M. The results contain answers to questions by Li, Li and Tu [Chaos26 (2016), 093103] and Moothathu [Topology Appl.158 (2011), 2232–2239] in the zero-dimensional case. A related counter-example on the chain components under shadowing is also given.

Structure of super strongly perfect graphs

Super strongly perfect graphs and their association with certain other classes of graphs are discussed in this paper. It is observed that every split graph is super strongly perfect. An existing result on super strongly perfect graphs is disproved providing a counter example. It is also established that if a graph [Formula: see text] contains a cycle of odd length, then its line graph [Formula: see text] is not always super strongly perfect. Complements of cycles of length six or above are proved to be non-super strongly perfect. If a graph is strongly perfect, then it is shown that they are super strongly perfect and hence all [Formula: see text]-free graphs are super strongly perfect.

Insights from Operando and Identical Location (IL) Techniques on the Activation of Electrocatalysts for the Conversion of CO2: A Mini-Review

In this mini-review we compare two prototypical metal foam electrocatalysts applied to the transformation of CO2 into value-added products (e.g. alcohols on Cu foams and formate on Bi foams). A substantial improvement in the catalyst performance is typically achieved through thermal annealing of the as-deposited foam materials, followed by the electro-reduction of the pre-formed oxidic precursors prior or during the actual CO2 electrolysis. Utilizing highly insightful and sensitive complementary operando analytical techniques (XAS, XRD, and Raman spectroscopy) we demonstrate that this catalyst pre-activation process is entirely accomplished in case of the oxidized Cu foams prior to the formation of hydrocarbons and alcohols from the CO2. The actually active catalyst is therefore the metallic Cu derived from the precursor by means of oxide electro-reduction. Conversely, in their oxidic form, the Cu-based foam catalysts are inactive towards the CO2 reduction reaction (denoted ec-CO2 RR). Oxidized Bi foams can be regarded as an excellent counter example to the above-mentioned Cu case as both metallic and the thermally derived oxidic Bi foams are highly active towards ec-CO2 RR (formate production). Indeed, operando Raman spectroscopy reveals that CO2 electrolysis occurs upon its embedment into the oxidic Bi2O3 foam precursor, which itself undergoes partial transformation into an active sub-carbonate phase. The potential-dependent transition of sub-carbonates/oxides into the corresponding metallic Bi foam dictates the characteristic changes of the ec-CO2 RR pathway. Identical location (IL) microscopic inspection of the catalyst materials, e.g. by means of scanning electron microscopy, demonstrates substantial morphological alterations on the nm length scale on the material surface as consequence of the sub-carbonate formation and the potential-driven oxide reduction into the metallic Bi foam. The foam morphology on a mesoscopic length scale (macroporosity) remains, by contrast, fully unaffected by these phase transitions.

Evidence for and against Zauner's MUB conjecture in C^6

The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case,~$d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~$d=6$ by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~$q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

A Detailed Account of The Inconsistent Labelling Problem of Stutter-Preserving Partial-Order Reduction

One of the most popular state-space reduction techniques for model checking is partial-order reduction (POR). Of the many different POR implementations, stubborn sets are a very versatile variant and have thus seen many different applications over the past 32 years. One of the early stubborn sets works shows how the basic conditions for reduction can be augmented to preserve stutter-trace equivalence, making stubborn sets suitable for model checking of linear-time properties. In this paper, we identify a flaw in the reasoning and show with a counter-example that stutter-trace equivalence is not necessarily preserved. We propose a stronger reduction condition and provide extensive new correctness proofs to ensure the issue is resolved. Furthermore, we analyse in which formalisms the problem may occur. The impact on practical implementations is limited, since they all compute a correct approximation of the theory. Comment: arXiv admin note: substantial text overlap with arXiv:1910.09829

The Annihilation of Space: A Bad (Historical) Concept

This article is a reminder that the concept of ‘annihilation of space’ or ‘spatial compression’, often used as a shorthand for referring to the cultural or economic consequences of industrial mobility, has a long intellectual history. The concept thus comes loaded with a specific outlook on the experience of modernity, which is – I argue – unsuitable for any cultural or social history of space. This article outlines the etymology of the concept and shows: first, that the historical phenomena it pretends to describe are too complex for such a simplistic signpost; and, second, that the term is never a neutral descriptor but always an engagement with a form of historical and cultural mediation on the nature of modernity in relation to space. In both cases this term obfuscates more than it reveals. As a counter-example, I look at the effect of the railways on popular representations of space and conclude that postmodern geography is a relative dead end for historians interested in the social and cultural history of space.