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The C++ programming language and its cousins lean towards a memory-inefficient storage of structs: The compiler inserts helper bits into the struct such that individual attributes align with bytes, and it adds additional bytes aligning attributes with cache lines, while it is not able to exploit knowledge about the range of integers, enums or bitsets to bring the memory footprint down. Furthermore, the language provides neither support for data exchange via MPI nor for arbitrary floating-point precision formats. If developers need to have a low memory footprint and MPI datatypes over structs which exchange only minimal data, they have to manipulate the data and to write MPI datatypes manually. We propose a C++ language extension based upon C++ attributes through which developers can guide the compiler what memory arrangements would be beneficial: Can multiple booleans be squeezed into one bit field, do floats hold fewer significant bits than in the IEEE standard, or does the code require a user-defined MPI datatype for certain subsets of attributes? The extension offers the opportunity to fall back to normal alignment and padding rules via plain C++ assignments, no dependencies upon external libraries are introduced, and the resulting code remains standard C++. Our work implements the language annotations within LLVM and demonstrates their potential impact, both upon the runtime and the memory footprint, through smoothed particle hydrodynamics (SPH) benchmarks. They uncover the potential gains in terms of performance and development productivity.

In this paper, we characterize supercyclic weighted composition operators on various function spaces. Moreover, we also characterize supercyclic adjoint of the weighted composition operator and supercyclic left multipliers acting on the space of compact operators. Finally, we illustrate our results with concrete examples. In addition, we introduce a class of non-porous subsets of the space of continuous functions on the real line vanishing at infinity. As an application, we consider the weighted composition operator on this space and we give sufficient conditions that ensure that the set of non-hypercyclic vectors for this operator is non-porous.

In this paper, we provide a representation of a certain class of C*-valued positive sesquilinear and linear maps on non-unital quasi *-algebras. Also, we illustrate our results on the concrete examples of non-unital Banach quasi *-algebras, such as the standard Hilbert module over a commutative C*-algebra, Schatten p-ideals, and noncommutative L2 spaces induced by a semifinite, nonfinite trace. As a consequence of our results, we obtain a representation of all bounded positive linear C*-valued maps on non-unital C*-algebras. We also deduce some norm inequalities for these maps. Finally, we consider a noncommutative L2 space equipped with the topology generated by a positive sesquilinear form and we construct a topologically transitive operator on this space., Comment: This version of the article has been accepted for publication, after peer review (when applicable) but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s11117-024-01079-8

The GNS construction for positive invariant sesquilinear forms on quasi *-algebras is generalized to a class of positive C*-valued sesquilinear maps on quasi *-algebras. The result is a *-representation taking values in a space of operators acting on a certain quasi-normed C-module., Comment: This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of the Record of this article is published in Mediterranean Journal of Mathematics and is available online at https://doi.org/10.1007/s00009-024-02704-2

In this paper, we extend Fredholm theory in von Neumann algebras established by Breuer in [5] and [6] to spectral Fredholm theory. We consider 2 by 2 upper triangular operator matrices with coefficients in a von Neumann algebra and give the relationship between the generalized essential spectra in the sense of Breuer of such matrices and of their diagonal entries. Next, we prove that if a generalized Fredholm operator in the sense of Breuer has 0 as an isolated point of its spectrum, then the corresponding spectral projection is finite. Finally, we define the generalized B-Fredholm operator in a von Neumann algebra as a generalization in the sense of Breuer of the classical B-Fredholm operators on Hilbert and Banach spaces. We provide sufficient conditions under which a sum of a generalized B-Fredholm operator and a finite operator in a von Neumann algebra is again a generalized B-Fredholm operator.

This research explores the reliability of deep learning, specifically Long Short-Term Memory (LSTM) networks, for estimating the Hurst parameter in fractional stochastic processes. The study focuses on three types of processes: fractional Brownian motion (fBm), fractional Ornstein-Uhlenbeck (fOU) process, and linear fractional stable motions (lfsm). The work involves a fast generation of extensive datasets for fBm and fOU to train the LSTM network on a large volume of data in a feasible time. The study analyses the accuracy of the LSTM network's Hurst parameter estimation regarding various performance measures like RMSE, MAE, MRE, and quantiles of the absolute and relative errors. It finds that LSTM outperforms the traditional statistical methods in the case of fBm and fOU processes; however, it has limited accuracy on lfsm processes. The research also delves into the implications of training length and valuation sequence length on the LSTM's performance. The methodology is applied by estimating the Hurst parameter in Li-ion battery degradation data and obtaining confidence bounds for the estimation. The study concludes that while deep learning methods show promise in parameter estimation of fractional processes, their effectiveness is contingent on the process type and the quality of training data.

In this paper, we study the dynamics of the adjoint of a weighted composition operator and we give necessary and sufficient conditions for this adjoint operator to be topologically hyper-transitive on the space of Radon measures on a locally compact Hausdorff space. Moreover, we provide sufficient conditions for this operator to be chaotic and we give concrete examples. Next, we consider the real Banach space of signed Radon measures and we give in this context the sufficient conditions for the convergence of Markov chains induced by the adjoint of an integral operator. Also, we illustrate this result with a concrete example. In addition, we obtain some structural results regarding the space of Radon measures. We characterize a class of cones whose complement is spaceable in the space of Radon measures.

Axiomatic Fredholm theory in unital C*-algebras was established by Keckic and Lazovic in [15]. Following the purely algebraic approach by Keckic and Lazovic, in [14] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [11] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and which is equivalent to the algebraic approach by Keckic and Lazovic. In this paper, we extend further this new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide a new proof of the results in [14].

Motivated by the Fredholm theory on the standard Hilbert module over an unital C*-algebra introduced by Mishchenko and Fomenko, we provide a new approach to axiomatic Fredholm theory in unital C*-algebras established by Keckic and Lazovic in [28]. Our approach is equivalent to the approach introduced by Keckic and Lazovic, however, we provide new proofs which are motivated by the proofs given by Mishchenko and Fomenko.

Numerical simulations have become one of the key tools used by theorists in all the fields of astrophysics and cosmology. The development of modern tools that target the largest existing computing systems and exploit state-of-the-art numerical methods and algorithms is thus crucial. In this paper, we introduce the fully open-source highly-parallel, versatile, and modular coupled hydrodynamics, gravity, cosmology, and galaxy-formation code SWIFT. The software package exploits hybrid shared- and distributed-memory task-based parallelism, asynchronous communications, and domain-decomposition algorithms based on balancing the workload, rather than the data, to efficiently exploit modern high-performance computing cluster architectures. Gravity is solved for using a fast-multipole-method, optionally coupled to a particle mesh solver in Fourier space to handle periodic volumes. For gas evolution, multiple modern flavours of Smoothed Particle Hydrodynamics are implemented. SWIFT also evolves neutrinos using a state-of-the-art particle-based method. Two complementary networks of sub-grid models for galaxy formation as well as extensions to simulate planetary physics are also released as part of the code. An extensive set of output options, including snapshots, light-cones, power spectra, and a coupling to structure finders are also included. We describe the overall code architecture, summarise the consistency and accuracy tests that were performed, and demonstrate the excellent weak-scaling performance of the code using a representative cosmological hydrodynamical problem with $\approx$$300$ billion particles. The code is released to the community alongside extensive documentation for both users and developers, a large selection of example test problems, and a suite of tools to aid in the analysis of large simulations run with SWIFT., Comment: 43 pages, 20 figures, accepted for publication in MNRAS. Code, documentation, and examples available at www.swiftsim.com

The development and implementation of GEAR-RT, a radiative transfer solver using the M1 closure in the open source code SWIFT, is presented, and validated using standard tests for radiative transfer. GEAR-RT is modeled after RAMSES-RT (Rosdahl et al. 2013) with some key differences. Firstly, while RAMSES-RT uses Finite Volume methods and an Adaptive Mesh Refinement (AMR) strategy, GEAR-RT employs particles as discretization elements and solves the equations using a Finite Volume Particle Method (FVPM). Secondly, GEAR-RT makes use of the task-based parallelization strategy of SWIFT, which allows for optimized load balancing, increased cache efficiency, asynchronous communications, and a domain decomposition based on work rather than on data. GEAR-RT is able to perform sub-cycles of radiative transfer steps w.r.t. a single hydrodynamics step. Radiation requires much smaller time step sizes than hydrodynamics, and sub-cycling permits calculations which are not strictly necessary to be skipped. Indeed, in a test case with gravity, hydrodynamics, and radiative transfer, the sub-cycling is able to reduce the runtime of a simulation by over 90%. Allowing only a part of the involved physics to be sub-cycled is a contrived matter when task-based parallelism is involved, and is an entirely novel feature in SWIFT. Since GEAR-RT uses a FVPM, a detailed introduction into Finite Volume methods and Finite Volume Particle Methods is presented. In astrophysical literature, two FVPM methods are written about: Hopkins (2015) have implemented one in their GIZMO code, while the one mentioned in Ivanova et al. (2013) isn't used to date. In this work, I test an implementation of the Ivanova et al. (2013) version, and conclude that in its current form, it is not suitable for use with particles which are co-moving with the fluid, which in turn is an essential feature for cosmological simulations., Comment: PhD Thesis

In this paper, we obtain several extensions of semi-Fredholm theory on Hilbert modules by generalizing in this setting their classical counterparts regarding Weyl operators and Drazin invertible operators., Comment: This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of Record of this article is published in Annals of Functional Analysis 14, 36 (2023) and is available online at https://doi.org/10.1007/s43034-023-00258-0. arXiv admin note: text overlap with arXiv:2002.04905

In this paper, we characterize hypercyclic generalized bilateral weighted shift operators on the standard Hilbert module over the C*-algebra of compact operators on the separable Hilbert space. Moreover, we give necessary and sufficient conditions for these operators to be chaotic and we provide concrete examples. In addition, we characterize a class of hypercyclic operators on non-unital C*-algebras and we provide concrete examples.

In this paper, we introduce several classes of non-{\sigma}-porous subsets of a general Lebesgue space. Also, we study some linear dynamics of operators and show that the set of all non-hypercyclic vectors of a sequence of weighted translation operators on Lp-spaces is not {\sigma}-porous.

In this paper, we study the linear dynamical properties of shift operators on some classes of Segal algebras. Moreover, we characterize hypercyclic generalized bilateral shift operators on the standard Hilbert module., Comment: This preprint has not undergone peer review (when applicable) or any post-submission improvements or corrections. The Version of the Record of this article is published in Complex Analysis and Operator Theory, and is available online at https://doi.org/10.1007/s11785-023-01376-2