Fast M\"obius and Zeta Transforms

M\"obius inversion of functions on partially ordered sets (posets) $\mathcal{P}$ is a classical tool in combinatorics. For finite posets it consists of two, mutually inverse, linear transformations called zeta and M\"obius transform, respectively. In this paper we provide novel fast algorithms for both that require $O(nk)$ time and space, where $n = |\mathcal{P}|$ and $k$ is the width (length of longest antichain) of $\mathcal{P}$, compared to $O(n^2)$ for a direct computation. Our approach assumes that $\mathcal{P}$ is given as directed acyclic graph (DAG) $(\mathcal{E}, \mathcal{P})$. The algorithms are then constructed using a chain decomposition for a one time cost of $O(|\mathcal{E}| + |\mathcal{E}_\text{red}| k)$, where $\mathcal{E}_\text{red}$ is the number of edges in the DAG's transitive reduction. We show benchmarks with implementations of all algorithms including parallelized versions. The results show that our algorithms enable M\"obius inversion on posets with millions of nodes in seconds if the defining DAGs are sufficiently sparse.Comment: 16 pages, 7 figures, submitted for revie

Fast Fourier Transforms for the Rook Monoid

We define the notion of the Fourier transform for the rook monoid (also called the symmetric inverse semigroup) and provide two efficient divide-and-conquer algorithms (fast Fourier transforms, or FFTs) for computing it. This paper marks the first extension of group FFTs to non-group semigroups

Fermion Low Modes in Lattice QCD: Topology, the η' Mass and Algorithm Development

Lattice gauge theory is an important approach to understanding quantum chromodynamics (QCD) due to the large coupling constant in the theory at low energy. In this thesis, we report our study of the topological properties of the gauge fields and we calculate _η and _η' which are related to the topology of the gauge fields. We also develop two algorithms to speed up the inversion of the Dirac equation which is computationally demanding in lattice QCD calculations. The topology of lattice gauge fields is important but difficult to study because of the large local fluctuations of the gauge fields. In chapter 2, we probe the topological properties of the gauge fields through the measurement of closed quark loops, field strength and low-lying eigenvectors of the Shamir domain wall operator. The closed quark loops suggest the slow evolution of topological modes during the generation of QCD configurations. The chirality of the low-lying eigenvectors is studied and the lattice eigenvectors are compared to the eigenvectors in the continuous theory. The topological charges are calculated from the eigenvectors and the results agree with the topological charges calculated from the smoothed gauge fields. The fermion correlators are also obtained from the eigenvectors. The non-trivial topological properties of QCD gauge fields are important to the mass of the η and η', _η and _η'. Lattice QCD is an area where _{\eta}$and _{\eta'}$ can be calculated by using gauge fields that are sampled over different topological sectors. We calculate _η and _η' in chapter 3 by including the fermion correlators and the topological charge density correlators. The errors of _η and _η' are reduced to the percent level and the mixing angle between the octet, singlet states in the SU(3) limit and the physical eigenstates is calculated. An algorithm that reduces communication and increases the usage of the local computational power is developed in chapter 4. The algorithm uses the multisplitting algorithm as a preconditioner in the preconditioned conjugate gradient method. It speeds up the inversion of the Dirac equation during the evolution phase. In chapter 5, we utilize two lattices, called the coarse lattice and the fine lattice, that lie on the renormalization group trajectory and have different lattice spacings. We find that the low-mode space of the coarse lattice corresponds to the low-mode space of the fine lattice. Because of the correspondence, the coarse lattice can be used to solve the low modes of the fine lattice. The coarse lattice is used in the restart algorithm and the preconditioned conjugate gradient algorithm where the latter is called the renormalization group based preconditioned conjugate gradient algorithm (RGPCG). By using the near-null vectors as the filter, RGPCG could reduce the operations of the matrix multiplications on the fine lattice by 33% to 44% for the inversion of Dirac equation. The algorithm works better than the conjugate gradient algorithm when multiple equations are solved

FarSense: pushing the range limit of WiFi-based respiration sensing with CSI ratio of two antennas

International audienceThe past few years have witnessed the great potential of exploiting channel state information retrieved from commodity WiFi devices for respiration monitoring. However, existing approaches only work when the target is close to the WiFi transceivers and the performance degrades significantly when the target is far away. On the other hand, most home environments only have one WiFi access point and it may not be located in the same room as the target. This sensing range constraint greatly limits the application of the proposed approaches in real life. This paper presents FarSense-the first real-time system that can reliably monitor human respiration when the target is far away from the WiFi transceiver pair. FarSense works well even when one of the transceivers is located in another room, moving a big step towards real-life deployment. We propose two novel schemes to achieve this goal: (1) Instead of applying the raw CSI readings of individual antenna for sensing, we employ the ratio of CSI readings from two antennas, whose noise is mostly canceled out by the division operation to significantly increase the sensing range; (2) The division operation further enables us to utilize the phase information which is not usable with one single antenna for sensing. The orthogonal amplitude and phase are elaborately combined to address the "blind spots" issue and further increase the sensing range. Extensive experiments show that FarSense is able to accurately monitor human respiration even when the target is 8 meters away from the transceiver pair, increasing the sensing range by more than 100%. 1 We believe this is the first system to enable through-wall respiration sensing with commodity WiFi devices and the proposed method could also benefit other sensing applications

АЛГОРИТМ ПРИБЛИЖЁННОГО ЦЕЛОЧИСЛЕННОГО ДИСКРЕТНОГО ПРЕОБРАЗОВАНИЯ ФУРЬЕ

Приводится алгоритм прямого дискретного преобразования Фурье с использованием целочисленной арифметики. Преобразование Фурье основывается на последовательности дискретных косинусных преобразований и аппроксимации отсчётов базисной гармонической функции знакопеременной конечной степенной суммой по основанию 2. Алгоритм построен таким образом, что погрешность преобразования определяется только погрешностью аппроксимации базисной функции. Обосновывается факт точного восстановления первоначальной выборки при приближённом прямом преобразовании

On the synthesis and processing of high quality audio signals by parallel computers

This work concerns the application of new computer architectures to the creation and manipulation of high-quality audio bandwidth signals. The configuration of both the hardware and software in such systems falls under consideration in the three major sections which present increasing levels of algorithmic concurrency. In the first section, the programs which are described are distributed in identical copies across an array of processing elements; these programs run autonomously, generating data independently, but with control parameters peculiar to each copy: this type of concurrency is referred to as isonomic}The central section presents a structure which distributes tasks across an arbitrary network of processors; the flow of control in such a program is quasi- indeterminate, and controlled on a demand basis by the rate of completion of the slave tasks and their irregular interaction with the master. Whilst that interaction is, in principle, deterministic, it is also data-dependent; the dynamic nature of task allocation demands that no a priori knowledge of the rate of task completion be required. This type of concurrency is called dianomic? Finally, an architecture is described which will support a very high level of algorithmic concurrency. The programs which make efficient use of such a machine are designed not by considering flow of control, but by considering flow of data. Each atomic algorithmic unit is made as simple as possible, which results in the extensive distribution of a program over very many processing elements. Programs designed by considering only the optimum data exchange routes are said to exhibit systolic^ concurrency. Often neglected in the study of system design are those provisions necessary for practical implementations. It was intended to provide users with useful application programs in fulfilment of this study; the target group is electroacoustic composers, who use digital signal processing techniques in the context of musical composition. Some of the algorithms in use in this field are highly complex, often requiring a quantity of processing for each sample which exceeds that currently available even from very powerful computers. Consequently, applications tend to operate not in 'real-time' (where the output of a system responds to its input apparently instantaneously), but by the manipulation of sounds recorded digitally on a mass storage device. The first two sections adopt existing, public-domain software, and seek to increase its speed of execution significantly by parallel techniques, with the minimum compromise of functionality and ease of use. Those chosen are the general- purpose direct synthesis program CSOUND, from M.I.T., and a stand-alone phase vocoder system from the C.D.P..(^4) In each case, the desired aim is achieved: to increase speed of execution by two orders of magnitude over the systems currently in use by composers. This requires substantial restructuring of the programs, and careful consideration of the best computer architectures on which they are to run concurrently. The third section examines the rationale behind the use of computers in music, and begins with the implementation of a sophisticated electronic musical instrument capable of a degree of expression at least equal to its acoustic counterparts. It seems that the flexible control of such an instrument demands a greater computing resource than the sound synthesis part. A machine has been constructed with the intention of enabling the 'gestural capture' of performance information in real-time; the structure of this computer, which has one hundred and sixty high-performance microprocessors running in parallel, is expounded; and the systolic programming techniques required to take advantage of such an array are illustrated in the Occam programming language

Improving the accuracy of Cu(ii)-nitroxide RIDME in the presence of orientation correlation in water-soluble Cu(ii)-nitroxide rulers.

Ritsch I, Hintz H, Jeschke G, Godt A, Yulikov M. Improving the accuracy of Cu(ii)-nitroxide RIDME in the presence of orientation correlation in water-soluble Cu(ii)-nitroxide rulers. Physical chemistry chemical physics : PCCP. 2019;21(19):9810-9830.Orientation selection is a challenge in distance determination with double electron electron resonance (DEER) spectroscopy of rigid molecules. The problem is reduced when applying the Relaxation-Induced Dipolar Modulation Enhancement (RIDME) experiment. Here we present an in-depth study on nitroxide-detected RIDME in Cu(ii)-nitroxide spin pairs using two Cu(ii)-nitroxide rulers that are both water soluble and have comparable spin-spin distances. They differ in the type of the ligand (TAHA and PyMTA) for the Cu(ii) ion which results in different contributions of exchange coupling. Both rulers feature substantial orientation correlation between the molecular frames of the Cu(ii) complex and the nitroxide. We discuss how the spin-spin couplings can be accurately measured and how they can be correlated to the nitroxide resonance frequencies. In that, we pay particular attention to the suppression of nuclear modulation and of echo crossing artefacts, to background correction, and to orientation averaging. With a nitroxide observer sequence based on chirp pulses, we achieve wideband detection of all nitroxide orientations. Two-dimensional Fourier transformation of data obtained in this manner affords observer-EPR correlated RIDME spectra that enable visual understanding of the orientation correlation. The syntheses of the Cu(ii)-nitroxide rulers are presented. The synthetic route is considered to be of general use for the preparation of [metal ion complex]-nitroxide rulers, including water soluble ones

Quantum algorithms for classical Boolean functions via adaptive measurements: Exponential reductions in space-time resources

The limited computational power of constant-depth quantum circuits can be boosted by adapting future gates according to the outcomes of mid-circuit measurements. We formulate computation of a variety of Boolean functions in the framework of adaptive measurement-based quantum computation using a cluster state resource and a classical side-processor that can add bits modulo 2, so-called $l2$-MBQC. Our adaptive approach overcomes a known challenge that computing these functions in the nonadaptive setting requires a resource state that is exponentially large in the size of the computational input. In particular, we construct adaptive $l2$-MBQC algorithms based on the quantum signal processing technique that compute the mod-$p$ functions with the best known scaling in the space-time resources (i.e., qubit count, quantum circuit depth, classical memory size, and number of calls to the side-processor). As the subject is diverse and has a long history, the paper includes reviews of several previously constructed algorithms and recasts them as adaptive $l2$-MBQCs using cluster state resources. Our results constitute an alternative proof of an old theorem regarding an oracular separation between the power of constant-depth quantum circuits and constant-depth classical circuits with unbounded fan-in NAND and mod-$p$ gates for any prime $p$.Comment: 15 + 14 pages, 4 figure