Exponential Space Research Articles

Weighted variable exponent grand Lebesgue spaces and inequalities of approximation

In this paper we discuss and investigate trigonometric approximation in weighted grand variable exponent Lebesgue spaces. We also prove the direct and inverse theorems in these spaces.

An annihilator-based strategy for the automatic detection of exponential polynomial spaces in subdivision

Exponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to reproduce a certain space of exponential polynomials, then it must be level-dependent, with rules depending on the frequencies (and eventual multiplicities) defining the considered space. This work discusses a general strategy that exploits annihilating operators to locally detect those frequencies directly from the given data and therefore to choose the correct subdivision rule to be applied. This is intended as a first step towards the construction of self-adapting subdivision schemes able to locally reproduce exponential polynomials belonging to different spaces. An application of the proposed strategy is shown explicitly on an example involving the classical butterfly interpolatory scheme. This particular example is the generalization of what has been done for the univariate case in Donat and López-Ureña (2019), which inspired this work.

Accurate simulation of surfaces and interfaces of ten FCC metals and steel using Lennard\u2013Jones potentials

The earlier integration of validated Lennard–Jones (LJ) potentials for 8 fcc metals into materials and biomolecular force fields has advanced multiple research fields, for example, metal–electrolyte interfaces, recognition of biomolecules, colloidal assembly of metal nanostructures, alloys, and catalysis. Here we introduce 12-6 and 9-6 LJ parameters for classical all-atom simulations of 10 further fcc metals (Ac, Ca (α), Ce (γ), Es (β), Fe (γ), Ir, Rh, Sr (α), Th (α), Yb (β)) and stainless steel. The parameters reproduce lattice constants, surface energies, water interfacial energies, and interactions with (bio)organic molecules in 0.1 to 5% agreement with experiment, as well as qualitative mechanical properties under standard conditions. Deviations are reduced up to a factor of one hundred in comparison to earlier Lennard–Jones parameters, embedded atom models, and density functional theory. We also explain a quantitative correlation between atomization energies from experiments and surface energies that supports parameter development. The models are computationally very efficient and applicable to an exponential space of alloys. Compatibility with a wide range of force fields such as the Interface force field (IFF), AMBER, CHARMM, COMPASS, CVFF, DREIDING, OPLS-AA, and PCFF enables reliable simulations of nanostructures up to millions of atoms and microsecond time scales. User-friendly model building and input generation are available in the CHARMM-GUI Nanomaterial Modeler. As a limitation, deviations in mechanical properties vary and are comparable to DFT methods. We discuss the incorporation of reactivity and features of the electronic structure to expand the range of applications and further increase the accuracy.

Game tree search for minimizing detectability and maximizing visibility

We introduce and study the problem of planning a trajectory for an agent to carry out a scouting mission while avoiding being detected by an adversarial opponent. This introduces a multi-objective version of classical visibility-based target search and pursuit-evasion problem. In our formulation, the agent receives a positive reward for increasing its visibility (by exploring new regions) and a negative penalty every time it is detected by the opponent. The objective is to find a finite-horizon path for the agent that balances the trade off between maximizing visibility and minimizing detectability. We model this problem as a discrete, sequential, two-player, zero-sum game. We use two types of game tree search algorithms to solve this problem: minimax search tree and Monte-Carlo search tree. Both search trees can yield the optimal policy but may require possibly exponential computational time and space. We first propose three pruning techniques to reduce the computational time while preserving optimality guarantees. When the agent and the opponent are located far from each other initially, we present a variable resolution technique with longer planning horizon to further reduce computational time. Simulation results show the effectiveness of the proposed strategies in terms of computational time.

Quadratic convective transport of Cu‐Al2O3hybrid nanoliquid with Hall current, variable suction, and exponential heat source

The time‐dependent slip flow and heat transport of the Cu‐Al2O3hybrid nanofluid through a vertical permeable plate with Hall current and variable suction is investigated. An exponential heat source and quadratic convection effects are considered. The dimensionless governing equations are solved analytically using the regular perturbation method. The graph of variation of the different flow fields with respect to the distance from the plate is drawn for the analysis. It is established that the exponential heat source aspect increased the temperature profile while the quadratic convection aspect decreased the temperature profile. The applied magnetic field and nonlinear convection aspects reduce the axial velocity field. The quadratic convection and exponential space‐dependent heat source phenomena are favorable to friction factors. Furthermore, the global heat transfer is increased in the presence of an exponential space dependent heat source and hybrid nanoparticles.

SSIA: A sensitivity-supervised interlock algorithm for high-performance microkinetic solving.

Microkinetic modeling has drawn increasing attention for quantitatively analyzing catalytic networks in recent decades, in which the speed and stability of the solver play a crucial role. However, for the multi-step complex systems with a wide variation of rate constants, the often encountered stiff problem leads to the low success rate and high computational cost in the numerical solution. Here, we report a new efficient sensitivity-supervised interlock algorithm (SSIA), which enables us to solve the steady state of heterogeneous catalytic systems in the microkinetic modeling with a 100% success rate. In SSIA, we introduce the coverage sensitivity of surface intermediates to monitor the low-precision time-integration of ordinary differential equations, through which a quasi-steady-state is located. Further optimized by the high-precision damped Newton's method, this quasi-steady-state can converge with a low computational cost. Besides, to simulate the large differences (usually by orders of magnitude) among the practical coverages of different intermediates, we propose the initial coverages in SSIA to be generated in exponential space, which allows a larger and more realistic search scope. On examining three representative catalytic models, we demonstrate that SSIA is superior in both speed and robustness compared with its traditional counterparts. This efficient algorithm can be promisingly applied in existing microkinetic solvers to achieve large-scale modeling of stiff catalytic networks.

A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales.

Network Compression via Mixed Precision Quantization Using a Multi-Layer Perceptron for the Bit-Width Allocation

Deep Neural Networks (DNNs) are a powerful tool for solving complex tasks in many application domains. The high performance of DNNs demands significant computational resources, which might not always be available. Network quantization with mixed-precision across the layers can alleviate this high demand. However, determining layer-wise optimal bit-widths is non-trivial, as the search space is exponential. This article proposes a novel technique for allocating layer-wise bit-widths for a DNN using a multi-layer perceptron (MLP). The Kullback-Leibler (KL) divergence of the softmax outputs between the quantized and full precision network is used as the metric to quantify the quantization quality. We explore the relationship between the KL-divergence and the network size, and from our experiments observe that more aggressive quantization leads to higher divergence, and vice versa. The MLP is trained with layer-wise bit-widths as labels and their corresponding KL-divergence as the input. The MLP training set, i.e. the pairs of the layer-wise bit-widths and their corresponding KL-divergence, is collected using a Monte Carlo sampling of the exponential search space. We introduce a penalty term in the loss to ensure that the MLP learns to predict bit-widths resulting in the smallest network size. We show that the layer-wise bit-width predictions from the trained MLP result in reduced network size without degrading accuracy while achieving better or comparable results with SOTA work but with less computational overhead. Our method achieves up to 6x, 4x, 4x compression on VGG16, ResNet50, and GoogLeNet respectively, with no accuracy drop compared to the original full precision pretrained model, on the ImageNet dataset.

An Extended Variational Theory for Nonlinear Evolution Equations via Modular Spaces

We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly non-reflexive and non-separable spaces. The pivoting point is to establish a novel variational structure, based on abstract modular spaces associated to a given convex function. Firstly, we show that the new variational triple is suited for framing the evolution, in the sense that a novel duality paring can be introduced and a generalised computational chain rule holds. Secondly, we prove well-posedness in an extended variational sense for evolution equations, without relying on any reflexivity assumption and any polynomial requirement on the nonlinearity. Finally, we discuss several important applications that can be addressed in this framework: these cover, but are not limited to, equations in Musielak-Orlicz-Sobolev spaces, such as variable exponent, Orlicz, weighted Lebesgue, and double-phase spaces.

Reduced Quotient Cube: Maximize Query Answering Capacity in OLAP

The data cube is a critical tool for accelerating online analysis in big data. Due to its exponential space overhead, the quotient cube, as the main data cube compression approach, was proposed to significantly reduce the number of data cells if they are aggregated over the same base tuple set, i.e. they are cover equivalent to form an equivalence class. Nevertheless, it still poses challenges to efficiently analyze massive data due to high storage space consumption. This paper proposes the reduced quotient cube (RQC) based on the following observation. (i) there are equivalence classes of various sizes in a quotient cube; (ii) the small equivalence classes usually dominate; (iii) the big equivalence classes are more capable of query answering since they can induce more data cells. Unlike the quotient cube, which preserves all the equivalence classes of equal priority, the reduced quotient cube preferentially does those with larger query answering capacity and smaller space occupied capacity. Further, we design its efficient constructing and querying algorithms. The extensive experimental results show that compared with the quotient cube, the reduced quotient cube space is only 11.3%, while the maximum query capacity is 95.9%. The query time of the reduced quotient cube is reduced by 51.24% on average compared to the quotient cube.

$ \theta $-type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces

$ \theta $-type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces

$ \theta $-type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces

$ \theta $-type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces

The Reachability Problem for Petri Nets Is Not Elementary

Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modeling and analysis of hardware, software, and database systems, as well as chemical, biological, and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and, currently, the best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from Symposium on Logic in Computer Science 2019. We establish a non-elementary lower bound, i.e., that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi, and other areas, which are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the current best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. We develop a construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R . At the heart of our proof is then a novel gadget, the so-called factorial amplifier that, assuming availability of counters that are zero testable and bounded by k , guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k . Repeatedly composing the factorial amplifier with itself by means of the former construction enables us to compute, in linear time, Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we, in fact, already establish hardness for h -exponential space for Petri nets with h + 13 counters.

A conceptual essay on the nature of spacetime

A conceptual essay on the nature of spacetime

Pixelor

We present the first competitive drawing agent Pixelor that exhibits humanlevel performance at a Pictionary-like sketching game, where the participant whose sketch is recognized first is a winner. Our AI agent can autonomously sketch a given visual concept, and achieve a recognizable rendition as quickly or faster than a human competitor. The key to victory for the agent's goal is to learn the optimal stroke sequencing strategies that generate the most recognizable and distinguishable strokes first. Training Pixelor is done in two steps. First, we infer the stroke order that maximizes early recognizability of human training sketches. Second, this order is used to supervise the training of a sequence-to-sequence stroke generator. Our key technical contributions are a tractable search of the exponential space of orderings using neural sorting; and an improved Seq2Seq Wasserstein (S2S-WAE) generator that uses an optimal-transport loss to accommodate the multi-modal nature of the optimal stroke distribution. Our analysis shows that Pixelor is better than the human players of the Quick, Draw! game, under both AI and human judging of early recognition. To analyze the impact of human competitors' strategies, we conducted a further human study with participants being given unlimited thinking time and training in early recognizability by feedback from an AI judge. The study shows that humans do gradually improve their strategies with training, but overall Pixelor still matches human performance. The code and the dataset are available at http://sketchx.ai/pixelor.

On compound Riemann Hilbert boundary value problem in the framework of variable exponent spaces

The aim of this paper is to study the compound Riemann Hilbert boundary value problem in the class of Cauchy-type integral with density in variable exponent Lebesgue spaces. The method is transforming the Riemann Hilbert boundary value problem into a certain Hilbert one. The equivalence of the transformation is discussed. At last the conditions of solvability as well as the explicit solutions are obtained.

Magnetohydrodynamics Eyring‐Powell fluid in a vertical porous microchannel with convective boundary condition subjected to entropy generation

AbstractThis investigation is concentrated on the second law analysis of a magnetohydrodynamics Eyring‐Powell fluid in a vertical microporous channel with the convective boundary conditions under the impacts of heat generation, viscous dissipation, exponential space, temperature dependence, and Joule heating. The reduced form of the governing equations is obtained with the aid of applying nondimensional variables and is resolved using Runge‐Kutta‐Fehlberg's fourth fifth‐order method. The various relevant parameters that affect the heat transfer and entropy have been discussed in detail through graphs. It is found that the impacts of suction/injection, viscous dissipation, and convective conditions are important and the thermal performance can be improved with these factors. The generation of entropy boosts up with the impacts of radiation, space/temperature‐dependent, and Biot number and slows down with Eyring‐Powel parameters. Furthermore, the heat transfer rate amplifies with the magnetic number and the drag force intensifies with the Brinkman parameters. The comparison of results has been performed and it provides an excellent agreement.

Norm inequalities on variable exponent vanishing Morrey type spaces for the rough singular type integral operators

Abstract In this paper, applying the properties of variable exponent analysis and rough kernel, we study the mapping properties of rough singular integral operators. Then, we show the boundedness of rough Calderón–Zygmund type singular integral operator, rough Hardy–Littlewood maximal operator, as well as the corresponding commutators in variable exponent vanishing generalized Morrey spaces on bounded sets. In fact, the results above are generalizations of some known results on an operator basis.

A Data-Driven Soft Sensor to Forecast Energy Consumption in Wastewater Treatment Plants: A Case Study

Energy consumption is vital to the global costs of wastewater treatment plants (WWTPs). With the increase of installed WWTPs worldwide, the modeling and forecast of their energy consumption have become a critical factor in WWTP design to meet environmental and economic requirements. The accurate and swift energy consumption forecasting soft-sensors are not only supportive to the daily electric and financial budgeting by WWTP practitioners on the micro-scale, but also beneficial to local municipal operation and fundamental to regional environmental impact estimation on the macro-scale. Energy consumption in WWTPs is influenced by different biological and environmental factors, making it complicated and challenging to build soft-sensors. This article intends to provide short-term forecasting of WWTP energy consumption based on data-driven soft sensors using traditional time-series and deep learning methods. Ten data-driven soft sensors, including the ordinary least square, exponential smoothing state space, local regression, auto-regressive integrated moving average (ARIMA), structural time series model, Bayesian structural time series, non-linear auto-regressive, long short-term memory with and without updates, and gated recurrent units have been investigated and compared for WWTP energy consumption forecasting. Energy consumption time-series data from a membrane bioreactor-based WWTP in the middle east is used to evaluate the performances of the proposed soft-sensors. Results showed that ARIMA achieved slightly improved performances, among others. The employment of adaptive deep learning-based soft sensors is expected to enhance the capabilities of the deep models to quickly and accurately follow the trend of future data.

Sequential Reinforced 360-Degree Video Adaptive Streaming With Cross-User Attentive Network

In the tile-based 360-degree video streaming, predicting user's future viewpoints and developing adaptive bitrate (ABR) algorithms are essential for optimizing user's quality of experience (QoE). Traditional single-user based viewpoint prediction methods fail to achieve good performance in long-term prediction, and the recently proposed reinforcement learning (RL) based ABR schemes applied in traditional video streaming can not be directly applied in the tile-based 360-degree video streaming due to the exponential action space. Therefore, we propose a sequential reinforced 360-degree video streaming scheme with cross-user attentive network. Firstly, considering different users may have the similar viewing preference on the same video, we propose a cross-user attentive network (CUAN), boosting the performance of long-term viewpoint prediction by selectively utilizing cross-user information. Secondly, we propose a sequential RL-based (360SRL) ABR approach, transforming action space size of each decision step from exponential to linear via introducing a sequential decision structure. We evaluate the proposed CUAN and 360SRL using trace-driven experiments and experimental results demonstrate that CUAN and 360SRL outperform existing viewpoint prediction and ABR approaches with a noticeable margin.