Lower Bound Research Articles

Refining Heisenberg’s principle: A greedy approximation of step functions with triangular waveform dictionaries

In this paper, we consider a step function characterized by a real-valued sequence and its linear expansion representation constructed via the matching pursuit (MP) algorithm. We utilize a waveform dictionary based on the triangular function as part of this algorithm and representation. The waveform dictionary is comprised of waveforms localized in the time–frequency domain. In view of this, we prove that the triangular waveforms are more efficient than the rectangular waveforms used in a prior study by achieving a product of variances in the time–frequency domain closer to the lower bound of the Heisenberg Uncertainty Principle. We provide a MP algorithm solvable in polynomial time, contrasting the common exponential time when using Gaussian windows. We apply this algorithm on simulated data and real GDP data from 1947–2024 to demonstrate its application and efficiency.

A Method for Calculating Passive Earth Pressure Under Finite Soil Conditions Based on Upper Bound Limit Analysis

A Method for Calculating Passive Earth Pressure Under Finite Soil Conditions Based on Upper Bound Limit Analysis

A single-dose, randomized, open-label, four-period, crossover equivalence trial comparing the clinical similarity of the proposed biosimilar rupatadine fumarate to reference Wystamm® in healthy Chinese subjects.

The aim of this study was to evaluate the bioequivalence of two formulations of rupatadine (10-mg tablets) under fasting and fed conditions in healthy Chinese subjects. A total of 72 subjects were randomly assigned to the fasting cohort (n = 36) and fed cohort (n = 36). Each cohort includes four single-dose observation periods and 7-day washout intervals. Blood samples were collected at several timepoints for up to 72h post-dose. The plasma concentration of rupatadine and the major active metabolites (desloratadine and 3-hydroxydesloratadine) were analyzed by a validated HPLC-MS/MS method. The non-compartmental analysis method was employed to determine the pharmacokinetic parameters. Based on the within-subject standard deviation of the reference formulation, a reference-scaled average bioequivalence or average bioequivalence method was used to evaluate the bioequivalence of the two formulations. For the fasting status, the reference-scaled average bioequivalence method was used to evaluate the bioequivalence of the maximum observed rupatadine concentration (Cmax; subject standard deviation > 0.294), while the average bioequivalence method was used to evaluate the bioequivalence of the area under the rupatadine concentration-time curve from time 0 to the last detectable concentration (AUC0-t) and from time 0 to infinity (AUC0-∞). The geometric mean ratio (GMR) of the test/reference for Cmax was 95.91%, and the upper bound of the 95% confidence interval was 95.91%. For AUC0-t and AUC0-∞ comparisons, the GMR and 90% confidence interval (CI) were 98.76% (93.88%-103.90%) and 98.71% (93.93%-103.75%), respectively. For the fed status, the subject standard deviation values of Cmax, AUC0-t, and AUC0-∞ were all <0.294; therefore, the average bioequivalence method was used. The GMR and 90% CI for Cmax, AUC0-t, and AUC0-∞ were 101.19% (91.64%-111.74%), 98.80% (94.47%-103.33%), and 98.63% (94.42%-103.03%), respectively. The two-sided 90% CI of the GMR for primary pharmacokinetic endpoints of desloratadine and 3-hydroxydesloratadine was also within 80%-125% for each cohort. These results met the bioequivalence criteria for highly variable drugs. All adverse events (AEs) were mild and transient. The test drug rupatadine fumarate showed a similar safety profile to the reference drug Wystamm® (J. Uriach y Compañía, S.A., Spain), and its pharmacokinetic bioequivalence was confirmed in healthy Chinese subjects based on fasting and postprandial status. http://www.chinadrugtrials.org.cn/index.html, identifier CTR20213217.

Novel highly permeable ultrathin polyamide composite membrane with interfacial polymerization of 4,6-diaminoresorcinol for molecular separation

Developing ultrathin polyamide thin-film composite (PA TFC) membrane with ultrahigh performance to overcome the perm-selectivity upper bound of molecular separation is still the major challenge for practical applications. Herein, we first designed an ultrahigh performance PA TFC membrane fabricated by interfacial polymerization of 4,6-diaminoresorcinol dihydrochloride (DAR) and trimethyl chloride (TMC) on the substrate. The structural morphology of the PA active layer and the separation performance of the resultant membrane were tuned by DAR dosage. This ultrathin PA TFC membrane breaks the trade-off between water permeance and dye (NOM) selectivity. Accordingly, the optimal membrane had a water flux of 212.5 LHM/bar with superior rejection of 98.6 % and 100 % for BBR and CR, respectively, surpassing the upper-bound limit of most reported polymeric membranes thus far. The membranes also exhibited a high salt passage (>95 %), achieving an impressive screening capacity for the dye/salt mixture as well as excellent antifouling and operation stability performance. This work sheds light on the regulation of the PA selective layer via new amine monomer featuring ultrahigh performance.

Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces

We compute the Hofer–Zehnder capacity of magnetic disc tangent bundles over constant curvature surfaces. We use the fact that the magnetic geodesic flow is totally periodic and can be reparametrized to obtain a Hamiltonian circle action. The oscillation of the Hamiltonian generating the circle action immediately yields a lower bound of the Hofer–Zehnder capacity. The upper bound is obtained from Lu’s bounds of the Hofer–Zehnder capacity using the theory of pseudo-holomorphic curves. In our case, the gradient spheres of the Hamiltonian H will give rise to the non-vanishing Gromov–Witten invariant.

The Reflexive H-Strength on Some Graphs

Let G be a connected, simple, and undirected graph with a vertex set V(G) and an edge set E(G). The irregular reflexive -labeling is defined by the function and such that if and if , where max . The irregular reflexive labeling is called an -irregular reflexive -labeling of the graph if every two different sub graphs and isomorphic to it holds , where for the sub graph . The minimum for graph which has an -irregular reflexive -labelling is called the reflexive strength of the graph and denoted by . In this paper we determine the lower bound of the reflexive strength of some subgraphs, on , the sub graph on the sub graph on and the sub graph on .

Single-molecule localization microscopy at 2.4-fold resolution improvement with optical lattice pattern illumination

Traditional camera-based single-molecule localization microscopy (SMLM), with its high imaging resolution and localization throughput, has made significant advancements in biological and chemical researches. However, due to the limitation of the fluorescence signal-to-noise ratio (SNR) of a single molecule, its resolution is difficult to reach to 5 nm. Optical lattice produces a nondiffracting beam pattern that holds the potential to enhance microscope performance through its high contrast and penetration depth. Here, we propose a new method named LatticeFLUX which utilizes the wide-field optical lattice pattern illumination for individual molecule excitation and localization. We calculated the Cramér-Rao lower bound of LatticeFLUX resolution and proved that our method can improve the single molecule localization precision by 2.4 times compared with the traditional SMLM. We propose a scheme using 9-frame localization, which solves the problem of uneven lattice light illumination. Based on the experimental single-molecule fluorescence SNR, we coded the image reconstruction software to further verify the resolution enhancement capability of LatticeFLUX on simulated punctate DNA origami, line pairs, and cytoskeleton. LatticeFLUX confirms the feasibility of using 2D structured light illumination to obtain high single-molecule localization precision under high localization throughput. It paves the way for further implementation of ultra-high resolution full 3D structured-light-illuminated SMLM.

The second Hankel determinant for logarithmic coefficients of inverse functions of bounded turning of a given order

The sharp bound of the second Hankel determinant of logarithmic coefficients of inverse functions of bounded turning of order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is computed.

On a finite sum of cosecants appearing in various problems

Finite trigonometric sums is a challenging and often quite difficult object of study. In this paper we investigate the finite sum of cosecants ∑csc⁡(φ+aπl/n), where the summation index l runs through 1 to n−1 and φ and a are arbitrary parameters, as well as several closely related sums, such as similar sums of a series of secants, of tangents and of cotangents. These trigonometric sums appear in various problems in mathematics, physics, and a variety of related disciplines. Their particular cases were fragmentarily considered in previous works, and it was noted that even a simple particular case ∑csc⁡(πl/n) does not have a closed–form, i.e. a compact summation formula (similar sums of sines, of cosines, of tangents, of cotangents and even of secants, if they exist, possess such expressions). In this paper, we derive several alternative representations for the above–mentioned sums, study their properties, relate them to many other finite and infinite sums, obtain their complete asymptotic expansions for large n and provide accurate upper and lower bounds (e.g. the typical relative error for the upper bound is lesser than 2×10−9 for n⩾10 and lesser than 7×10−14 for n⩾50, which is much better than the bounds we could find in previous works). Our researches reveal that these sums are deeply related to several special numbers and functions, especially to the digamma function (furthermore, as a by-product, we obtain several interesting summation formulæ for the digamma function). Asymptotical studies show that these sums may have qualitatively different behaviour depending on the choice of φ and a. In particular, as n increases some of them may become sporadically large and we identify the terms of the asymptotics responsible for such a behaviour. Finally, we also provide several historical remarks related to various sums considered in the paper. We show that some results in the field either were rediscovered several times or can easily be deduced from various known formulæ, including some formulæ dating back to the XVIIIth century.

De‐carboxylation of cross‐linkable triptycene‐based polyimides for CO2 separation

AbstractPolymer‐based membrane technology holds immense promise for CO2 separation. However, it faces persistent challenges, including the high CO2 pressure‐induced plasticization and permeability‐selectivity trade‐offs, which significantly hinder the development of polymeric membranes. To tackle this issue, we synthesized a novel polyimide 6FDA‐DAT:DABA(6FDD) containing triptycene and carboxylic groups. Upon de‐carboxylation induced cross‐linking, the membrane demonstrated a simultaneous enhancement of gas permeability and selectivity. Specifically, compared to the uncross‐linked 6FDD, the 400°C‐24 h cross‐linked membrane exhibited a remarkable increase in CO2 permeability by 177% (93.1 Barrer) and a significant rise in CO2/CH4 selectivity by 47% (57.5), reaching the CO2/CH4 upper bound. More importantly, the cross‐linked membrane displayed vastly improved CO2 plasticization resistance, withstanding up to 42 bar of CO2 feed pressure. The design of decarboxylated cross‐linked membranes in this work paves the way for creating high‐performing and plasticization‐resistant membranes with potential applications in high‐pressure CO2 separations.

Machine Learning Assisted Exploration for Affine Deligne–Lusztig Varieties

AbstractThis paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne–Lusztig varieties (ADLV). The primary objective is to investigate the non-emptiness pattern, dimension, and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a full mathematical proof of a newly identified problem concerning a certain lower bound of dimension. Furthermore, we extend an open invitation to the readers by providing the source code for computing ADLV and the ML models, promoting further explorations. This paper concludes by sharing valuable experiences and highlighting lessons learned from this collaboration.

Shannon’s entropy and its bounds for some a priori known equiprobable states

AbstractIt is known that Shannon’s entropy is nonnegative and its maximum value is reached for equiprobable events. Adding or removing impossible events does not affect Shannon’s entropy. However, if we increase the number of events and consider not necessarily all of them equiprobable, but at least as many of them as the initial number of equiprobable events, how does Shannon’s entropy change? We study the lower bound of the interval where the probability value of the a priori assumed equiprobable states must belong when the entropy increases.

Exponential H∞ Output Control for Switching Fuzzy Systems via Event-Triggered Mechanism and Logarithmic Quantization

This paper investigates the problem of exponential H∞ output control for switching fuzzy systems, considering both impulse and non-impulse scenarios. Unlike previous research, where the average dwell time (ADT: τa) and the upper bound of inter-event intervals (IEIs: T) satisfy the condition τa≥lnμ+(α+β)Tα=lnμ+βTα+T, implying that frequent switching is difficult to achieve, this paper demonstrates that by adopting the mode-dependent event-triggered mechanism (ETM) and a switching law, frequent switching is indeed achieved. Moreover, the question of deriving the normal L2 norm constraint is solved through the ADT method, although only a weighted L2 norm constraint was obtained previously. Additionally, by constructing a controller-mode-dependent Lyapunov function and adopting logarithmic quantizers, the sufficient criteria of exponential H∞ output control problem are presented. The validity of established results is demonstrated by a given numerical simulation.

Upper Bound Finite Element Limit Analysis of the Response of Shallow Foundations Subjected to Dip-Slip Reverse Fault Rupture Outcrop

Upper Bound Finite Element Limit Analysis of the Response of Shallow Foundations Subjected to Dip-Slip Reverse Fault Rupture Outcrop

Model-Free Nonstationary Reinforcement Learning: Near-Optimal Regret and Applications in Multiagent Reinforcement Learning and Inventory Control

We consider model-free reinforcement learning (RL) in nonstationary Markov decision processes. Both the reward functions and the state transition functions are allowed to vary arbitrarily over time as long as their cumulative variations do not exceed certain variation budgets. We propose Restarted Q-Learning with Upper Confidence Bounds (RestartQ-UCB), the first model-free algorithm for nonstationary RL, and show that it outperforms existing solutions in terms of dynamic regret. Specifically, RestartQ-UCB with Freedman-type bonus terms achieves a dynamic regret bound of [Formula: see text], where S and A are the numbers of states and actions, respectively, [Formula: see text] is the variation budget, H is the number of time steps per episode, and T is the total number of time steps. We further present a parameter-free algorithm named Double-Restart Q-UCB that does not require prior knowledge of the variation budget. We show that our algorithms are nearly optimal by establishing an information-theoretical lower bound of [Formula: see text], the first lower bound in nonstationary RL. Numerical experiments validate the advantages of RestartQ-UCB in terms of both cumulative rewards and computational efficiency. We demonstrate the power of our results in examples of multiagent RL and inventory control across related products. This paper was accepted by Omar Besbes, revenue management and market analytics. Funding: The research of D. Simchi-Levi and R. Zhu was supported by the MIT Data Science Laboratory. The research of W. Mao, K. Zhang, and T. Başar was supported in part by the U.S. Army Research Laboratory (ARL) Cooperative Agreement W911NF-17-2-0196, in part by the Office of Naval Research (ONR) [MURI Grant N00014-16-1-2710], and in part by the Air Force Office of Scientific Research (AFOSR) [Grant FA9550-19-1-0353]. K. Zhang also acknowledges support from U.S. Army Research Laboratory (ARL) [Grant W911NF-24-1-0085]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.02533 .

On a lower bound of Hausdorff dimension of weighted singular vectors

AbstractLet be a ‐tuple of positive real numbers such that and . A ‐dimensional vector is said to be ‐singular if for every , there exists such that for all , the system of inequalities has an integer solution . We prove that the Hausdorff dimension of the set of ‐singular vectors in is bounded below by . Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in , J. Eur. Math. Soc. 22 (2020), 833–875].

Drone scheduling in a hexagon‐based delivery network with multiple fulfillment centers

AbstractAs an emerging technology, drones have great potential in small parcel delivery, reducing operational costs, and improving efficiency. However, battery technologies restrict the endurance of drones. Inspired by the cellular networks used in mobile communications, a hexagon‐based delivery network (HBDN) is proposed to overcome such restrictions. Battery‐swapping stations are incorporated into the HBDN to expand the delivery service range of drones. Based on the proposed HBDN, the multidrone scheduling problem in multiple fulfillment centers (FCs) is addressed. The objective is to minimize the makespan. A mixed‐integer programming model is constructed for the problem. A memetic algorithm with hill climbing (MAHC) is developed to address the problem. A strategy is proposed to prevent inbreeding in the crossover procedure. A local search based on hill climbing is combined with the mutation operation. A lower bound of the problem is proposed to measure the performance of MAHC. Furthermore, load balancing and nearest‐priority strategies are proposed to select FCs. Comparative experiments show that MAHC performs competitively in terms of running time and solution quality.

Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants

Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and Variants

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

Quantum knot mosaics and bounds of the growth constant

The discovery of the Jones polynomial made an important connection between quantum physics and knot theory. Kauffman and Lomonaco introduced the knot mosaic system to define the quantum knot system for the purpose of representing an actual physical quantum system. This paper is inspired by an open question about the knot mosaic enumeration suggested by them. A knot [Formula: see text]-mosaic is an [Formula: see text] array of 11 mosaic tiles representing a knot diagram by adjoining properly. The total number [Formula: see text] of knot [Formula: see text]-mosaics, which indicates the dimension of the Hilbert space of the quantum knot system, is known to grow in a quadratic exponential rate. Recently, Oh et al. developed the state matrix recursion method producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. Furthermore, they showed the existence of the knot mosaic constant [Formula: see text] and found its upper and lower bounds in a series of papers. The latest upper bound was obtained through two new concepts: quasimosaics and cling mosaics. As a sequel to this research program, we adjust the state matrix recursion method to handle cling mosaics inside a quasimosaic, which is called the progressive state matrix recursion method. This method provides recursive matrix-relations producing a sharper bound of the knot mosaic constant: [Formula: see text]