Lower Bound Research Articles

Product of polynomial values being large power

Abstract Erdös and Selfridge first showed that the product of consecutive integers cannot be a perfect power. Later, this result was generalized to polynomial values by various authors. They demonstrated that the product of consecutive polynomial values cannot be the perfect power for a suitable polynomial. In this article, we consider a related problem to the product of consecutive integers. We consider all sequences of polynomial values from a given interval whose products are almost perfect powers. We study the size of these powers and give an asymptotic result. We also define a group theoretic invariant, which is a natural generalization of the Davenport constant. We provide a non-trivial upper bound of this group theoretic invariant.

Selecting Balls From Urns With Partial Replacement Rules

Consider an urn with an initial state of R red balls and W white balls. Draw a ball from the urn, uniformly at random, and note its color. If the ball is white, do not replace it; if the ball is red, do replace it. Define this sampling rule to be "Preferential". We study the random variable X denoting the number of white balls drawn under the Preferential sampling rule for a sample size n. It is known that the expected number of X is bounded below by 3nW/(4N), and bounded above by nW/N. In this paper we improve the lower bound, give a heuristic for the best possible lower bound, and we explore some properties of a generalization of this sampling rule, we call "Super-Preferential", where the probability of retaining a white ball is w and the probability of retaining a red ball is r.

Maximum non-bounded difference method for overbounding Global Navigation Satellite System errors

AbstractGlobal Navigation Satellite Systems (GNSS) support numerous applications, including mission-critical ones that require high integrity for safe operations such as air, maritime, and land-based navigation. These applications require ensuring the upper bound of the positioning computation, including overbounding any assumptions made in positioning algorithms. One of the main assumptions in the GNSS positioning algorithm is that the error follows a Probability Density Function (PDF), widely assumed to be Gaussian. This assumption should be overbounded to ensure safety. Two mathematical definitions of distribution overbounding have been proposed: Cumulative Distribution Function (CDF) overbounding and paired overbounding. Achieving these definitions has been an open problem for the past 20 years, and to date, no methods have effectively achieved either definition in multimode datasets. This paper proposes the first overbounding method, the Maximum non-Bounded Difference (MnBD) method, capable of achieving both definitions. Its effectiveness is assessed by comparison with the combined CDF bounding and paired overbounding approach. The results show that the MnBD method successfully achieves overbounding in both real datasets and 1000 simulated multimode data, tested on Gaussian and Generalized Extreme Value (GEV) distributions. The latter demonstrates MnBD’s ability to handle asymmetric distributions. The developed approach can be utilized in overbounding the GNSS error at the system level and user level to ensure the system integrity.

Improved Lower Bounds for Queen’s Domination via an Exactly-Solvable Relaxation

The Queen's Domination problem, studied for over 160 years, poses the following question: What is the least number of queens that can be arranged on a m × n chessboard so that they either attack or occupy every cell? We propose a relaxation of the Queen's Domination problem and solve it exactly for rectangular chessboards. As a consequence, we improve on the best known lower bound for rectangular chessboards for one-eighth of the nontrivial cases. As another consequence, we generalize and provide a new interpretation of the best known lower bounds for Queen's Domination of square n × n chessboards for n ≡ {0, 1, 2} mod 4.Finally, we show some results and make some conjectures towards the goal of simplifying the long complicated proof for the best known lower bound for square boards when n ≡ 3 mod 4 (and n > 11). These simply stated conjectures may also be of independent interest.

Some Lower Bounds for a Double Integral Depending on Six Adaptable Functions

This paper is devoted to sharp lower bounds of a special double integral depending on six adaptable functions. The bounds obtained depend on simple integrals. Some connections with the famous Hilbert integral inequality and its variants are made. Several numerical examples illustrate the results.

A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate

A Lower Bound on the Critical Momentum of an Impurity in a Bose–Einstein Condensate

Disseminated intravascular coagulation is an underestimated but fatal adverse event associated with blinatumomab therapy: A pharmacovigilance analysis of FAERS.

Hematologic adverse events (AEs) are common and serious toxicities in patients with hematologic malignancies undergoing blinatumomab therapy. However, restrictive selection criteria in pivotal clinical trials can lead to an underestimation of rare but fatal toxicities. In this study, we systematically analyzed hematologic AEs associated with blinatumomab using the Food and Drug Administration Adverse Event Reporting System (FAERS) from October 2014 to December 2023. Disproportionate analysis was performed to identify overreported AEs, with a reporting odds ratio (ROR), and a lower bound of the 95% confidence interval (ROR025) exceeding one considered significant. Additionally, adjusted mortality rates and risk ratios (RR) of the top 10 reported hematologic AEs were calculated using a logistic regression model. Among 4745 blinatumomab-related cases, 418 (8.81%) involved hematologic AEs. We identified 22 significantly overreporting hematologic AEs compared to the full database, with myelosuppression (n = 39 [9.33%], ROR025 = 8.04), disseminated intravascular coagulation (DIC, n = 31 [7.42%], ROR025 = 15.14), and bone marrow failure (n = 14 [3.35%], ROR025 = 3.41) notably underestimated in clinical trials. DIC resulted in a substantial mortality rate of 45.16%. Finally, DIC was found to be independently associated with death in a multivariable logistic regression analysis (RR = 2.47 [95% CI: 1.11-3.83]). These findings could aid clinicians in the early detection of these rarely reported but fatal hematologic AEs, thereby reducing the risk of severe toxicities in blinatumomab recipients.

Online Learning and Decision Making Under Generalized Linear Model with High-Dimensional Data

We propose a minimax concave penalized multiarmed bandit algorithm under the generalized linear model (G-MCP-Bandit) for decision-makers facing high-dimensional data in an online learning and decision-making environment. We demonstrate that in the data-rich regime, the G-MCP-Bandit algorithm attains the optimal cumulative regret in the sample size dimension and a tight bound in the covariate dimension and the significant covariate dimension. In the data-poor regime, the G-MCP-Bandit algorithm maintains a tight regret upper bound. In addition, we develop a local linear approximation method, the two-step weighted Lasso procedure, to identify the minimax concave penalty (MCP) estimator for the G-MCP-Bandit algorithm when samples are not independent and identically distributed. Under this procedure, the MCP estimator can match the oracle estimator with high probability and converge to the true parameters at the optimal convergence rate. Finally, through experiments based on both synthetic and real data sets, we show that the G-MCP-Bandit algorithm outperforms other benchmarking algorithms in terms of cumulative regret and that the benefits of the G-MCP-Bandit algorithm increase in the data’s sparsity level and the size of the decision set. This paper was accepted by J. George Shanthikumar, big data analytics. Funding: This work was supported by the National Natural Science Foundation of China (NSFC) [Grants W2441021, 72371172, 72342023, and 71929101] and the National Science Foundation (NSF) [Grant DMS 1820702]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.01557 .

Global attractor for the weakly damped forced Kawahara equation on the torus

We study the long-time behaviour of solutions for the weakly damped forced Kawahara equation on the torus. More precisely, we prove the existence of a global attractor in L 2 , to which as time passes all solutions draw closer. In fact, we show that the global attractor turns out to lie in a smoother space H 2 and be bounded therein. Further, we give an upper bound of the size of the attractor in H 2 that depends only on the damping parameter and the norm of the forcing term.

Tighter sum unitary uncertainty relation

Abstract Unitary uncertainty relations provide a theoretical framework that enhances our understanding of the principles underlying quantum mechanics and its applications in quantum information science. In this study, we derive both the unitary uncertainty relation and the weighted unitary uncertainty relation based on the sum variance for arbitrary pairs of unitary operators. By applying the arithmetic geometric mean inequality, we obtain a lower bound that is tighter than the one provided by Bagchi and Pati. [Physical Review A 94,042104] for two unitary operators. To illustrate our results, we include examples of both the unitary uncertainty relation and the weighted uncertainty relation based on sum variance.

Near-optimal quantum kernel principal component analysis

Abstract Kernel principal component analysis (kernel PCA) is a nonlinear dimensionality reduction technique that employs kernel functions to map data into a high-dimensional feature space, thereby extending the applicability of linear PCA to nonlinear data and facilitating the extraction of informative principal components. However, kernel PCA necessitates the manipulation of large-scale matrices, leading to high computational complexity and posing challenges for efficient implementation in big data environments. Quantum computing has recently been integrated with kernel methods in machine learning, enabling effective analysis of input data within intractable feature spaces. Although existing quantum kernel PCA proposals promise exponential speedups, they impose stringent requirements on quantum hardware that are challenging to fulfill. In this work, we propose a quantum algorithm for kernel PCA by establishing a connection between quantum kernel methods and block encoding, thereby diagonalizing the centralized kernel matrix on a quantum computer. The query complexity is logarithmic with respect to the size of the data vector, $D$, and linear with respect to the size of the dataset. An exponential speedup could be achieved when the dataset consists of a few high-dimensional vectors, wherein the dataset size is polynomial in $\log(D)$, with $D$ being significantly large. In contrast to existing work, our algorithm enhances the efficiency of quantum kernel PCA and reduces the requirements for quantum hardware. Furthermore, we have also demonstrated that the algorithm based on block encoding matches the lower bound of query complexity, indicating that our algorithm is nearly optimal. Our research has laid down new pathways for developing quantum machine learning algorithms aimed at addressing tangible real-world problems and demonstrating quantum advantages within machine learning.

Improved Lower Bound, and Proof Barrier, for Constant Depth Algebraic Circuits

We show that any product-depth Δ algebraic circuit for the Iterated Matrix Multiplication polynomial IMM n, d (when d = O (log n /log log n ) must be of size at least \(n^{\Omega (d^{1/(\varphi ^2)^{\Delta }})}\) , where φ = 1.618 … is the golden ratio. This improves the recent breakthrough result of Limaye, Srinivasan, and Tavenas (FOCS’21), who showed a super polynomial lower bound of the form \(n^{\Omega (d^{1/4^{\Delta }})}\) for constant-depth circuits. One crucial idea of the (LST21) result was to use set-multilinear polynomials where each set in the variables’ underlying partition could be of different sizes. By picking the set sizes more carefully (depending on the depth we are working with), we first show that any product-depth \(\Delta\) set-multilinear circuit for \(\mathrm{IMM}_{n,d}\) (when \(d = O(\log n)\) ) needs size at least \(n^{\Omega (d^{1/\varphi ^{\Delta }})}\) . This improves the \(n^{\Omega (d^{1/2^{\Delta }})}\) lower bound of (LST21). We then use their Hardness Escalation technique to lift this to general circuits. We also show that these techniques cannot improve our lower bound significantly. For the specific two set sizes used in (LST21), they showed that their lower bound cannot be improved. We show that for any \(d^{o(1)}\) set sizes (out of maximum possible d ), the scope for improving our lower bound is minuscule. There exists a set-multilinear circuit that has product-depth \(\Delta\) and size almost matching our lower bound such that the value of the measure used to prove the lower bound is maximum for this circuit. This results in a barrier to further improvement using the same measure.

Gill and Massar type bound for estimation of SU(2) channel

Abstract In the estimation for a parametric family of quantum state on a Hilbert space H, the Gill and Massar bound is known as a lower bound of weighted traces of covariances of unbiased estimators. The Gill and Massar bound is derived by considering the convexity of the set of classical Fisher information matrices, and the bound is locally achievable by using randomized strategies when H=C2. In this paper, we show that estimation for a parametric SU(2) unitary channel model has a similar convex structure as qubit state model, and a Gill and Massar type lower bound of weighted traces of covariances of unbiased estimators can be derived for any weight matrix. We show that the Gill and Massar type lower bound is achievable by using randomized strategies when certain conditions are satisfied. To derive a convex strucutre of the set of classical Fisher information matrices, we introduce a Fisher information matrix J(U) for a SU(2) unitary channel model, and we show a upper bound of inverse J(U) weighted trace of classical Fisher information matrix. The optimal randomized strategy we construct in this paper does not require ancilla systems in many cases.

Fixed-time adaptive sliding mode-based active rollover prevention control of automated guided vehicles via improved super-twisting observer

As an important component of automated guided vehicle (AGV) active safety systems, the active rollover prevention (ARP) control has been widely studied. In this paper, a fixed-time adaptive sliding mode (FTASM) control strategy based on nested adaptive law (NAL) and improved super-twisting observer (ISTO) is developed for the AGV. Firstly, the lateral and roll motion dynamics of AGV are established, and the ARP problem is transformed into the yaw stability control with constraints. Subsequently, a faster fixed-time stable system-based FTASM composite controller is proposed to guarantee a fixed and faster convergence time of the yaw rate tracking error, such that the rollover stability will reach a region of permissible risk. The adoption of NAL enables the adaptive controller to automatically search the minimum switching gain satisfying the reaching condition of sliding mode, thus the conservative assumption constraint of the disturbance upper bound is released. In addition, an ISTO is developed to estimate the unknown disturbance to further improve the control robustness and attenuate the control chattering. The closed-loop stability analysis is rigorously given by Lyapunov theory. Finally, the performance of the proposed control strategy is validated by hardware-in-loop simulations with Adams and Matlab software and an actual steer-by-wire plant.

Identifying the Most Probable Transition Path with Constant Advance Replicas.

Locating plausible transition paths and enhanced sampling of rare events are fundamental to understanding the functional dynamics of biomolecules. Here, a constraint-based constant advance replicas (CAR) formalism of reaction paths is reported for identifying the most probable transition path (MPTP) between two given states. We derive the temporal-integrated effective dynamics governing the projected subsystem under the holonomic CAR path constraints and show that a dynamical action functional can be defined and used for optimizing the MPTP. We further demonstrate how the CAR MPTP can be located by asymptotically minimizing an upper bound of the CAR action functional using a variational expectation-maximization framework. Essential thermodynamics and kinetic observables are retrieved by integrating the boxed molecular dynamics on the CAR MPTP using a newly proposed adaptive reflecting boundary condition. The efficiency of the proposed method is demonstrated for the Müller potential, the alanine dipeptide isomerization, and the DNA base pairing transition (Watson-Crick to Hoogsteen) in explicit solvent. The CAR representation of transition paths constitutes a robust and extensible platform that can be combined with diverse enhanced sampling methods to aid future flexible and reliable biomolecular simulations.

A Comparison of Battery Equivalent Circuit Model Parameter Extraction Approaches Based on Electrochemical Impedance Spectroscopy

This paper presents three approaches to estimating the battery parameters of the electrical equivalent circuit model (ECM) based on electrochemical impedance spectroscopy (EIS); these approaches are referred to as (a) least squares (LS), (b) exhaustive search (ES), and (c) nonlinear least squares (NLS). The ES approach is assisted by the LS method for the rough determination of the lower and upper bound of the ECM parameters, and the NLS approach is incorporated with the Monte Carlo run such that different initial guesses can be assigned to improve the goodness of EIS fitting. The proposed approaches are validated using both simulated and real EIS data. Compared to the LS approach, the ES and NLS approaches show better fitting accuracy at various noise levels, whereas in both the validation using simulated EIS data and actual EIS data collected from LG 18650 and Molicel 21700 batteries, the NLS approach shows better fitting accuracy than that of LS and ES approaches. In all cases, compared with the ES approach, the computational time of the NLS approach is significantly faster, and compared with the LS approach, the NLS approach shows a minimal difference in computational time and considerably better fitting performance.

On hyperelliptic curves of odd degree and genus g with 6 torsion points of order 2g + 1

Let a hyperelliptic curve C of genus g defined over an algebraically closed field K of characteristic 0, given by the equation y2=f(x), where the polynomial f(x)∈K[x]is square-free and has odd degree 2g+1. The curve Ccontains a single “infinite” point O, which is the Weierstrass point. There is a classical embedding of C(K)into the group of K-points J(K)of the Jacobian variety Jof the curve C, identifying the point Owith the unit element of the group J(K). For 2≤g≤5, the article explicitly found representatives of birational equivalence classes such hyperelliptic curves Cwith a marked unique point at infinity Othat the set C(K)∩J(K)contains at least 6 torsion points of order 2g+1. It was previously known that for g=2there are exactly 5 such equivalence classes, and for g≥3an upper bound was known that depended only on the genus of g. We improve the previously known upper bound by almost 36 times.

Lightcone bounds for quantum circuit mapping via uncomplexity

Efficiently mapping quantum circuits onto hardware is integral for the quantum compilation process, wherein a circuit is modified in accordance with a quantum processor’s connectivity. Many techniques currently exist for solving this problem, wherein SWAP-gate overhead is usually prioritized as a cost metric. We reconstitute quantum circuit mapping using tools from quantum information theory, showing that a lower bound, which we dub the lightcone bound, emerges for a circuit executed on hardware. We also develop an initial placement algorithm based on graph similarity search, aiding us in optimally placing circuit qubits onto a device. 600 realistic benchmarks using the IBM Qiskit compiler and a brute-force method are then tested against the lightcone bound, with results unambiguously verifying the veracity of the bound, while permitting trustworthy estimations of minimal overhead in near-term realizations of quantum algorithms. This work constitutes the first use of quantum circuit uncomplexity to practically-relevant quantum computing.

Predefined-Time Hybrid Tracking Control for Dynamic Positioning Vessels Based on Fully Actuated Approach

This study investigates the problem of tracking the trajectory of a dynamic positioning (DP) ship under sudden surges of elevated sea states. First, the tracking problem is reformulated as an error calibration problem through the introduction of fully actuated system (FAS) approaches, thereby simplifying controller design. Second, a predefined-time control term is designed to maintain the convergence time of the trajectory tracking error within a specified range; however, the upper bound of the perturbation must be estimated in advance. The high sea state during operation can result in an abrupt change in the upper bound of disturbance, thereby affecting the control accuracy and stability of the system. Therefore, a linear control matrix is developed to eliminate the system’s dependence on the estimation of the upper bound of disturbance following smooth switching, thereby achieving control decoupling and providing a conservative switching time. Additionally, a nonlinear reduced-order expansion observer (RESO) is constructed for feedforward compensation. The stability of the system is demonstrated using the Lyapunov function, indicating that the selection of appropriate poles can theoretically enhance the system’s convergence with greater control accuracy and robustness after switching. Finally, the effectiveness of the proposed method is validated through simulations and comparative experiments.

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. ’99] also on proof size. [Alekhnovich and Razborov’03] established that if the clause-variable incidence graph of a conjunctive normal form (CNF) formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop their techniques to show that if one can “cluster” clauses and variables in a way that “respects the structure” of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. We also show how a weaker structural condition is sufficient to obtain lower bounds on width for the resolution proof system, and give a unified treatment that highlights similarities and differences between resolution and polynomial calculus (PC) lower bounds. As a corollary of our main technical theorem, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov’02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution (PCR). Thus, while onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis’93], both FPHP and onto-PHP formulas are hard even when restricted to bounded-degree expanders.