Benchmark instances for the unbounded knapsack problem are typically generated according to specific criteria within a given constant range $R$, and these instances can be referred to as the unbounded knapsack problem with bounded coefficients (UKPB). In order to increase the difficulty of solving these instances, the knapsack capacity $C$ is usually set to a very large value. Therefore, an exact algorithm that neither time complexity nor space complexity includes the capacity coefficient $C$ is highly anticipated. In this paper, we propose an exact algorithm with time complexity of $O(R^4)$ and space complexity of $O(R^3)$. The algorithm initially divides the multiset $N$ into two multisubsets, $N_1$ and $N_2$, based on the profit density of their types. For the multisubset $N_2$ composed of types with profit density lower than the maximum profit density type, we utilize a recent branch and bound (B\&B) result by Dey et al. (Math. Prog., pp 569-587, 2023) to determine the maximum selection number for types in $N_2$. We then employ the Unbounded-DP algorithm to exactly solve for the types in $N_2$. For the multisubset $N_1$ composed of the maximum profit density type and its counterparts with the same profit density, we transform it into a linear Diophantine equation and leverage relevant conclusions from the Frobenius problem to solve it efficiently. In particular, the proof techniques required by the algorithm are primarily covered in the first-year mathematics curriculum, which is convenient for subsequent researchers to grasp.
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We present a novel solution procedure for initial boundary value problems. The procedure is based on an action principle, in which coordinate maps are included as dynamical degrees of freedom. This reparametrization invariant action is formulated in an abstract parameter space and an energy density scale associated with the space-time coordinates separates the dynamics of the coordinate maps and of the propagating fields. Treating coordinates as dependent, i.e. dynamical quantities, offers the opportunity to discretize the action while retaining all space-time symmetries and also provides the basis for automatic adaptive mesh refinement (AMR). The presence of unbroken space-time symmetries after discretization also ensures that the associated continuum Noether charges remain exactly conserved. The presence of coordinate maps in addition provides new freedom in the choice of boundary conditions. An explicit numerical example for wave propagation in $1+1$ dimensions is provided, using recently developed regularized summation-by-parts finite difference operators.
Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise evaluations, which in turn result in redundant function terms or erroneous equations. This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a novel reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with higher rewards are utilized to iteratively optimize the generator via the RL strategy and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.
The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by quantum shallow circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice $\mathsf{QNC}^0/\mathsf{qpoly}$, and $\mathsf{AC}^0[p]$, which is the class of classical constant-depth circuits with unbounded-fan in and $\pmod{p}$ gates. In addition, we show a separation between $\mathsf{QAC}^0$, which additionally has Toffoli gates with unbounded control, and $\mathsf{AC}^0[p]$. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fan-out gates. Secondly, we consider $\mathsf{QNC}^0$ circuits with infinite-size gate sets. We show that these circuits, along with (classical or quantum) prime modular gates, can implement threshold gates, showing that $\mathsf{QNC}^0[p]=\mathsf{QTC}^0$. Finally, we also show that in the infinite-size gateset case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.
Quantum hypothesis testing (QHT) has been traditionally studied from the information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of samples of an unknown state. In this paper, we study the sample complexity of QHT, wherein the goal is to determine the minimum number of samples needed to reach a desired error probability. By making use of the wealth of knowledge that already exists in the literature on QHT, we characterize the sample complexity of binary QHT in the symmetric and asymmetric settings, and we provide bounds on the sample complexity of multiple QHT. In more detail, we prove that the sample complexity of symmetric binary QHT depends logarithmically on the inverse error probability and inversely on the negative logarithm of the fidelity. As a counterpart of the quantum Stein's lemma, we also find that the sample complexity of asymmetric binary QHT depends logarithmically on the inverse type II error probability and inversely on the quantum relative entropy. We then provide lower and upper bounds on the sample complexity of multiple QHT, with it remaining an intriguing open question to improve these bounds. The final part of our paper outlines and reviews how sample complexity of QHT is relevant to a broad swathe of research areas and can enhance understanding of many fundamental concepts, including quantum algorithms for simulation and search, quantum learning and classification, and foundations of quantum mechanics. As such, we view our paper as an invitation to researchers coming from different communities to study and contribute to the problem of sample complexity of QHT, and we outline a number of open directions for future research.
We propose a quantum soft-covering problem for a given general quantum channel and one of its output states, which consists in finding the minimum rank of an input state needed to approximate the given channel output. We then prove a one-shot quantum covering lemma in terms of smooth min-entropies by leveraging decoupling techniques from quantum Shannon theory. This covering result is shown to be equivalent to a coding theorem for rate distortion under a posterior (reverse) channel distortion criterion by two of the present authors. Both one-shot results directly yield corollaries about the i.i.d. asymptotics, in terms of the coherent information of the channel. The power of our quantum covering lemma is demonstrated by two additional applications: first, we formulate a quantum channel resolvability problem, and provide one-shot as well as asymptotic upper and lower bounds. Secondly, we provide new upper bounds on the unrestricted and simultaneous identification capacities of quantum channels, in particular separating for the first time the simultaneous identification capacity from the unrestricted one, proving a long-standing conjecture of the last author.
Logistic regression is widely used in many areas of knowledge. Several works compare the performance of lasso and maximum likelihood estimation in logistic regression. However, part of these works do not perform simulation studies and the remaining ones do not consider scenarios in which the ratio of the number of covariates to sample size is high. In this work, we compare the discrimination performance of lasso and maximum likelihood estimation in logistic regression using simulation studies and applications. Variable selection is done both by lasso and by stepwise when maximum likelihood estimation is used. We consider a wide range of values for the ratio of the number of covariates to sample size. The main conclusion of the work is that lasso has a better discrimination performance than maximum likelihood estimation when the ratio of the number of covariates to sample size is high.
This work is concerned with implementing the hybridizable discontinuous Galerkin (HDG) method to solve the linear anisotropic elastic equation in the frequency domain. First-order formulation with the compliance tensor and Voigt notation are employed to provide a compact description of the discretized problem and flexibility with highly heterogeneous media. We further focus on the question of optimal choices of stabilization in the definition of HDG numerical traces. For this purpose, we construct a hybridized Godunov-upwind flux for anisotropic elastic media possessing three distinct wavespeeds. This stabilization removes the need to choose a scaling factor, contrary to the identity and Kelvin-Christoffel based stabilizations which are popular choices in the literature. We carry out comparisons among these families for isotropic and anisotropic material, with constant background and highly heterogeneous ones, in two and three dimensions. These experiments establish the optimality of the Godunov stabilization which can be used as a reference choice for a generic material in which different types of waves propagate.
We discuss finitely generated infinite groups on which natural random walks are noise sensitive in total variation as well as ones on which natural random walks are noise stable in total variation.
We describe two families of statistical tests to detect partial correlation in vectorial timeseries. The tests measure whether an observed timeseries Y can be predicted from a second series X, even after accounting for a third series Z which may correlate with X. They do not make any assumptions on the nature of these timeseries, such as stationarity or linearity, but they do require that multiple statistically independent recordings of the 3 series are available. Intuitively, the tests work by asking if the series Y recorded on one experiment can be better predicted from X recorded on the same experiment than on a different experiment, after accounting for the prediction from Z recorded on both experiments.
Replication studies are increasingly conducted to assess the credibility of scientific findings. Most of these replication attempts target studies with a superiority design, but there is a lack of methodology regarding the analysis of replication studies with alternative types of designs, such as equivalence. In order to fill this gap, we propose two approaches, the two-trials rule and the sceptical TOST procedure, adapted from methods used in superiority settings. Both methods have the same overall Type-I error rate, but the sceptical TOST procedure allows replication success even for non-significant original or replication studies. This leads to a larger project power and other differences in relevant operating characteristics. Both methods can be used for sample size calculation of the replication study, based on the results from the original one. The two methods are applied to data from the Reproducibility Project: Cancer Biology.
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