Abstract: We establish a list of characterizations of bounded twin-width for hereditary classes of totally ordered graphs: as classes of at most exponential growth studied in enumerative combinatorics, as monadically NIP classes studied in model theory, as classes that do not transduce the class of all graphs studied in finite model theory, and as classes for which model checking first-order logic is fixed-parameter tractable studied in algorithmic graph theory. This has several consequences. First, it allows us to show that every hereditary class of ordered graphs either has at most exponential growth, or has at least factorial growth. This settles a question first asked by Balogh, Bollobás, and Morris [Eur. J. Comb. ’06] on the growth of hereditary classes of ordered graphs, generalizing the Stanley-Wilf conjecture/Marcus-Tardos theorem. Second, it gives a fixed-parameter approximation algorithm for twin-width on ordered graphs. Third, it yields a full classification of fixed-parameter tractable first-order model checking on hereditary classes of ordered binary structures. Fourth, it provides a model-theoretic characterization of classes with bounded twin-width. Finally, it settles the small conjecture [SODA ’21] in the case of ordered graphs.

Abstract: We study an information theoretic privacy mechanism design problem for two scenarios where the private data is either observable or hidden.In the hidden private data scenario, an agent observes useful data Y that is correlated with private data X, and generate disclosed data U which maximizes the revealed information about Y while satisfying a bounded privacy leakage constraint.Considering the other scenario, the agent has additional access to X.To design the privacy mechanism, we first extend the Functional Representation Lemma and Strong Functional Representation Lemma by relaxing the independence condition and thereby allowing a certain leakage.We then find lower and upper bounds on the privacy-utility trade-offs in both scenarios.In particular, for the case where no leakage is allowed and X is observable, our upper and lower bounds improve previous bounds.Considering bounded mutual information as privacy constraint and the observable private data scenario we show that if the common information and mutual information between X and Y are equal, then the attained upper bound is tight.Finally, the privacy-utility trade-off with prioritized private data is studied where part of X is more private than the remaining part.

Abstract: Notions of nonstabilizerness, or "magic," quantify how nonclassical quantum states are in a precise sense: states exhibiting low nonstabilizerness preclude quantum advantage. We introduce "pseudomagic" ensembles of quantum states that, despite low nonstabilizerness, are computationally indistinguishable from those with high nonstabilizerness. Previously, such computational indistinguishability has been studied with respect to entanglement, introducing the concept of pseudoentanglement. However, we demonstrate that pseudomagic neither follows from pseudoentanglement nor implies it. In terms of applications, the study of pseudomagic offers fresh insights into the theory of quantum scrambling: it uncovers states that, even though they originate from nonscrambling unitaries, remain indistinguishable from scrambled states to any physical observer. Additional applications include new lower bounds on state synthesis problems, property testing protocols, and implications for quantum cryptography. Our Letter is driven by the observation that only quantities measurable by a computationally bounded observer-intrinsically limited by finite-time computational constraints-hold physical significance. Ultimately, our findings suggest that nonstabilizerness is a "hide-able" characteristic of quantum states: some states are much more magical than is apparent to a computationally bounded observer.

Abstract: Owing to the development and popularization of social media, multi-attribute group decision-making (MAGDM) adopting social network analysis (SNA) has appealed to widespread focus for a few years. However, most existing research on SNA has not simultaneously considered the expert's weight manipulation behavior and bounded rationality during the consensus-reaching process (CRP). To overcome this limitation, an original consensus model predicated on prospect theory (PT) and preventing individual weight manipulation is proposed in this paper. First, the experts' behavior under certain risks is characterized by using PT, where a dynamic reference point is proposed to provide a more objective description of the experts' psychological behavior. Then, the PT is integrated into the CRP, and the three-level prospect consensus measurement and a three-stage feedback identification mechanism are investigated. Subsequently, to safeguard against individual weight manipulation and obtain the appropriate expert weights, a feedback model incorporating minimum adjustment and maximum entropy rooted in PT is developed. This model is established from the perspectives of "efficiency" and "equity." At last, the proposed method is applied to evaluate low-carbon investment projects in a power generation enterprise to verify its effectiveness and applicability. The superiority and robustness of the proposed method are also demonstrated through comparative analysis and sensitivity analysis.

Abstract: Abstract Despite its great scientific and technological importance, wall-bounded turbulence is an unresolved problem in classical physics that requires new perspectives to be tackled. One of the key strategies has been to study interactions among the energy-containing coherent structures in the flow. Such interactions are explored in this study using an explainable deep-learning method. The instantaneous velocity field obtained from a turbulent channel flow simulation is used to predict the velocity field in time through a U-net architecture. Based on the predicted flow, we assess the importance of each structure for this prediction using the game-theoretic algorithm of SHapley Additive exPlanations (SHAP). This work provides results in agreement with previous observations in the literature and extends them by revealing that the most important structures in the flow are not necessarily the ones with the highest contribution to the Reynolds shear stress. We also apply the method to an experimental database, where we can identify structures based on their importance score. This framework has the potential to shed light on numerous fundamental phenomena of wall-bounded turbulence, including novel strategies for flow control.

Abstract: We study the mapping properties of metaplectic operators Sˆ∈Mp(2d,R) on modulation spaces of the type Mmp,q(Rd). Our main result is a full characterisation of the pairs (Sˆ,Mp,q(Rd)) for which the operator Sˆ:Mp,q(Rd)→Mp,q(Rd) is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that Sˆ is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of Sˆ:Mp,q(Rd)→Mp,q(Rd) transfers to Sˆ:Mmp,q(Rd)→Mmp,q(Rd).

Abstract: We consider a multi-agent network where each node has a stochastic (local) cost function that depends on the decision variable of that node and a random variable, and further, the decision variables of neighboring nodes are pairwise constrained. There is an aggregated objective function for the network, composed additively of the expected values of the local cost functions at the nodes, and the overall goal of the network is to obtain the minimizing solution to this aggregate objective function subject to all the pairwise constraints. This is to be achieved at the level of the nodes using decentralized information and local computation, with exchanges of only compressed information allowed by neighboring nodes. The paper develops algorithms and obtains performance bounds for two different models of local information availability at the nodes: (i) sample feedback, where each node has direct access to samples of the local random variable to evaluate its local cost, and (ii) bandit feedback, where samples of the random variables are not available, but only the values of the local cost functions at two random points close to the decision are available to each node. For both models, with compressed communication between neighbors, we have developed decentralized saddle-point algorithms that deliver performances no different (in order sense) from those without communication compression; specifically, we show that deviation from the global minimum value and violations of the constraints are upper-bounded by O(T−12) and O(T−14), respectively, where T is the number of iterations. Numerical examples provided in the paper corroborate these bounds.

Abstract: In a bounded domain $ \Omega \subset \mathbb{R}^d $, $ d\geq 1 $, over a time interval $ (0,T) $, we consider mean field game equations whose principal coefficients depend on the time and the state variables with a general Hamiltonian. We attach a non-zero Robin boundary condition. We first prove the Lipschitz stability in $ \Omega \times ( \varepsilon, T- \varepsilon) $ with given $ \varepsilon>0 $ for the determination of the solutions by the associated Dirichlet data on an arbitrarily chosen subboundary of $ \partial \Omega $. Next we prove the Lipschitz stability for an inverse problem of determining spatially varying factors of source terms and a coefficient by extra boundary data and spatial data at an intermediate time.

Abstract: We consider degenerate Kolmogorov-Fokker-Planck operatorsLu=∑i,j=1qaij(x,t)∂xixj2u+∑k,j=1Nbjkxk∂xju−∂tu,(x,t)∈RN+1,N≥q≥1 such that the corresponding model operator having constant aij is hypoelliptic, translation invariant w.r.t. a Lie group operation in RN+1 and 2-homogeneous w.r.t. a family of nonisotropic dilations. The coefficients aij are bounded and Hölder continuous in space (w.r.t. some distance induced by L in RN) and only bounded measurable in time; the matrix {aij}i,j=1q is symmetric and uniformly positive on Rq. We prove "partial Schauder a priori estimates" of the kind∑i,j=1q‖∂xixj2u‖Cxα(ST)+‖Yu‖Cxα(ST)≤c{‖Lu‖Cxα(ST)+‖u‖C0(ST)} for suitable functions u, where Yu=∑k,j=1Nbjkxk∂xju−∂tu and‖f‖Cxα(ST)=supt≤T⁡supx1,x2∈RN,x1≠x2⁡|f(x1,t)−f(x2,t)|‖x1−x2‖α+‖f‖L∞(ST). We also prove that the derivatives ∂xixj2u are locally Hölder continuous in space and time while ∂xiu and u are globally Hölder continuous in space and time.

Abstract: The Knapsack problem is one of the most fundamental NP-complete problems at the intersection of computer science, optimization, and operations research. A recent line of research worked towards understanding the complexity of pseudopolynomial-time algorithms for Knapsack parameterized by the maximum item weight wmax and the number of items n. A conditional lower bound rules out that Knapsack can be solved in time O((n+wmax)2−δ) for any δ > 0 [Cygan, Mucha, Wegrzycki, Wlodarczyk'17, Künnemann, Paturi, Schneider'17]. This raised the question whether Knapsack can be solved in time Õ((n+wmax)2). This was open both for 0-1-Knapsack (where each item can be picked at most once) and Bounded Knapsack (where each item comes with a multiplicity). The quest of resolving this question lead to algorithms that solve Bounded Knapsack in time Õ(n3 wmax2) [Tamir'09], Õ(n2 wmax2) and Õ(n wmax3) [Bateni, Hajiaghayi, Seddighin, Stein'18], O(n2 wmax2) and Õ(n wmax2) [Eisenbrand and Weismantel'18], O(n + wmax3) [Polak, Rohwedder, Wegrzycki'21], and very recently Õ(n + wmax12/5) [Chen, Lian, Mao, Zhang'23]. In this paper we resolve this question by designing an algorithm for Bounded Knapsack with running time Õ(n + wmax2), which is conditionally near-optimal. This resolves the question both for the classic 0-1-Knapsack problem and for the Bounded Knapsack problem.

Abstract: Using the polar decomposition of a bounded linear operator A defined on a complex Hilbert space, we obtain several numerical radius inequalities of the operator A, which generalize and improve the earlier related ones. Among other bounds, we show that if w(A) is the numerical radius of A, thenw(A)≤12‖A‖1/2‖|A|t+|A⁎|1−t‖, for all t∈[0,1]. Also, we obtain some upper bounds for the numerical radius involving the spectral radius and the Aluthge transform of operators. It is shown thatw(A)≤‖A‖1/2(12‖|A|+|A⁎|2‖+12‖A˜‖)1/2, where A˜=|A|1/2U|A|1/2 is the Aluthge transform of A and A=U|A| is the polar decomposition of A. Other related results are also provided.

Abstract: In the domain of machine learning, the significance of the loss function is paramount, especially in supervised learning tasks. It serves as a fundamental pillar that profoundly influences the behavior and efficacy of supervised learning algorithms. Traditional loss functions, though widely used, often struggle to handle outlier-prone and high-dimensional data, resulting in suboptimal outcomes and slow convergence during training. In this paper, we address the aforementioned constraints by proposing a novel robust, bounded, sparse, and smooth (RoBoSS) loss function for supervised learning. Further, we incorporate the RoBoSS loss within the framework of support vector machine (SVM) and introduce a new robust algorithm named L

Abstract: We investigate pseudopolynomial-time algorithms for Bounded Knapsack and Bounded Subset Sum. Recent years have seen a growing interest in settling their fine-grained complexity with respect to various parameters. For Bounded Knapsack, the number of items n and the maximum item weight wmax are two of the most natural parameters that have been studied extensively in the literature. The previous best running time in terms of n and wmax is O(n + wmax3) [Polak, Rohwedder, Węgrzycki ‘21]. There is a conditional lower bound of (n + wmax)2-o(1) based on (min, +)-convolution hypothesis [Cygan, Mucha, Węgrzycki, Włodarczyk ‘17]. We narrow the gap significantly by proposing an -time algorithm. Our algorithm works for both 0-1 Knapsack and Bounded Knapsack. Note that in the regime where wmax ≈ n, our algorithm runs in Õ(n12/5) time, while all the previous algorithms require Ω(n3) time in the worst case.For Bounded Subset Sum, we give two algorithms running in Õ(nwmax) and time, respectively. These results match the currently best running time for 0-1 Subset Sum. Prior to our work, the best running times (in terms of n and wmax) for Bounded Subset Sum are [Polak, Rohwedder, Węgrzycki ‘21] and [implied by Bringmann ‘19 and Bringmann, Wellnitz ‘21], where µmax refers to the maximum multiplicity of item weights.

Abstract: This paper develops a framework for privatizing the spectrum of the Laplacian of an undirected graph using differential privacy. We consider two privacy formulations. The first obfuscates the presence of edges in the graph and the second obfuscates the presence of nodes. We compare these two privacy formulations and show that the privacy formulation that considers edges is better suited to most engineering applications. We use the bounded Laplace mechanism to provide $(\epsilon,\delta)$ -differential privacy to the eigenvalues of a graph Laplacian, and we pay special attention to the algebraic connectivity, which is the Laplacian's the second smallest eigenvalue. Analytical bounds are presented on the accuracy of the mechanisms and on certain graph properties computed with private spectra. A suite of numerical examples confirms the accuracy of private spectra in practice.

Abstract: While quantum state tomography is notoriously hard, most states hold little interest to practically minded tomographers. Given that states and unitaries appearing in nature are of bounded gate complexity, it is natural to ask if efficient learning becomes possible. In this work, we prove that to learn a state generated by a quantum circuit with G two-qubit gates to a small trace distance, a sample complexity scaling linearly in G is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by G gates to a small average-case error scales linearly in G. While sample-efficient learning can be achieved, we show that under reasonable cryptographic conjectures, the computational complexity for learning states and unitaries of gate complexity G must scale exponentially in G. We illustrate how these results establish fundamental limitations on the expressivity of quantum machine-learning models and provide new perspectives on no-free-lunch theorems in unitary learning. Together, our results answer how the complexity of learning quantum states and unitaries relate to the complexity of creating these states and unitaries. Published by the American Physical Society 2024

Abstract: Trade-off relations place fundamental limits on the operations that physical systems can perform. This Letter presents a trade-off relation that bounds the correlation function, which measures the relationship between a system's current and future states, in Markov processes. The obtained bound, referred to as the thermodynamic correlation inequality, states that the change in the correlation function has an upper bound comprising the dynamical activity, a thermodynamic measure of the activity of a Markov process. Moreover, by applying the obtained relation to the linear response function, it is demonstrated that the effect of perturbation can be bounded from above by the dynamical activity.

Abstract: How multiple observables mutually influence their dynamics has been a crucial issue in statistical mechanics. We here introduce a new concept, ``quantum velocity limits,'' to establish a quantitative and rigorous theory for nonequilibrium quantum dynamics for multiple observables. Quantum velocity limits are universal inequalities for a vector that describes velocities of multiple observables. They elucidate that the speed of an observable of our interest can be tighter bounded when we have knowledge of other observables, such as experimentally accessible ones or conserved quantities, compared with conventional speed limits for a single observable. Moreover, quantum velocity limits are conceptually distinct from the conventional speed limits because we need to introduce the velocity vector and solve an optimization problem for multiple variables to obtain them. We first derive an information-theoretical velocity limit in terms of the generalized correlation matrix of the observables and the quantum Fisher information. The velocity limit has various novel consequences: (i) conservation law in the system, a fundamental ingredient of quantum dynamics, can improve the velocity limits through the correlation between the observables and conserved quantities; (ii) speed of an observable can be bounded by a nontrivial lower bound from the information on another observable, while most of the previous speed limits provide only upper bounds; (iii) there exists a notable nonequilibrium tradeoff relation, stating that speeds of uncorrelated observables, e.g., anticommuting observables, cannot be simultaneously large; (iv) velocity limits for local observables in locally interacting many-body systems are described by the fluctuation of a local Hamiltonian, which is convergent even in the thermodynamic limit, with a nontrivial finite-size correction. Moreover, we discover another distinct velocity limit for multiple observables on the basis of the local conservation law of probability current, which becomes advantageous for macroscopic transitions of multiple quantities. Our newly found velocity limits ubiquitously apply not only to unitary quantum dynamics but to classical and quantum stochastic dynamics, offering a key step towards universal theory of far-from-equilibrium dynamics for multiple observables.

Abstract: In this paper we consider an abstract Cauchy problem for a Maxwell system modeling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain.

Abstract: Let $k$ be a field of characteristic $0$ endowed with a fixed field embedding $\sigma: k \hookrightarrow \mathbb{C}$. In this paper we complete the construction of the six functor formalism on perverse Nori motives over quasi-projective $k$-varieties initiated by F. Ivorra and S. Morel. Our main contribution is the construction of a canonical closed unitary symmetric monoidal structure on the bounded derived categories of perverse Nori motives compatible with the analogous structure on the underlying constructible derived categories; along the way, we extend some of Nori's original results to perverse Nori motives. As a consequence, we obtain well-behaved Tannakian categories of motivic local systems over smooth, geometrically connected $k$-varieties. Our constructions do not depend on the chosen complex embedding of $k$ and, in fact, our results generalize to arbitrary base fields of characteristic $0$.

Abstract: In this paper, we derive novel families of inclusion sets for the spectra and pseudospectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate.Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated.More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space.In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020).As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [

Abstract: Accurate local state measurement is important to ensure the reliable operation of distributed multi-agent systems (MAS). Existing fault-tolerant control strategies generally assume the sensor faults to be bounded and uncorrelated. In this paper, we study the ramifications of allowing the sensor attack injections to be unbounded and correlated. These malicious sensor attacks may bypass the conventional attack-detection methods and compromise the cooperative performance and even stability of the distributed networked MAS. Moreover, the attackers may gain access to the actuation computing channels and manipulate the control input commands. To this end, we consider the resilient containment control problem of general linear heterogeneous MAS in the face of correlated and unbounded sensor attacks, as well as general unbounded actuator attacks. We propose an attack-resilient control framework to guarantee the uniform ultimate boundedness of the closed-loop dynamical systems and preserve the bounded containment performance. Compared with existing literature addressing bounded faults and/or disturbances that are unintentionally caused in the sensor and actuator channels, the proposed control protocols are resilient against unknown and unbounded attack signals simultaneously injected into sensor and actuator channels, and hence are more practical in the real-world security applications. A numerical example illustrates the efficacy of the proposed result, by highlighting the resilience improvement over the conventional cooperative control method.

Abstract: In this paper, we study the problem of the distributed Nash equilibrium seeking of $N$ -player games over jointly strongly connected switching networks. The action of each player is governed by a class of uncertain nonlinear systems. Our approach integrates the consensus algorithm, the distributed estimator over jointly strongly connected switching networks, and some adaptive control technique. Furthermore, we also consider the disturbance rejection problem for bounded disturbances with unknown bounds. A special case of our results gives the solution of the distributed Nash equilibrium seeking for high-order integrator systems.

Abstract: In this paper, we develop a fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme for simulating the multicomponent, chemically reacting, compressible Euler equations with complex thermodynamics. The proposed formulation is an extension of the fully conservative, high-order numerical method previously developed by Johnson and Kercher (2020) [14] that maintains pressure equilibrium between adjacent elements. In this first part of our two-part paper, we focus on the one-dimensional case. Our methodology is rooted in the minimum entropy principle satisfied by entropy solutions to the multicomponent, compressible Euler equations, which was proved by Gouasmi et al. (2020) [16] for nonreacting flows. We first show that the minimum entropy principle holds in the reacting case as well. Next, we introduce the ingredients, including a simple linear-scaling limiter, required for the discrete solution to have nonnegative species concentrations, positive density, positive pressure, and bounded entropy. We also discuss how to retain the aforementioned ability to preserve pressure equilibrium between elements. Operator splitting is employed to handle stiff chemical reactions. To guarantee discrete satisfaction of the minimum entropy principle in the reaction step, we develop an entropy-stable discontinuous Galerkin method based on diagonal-norm summation-by-parts operators for solving ordinary differential equations. The developed formulation is used to compute canonical one-dimensional test cases, namely thermal-bubble advection, advection of a low-density Gaussian wave, multicomponent shock-tube flow, and a moving hydrogen-oxygen detonation wave with detailed chemistry. We demonstrate that the formulation can achieve optimal high-order convergence in smooth flows. Furthermore, we find that the enforcement of an entropy bound can considerably reduce the large-scale nonlinear instabilities that emerge when only the positivity property is enforced, to an even greater extent than in the monocomponent, calorically perfect case. Finally, mass, total energy, and atomic elements are shown to be discretely conserved.

Abstract: Imbalanced datasets often lead to biased machine learning models that underperform on minority classes. The Synthetic Minority Over-sampling Technique (SMOTE) addresses this by generating synthetic samples for the minority class. Despite its empirical success, the theoretical properties of SMOTE remain underexplored. This paper investigates the asymptotic behavior of SMOTE-generated random variables using order statistics. We establish that, as the sample size n increases, the SMOTE-generated variable Z conditioned on the k-th order statistic X(k) converges in mean to X(k). Additionally, the expected value of Z converges to the expected value of the original random variable X as n approaches infinity. The results are derived under the assumption of left-bounded support for X. Our findings provide a theoretical foundation for SMOTE, demonstrating that it preserves key statistical properties of the original data distribution in large samples.