Induction Argument Research Articles

Global-in-time error estimates of non-relativistic limits for Euler–Maxwell system near non-constant equilibrium

It was proved that Euler–Maxwell systems converge globally-in-time to Euler–Poisson systems near non-constant equilibrium states when the speed of light c→∞. In this paper, we establish the global-in-time error estimates between smooth solutions of Euler–Maxwell systems and those of Euler–Poisson systems near non-constant equilibrium states. The main difficulty lies in the singularity of the error variable for the electric field E, so that more careful estimates for the time derivatives of error variables should be established. The proof takes good advantage of the anti-symmetric structure of the error system and an induction argument on the order of the derivatives.

Numerical solutions of Schrödinger–Boussinesq system by orthogonal spline collocation method

This paper is concerned with the numerical solutions of Schrödinger–Boussinesq (SBq) system by an orthogonal spline collocation (OSC) discretization in space and Crank–Nicolson (CN) type approximation in time. By using the mathematical induction argument and standard energy method, the proposed CN+OSC scheme is proved to be unconditionally convergent at the order O(τ2+h4) with mesh-size h and time step τ in the discrete L2-norm. We devise a new computation method based on the orthogonal diagonalization techniques (ODT) to realize the proposed CN+OSC scheme. In order to compare the performance of ODT, we devise an alternating direction implicit (ADI) method to compute the CN+OSC scheme for high spatial dimension SBq system. As an alternative implementation, the new method ODT not only exhibits more accurate numerical results, but also demonstrates stronger invariance preserving ability. Numerical results are reported to verify the error estimates and the discrete conservation laws.

Multidimensional polynomial patterns over finite fields: Bounds, counting estimates and Gowers norm control

We examine multidimensional polynomial progressions involving linearly independent polynomials over finite fields, proving power saving bounds for sets lacking such configurations. This jointly generalises earlier results of Peluse (for the single dimensional case) and the author (for distinct degree polynomials). In contrast to the cases studied in the aforementioned two papers, a usual PET induction argument does not give Gowers norm control over multidimensional progressions that involve polynomials of the same degrees. The main challenge is therefore to obtain Gowers norm control, and we accomplish this for all multidimensional polynomial progressions with pairwise independent polynomials. The key inputs are: (1) a quantitative version of a PET induction scheme developed in ergodic theory by Donoso, Koutsogiannis, Ferré-Moragues and Sun, (2) a quantitative concatenation result for Gowers box norms in arbitrary finite abelian groups, motivated by (but different from) earlier results of Tao, Ziegler, Peluse and Prendiville; (3) an adaptation to combinatorics of the box norm smoothing technique, recently developed in the ergodic setting by the author and Frantzikinakis; and (4) a new version of the multidimensional degree lowering argument.

On the role of advection in a spatial epidemic model with general boundary conditions

This paper is concerned with a reaction-diffusion-advection model for vector-borne disease with general boundary conditions and general incidences. Due to the boundary conditions, we first apply the eigenvalue theory of elliptic system to prove the existence and uniqueness of steady state for the model. The well-posedness of the model is established using an induction argument. By overcoming the difficulty of the associated elliptic eigenvalue problem, we originally derive the variational expression of the basic reproduction ratio R0. The asymptotic profiles and monotonicity of R0 with respect to the mobility and advection rates are investigated following the variational characterization of R0. Furthermore, the spatial dynamics of the model with Robin type boundary condition are categorized via classifying the level set of R0. This work provides new clues for further research on the spread of epidemics in open advective environments.

Linearly implicit energy-preserving integrating factor methods and convergence analysis for the 2D nonlinear Schrödinger equation with wave operator

Abstract In this paper, we develop a novel class of linearly implicit and energy-preserving integrating factor methods for the 2D nonlinear Schrödinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating factor methods. To begin, a second-order scheme is proposed, which is rigorously proved to be energy-preserving. By using the energy methods, we analyze its optimal convergence without any restrictions on the grid ratio, where a novel technique and an improved induction argument are proposed to circumvent the difficulty arising from the unavailability of a priori$L^{\infty }$ estimates of numerical solutions. Based on the integrating factor Runge–Kutta methods, we extend the proposed scheme to arbitrarily high order, which is also linearly implicit and conservative. Numerical experiments are presented to confirm the theoretical analysis and demonstrate the advantages of the proposed methods.

Inductive reasoning in minds and machines.

Induction-the ability to generalize from existing knowledge-is the cornerstone of intelligence. Cognitive models of human induction are largely limited to toy problems and cannot make quantitative predictions for the thousands of different induction arguments that have been studied by researchers, or to the countless induction arguments that could be encountered in everyday life. Leading large language models (LLMs) go beyond toy problems but fail to mimic observed patterns of human induction. In this article, we combine rich knowledge representations obtained from LLMs with theories of human inductive reasoning developed by cognitive psychologists. We show that this integrative approach can capture several benchmark empirical findings on human induction and generate human-like responses to natural language arguments with thousands of common categories and properties. These findings shed light on the cognitive mechanisms at play in human induction and show how existing theories in psychology and cognitive science can be integrated with new methods in artificial intelligence, to successfully model high-level human cognition. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Global convergence in non-relativistic limits for Euler-Maxwell system near non-constant equilibrium

In this paper, we study the global-in-time convergence of non-relativistic limits from Euler-Maxwell systems to Euler-Poisson systems near non-constant equilibrium states by letting the reciprocal of the speed of light ν:=1/c→0. In previous studies, the dissipative estimates for the electric field E appear to be singular and the orders of singularities for divE and ∇×E are different, hence a div-curl decomposition should be considered in our case, and this makes it unclear the preservation of the anti-symmetric structure of the system and the global-in-time L2–estimate of the smooth solutions. To overcome these difficulties, we find a strictly convex entropy of the system that is valid for the case of non-constant equilibrium to obtain the global L2–estimate and use some induction arguments to close the estimates. It is worth mentioning that in our proof, very careful and accurate estimates of solutions are needed and we show that the electric field E is actually non-singular, which is vital and necessary for our proof.

Stability of stochastic reaction-diffusion equation under random influences in high regular spaces

In this paper, we systematically study the high-order stability of the stochastic reaction-diffusion equation driven by additive noise as the noise intensity vanishes. First, with a general assumption on the nonlinear term, we obtain the convergence of solutions and upper semi-continuity of random attractors in L2(RN). Second, by using the nonlinear decomposition method, we technically establish the convergence of solutions in Lp(RN)∩H1(RN)(p>2), and therefore, the upper semi-continuity of random attractors is proved, where p is the growth exponent of the nonlinearity. Finally, by induction argument, we prove that the solution is uniformly bounded near the initial time in Lδ(RN) for arbitrary δ > p, in which space the convergence of solutions and the upper semi-continuity of random attractors are also established.

Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms

In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory (BAM).

Constant rank theorems for curvature problems via a viscosity approach

An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map Brendle et al. (Acta Math 219:1–16, 2017) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument.

Shift-invariance of the colored TASEP and finishing times of the oriented swap process

We prove a new shift-invariance property of the colored TASEP. From the shift-invariance of the colored stochastic six-vertex model (proved in Borodin-Gorin-Wheeler or Galashin), one can get a shift-invariance property of the colored TASEP at one time, and our result generalizes this to multiple times. Our proof takes the single-time shift-invariance as an input, and uses analyticity of the probability functions and induction arguments. We apply our shift-invariance to prove a distributional identity between the finishing times of the oriented swap process and the point-to-line passage times in exponential last-passage percolation, which is conjectured by Bisi-Cunden-Gibbons-Romik and Bufetov-Gorin-Romik, and is also equivalent to a purely combinatorial identity related to the Edelman-Greene correspondence. With known results from last-passage percolation, we also get new asymptotic results on the colored TASEP and the finishing times of the oriented swap process.

Two energy-preserving Fourier pseudo-spectral methods and error estimate for the Klein–Gordon–Dirac system

We devote the present paper to the convergency of conservative Fourier pseudo-spectral methods for three-dimensional Klein–Gordon–Dirac (KGD) system. By adopting the mathematical induction argument, standard energy method and inverse inequality, the error bound is established under the condition τ<h with time step τ and mesh size h. More specifically, the scalar ϕ and 4-spinor ψ are proved to be convergent in H2 and H1-norm, respectively. In addition, we establish a frame to compute the numerical solutions of three-dimensional KGD system with periodic conditions, and a faster solver is designed to accelerate the computation by means of orthogonal diagonalization technique. Numerical results are reported to verify the error estimate and discrete conservation laws.

Entropic curvature on graphs along Schrödinger bridges at zero temperature

Lott-Sturm-Villani theory of curvature on geodesic spaces has been extended to discrete graph spaces by C. L{\'e}onard by replacing W2-Wasserstein geodesics by Schr{\"o}odinger bridges in the definition of entropic curvature [23, 25, 24]. As a remarkable fact, as a temperature parameter goes to zero, these Schr{\"o}dinger bridges are supported by geodesics of the space. We analyse this property on discrete graphs to reach entropic curvature on discrete spaces. Our approach provides lower bounds for the entropic curvature for several examples of graph spaces: the lattice Z n endowed with the counting measure, the discrete cube endowed with product probability measures, the circle, the complete graph, the Bernoulli-Laplace model. Our general results also apply to a large class of graphs which are not specifically studied in this paper. As opposed to Erbar-Maas results on graphs [27, 10, 11], entropic curvature results of this paper imply new Pr{\'e}kopa-Leindler type of inequalities on discrete spaces, and new transport-entropy inequalities related to refined concentration properties for the graphs mentioned above. For example on the discrete hypercube {0, 1} n and for the Bernoulli Laplace model, a new W2 -- W1 transport-entropy inequality is reached, that can not be derived by usual induction arguments over the dimension n. As a surprising fact, our method also gives improvements of weak transport-entropy inequalities (see [28, 15]) associated to the so-called convex-hull method by Talagrand [38].

Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity

Abstract In this article, we will develop an analytical approach to construct the global bounded weak solutions to the initial-boundary value problem of a three-dimensional chemotaxis-Stokes system with porous medium cell diffusion Δ n m \Delta {n}^{m} for m ≥ 65 63 m\ge \frac{65}{63} and general sensitivity. In particular, this extended the precedent results which asserted global solvability within the larger range m &gt; 7 6 m\gt \frac{7}{6} for general sensitivity (M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. 54 (2015), 3789–3828) or m &gt; 9 8 m\gt \frac{9}{8} for scalar sensitivity (M. Winkler, Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement, J. Differ. Equ. 264 (2018), 6109–6151). Our proof is based on a new observation on the quasi-energy-type functional and on an induction argument.

Payment schemes for sustaining cooperation in dynamic games

It is a challenge to sustain cooperation in a finite-horizon dynamic game. Since players generally have an incentive to deviate to their noncooperative strategies in the last stage, a backward induction argument leads them to defect from cooperation in all stages. In this paper, we propose two payment schemes having some desirable properties, namely, individual rationality and stability, which ensure that the players cooperate throughout the entire planning horizon. The setup and the results are general, that is, they do not rest on particular specifications of the payoff functionals or the state dynamics. We illustrate our results with a linear-quadratic dynamic game of pollution control.

Strong Traces to Degenerate Parabolic Equations

We prove existence of strong traces at $t=0$ for quasi-solutions to (multidimensional) degenerate parabolic equations with no non-degeneracy conditions. In order to solve the problem, we combine the blow up method and a strong precompactness result for quasi-solutions to degenerate parabolic equations with the induction argument with respect to the space dimension.

Structural losses, structural realism and the stability of Lie algebras

One of the key assumptions associated with structural realism is the claim that successful scientific theories approximately preserve their structurally based content as they are progressively developed and that this content alone can explain their relevant predictions. The precise way in which these theories are preserved is not trivial but, according to this realist thesis, any kind of structural loss should not occur among theoretical transitions. Although group theory has been proven effective in accounting for preserved structures in the context of physics, structural realists are confronted with the fact that even group-theoretic structures are not immune to these structural discontinuities. Under such circumstances, my contribution consists in a two-fold task. Firstly, I will establish a general condition at the level of the group-theoretic structure to avoid the pessimistic induction argument by appealing to Lie algebra deformation and stability theory; and secondly, I will provide a case study associated with quantum-relativistic kinematics to demonstrate that this condition is actually satisfied. Specifically, through this case study I will support the claim that if the full Lie algebras of our current successful theories are stable, it is possible to disregard any kind of structural loss in the future and explain the relevant successful predictions in a way that we can support structural realism accordingly.

Delegation Using Forward Induction

This paper explores how delegation can be used as a signal to sustain cooperation. I consider a static principal–agent model with two tasks, one resembling a coordination game. If there is asymmetric information about the agent’s type, the principal with high private belief can delegate the first task as a signal. This is also supported by the forward induction argument. However, in the laboratory setting, this equilibrium is chosen only sometimes. When the subjects have information about past sessions, there is a significant increase in the use of delegation. This finding sheds light on equilibrium selection in Bayesian games.

Global Controllability for Quasilinear Nonnegative Definite System of ODEs and SDEs

We consider exact and averaged control problem for a system of quasilinear ODEs and SDEs with a nonnegative definite symmetric matrix of the system. The strategy of the proof is the standard linearization of the system by fixing the function appearing in the nonlinear part of the system and then applying the Leray–Schauder fixed point theorem. We shall also need the continuous induction arguments to prolong the control to the final state which is a novel approach in the field. This enables us to obtain controllability for arbitrarily large initial data (so-called global controllability).

A Problem with the Traveller’s Dilemma

AbstractIn the traveller’s dilemma, a compelling argument seems to prescribe an intuitively irrational course of action. However, I argue all is not as it appears in the traveller’s dilemma; standard game‐theoretic agents have a way out. To show this, I detail the key characteristics of the traveller’s dilemma, its underlying assumptions, and explain why we have an apparent conflict with what an idealized agent would intuitively do. I then argue that the backwards induction argument fails here because of the unique structure of the game. I conclude that we ought not focus our attention on the traveller’s dilemma.