Symbol Of Operator Research Articles

Five ways to recover the symbol of a non-binary localization operator

Five constructive methods for recovering the symbol of a time-frequency localization operator with non-binary symbol are presented, two based on earlier work and three novel methods. For the two derivative methods which have previously been applied to binary symbols, we propose a changed symbol estimator and provide additional estimates that show how we can recover non-binary symbols. The three novel methods each have their own advantages and are all applicable to non-binary symbols. Two of them rely on prescribing the input of the localization operator and examining the output, allowing for targeting of the part of the symbol one wishes to recover while the last one relies on spectral information about the operator. All five methods are also implemented numerically and evaluated with the code available.

Topology of shallow-water waves on a rotating sphere

Topological properties of the spectrum of shallow-water waves on a rotating spherical body are established. Particular attention is paid to spectral flow, i.e. the modes whose frequencies transit between the Rossby and inertia–gravity wavebands as the zonal wavenumber is varied. Organising the modes according to the number of zeros of their meridional velocity, we conclude that the net number of modes transiting between the shallow-water wavebands on the sphere is null, in contrast to the Matsuno spectrum. This difference can be explained by a miscount of zeros under the $\beta$ -plane approximation. We corroborate this result with the analysis of Delplace et al. (Science, vol. 358, 2017, pp. 1075–1077) by showing that the curved metric discloses a pair of degeneracy points in the Weyl symbol of the wave operator, non-existent under the $\beta$ -plane approximation, each of them bearing a Chern number of $-1$ .

Physics-guided symbolic neural network reveals optimal functional forms describing ground motions

This study presents a novel framework for ground motion modelling utilizing Physics-Guided Symbolic Neural Networks (PGSNN). Symbolic neural networks offer a new method for knowledge discovery, providing a unique perspective for automatically uncovering predictive functional forms from data. This approach differs from traditional methods as it does not rely on predefined equations. Instead, it employs symbolic operators to freely combine input parameters in a high-dimensional space. This method addresses the problem of data imbalance by incorporating physical guidance to ensure that the model produces results that are consistent with established physical principles. The resulting equations align with the expectations of the engineering seismology community, particularly within the magnitude-distance ranges, where classical equations are well calibrated. The prediction performance of the PGSNN, evaluated across different intensity measures (PGA, PGV, and PSA), was assessed by calculating the residuals between measured and predicted values and their standard deviations. The predictive capability of this model was verified using new event records. The results indicate that the prediction performance of the PGSNN is comparable to those of traditional methods.

Notes on peculiarities of the Schwinger-DeWitt technique: One-loop double poles, total-derivative terms, and determinant anomalies

We discuss peculiarities of the Schwinger-DeWitt technique for quantum effective action, associated with the origin of dimensionally regularized double-pole divergences of the one-loop functional determinant for massive Proca model in a curved spacetime. These divergences have the form of the total-derivative term generated by integration by parts in the functional trace of the heat kernel for the Proca vector field operator. Because of the nonminimal structure of second-order derivatives in this operator, its vector field heat kernel has a nontrivial form, involving the convolution of the scalar d’Alembertian Green’s function with its heat kernel. Moreover, its asymptotic expansion is very different from the universal predictions of Gilkey-Seeley heat kernel theory because the Proca operator violates one of the basic assumptions of this theory—the nondegeneracy of the principal symbol of an elliptic operator. This modification of the asymptotic expansion explains the origin of double-pole total-derivative terms. Another hypostasis of such terms is in the problem of multiplicative determinant anomalies—lack of factorization of the functional determinant of a product of differential operators into the product of their individual determinants. We demonstrate that this anomaly should have the form of total-derivative terms and check this statement by calculating divergent parts of functional determinants for products of minimal and nonminimal second-order differential operators in curved spacetime. Published by the American Physical Society 2024

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} consists on those Dirichlet series convergent in the right half-plane C+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {C}}}_+$$\\end{document} and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols Φ:C+→C+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi :{{\\mathbb {C}}}_+\\rightarrow {{\\mathbb {C}}}_+$$\\end{document} giving rise to bounded composition operators CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document} in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} and denote this class by GA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}_{\\mathcal {A}}$$\\end{document}. We also characterise when the operator CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document} is compact in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document}. As a byproduct, we show that the weak compactness is equivalent to the compactness for CΦ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\Phi }$$\\end{document}. Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions {Φt}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{\\Phi _t\\}$$\\end{document} in the class GA\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {G}_{\\mathcal {A}}$$\\end{document} and strongly continuous semigroups of composition operators {Tt}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{T_t\\}$$\\end{document}, Ttf=f∘Φt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_tf=f\\circ \\Phi _t$$\\end{document}, f∈A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\in \\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document}. We conclude providing examples showing the differences between the symbols of bounded composition operators in A(C+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {A}({{\\mathbb {C}}}_+)$$\\end{document} and the Hardy spaces of Dirichlet series Hp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^p$$\\end{document} and H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}^{\\infty }$$\\end{document}.

Optimal tree tensor network operators for tensor network simulations: Applications to open quantum systems.

Tree tensor network states (TTNS) decompose the system wavefunction to the product of low-rank tensors based on the tree topology, serving as the foundation of the multi-layer multi-configuration time-dependent Hartree method. In this work, we present an algorithm that automatically constructs the optimal and exact tree tensor network operators (TTNO) for any sum-of-product symbolic quantum operator. The construction is based on the minimum vertex cover of a bipartite graph. With the optimal TTNO, we simulate open quantum systems, such as spin relaxation dynamics in the spin-boson model and charge transport in molecular junctions. In these simulations, the environment is treated as discrete modes and its wavefunction is evolved on equal footing with the system. We employ the Cole-Davidson spectral density to model the glassy phonon environment and incorporate temperature effects via thermo-field dynamics. Our results show that the computational cost scales linearly with the number of discretized modes, demonstrating the efficiency of our approach.

Asymptotics of solutions to systems of (pseudo)differential equations with localized right-hand sides

Abstract The article describes a method for constructing semiclassical asymptotic solutions of multidimensional (pseudo)differential partial differential equations with localized right-hand sides. The method is based on the asymptotic reduction of vector problems (under certain additional conditions) to scalar (pseudo)differential problems, for which efficient asymptotic formulas have recently been constructed. The method is based on the Feynman–Maslov operator calculus. Even if the original problems are stated for differential equations, the operators in reduced problems, as a rule, turn out to be pseudodifferential. Further, nontrivial problems of calculating operator symbols in the reduced scalar problems arise in the reduction process. We discuss approaches to the transformation of these symbols so as to enable their effective use when solving real problems. The method is illustrated by several physical examples.

Axiomatization of Non-Recursive Aggregates in First-Order Answer Set Programming

This paper contributes to the development of theoretical foundations of answer set programming. Groundbreaking work on the SM operator by Ferraris, Lee, and Lifschitz proposed a definition/semantics for logic (answer set) programs based on a syntactic transformation similar to parallel circumscription. That definition radically differed from its predecessors by using classical (second-order) logic and avoiding reference to either grounding or fixpoints. Yet, the work lacked the formalization of crucial and commonly used answer set programming language constructs called aggregates. In this paper, we present a characterization of logic programs with aggregates based on a many-sorted generalization of the SM operator. This characterization introduces new function symbols for aggregate operations and aggregate elements, whose meaning can be fixed by adding appropriate axioms to the result of the SM transformation. We prove that our characterization coincides with the ASP-Core-2 semantics for logic programs and, if we allow non-positive recursion through aggregates, it coincides with the semantics of the answer set solver CLINGO.

Crystallographic Quaternions

Symmetry transformations in crystallography are traditionally represented as equations and matrices, which can be suitable both for orthonormal and crystal reference systems. Quaternion representations, easily constructed for any orientations of symmetry operations, owing to the vector structure based on the direction of the rotation axes or of the normal vectors to the mirror plane, are known to be advantageous for optimizing numerical computing. However, quaternions are described in Cartesian coordinates only. Here, we present the quaternion representations of crystallographic point-group symmetry operations for the crystallographic reference coordinates in triclinic, monoclinic, orthorhombic, tetragonal, cubic and trigonal (in rhombohedral setting) systems. For these systems, all symmetry operations have been listed and their applications exemplified. Owing to their concise form, quaternions can be used as the symbols of symmetry operations, which contain information about both the orientation and the rotation angle. The shortcomings of quaternions, including different actions for rotations and improper symmetry operations, as well as inadequate representation of the point symmetry in the hexagonal setting, have been discussed.

On Hyperbolic Equations with a Translation Operator in Lowest Derivatives

In the half-plane, a solution to a two-dimensional hyperbolic equation with a translation operator in the lowest derivative with respect to a spatial variable varying along the entire real axis is constructed in an explicit form. It is proven that the solutions obtained are classical if the real part of the symbol of a differential-difference operator in the equation is positive.

On Global Solutions of Two-Dimensional Hyperbolic Equations with General-Kind Nonlocal Potentials

In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not assumed that the real part of the symbol of the differential-difference operator contained in the equation has a constant sign. Previously, it was possible to remove that substantial restriction (i.e., the specified sign constancy) only for the case where the nonlocal term (i.e., the translated potential) is unique. In the present paper, we consider the case of the general-kind one-variable nonlocal potential, i.e., equations with an arbitrary amount of translated terms. No commensurability assumptions are imposed on the translation lengths. The following results are presented: We find a condition relating the coefficients at the nonlocal terms of the investigated equation and the length of the translations, providing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation.

Monumentalizing Landscapes: Ontologies of Scale and Guatemalan National Consolidation

ABSTRACT: In 1905, Manuel Estrada Cabrera, one of Guatemala's most infamous dictators, inaugurates a gargantuan relief map, one of Latin American's most dramatic cartographic projects of the time. In 1882, the nation had relinquished vast regions of its territory to Mexico. This exemplified the region's fragility in the face of continuous Central American civil wars and a commercial weakness that rendered Guatemala vulnerable to foreign intervention, both political and economic. In this context, Guatemalan was faced with foundational narrative voids and a lack of any semblance of national identity. In this essay I contend that the state turned to the nation's volcanic geography as a solution: its dramatic landscapes could serve as both settings and protagonists for an alternative spatial narrative. I argue that the relief map can be read as an ideal fossilized artifact capturing these symbolic operations imposed by Guatemala's creole liberal elite within the complex global and national spatial dynamics of the time. In its monumentality, the map attempted to symbolically tap into and domesticate the power of the nation's landscape for the sake of national consolidation: a case of the natural sublime turned cultural/political sublime. In the process, and underlying these symbolic spatial operations, the State inflicted unprecedented violence on nature and rendered invisible entire sections of the population. The project ultimately fails, however, because the semantic power of a volcanic landscape is predicated on a chaotic and volatile land that had seeped into popular and political imagination through a long history of natural disasters and corresponding social turmoil.

Varieties defined by basic equations have the amalgamation property

A variety is a class of algebraic structures axiomatized by a set of equations. An equation is basic if there is at most one occurrence of a nonnullary operation symbol on each side. We show that a variety axiomatized by basic equations has the strong amalgamation property. Suppose further that the language has no constant symbol and, for each equation, either one side is operation-free, or exactly the same variables appear on both sides. Then also the joint embedding property holds, hence if the language is finite we get the existence of Fraïssé limits for finite algebras. Examples include virtually all the varieties defining classical Maltsev conditions. In a few special cases, the above properties are preserved when further unary operation symbols appear in the equations.

Płonka adjunction

Abstract Let $\varSigma $ be a signature without $0$-ary operation symbols and $\textsf{Sl}$ the category of semilattices. Then, after defining and investigating the categories $\int ^{\textsf{Sl}}\textrm{Isys}_{\varSigma }$, of inductive systems of $\varSigma $-algebras over all semilattices, which are ordered pairs $\mathscr{A}= (\textbf{I},\mathscr{A})$ where $\textbf{I}$ is a semilattice and $\mathscr{A}$ an inductive system of $\varSigma $-algebras relative to $\textbf{I}$, and PłAlg$ (\varSigma )$, of Płonka $\varSigma $-algebras, which are ordered pairs $(\textbf{A},D)$ where $\textbf{A}$ is a $\varSigma $-algebra and $D$ a Płonka operator for $\textbf{A}$, i.e. in the traditional terminology, a partition function for $\textbf{A}$, we prove the main result of the paper: There exists an adjunction, which we call the Płonka adjunction, from $\int ^{\textsf{Sl}}\textrm{Isys}_{\varSigma }$ to PłAlg$ (\varSigma )$.

Symbols of mixed unitary quantum channels generated by finite unitary groups

We study the symbols of linear operators generated by a finite unitary group. Under some assumptions, it is shown that the operator can be restored from its symbol. We also introduce symbols of mixed unitary quantum channels generated by finite unitary groups. An example illustrating the technique is given.

There are no post-quantum weakly pseudo-free families in any nontrivial variety of expanded groups

Let [Formula: see text] be a finite set of finitary operation symbols and let [Formula: see text] be a nontrivial variety of [Formula: see text]-algebras. Assume that for some set [Formula: see text] of group operation symbols, all [Formula: see text]-algebras in [Formula: see text] are groups under the operations associated with the symbols in [Formula: see text]. In other words, [Formula: see text] is assumed to be a nontrivial variety of expanded groups. In particular, [Formula: see text] can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in [Formula: see text], even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families [Formula: see text] of computational and black-box [Formula: see text]-algebras (where [Formula: see text]) such that for every [Formula: see text], each element of [Formula: see text] is represented by a unique bit string of length polynomial in the length of d. In our main result, we use straight-line programs to represent nontrivial relations between elements of [Formula: see text]-algebras. Note that under certain conditions, this result depends on the classification of finite simple groups. Also, we define and study some types of post-quantum weak pseudo-freeness for families of computational and black-box [Formula: see text]-algebras.

ПСИХОЛОГО-ПЕДАГОГІЧНІ ПІДХОДИ ДО НАВЧАЛЬНОЇ ДІЯЛЬНОСТІ ШКОЛЯРІВ З РІЗНОЮ ЛАТЕРАЛІЗАЦІЄЮ ПІВКУЛЬ ГОЛОВНОГО МОЗКУ

The study of the functional asymmetry of the cerebral hemispheres is an important problem of modern science. Teaching that takes into account functional asymmetry can be more effective and contribute to the development of the child’s full potential, including students with special educational needs. It has been proven that school teaching methods are usually focused on the development of the left hemisphere. This is due to the fact that language is the main means of learning at school. In the process of learning, the functional asymmetry increases: the left hemisphere specialises in symbolic operations, and the right hemisphere in figurative ones. However, teaching left- and right-brained students should not be the same. The predominance of left or right brain activity determines the type of thinking a person has, their character, temperament, memory, hand dominance and other psychophysiological features that must be taken into account for effective teaching. To study the peculiarities of the distribution of functional asymmetry of the brain, we selected 14-15 aged students, in which the process of lateralisation is considered to be completed. The results of the study of the coefficient of functional asymmetry of the brain and the type of temperament of 9th grade students of the experimental and control groups are presented. The obtained results correspond to the average world indicators. General recommendations for improving the academic achievements of adolescents with different lateralisation of the hemispheres, taking into account their individual characteristics, in particular the dominant type of temperament. The psychological and pedagogical approaches to the effective implementation of the educational process for right- and left-hemispheric students and ambidextrous students are proposed, which contain methodological recommendations for the selection of forms, methods and means of teaching students, taking into account their functional asymmetry of the hemispheres.

TinyNS: Platform-Aware Neurosymbolic Auto Tiny Machine Learning.

Machine learning at the extreme edge has enabled a plethora of intelligent, time-critical, and remote applications. However, deploying interpretable artificial intelligence systems that can perform high-level symbolic reasoning and satisfy the underlying system rules and physics within the tight platform resource constraints is challenging. In this paper, we introduce TinyNS, the first platform-aware neurosymbolic architecture search framework for joint optimization of symbolic and neural operators. TinyNS provides recipes and parsers to automatically write microcontroller code for five types of neurosymbolic models, combining the context awareness and integrity of symbolic techniques with the robustness and performance of machine learning models. TinyNS uses a fast, gradient-free, black-box Bayesian optimizer over discontinuous, conditional, numeric, and categorical search spaces to find the best synergy of symbolic code and neural networks within the hardware resource budget. To guarantee deployability, TinyNS talks to the target hardware during the optimization process. We showcase the utility of TinyNS by deploying microcontroller-class neurosymbolic models through several case studies. In all use cases, TinyNS outperforms purely neural or purely symbolic approaches while guaranteeing execution on real hardware.

The Liouville theorem for a class of Fourier multipliers and its connection to coupling

AbstractThe classical Liouville property says that all bounded harmonic functions in , that is, all bounded functions satisfying , are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator , such that the solutions to are Lebesgue a.e. constant (if is bounded) or coincide Lebesgue a.e. with a polynomial (if is polynomially bounded). The class of Fourier multipliers includes the (in general non‐local) generators of Lévy processes. For generators of Lévy processes, we obtain necessary and sufficient conditions for a strong Liouville theorem where is positive and grows at most exponentially fast. As an application of our results above, we prove a coupling result for space‐time Lévy processes.