Representation Theorem Research Articles

Parsing as a lifting problem and the Chomsky-Sch\"utzenberger representation theorem

We begin by explaining how any context-free grammar encodes a functor of operads from a freely generated operad into a certain "operad of spliced words". This motivates a more general notion of CFG over any category $C$, defined as a finite species $S$ equipped with a color denoting the start symbol and a functor of operads $p : Free[S] \to W[C]$ into the operad of spliced arrows in $C$. We show that many standard properties of CFGs can be formulated within this framework, and that usual closure properties of CF languages generalize to CF languages of arrows. We also discuss a dual fibrational perspective on the functor $p$ via the notion of "displayed" operad, corresponding to a lax functor of operads $W[C] \to Span(Set)$. We then turn to the Chomsky-Sch\"utzenberger Representation Theorem. We describe how a non-deterministic finite state automaton can be seen as a category $Q$ equipped with a pair of objects denoting initial and accepting states and a functor of categories $Q \to C$ satisfying the unique lifting of factorizations property and the finite fiber property. Then, we explain how to extend this notion of automaton to functors of operads, which generalize tree automata, allowing us to lift an automaton over a category to an automaton over its operad of spliced arrows. We show that every CFG over a category can be pulled back along a ND finite state automaton over the same category, and hence that CF languages are closed under intersection with regular languages. The last important ingredient is the identification of a left adjoint $C[-] : Operad \to Cat$ to the operad of spliced arrows functor, building the "contour category" of an operad. Using this, we generalize the C-S representation theorem, proving that any context-free language of arrows over a category $C$ is the functorial image of the intersection of a $C$-chromatic tree contour language and a regular language.

Least-squares collocation: a spherical harmonic representer theorem

SUMMARY The functional analysis of the least-squares collocation (LSC) for gravity potential modelling using m measurements is revisited starting from an explicit spherical harmonic expansion. A spherical harmonic representer theorem (SHRT) is given: the model of the potential is a linear combination of m kernels or covariances. This theorem is independent of the specific forms of the data-fitting loss and the regularizer, showing that it is a stronger result than the LSC theory. The corresponding reproducing kernel Hilbert space is explicitly specified. When the least-squares data-fitting loss and the quadratic regularizer are employed, the SHRT gives exactly the LSC method for variable prediction. The nominal prediction precision assessment of the SHRT and that of the LSC are also explicitly compared; this contributes to the unification of the deterministic and stochastic analyses of the LSC theory.

Cashing In or Paying Out? Analysis of the Relation Between Corporate Earnings and Cash

This paper examines the relation between corporate earnings and cash in the US on annual basis in the period 2002 to 2022. We document that both earnings and cash holdings are non-stationary and therefore we use the Granger representation theorem and the methods of cointegration analysis and make an attempt to model the relation between these company variables. We document a negative sign of the cointegration coefficient estimate that is statistically significant. The statistical significance confirms that cash and earnings are cointegrated, and depend on each other. However, the significance of the models with earnings as the dependent variable is larger, and the coefficient estimates are more reliably negative. These results indicate that earnings depend on cash to a larger degree than cash depends on earnings.

Lindenbaum-Type Logical Structures

In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the Tarski- and the Lindenbaum-types. The characterization theorems for the Tarski- and two of the four different Lindenbaum-type logical structures have been proved as well. The separations between the five classes of logical structures, viz., the four Lindenbaum-types and the Tarski-type have been established via examples. Finally, we study the logical structures that are of both Tarski- and a Lindenbaum-type, show their separations, and end with characterization, adequacy, minimality, and representation theorems for one of the Tarski–Lindenbaum-type logical structures.

Ranked soft sets

AbstractThis article defines ranked soft sets and establishes their fundamental theory. This new model of uncertain knowledge is a non‐numerical yet powerful improvement of soft sets. The model relies on a qualitative improvement of the basic parameterized description posed by soft sets. We define relations between ranked soft sets and some existing models (N‐soft sets, fuzzy soft sets, probabilistic soft sets) that enhance the soft set spirit with the help of additional quantities. Primary contributions to their development include set‐theoretic operations and representation theorems, at a theoretical level; and scores and aggregation operators, at a practical level. Finally we design a multi‐person decision‐making strategy for data in the form of ranked soft sets that takes advantage of these elements.

B-frame duality

This paper introduces the category of b-frames as a new tool in the study of complete lattices. B-frames can be seen as a generalization of posets, which play an important role in the representation theory of Heyting algebras, but also in the study of complete Boolean algebras in forcing. This paper combines ideas from the two traditions in order to generalize some techniques and results to the wider context of complete lattices. In particular, we lift a representation theorem of Allwein and MacCaull to a duality between complete lattices and b-frames, and we derive alternative characterizations of several classes of complete lattices from this duality. This framework is then used to obtain new results in the theory of complete Heyting algebras and the semantics of intuitionistic propositional logic.

Reducible Theories and Amalgamations of Models

Within knowledge representation in artificial intelligence, a first-order ontology is a theory in first-order logic that axiomatizes the concepts in some domain. Ontology verification is concerned with the relationship between the intended models of an ontology and the models of the axiomatization of the ontology. In particular, we want to characterize the models of an ontology up to isomorphism and determine whether or not these models are equivalent to the intended models of the ontology. Unfortunately, it can be quite difficult to characterize the models of an ontology up to isomorphism. In the first half of this article, we review the different metalogical relationships between first-order theories and identify which relationship is needed for ontology verification. In particular, we will demonstrate that the notion of logical synonymy is needed to specify a representation theorem for the class of models of one first-order ontology with respect to another. In the second half of the article, we discuss the notion of reducible theories and show we can specify representation theorems by which models are constructed by amalgamating models of the constituent ontologies.

Ext-multiplicity theorem for standard representations of $$(\textrm{GL}_{n+1},\textrm{GL}_n)$$

Let $\pi_1$ be a standard representation of $\mathrm{GL}_{n+1}(F)$ and let $\pi_2$ be the smooth dual of a standard representation of $\mathrm{GL}_n(F)$. When $F$ is non-Archimedean, we prove that $\mathrm{Ext}^i_{\mathrm{GL}_n(F)}(\pi_1, \pi_2)$ is $\cong \mathbb C$ when $i=0$ and vanishes when $i \geq 1$. The main tool of the proof is a notion of left and right Bernstein-Zelevinsky filtrations. An immediate consequence of the result is to give a new proof on the multiplicity at most one theorem. Along the way, we also study an application of an Euler-Poincar\'e pairing formula of D. Prasad on the coefficients of Kazhdan-Lusztig polynomials. When $F$ is an Archimedean field, we use the left-right Bruhat-filtration to prove a multiplicity result for the equal rank Fourier-Jacobi models of standard principal series.

Algebras of generalized tree languages with fixed variables

Tree languages which are sets of terms always play a prominent role in the first-order languages and theoretical computer science. In this paper, tree languages induced by terms with fixed variables are considered. Under the applications of an operation on tree languages, we construct the algebra of such languages having many properties of abstract clones. A strong connection with theory of general functions is given through a representation theorem. Additionally, the semigroup of mappings of which their images are tree languages with fixed variables is given.

Representation theorem and viability property for multidimensional BSDEs and their applications

Representation theorem and viability property for multidimensional BSDEs and their applications

Blind Nonparametric Estimation of SISO Continuous-time Systems

Blind system identification is aimed at finding parameters of a system model when the input is inaccessible. In this paper, we propose a blind system identification method that delivers a single-input single-output, continuous-time model in a nonparametric kernel form. We take advantage of the representer theorem to form a joint maximum a posteriori estimator of the input and system impulse response. The identified system model and input are optimised in sequence to overcome the blind problem with generalised cross validation used to select appropriate hyperparameters given some fixed input sequence. We demonstrate via Monte Carlo simulations the accuracy of the method in terms of estimating the input.

Understanding neural networks with reproducing kernel Banach spaces

Characterizing the function spaces corresponding to neural networks can provide a way to understand their properties. In this paper we discuss how the theory of reproducing kernel Banach spaces can be used to tackle this challenge. In particular, we prove a representer theorem for a wide class of reproducing kernel Banach spaces that admit a suitable integral representation and include one hidden layer neural networks of possibly infinite width. Further, we show that, for a suitable class of ReLU activation functions, the norm in the corresponding reproducing kernel Banach space can be characterized in terms of the inverse Radon transform of a bounded real measure, with norm given by the total variation norm of the measure. Our analysis simplifies and extends recent results in [45], [36], [37].

The Sparsity of Cycle Spinning for Wavelet-Based Solutions of Linear Inverse Problems

The usual explanation of the efficacy of wavelet-based methods hinges on the sparsity of many real-world objects in the wavelet domain. Yet, standard wavelet-shrinkage techniques for sparse reconstruction are not competitive in practice, one reason being that the lack of shift-invariance of the wavelet transform produces blocky artifacts. The standard remedy is cycle spinning, which results in a substantial reduction of these artifacts. In this letter, we propose a theoretical investigation of the sparsity of solutions to the cycle-spinning variant of wavelet-based resolutions of linear inverse problems. We derive a representer theorem that provides a complete characterization of the solution set. Our theorem indicates that the solutions are typically not sparse, where sparsity is measured with respect to the wavelet dictionary. This exposes that the role of sparsity in the success of wavelet-based solutions of linear inverse problems requires further investigation. We corroborate our theoretical results with numerical examples for the problem of image denoising.

Tectonic Earthquake Source Mechanism Model Based on Moment Theory

During the construction of strategic facilities, micro zoning is required to determine the peak values of terrain acceleration due to possible earthquakes. The acceleration of the soil in this case strongly depends not only on the power of the earthquake, but also on the earthquake source mechanism and the geodynamic state of the terrain. The known dislocation models of a single dipole with a moment and a pair of dipoles without a moment of the earthquake source mechanism satisfactorily describe the observed effects of the quadrant stress distribution on the Earth’s surface during earthquakes. When carrying out calculations within the framework of the theory of elasticity, the actions of the dipoles are expressed through volumetric forces. There are two known models of replacing the moments with equivalent forces: one of them is based on the equilibrium equations for an infinitely elastic space (Landau and Lifshitz,1965; Maruyama,1963), and the other is based on the representation theorem for elastic bodies, by introducing a singular internal volume, at the boundary of which there are dislocations (Vvedenskaya, 1969; Aki and Richards, 1983). Although these models involve moment effects, they themselves are derived from the momentless theory of elasticity. In our work, we propose a double dipole effect without a moment based on the moment theory of elasticity. The proposed model of the earthquake source mechanism is applied to solve the problem of stress variations in the Earth’s crust in Central Asia using the example of a particular earthquake with a simplified orientation of the rupture plane. Stress variation is understood as the difference in stresses in problems with and without an earthquake mechanism. Static stresses are obtained by solving the inverse elasticity problem with partially unknown boundary conditions. The lithosphere is a prismatic body consisting of several homogeneous blocks, the upper surfaces of which correspond to the relief of Central Asia. Verification of the results of the numerical solution is carried out by comparing the obtained stresses with previously established empirical values. As a priori stresses for solving the inverse problem, we used the solution of the elastic plane problem, the boundary conditions in which correspond to the lateral compression of the lithosphere of the region of the Indian and Arabian plates on the one hand, and the Eurasian plate on the other hand. The obtained solutions of the problem were used to analyze the geodynamic state of Central Asia. Based on the results of laboratory experiments, the unambiguity of the conclusions about the geodynamic state of the Earth’s crust (compression, extension) according to the Lode-Nadai coefficient, which are currently used by many researchers, is questioned. It is shown, contrary to earlier statements, that the values μσ = +1 and μσ = –1 can correspond simultaneously to both tension and compression cases, depending on the spatial form (ellipsoid) of the stress state. Geodynamic analysis of the Earth’s crust is carried out according to the Anderson method.

Fredholm inversion around a singularity: Application to autoregressive time series in Banach space

<abstract><p>This paper considers inverting a holomorphic Fredholm operator pencil. Specifically, we provide necessary and sufficient conditions for the inverse of a holomorphic Fredholm operator pencil to have a simple pole and a second order pole. Based on these results, a closed-form expression of the Laurent expansion of the inverse around an isolated singularity is obtained in each case. As an application, we also obtain a suitable extension of the Granger-Johansen representation theorem for random sequences taking values in a separable Banach space. Due to our closed-form expression of the inverse, we may fully characterize solutions to a given autoregressive law of motion except a term that depends on initial values.</p></abstract>

Unitless Frobenius Quantales

It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices \(M_n\) and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.

On a class of analytic functions closely related to a class defined by Silverman and Silvia

UDC 517.5 We define and study a class of analytic functions in the unit disc by using the modification of the well-known Silverman and Silvia's analytic formula for starlike functions with respect to a boundary point. The representation theorem, as well as growth and distortion theorems are established for the new class of functions. Further, early coefficients of the new class of functions are also estimated.

Lipschitz Bernoulli Utility Functions

We obtain several variants of the classic von Neumann–Morgenstern expected utility theorem with and without the completeness axiom in which the derived Bernoulli utility functions are Lipschitz. The prize space in these results is an arbitrary separable metric space, and the utility functions are allowed to be unbounded. The main ingredient of our results is a novel (behavioral) axiom on the underlying preference relations, which is satisfied by virtually all stochastic orders. The proof of the main representation theorem is built on the fact that the dual of the Kantorovich–Rubinstein space is (isometrically isomorphic to) the Banach space of Lipschitz functions that vanish at a fixed point. An application to the theory of nonexpected utility is also provided.

Lions' representation theorem and applications

The Lions' Representation Theorem (LRT) is a version of the Lax–Milgram Theorem where completeness of one of the spaces is not needed. In this paper, LRT is deduced from an operator-theoretical result on normed spaces, which is of independent interest. As an example, we give a new characterization of dissipativity. The main part of the paper is a theory of derivations, based on LRT, which we develop. Its aim is to establish well-posedness results, not only for evolution in time but also for more general settings in terms of this new notion of derivation. One application concerns non-autonomous evolution equations with a new kind of boundary condition where values at the initial and final time are mixed.

Convergence and error analysis of compressible fluid flows with random data: Monte Carlo method

The goal of this paper is to study convergence and error estimates of the Monte Carlo method for the Navier–Stokes equations with random data. To discretize in space and time, the Monte Carlo method is combined with a suitable deterministic discretization scheme, such as a finite volume (FV) method. We assume that the initial data, force and the viscosity coefficients are random variables and study both the statistical convergence rates as well as the approximation errors. Since the compressible Navier–Stokes equations are not known to be uniquely solvable in the class of global weak solutions, we cannot apply pathwise arguments to analyze the random Navier–Stokes equations. Instead, we have to apply intrinsic stochastic compactness arguments via the Skorokhod representation theorem and the Gyöngy–Krylov method. Assuming that the numerical solutions are bounded in probability, we prove that the Monte Carlo FV method converges to a statistical strong solution. The convergence rates are discussed as well. Numerical experiments illustrate theoretical results.