Predicate Calculus Research Articles

Basic Predicate Calculus is not Sound with Respect to the Strong Variant of Strictly Primitive Recursive Realizability

Basic Predicate Calculus is not Sound with Respect to the Strong Variant of Strictly Primitive Recursive Realizability

Carleman type inequalities for fractional relativistic operators

In this paper we derive Carleman estimates for the fractional relativistic operator. We consider changing-sign solutions to the heat equation for such operators. We prove monotonicity inequalities and convexity of certain energy functionals to deduce Carleman estimates with linear exponential weight. Our approach is based on spectral methods and functional calculus.

Maximal Hörmander Functional Calculus onLpSpaces and UMD Lattices

AbstractLet $A$ be a generator of an analytic semigroup having a Hörmander functional calculus on $X = L^p(\Omega ,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for Hörmander spectral multipliers decaying sufficiently fast at $\infty $, there holds a maximal estimate $\| \sup _{t> 0} |m(tA)f|\, \|_{L^p(\Omega ,Y)} \lesssim \|f\|_{L^p(\Omega ,Y)}$. We also show square function estimates $\left \| \left ( \sum _k \sup _{t> 0} |m_k(tA)f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)} \lesssim \left \| \left ( \sum _k |f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\ensuremath {{\mathbb {R}}}^d)$. As corollaries, we obtain maximal estimates for wave propagators and Bochner–Riesz means. Finally, we illustrate the results by giving several examples of operators $A$ that admit a Hörmander functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.

A Lichnerowicz vanishing theorem for the maximal Roe algebra

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher index of the Dirac operator vanishes in K-theory of the maximal equivariant Roe algebra. The group action is not assumed to be cocompact. A key step in the proof is to establish a functional calculus for the Dirac operator in the maximal equivariant uniform Roe algebra. This allows us to prove vanishing of the index of the Dirac operator in K-theory of this algebra, which in turn yields the result for the maximal higher index.

A fine-grained computational interpretation of Girard’s intuitionistic proof-nets

This paper introduces a functional term calculus, called pn, that captures the essence of the operational semantics of Intuitionistic Linear Logic Proof-Nets with a faithful degree of granularity, both statically and dynamically. On the static side, we identify an equivalence relation on pn-terms which is sound and complete with respect to the classical notion of structural equivalence for proof-nets. On the dynamic side, we show that every single (exponential) step in the term calculus translates to a different single (exponential) step in the graphical formalism, thus capturing the original Girard’s granularity of proof-nets but on the level of terms. We also show some fundamental properties of the calculus such as confluence, strong normalization, preservation of β-strong normalization and the existence of a strong bisimulation that captures pairs of pn-terms having the same graph reduction.

Functions of class $C^\infty$ in non-commuting variables in the context of triangular Lie algebras

We construct a certain completion $C^\infty_\mathfrak{g}$ of the universal enveloping algebra of a triangular real Lie algebra $\mathfrak{g}$. It is a Fréchet-Arens-Michael algebra that consists of elements of polynomial growth and satisfies to the following universal property: every Lie algebra homomorphism from $\mathfrak{g}$ to a real Banach algebra all of whose elements are of polynomial growth has an extension to a continuous homomorphism with domain $C^\infty_\mathfrak{g}$. Elements of this algebra can be called functions of class $C^\infty$ in non-commuting variables. The proof is based on representation theory and employs an ordered $C^\infty$-functional calculus. Beyond the general case, we analyze two simple examples. As an auxiliary material, the basics of the general theory of algebras of polynomial growth are developed. We also consider local variants of the completion and obtain a sheaf of non-commutative functions on the Gelfand spectrum of $C^\infty_\mathfrak{g}$ in the case when $\mathfrak{g}$ is nilpotent. In addition, we discuss the theory of holomorphic functions in non-commuting variables introduced by Dosi and apply our methods to prove theorems strengthening some his results.

Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics

We show for a certain class of operators A and holomorphic functions f that the functional calculus $$A\mapsto f(A)$$ is holomorphic. Using this result we are able to prove that fractional Laplacians $$(1+\Delta ^g)^p$$ depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.

Calculus of Several Harmonic Functions

Calculus of Several Harmonic Functions

Extraction of common properties of objects for creation of a logic ontology

The paper describes an approach to the formation of ontology based on descriptions of objects in terms of the predicate calculus language. With this approach, an object is represented as a set of its elements, on which a set of predicates is defined that defines the properties of these elements and the relationship between them. A description of an object is a conjunction of literals that are true on elements of an object. In the present work, ontology is understood as an oriented graph with descriptions of subsets as nodes and such that the elements of a set at the end of an oriented edge have the properties of the elements of the set at the beginning of this edge. Three settings of an ontology construction problem are considered: 1) all predicates are binary and subsets of the original set of objects are given; 2) all predicates are binary and it is required to find subsets of the original set; 3) among the predicates there are many-valued ones and subsets of the original set of objects are given. The main tool for construction such a description is to extract an elementary conjunction of literals of predicate formulas that is isomorphic to subformulas of some formulas. The definition of an isomorphism of elementary conjunctions of atomic predicate formulas is given. The method of ontology construction is formulated. An illustrative example is provided.

Spectral order isomorphisms and AW$AW$*‐factors

AbstractThe paper deals with spectral order isomorphisms in the framework of *‐algebras. We establish that every spectral order isomorphism between sets of all self‐adjoint elements (or between sets of all effects, or between sets of all positive elements) in *‐factors of Type I has a canonical form induced by a continuous function calculus and an isomorphism between projection lattices. In particular, this solves an open question about spectral order automorphisms of the set of all (bounded) self‐adjoint operators on an infinite‐dimensional Hilbert space. We also discuss spectral order isomorphisms preserving, in addition, orthogonality in both directions.

Itô's formula for noncommutative $C^2$ functions of free Itô processes

In a recent paper, the author introduced a rich class NC^k(\mathbb{R}) of “noncommutative C^k ” functions \mathbb{R} \to \mathbb{C} whose operator functional calculus is k -times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative C^2 functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher's theory of free stochastic calculus – including their free Itô formula for polynomials – to allow free Itô processes driven by multiple freely independent semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space NC^2(\mathbb{R}) . Along the way, we also obtain a useful “traced” Itô formula for arbitrary C^2 scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for C^2 scalar functions of N \times N Hermitian matrix Itô processes.

Virtual Algebraic Isomorphisms between Predicate Calculi of Finite Rich Signatures

Virtual Algebraic Isomorphisms between Predicate Calculi of Finite Rich Signatures

The spectral theorem for normal operators on a Clifford module

In this paper, using the recently discovered notion of the S-spectrum, we prove the spectral theorem for a bounded or unbounded normal operator on a Clifford module (i.e., a two-sided Hilbert module over a Clifford algebra based on units that all square to be \(-1\)). Moreover, we establish the existence of a Borel functional calculus for bounded or unbounded normal operators on a Clifford module. Towards this end, we have developed many results on functional analysis, operator theory, integration theory and measure theory in a Clifford setting which may be of an independent interest. Our spectral theory is the natural spectral theory for the Dirac operator on manifolds in the non-self adjoint case. Moreover, our results provide a new notion of spectral theory and a Borel functional calculus for a class of n-tuples of commuting or non-commuting operators on a real or complex Hilbert space. Moreover, for a special class of n-tuples of operators on a Hilbert space our results provide a complementary functional calculus to the functional calculus of J. L. Taylor.

Non-linear monotone positive maps

e study several classes of general non-linear positive maps between C∗-algebras, which are not necessary completely positive maps. We characterize the class of the compositions of ∗-multiplicative maps and positive linear maps as the class of non-linear maps of boundedly positive type abstractly. We consider three classes of non-linear positive maps defined only on the positive cones, which are the classes of being monotone, supercongruent or concave. Any concave maps are monotone. The intersection of the monotone maps and the supercongruent maps characterizes the class of monotone Borel functional calculus. We give many examples of non-linear positive maps, which show that there exist no other relations among these three classes in general.

How can we illustrate the usefulness of the differential calculus course taught in a technical university

Working in a challenging academic environment as mathematics professors for the Technical University of Civil Engineering Bucharest, we thought to find a teaching strategy that, in addition to the required standard, but also has the standard of attractiveness and accessibility. Why? This is due to the fact that our students have a very nonhomogeneous level of knowledge and logical-mathematical skills. That's how we came up with the idea of organizing each lesson in our courses in a gradual way, and completing them with an informal part. On the other hand, we set out to collaborate with professors working in the departments of other natural sciences or in engineering departments, enriching the mathematics courses with technical applications. This has led to write several didactic works, which also include such applications. Proving the usefulness of this courses, our new work offers two such technical applications for the mathematical analysis course, taught in the first semester and dedicated to the differential calculus of functions having one or several variables. More precisely, we present in a gradual way, two applications solved by using the mathematical modelling: a problem belonging to electricity and then, the cruising speed problem. The gradual presentation begins, for each of these problems, with the necessary notions (organized in the form of two dictionaries, for Math and for Physics, respectively), continues with the statement of the problem, with the solution methodology, and finally, with the solution itself. Our presentation will provide students with a model of logical (mathematical) approach, useful to them in the courses of other natural sciences and of engineering disciplines that they will study later. In addition, it will prove them why mathematics is a fundamental discipline for the engineering education.

Pusz–Woronowicz Functional Calculus and Extended Operator Convex Perspectives

In this article, we first study, in the framework of operator theory, Pusz and Woronowicz’s functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $$[0,\infty )^2$$ . Our construction has special features that functions on $$[0,\infty )^2$$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in $$(-\infty ,\infty ]$$ . Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on $$(0,\infty )$$ . We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Ando’s theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.

A spectral projection based method for the numerical solution of wave equations with memory

In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin–Pipkin type of integro-differential equations. Both methods are based on a high-order Discontinous Galerkin approximation in space and the numerical inverse Laplace transform. In the first approach, we use functional calculus and the inverse Laplace transform to represent the solution. The spectral projections are then numerically computed and the approximation of the solution of the time-dependent problem is given by a summation of terms that are the product of projections of the data and the inverse Laplace transform of scalar functions. The second approach is the standard inverse Laplace transform technique. We show that the approach based on spectral projections can be very efficient when several time points are computed, and it is particularly interesting for parameter-dependent problems where the data or the kernel depends on a parameter.

Pusz–Woronowicz’s functional calculus revisited

This note is a complement to Pusz--Woronowicz's works on functional calculus for two positive forms from the viewpoint of operator theory. Based on an elementary, self-contained and purely Hilbert space operator explanation of their functional calculus, we show that any operator connection type operations (including any operator perspectives) are captured by their functional calculus.

Polynomial Embeddings of Unilateral Weighted shifts in 2–Variable Weighted Shifts

Given a bounded sequence $$\omega $$ of positive numbers, and its associated unilateral weighted shift $$W_{\omega }$$ acting on the Hilbert space $$\ell ^2(\mathbb {Z}_+)$$ , we consider natural representations of $$W_{\omega }$$ as a 2–variable weighted shift, acting on the Hilbert space $$\ell ^2(\mathbb {Z}_+^2)$$ . Alternatively, we seek to examine the various ways in which the sequence $$\omega $$ can give rise to a 2–variable weight diagram, corresponding to a 2–variable weighted shift. Our best (and more general) embedding arises from looking at two polynomials p and q nonnegative on a closed interval $$I \subseteq \mathbb {R}_+$$ and the double-indexed moment sequence $$\{\int p(r)^k q(r)^{\ell } d\sigma (r)\}_{k,\ell \in \mathbb {Z}_+}$$ , where $$W_{\omega }$$ is assumed to be subnormal with Berger measure $$\sigma $$ such that $${\text {supp}}\; \sigma \subseteq I$$ ; we call such an embedding a (p, q)–embedding of $$W_{\omega }$$ . We prove that every (p, q)–embedding of a subnormal weighted shift $$W_{\omega }$$ is (jointly) subnormal, and we explicitly compute its Berger measure. We apply this result to answer three outstanding questions: (i) Can the Bergman shift $$A_2$$ be embedded in a subnormal 2–variable spherically isometric weighted shift $$W_{(\alpha ,\beta )}$$ ? If so, what is the Berger measure of $$W_{(\alpha ,\beta )}$$ ? (ii) Can a contractive subnormal unilateral weighted shift be always embedded in a spherically isometric 2–variable weighted shift? (iii) Does there exist a (jointly) hyponormal 2–variable weighted shift $$\Theta (W_{\omega })$$ (where $$\Theta (W_{\omega })$$ denotes the classical embedding of a hyponormal unilateral weighted shift $$W_{\omega }$$ ) such that some integer power of $$\Theta (W_{\omega })$$ is not hyponormal? As another application, we find an alternative way to compute the Berger measure of the Agler j–th shift $$A_{j}$$ ( $$j\ge 2$$ ). Our research uses techniques from the theory of disintegration of measures, Riesz functionals, and the functional calculus for the columns of the moment matrix associated to a polynomial embedding.

Norm inequalities for the generalised commutator in Banach algebras

In this paper, by utilising the Riesz functional calculus in a Banach algebra $\mathcal{B}$, we provide some norm inequalities for the generalized commutator $$f(y)z-zf(x)$$ where $x,$ $y,$ $z\in \mathcal{B}$ and $f$ is an analytic function for which the elements $f(y)$ and $f(x)$ exist. Some examples for the resolvent and exponential functions are also given.