Proof Of Invariance Research Articles

On the orbit of invariant subspaces of linear operators in finite-dimensional spaces (new proof of a Halmos's result)

P.R. Halmos proved that for a linear operator A over a finite-dimensional complex vector space E, every A-invariant subspace of E is the range of a commutant of A. His proof was based on a generalization of the concept of eigenvector. In this note, we give an invariant proof of this Halmos's theorem.

Verification of clocked and hybrid systems

This paper presents a new computational model for real-time systems, called the clocked transition system (CTS) model. The CTS model is a development of our previous timed transition model, where some of the changes are inspired by the model of timed automata. The new model leads to a simpler style of temporal specification and verification, requiring no extension of the temporal language. We present verification rules for proving safety a nd liveness properties of clocked transition systems. All rules are associated with verification diagrams. The verification of response properties requires adjustments of the proof rules developed for untimed systems, reflecting the fact that progress in the real time systems is ensured by the progress of time and not by fairness. The style of the verification rules is very close to the verification style of untimed systems which allows the (re)use of verification methods and tools, developed for u ntimed reactive systems, for proving all interesting properties of real-time systems. We conclude with the presentation of a branching-time based approach for verifying that an arbitrary given CTS isnon-zeno. Finally, we present an extension of the model and the invariance proof rule for hybrid systems.

Adiabatic anholonomy and canonical transformations

Biswas and Soni [4] have surmised a semiclassical formula for Berry’s phase in terms of a generating function. We derive this formula apart from showing that it is not true in general and investigate its domain of validity. We also derive transformation formulae for Berry’s phase (Hannay’s angle) under general canonical transformations. A simpler proof for total angle invariance than hitherto available, is given.

The logic of “initially” and “next”: Complete axiomatization and complexity

Andréka et al. (1995) obtained a large number of completeness results for discrete linear-time temporal logics. One of them is left open: the completeness of the logic of “initially” and “next”, for which a deductive system is proposed. This simple logic is of practical importance, since the proof of program invariants only require these modalities. We show here that the conjectured medium completeness of this system indeed holds; further, we show that deciding this entailment is PSPACE-complete, while deciding validity is only coNP-complete.

Charged particles, gauge invariance, and molecular electrodynamics

A general account of the interaction of charges with electromagnetic radiation is presented; the charges are treated in the nonrelativistic approximation for simplicity (not an essential assumption). The two main approaches in the literature based on either the potentials of the electromagnetic field (minimal coupling) or the polarization fields of the charges (e.g., the multipolar Hamiltonian) are presented in a unified framework that characterizes them as simply different viewpoints of the same general structure. This structure is very suitable for a discussion of the gauge invariance of the theory. The final quantum mechanical Hamiltonian derived here has recently been shown to provide the basis for a proof of the gauge invariance of its associated S-matrix to all orders of perturbation theory, provided the S-matrix is taken on the energy shell. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 531–545, 1999

On the cobordism invariance of the index of Dirac operators

We describe a “tunneling” proof of the cobordism invariance of the index of Dirac operators. The goal of this note is to present a very short proof of the cobordism invariance of the index. More precisely, if D is a Dirac operator on an odd dimensional manifold M with boundary ∂M = M then we show that the index of its restriction D to M is zero. The novelty of this proof consists in the fact that we provide an explicit isomorphism between the kernel and the cokernel of D. This map can be viewed as a sort of “propagator” (see Sect. 4). 1. The setting Consider the following collection of data. (a) A compact, oriented, (2n + 1)-dimensional Riemann manifold (M, ĝ) with boundary ∂M = M such that ĝ is a product metric near the boundary. We denote by s the longitudinal coordinate on a collar neighborhood of M . The various orientations are defined as in Figure 1. (b) A bundle of complex self-adjoint Clifford modules E → M (in the sense of [BGV]). The Clifford multiplication is denoted by ĉ : T ∗M → End (E).

ON ROBERTS’ PROOF OF THE TURAEV-WALKER THEOREM

In this paper we discuss the beautiful idea of Justin Roberts [7] (see also [8]) to re-obtain the Turaev-Viro invariants [11] via skein theory, and re-prove elementarily the Turaev-Walker theorem [9], [10], [13]. We do this by exploiting the presentation of 3-manifolds introduced in [1], [4]. Our presentation supports in a very natural way a formal implementation of Roberts’ idea. More specifically, what we show is how to explicitly extract from an o-graph (the object by which we represent a manifold, see below), one of the framed links in S3 which Roberts uses in the construction of his invariant, and a planar diagrammatic representation of such a link. This implies that the proofs of invariance and equality with the Turaev-Viro invariant can be carried out in a completely “algebraic” way, in terms of a planar diagrammatic calculus which does not require any interpretation of 3-dimensional figures. In particular, when proving the “term-by-term” equality of the expansion of the Roberts invariant with the state sum which gives the Turaev-Viro invariant, we simultaneously apply several times the “fusion rule” (which is formally defined, strictly speaking, only in diagrammatic terms), showing that the “braiding and twisting” which a priori may exist on tetrahedra is globally dispensable. In our point of view the success of this formal “algebraic” approach witnesses a certain efficiency of our presentation of 3-manifolds via o-graphs. In this work we will widely use recoupling theory which was very clearly exposed in [2], and therefore we will avoid recalling notations. Actually, for the purpose of stating and proving our results we will need to slightly extend the class of trivalent ribbon diagrams on which the bracket can be computed. We also address the reader to the references quoted in [2], in particular for the fundamental contributions of Lickorish to this area. In our approach it is more natural to consider invariants of compact 3-manifolds with non-empty boundary. The case of closed 3-manifolds is included by introducing a correction factor corresponding to boundary spheres, as explained in §2. Our main result is actually an extension to manifolds with boundary of the Turaev-Walker theorem: we show that the Turaev-Viro invariant of such a manifold coincides (up to a factor which depends on the Euler characteristic) with the Reshetikhin-Turaev-Witten invariant of the manifold mirrored in its boundary.

Properties of the geometric phase of a de Broglie-Bohm causal quantum mechanical trajectory

In a manner analogous to that used to define the geometric phase of a wavefunction, one can define the geometric phase of a periodic trajectory associated with the de Broglie-Bohm causal interpretation of quantum mechanics. This phase is proportional to the action integral over the trajectory. Numerical evidence indicates that this is an adiabatic invariant, despite the fact that the equations of motion are not separable and that the effective classical Hamiltonian depends explicitly on time, two conditions excluded in the standard proof of adiabatic invariance. The associated concept of geometric frequency, proportional to the time average of the kinetic energy, can be defined for both periodic and aperiodic trajectories. Numerical evidence indicates that this also has remarkable properties.

A temporal logic approach to object certification

A brief overview is made of the use of temporal logic formalisms for specifying and verifying concurrent systems in general and information systems in particular. The requirements imposed by object-orientation on such formalisms are examined. A logic is proposed fulfilling those requirements (except concerning non-monotonic features), allowing the uniform treatment of both local and global properties of systems with concurrent, interacting components organized in classes, and supporting specialization. A semantics and a calculus (following an axiomatic, Hilbert style) are presented in detail. The calculus includes rules for the sound inheritance and reflection of theorems between classes. Practical aspects of the usage of such a logic for both specification and verification are considered. To this end a set of metatheorems is provided for expediting the proof of invariants. Finally, the need and availability of automatic theorem proving for systems querying is briefly discussed.

Nonabelian Gauge Theories: The Causal Approach

We present the causal construction of perturbative Yang-Mills theories in four (3 + 1)-dimensional space-time. We work with free quantum fields throughout. The inductive causal method by Epstein and Glaser leads directly to a finite perturbation series and does not rely on an intermediary regularization of the theory. The causal method naturally separates the physical infrared problem of massless theories from ultraviolet-sensitive features like normalizability by regarding the distributional character of the S-matrix. We prove the normalizability of the Yang-Mills theory with fermionic matter fields and study the discrete symmetry transformations in the causal formalism. We introduce a definition of nonabelian gauge invariance which only involves the free asymptotic field operators and give mathematically rigorous and conceptually simple proofs of nonabelian gauge invariance and of the physical unitarity of the S-matrix in all orders of perturbation theory.

Controlled K-theory

The controlled finiteness obstruction and torsion are defined using controlled algebra, giving a more algebraic proof of the topological invariance of torsion and the homotopy finiteness of compact ANRs.

The prolongation formula for tensor fields

We derive a generalization of the prolongation formula for tensor-valued functions, by taking into account the natural action of diffeomorphisms on tensor fields. This differs from the usual procedure, where such fields are viewed as defining 'multi-component scalars'; it replaces the ordinary derivative in the expression of the 'characteristic' by a Lie derivative. The resulting version of Noether's theorem takes a form more familiar to theoretical physicists. A very simple proof of the conformal invariance of the Maxwell Lagrangian also follows from this procedure. The Eshelby tensor is also readily obtained, as a further illustration of the practical use of this new prolongation formula.

Spectral asymptotics of foliated manifolds

In this paper we study the spectral distribution functions for the analytic and combinatorial leafwise Laplacians on a foliated manifold admitting a transverse invariant measure. We show (Theorem 4.1) that the dilational equivalence classes near zero of the two spectral distribution functions are the same. An immediate consequence is that this dilational equivalence class is independent of the metric (in the analytic case) and the bounded triangulation (in the combinatorial case) used to define it. Our main result (Theorem 2.6) is that this dilational equivalence class is invariant under a measure preserving leafwise homotopy equivalence. This result is equivalent to the invariance of the dilational equivalence class near infinity of the trace of the leafwise heat kernels. Our results are motivated by those of [Ef-Sh], [G-Sh] and [El] concerning the equivariant homotopy invariance of the asymptotic behaviour of the spectral distribution function for Riemannian manifolds with a free isometric action of a discrete group. We extend many of the ideas of these papers to our situation. The main techniques used in the proof of the homotopy invariance, however, are those developed in [H-L2] (where we prove the leafwise homotopy invariance of the foliation betti numbers), the simplicial techniques of [D] and [W], and those of [H-L1].

Subcritical closed string field theory in less than 26 dimensions.

We construct the second quantized action for sub-critical closed string field theory with zero cosmological constant in dimensions $ 2 \leq D < 26$, generalizing the non-polynomial closed string field theory action proposed by the author and the Kyoto and MIT groups for $D = 26$. The proof of gauge invariance is considerably complicated by the presence of the Liouville field $\phi$ and the non-polynomial nature of the action. However, we explicitly show that the polyhedral vertex functions obey BRST invariance to all orders. By point splitting methods, we calculate the anomaly contribution due to the Liouville field, and show in detail that it cancels only if $D - 26 + 1 + 3 Q ^ 2 = 0 $, in both the bosonized and unbosonized polyhedral vertex functions. We also show explicitly that the four point function generated by this action reproduces the shifted Shapiro-Virasoro amplitude found from $c=1$ matrix models and Liouville theory in two dimensions. LATEX file.

Gauge invariance of massless QED

A simple general proof of gauge invariance in QED is given in the framework of causal perturbation theory. It illustrates a method which can also be used in non-abelian gauge theories.

On the Interplay Between Interval Dimension and Dimension

This paper investigates a transformation $P \to Q$ between partial orders $P,Q$ that transforms the interval dimension of P to the dimension of Q, i.e., $\text{idim} ( P ) = \dim ( Q )$. Such a construction has been shown before in the context of Ferrer’s dimension by Cogis [Discrete Math., 38 (1982), pp. 47–52]. The construction in this paper can be shown to be equivalent to his, but it has the advantage of (1) being purely order-theoretic, (2) providing a geometric interpretation of interval dimension similar to that of Ore [Amer. Math. Soc. Colloq. Publ., Vol. 38, 1962] for dimension, and (3) revealing several somewhat surprising connections to other order-theoretic results. For instance, the transformation $P \to Q$ can be seen as almost an inverse of the well-known split operation; it provides a theoretical background for the influence of edge subdivision on dimension (e.g., the results of Spinrad [Order, 5 (1989), pp. 143–147]) and interval dimension, and it turns out to be invariant with respect to changes of P that do not alter its comparability graph, thus also providing a simple new proof for the comparability invariance of interval dimension.

Recursive definition of global cellular-automata mappings.

A method for a recursive definition of global cellular-automata mappings is presented. The method is based on a graphical representation of global cellular-automata mappings. For a given cellular-automaton rule the recursive algorithm defines the change of the global cellular-automaton mapping as the number of lattice sites is incremented. A proof of lattice size invariance of global cellular-automata mappings is derived from an approximation to the exact recursive definition. The recursive definitions are applied to calculate the fractal dimension of the set of reachable states and of the set of fixed points of cellular automata on an infinite lattice.

A simple and unified proof of dyadic shift invariance and the extension to cyclic shift invariance

A simple and unified proof of the dyadic shift invariance and an extension to cyclic shift invariance are presented. First, the concept of the dyadic shift invariance (DSI) and cyclic shift invariant (CSI) functions is proposed. Basic properties of the DSI and CSI functions are considered. Then it is shown that the Walsh-Hadamard transform (WHT) and discrete Fourier transform (DFT) are, in fact, special cases of the DSI and CSI functions, respectively. Many properties of the WHT and DFT can then be obtained easily from DSI and CSI points of view. The proposed unified approach is simple and rigorous. It is also shown that the properties of the WHT and DFT are the consequence of the basic principles of the DSI and CSI functions. >

Proof of conformal invariance in the critical regime for models of Gross-Neveu type

It is shown that for all multiplicatively renormalizable fermionic models without derivativs in the interaction, in particular, for the Gross—Neveu model, conformal invariance of the Green's functions of the fields in the critical regime (massless model at fixed renormalization-group point) automatically follows from the scale invariance for arbitrary space dimension.

Bimodules, the Brauer group, Morita equivalence, and cohomolgy

The purpose of this note is to focus attention on an equivalence relation which we call ‘congruence of k-algebras’ and define in terms of the categories of bimodules. (Here k is a commutative unital ring.) Many familiar Morita invariant properties are trivially seen to be congruence invariant. Examples include the center, the lattice of two-sided ideals, and the Hochschild and cyclic cohomologies. In Section 3 we prove that Morita equivalence implies congruence, thereby obtaining new, uniform, and more efficient proofs of the Morita invariance of these properties. (Actually, the invariance of cohomology seems to have been previously noticed for only one type of Morita equivalence: that between an algebra A and the r X r matrix algebra hlr(A).) We also give new cochain maps inducing the cohomology isomorphism. In Section 2 we use congruence to answer Zelinsky’s concluding question in [ll] and to give an economical description of the Brauer group as consisting of the Morita equivalence classes of congruence-trivial algebras (under tensor product). This definition provides a categorical interpretation of the Brauer group and is available, without introducing separability, immediately after proving the Eilenberg-Watts theorem characterizing functors between module categories. (The Morita theorems, which are invoked, at least tacitly, in most treatments of the Brauer group, follow easily