Infinite Vector Research Articles

Universal deformation rings and dihedral defect groups

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal 2-modular block of a finite simple group. We determine the universal deformation ring R(G,V) for every kG-module V which belongs to B and has stable endomorphism ring k. It follows that R(G,V) is always isomorphic to a subquotient ring of WD. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.

Universal deformation rings and dihedral 2-groups

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose that D is a dihedral 2-group. We prove that the universal deformation ring R(D, V) of an endo-trivial kD-module V is always isomorphic to W [ℤ/2×ℤ/2]. As a consequence, we obtain a similar result for modules V with stable endomorphism ring k belonging to an arbitrary nilpotent block with defect group D. This confirms, for such V, conjectures on the ring structure of the universal deformation ring of V that had previously been shown for V belonging to cyclic blocks or to blocks with Klein four defect groups.

Computable embeddings and strongly minimal theories

AbstractHere we prove that if T and T′ are strongly minimal theories, where T′ satisfies a certain property related to triviality and T does not, and T′ is model complete, then there is no computable embedding of Mod(T) into Mod(T′). Using this, we answer a question from [4], showing that there is no computable embedding of VS into ZS, where VS is the class of infinite vector spaces over ℚ, and ZS is the class of models of Th(ℤ, S). Similarly, we show that there is no computable embedding of ACF into ZS, where ACF is the class of algebraically closed fields of characteristic 0.

Fusion over sublanguages

AbstractGeneralising Hrushovski's fusion technique we construct the free fusion of two strongly minimal theoriesT1.T2intersecting in a totally categorical sub-theoryT0. We show that if. e.g.,T0is the theory of infinite vector spaces over a finite field then the fusion theoryTω, exists, is complete and ω-stable of rank ω. We give a detailed geometrical analysis ofTω, proving that if bothT1,T2are 1-based then.Tω can be collapsed into a strongly minimal theory, if some additional technical conditions hold—all trivially satisfied ifT0is the theory of infinite vector spaces over a finite field.

On the distribution of the residual cross-correlations of infinite order vector autoregressive series and applications

Here we derive the asymptotic distribution of an arbitrary vector of residual cross-correlations resulting from the fitting of finite autoregressions to two uncorrelated infinite order vector autoregressive series. Its asymptotic distribution is the same multivariate normal as the one of the corresponding vector of cross-correlations between the two innovation series. The application of that result for testing the uncorrelatedness of two series is also discussed.

All Fundamental Particular Solutions are Needed to Express an Arbitrary Firing Count Vector in Petri Nets

For fixed initial and destination states (i.e., markings), M 0 and M d , there exist generally infinite firing count vectors in a Petri net. In this letter, it is shown that all fundamental particular solutions as well as all minimal T-invariants w.r.t. firing count vectors are needed to express an arbitrary firing count vector for the fixed M 0 and M d . An algorithm for finding a special firing count vector which is expressed by using the only one specified fundamental particular solution is also given.

Power factor definition in single phase system with arbitrary waveform according to similarity

A new definition of power factor, which is suitable for an arbitrary waveform of voltage and current in a single‐phase system, is presented according to similarity between voltage and current waveforms. The similarity is described in infinite dimensions real vector space, and the power factor is defined as the multiplication of rootmean‐ square (RMS) value factor λ RMS and phase factor λϕ. The new definition is compatible with IEEE Trial‐Use Standard 1459–2000, and can be a useful reference for power factor measurement, power factor correction and power compensation.

On certain Heun functions, associated functions of discrete variables, and applications

Two sets of the Heun functions are introduced via integrals. Theorems about expanding functions with respect to these sets are proven. A number of integral and series representations as well as integral equations and asymptotic formulas are obtained for these functions. Some of the coefficients of the series are orthogonal (J‐orthogonal) functions of discrete variables and may be interpreted as orthogonal polynomials. Other sets of the coefficients are biorthonormal. Expanding infinite vectors to series with respect to the coefficients is discussed. Certain Legendre functions of complex degree are limiting cases of the studied functions. This leads to new relations for Legendre functions and associated integral transforms. The treated Heun functions find a use for solving dual Fourier series equations which are reduced to the Fredholm integral equations of the second kind. Explicit solutions are obtained in a special case.

Complex Jacobi matrices and quadrature rules

Given any sequence of orthogonal polynomials, satisfying the three term recurrence relation xpn(x) = ?n+1pn+1(x) + ?npn(x) + ?npn-1(x), p-1(x)=0 po(x)=1 with ?n ? 0, n ? N, ?0 = 1, an infinite Jacobi matrix can be associated in the following way ? ? ??0 ?1 0...? ? ? ??1 ?1 ?2...? ? ? J = ?0 ?2 ?2...? ?.... ? ?.... ? ?....? ? ? In the general case if the sequences {?n} or {?n} are complex the associated Jacobi matrix is complex. Under the condition that both sequences {?n}and {?n} are uniformly bounded, the associated Jacobi matrix can be understood as a linear operator J acting on ?2, the space of all complex square-summable sequences, where the value of the operator J at the vector x is a product of an infinite vector x and an infinite matrix J in the matrix sense. The case when the sequences {?n} and {?n} are not uniformly bounded, an operator acting on ?2 can not be defined that easily. Additional properties of the sequence of orthogonal polynomials are needed in order to be able to define the operator uniquely. The case when the sequences ?n and ?n are real is very well understood. The spectra of the Jacobi matrix J equals the support of the measure of orthogonality for the given sequence of orthogonal polynomials. All zeros of orthogonal polynomials are real, simple and interlace, contained in the convex hull of the spectra of the Jacobi operator associated with the infinite Jacobi matrix J. Every point in ?(J) attracts zeros of orthogonal polynomials. An application of orthogonal polynomials is the construction of quadrature rules for the approximation of integration with respect to the measure of orthogonality. For arbitrary sequences {?n} and {?n} the situation is changed dramatically. Zeros of orthogonal polynomials need not be simple; they are not real and they do not necessarily lie in the convex hull of ?(J). There is also a little known about convergence results of related quadrature rule. Only in recent years a connection between complex Jacobi matrices and related orthogonal polynomials is interesting again (see [2]). Studies of complex Jacobi operators should lead to a better understanding of related orthogonal polynomials, but also the study of orthogonal polynomials with the complex Jacobi matrices should put more light on the non-hermitian banded symmetric matrices. In this lecture some results are given about complex Jacobi matrices and related quadrature rules, and also some interesting examples are presented.

Estimating Transactions Given Balance Sheets and an Income Statement

One way to develop an appreciation for the power and beauty of doubleentry is to derive consistent transaction vectors, i.e., start with financial statements and derive the transaction amounts that could have generated the statements. Working backward to uncover transactions (the inverting exercise) complements the more traditional approach of working forward from transactions to financial statements. In addition, the approach highlights a fundamental accounting activity: aggregation. Financial statements summarize a firm's transactions using only a relatively small number of account balances. A consequence of aggregation is that there are infinite consistent transaction vectors. However, because these infinite solutions are all prepared in accordance with double-entry, they are linked to each other in a systematic fashion. We show how a directed graph representation of the accounting system can be used as a parsimonious means of characterizing all consistent transaction vectors. As the number of accounts and transactions are increased, the inverting exercise becomes tedious if one does not make use of the directed graph. To emphasize this point, the inverting exercise is conducted using a set of published financial statements. We also discuss the issue of picking the most likely transaction vector from the set of consistent transaction vectors. The authors have used this note in undergraduate, M.B.A., and Ph.D. classes. For the Ph.D. class, the note has been supplemented by the more rigorous analyses in the linear algebra literature and the accounting literature.

Model completeness of the new strongly minimal sets

Boris Zil'ber conjectured that all strongly minimal theories are bi-interpretable with one of the “classical” sorts: theories of algebraically closed fields, theories of infinite vector spaces over division rings and theories with trivial algebraic closure relations. Hrushovski produced the first two classes of counterexample to this conjecture in [10] and [9]. Subsequently, in [8], the author gave an explicit axiomatization of a special case of [9] from which model completeness could quickly be deduced. It was unclear at that writing whether the model completeness result was true in the general case or was due to peculiarities of the case under consideration. The main new result of this paper is model completeness, not only of the general case in [9], but also of the theories described in [10]. Specifically, we present a general framework in which producing a strongly minimal theory is reduced to finding an elementary class of theories satisfying certain requirements (see below). We present the theories of [10] and [9] as special instances of such theories, giving an explicit axiomatization from which model completeness immediately follows in each case.We hope by presenting these constructions in parallel, using common language and extracting common elements, to make easier both the exploitation of the ideas involved in their making and their comparison with other recent constructions of a similar flavor. For a selection of such constructions, see [6], [1], [2] and [3]. For more general background, see [2], [4] and [11].

Congruences between the coefficients of the Tate curve via formal groups

Let Eq: y2 + XY = X3 + h4X + h6 be the Tate curve with canonical differential, w = dX/(2Y + X). If the characteristic is p > 0, then the Hasse invariant, H, of the pair (Eq, w) should equal one. If p > 3, then calculation of H leads to a nontrivial separable relation between the coefficients h4 and h6. If p = 2 or p = 3, Thakur related h4 and h6 via elementary methods and an identity of Ramanujan. Here, we treat uniformly all characteristics via explicit calculation of the formal group law of Eq. Our analysis was motivated by the study of the invariant A which is an infinite Witt vector generalizing the Hasse invariant.

A Matrix Representation for Euler's Constant, γ

Euler's constant, y= 0.5772156649... can be represented as the product of an infinite row vector, the inverse of a +x E+ lower triangular mnatrix, and an infinite 7X 1 column vector, all with entries that are either zero or simple unit fractions. Observe that for Z+x E+ lower triangular matrices, the end result of all arithmetic matrix operations, matrix inversion, application to infinite column vectors, etc., has appropriate n-th truncation equal to that obtained by first truncating all matrices, and then carrying out the operations. Hence these operations may be performed and the familiar identities of linear algebra continue to hold in this context.

Testing cointegration in infinite order vector autoregressive processes

This paper studies test procedures which can be used to determine the cointegrating rank in infinite order vector autoregressive processes. The considered tests are analogs or close versions of previous likelihood ratio tests obtained for finite-order Gaussian vector autoregressive processes. It is shown that the use of the likelihood ratio tests is justified even when the data are generated by an infinite order non-Gaussian vector autoregressive process. New tests are also developed for cases where intercept terms are included in the cointegrating relations. These tests are based on a new approach of estimating the intercept terms. They have the property that, under the null hypothesis, the same asymptotic distribution theory applies as in the case where the values of the intercept terms are a priori known and not estimated. A limited simulation study indicates that the new tests can be considerably more powerful than their previous counterparts.

Bootstrapping Autoregressive and Moving Average Parameter Estimates of Infinite Order Vector Autoregressive Processes

We consider anr-dimensional multivariate time series {yt, t∈Z} which is generated by an infinite order vector autoregressive process. We show that a bootstrap procedure which works by generating time series replicates via an estimated finitek-order vector autoregressive process (k→∞ at an appropriate rate with the sample size) gives asymptotically valid approximations to the joint distribution of the growing set of estimated autoregressive coefficients and to the corresponding set of estimated moving average coefficients (impuls responses).

Skyrmion excitations in quantum Hall systems

Using finite size calculations on the surface of a sphere we study the topological (skyrmion) excitation in quantum Hall system with spin degree of freedom at filling factors around $\nu=1$. In the absence of Zeeman energy, we find, in systems with one quasi-particle or one quasi-hole, the lowest energy band consists of states with $L=S$, where $L$ and $S$ are the total orbital and spin angular momentum. These different spin states are almost degenerate in the thermodynamic limit and their symmetry-breaking ground state is the state with one skyrmion of infinite size. In the presence of Zeeman energy, the skyrmion size is determined by the interplay of the Zeeman energy and electron-electron interaction and the skyrmion shrinks to a spin texture of finite size. We have calculated the energy gap of the system at infinite wave vector limit as a function of the Zeeman energy and find there are kinks in the energy gap associated with the shrinking of the size of the skyrmion. breaking ground state is the state with one skyrmion of infinite size. In the presence of Zeeman energy, the skyrmion size is determined by the interplay of the Zeeman energy and electron-electron

Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization

In this paper, we establish necessary and sufficient dual conditions for weak and proper minimality of infinite dimensional vector convex programming problems without any regularity conditions. The optimality conditions are given in asymptotic forms using epigraphs of conjugate functions and sub differentials, It is shown how these asymptotic conditions yield standard Lagrangian conditions under appropriate regularity conditions. The main tool, used to obtain these results, is a new solvability result of Motzkin type for cone convex systems. We also provide local Lagrangian necessary conditions for certain non—convex problems using convex approximations

Confinement and infinite potential

We discuss the confinement problem with infinite vector and scalar potentials. Infinite potentials like harmonic oscillator potential and linear one are investigated. We show the particle cannot be confined when the vector potential is larger than the scalar one. It is also shown the existence of a bound state in 3-dimensional square well potentials, but this does not mean confinement of the particle.

Qualification conditions for multivalued functions in Banach spaces with applications to nonsmooth vector optimization problems

In this paper we introduce qualification conditions for multivalued functions in Banach spaces involving the A-approximate subdifferential, and we show that these conditions guarantee metric regularity of multivalued functions. The results are then applied for deriving Lagrange multipliers of Fritz—John type and Kuhn—Tucker type for infinite non-smooth vector optimization problems.

Acoustic radiation from two opposed semi-infinite coaxial cylindrical waveguides. II: separated ducts

The radiation of sound waves from two coaxial circular cylindrical waveguides is discussed in this article. Both ducts are semi-infinite in length, one extending from z = ℓ ‖ to z = -∞ and the other from z = 0 to z = +∞, where z is a coordinate measured along the cylinders' axis. Waves are incident from z = -∞ inside the smaller radius duct and the reflected, transmitted and radiated fields are calculated here for the case when the waveguides are separated by a gap of arbitrary length ℓ (ℓ < 0). In the first part of this study (Lawrie et al., Wave Motion 18, 121–142 (1993)) the model was solved for overlapping pipes ℓ > 0 by reducing the problem to a matrix Wieber-Hopf equation, which was solved by the introduction of an infinite constant vector satisfying a simple algebraic system of equations. This procedure does not follow through for the case of a gap, due to the presence of exponentially growing elements, and further analysis is required to solve for the Wiener-Hopf unknowns. Numerical results are presented here for a range of values of gap length and duct radii.