Multidimensional constant linear systems

A continuous resp. discrete r-dimensional (r1) system is the solution space of a system of linear partial differential resp. difference equations with constant coefficients for a vector of functions or distributions in r variables resp. of r-fold indexed sequences. Although such linear systems, both multidimensional and multivariable, have been used and studied in analysis and algebra for a long time, for instance by Ehrenpreis et al. thirty years ago, these systems have only recently been recognized as objects of special significance for system theory and for technical applications. Their introduction in this context in the discrete one-dimensional (r=1) case is due to J. C. Willems. The main duality theorem of this paper establishes a categorical duality between these multidimensional systems and finitely generated modules over the polynomial algebra in r indeterminates by making use of deep results in the areas of partial differential equations, several complex variables and algebra. This duality theorem makes many notions and theorems from algebra available for system theoretic considerations. This strategy is pursued here in several directions and is similar to the use of polynomial algebra in the standard one-dimensional theory, but mathematically more difficult. The following subjects are treated: input-output structures of systems and their transfer matrix, signal flow spaces and graphs of systems and block diagrams, transfer equivalence and (minimal) realizations, controllability and observability, rank singularities and their connection with the integral respresentation theorem, invertible systems, the constructive solution of the Cauchy problem and convolutional transfer operators for discrete systems. Several constructions on the basis of the Gröbner basis algorithms are executed. The connections with other approaches to multidimensional systems are established as far as possible (to the author).Partially supported by US Air Force Grant AFOSR-87-0249 and by Office of Naval Research Grant N 00014-86-K-0538 through the Center for Mathematical System Theory, University of Florida, Gainesville, Florida, U.S.A.     

Über das dynamisch-elastische Verhalten von Polyamiden und Polyurethanen in Abhängigkeit von Frequenz,Temperatur und Wassergehalt

Zusammenfassung Die Abh?ngigkeit des dynamisch-elastischen Verhaltens verschiedener Polyamide und Polyurethane von der Frequenz, der Temperatur und insbesondere vom Wassergehalt wurde untersucht. Gemessen wurden zu diesem Zweck der dynamische Elastizit?tsmodul und der zugeh?rige Verlustfaktor im Bereich 10 ... 1000 Hz und −30...+90 °C. Es wurde auf molekularen Relaxationsprozessen beruhende mechanische Dispersion mit der zugeh?rigen maximalen Absorption in Abh?ngigkeit von allen drei Ver?nderlichen beobachtet. Das Wasser erh?ht die Beweglichkeit der relaxierenden Molekülkettensegmente. Wenn man die Stoffe bei gleichen relativen Wassergehalten (bezogen auf die S?ttigungswerte) vergleicht, sind die Unterschiede ihres Verhaltens gering; dies gilt sowohl für die Lage der Dispersionsgebiete als auch für die H?he der Dispersionsstufen und die Betr?ge der Verlustfaktormaxima. Das molekulare Verhalten, insbesondere dessen Abh?ngigkeit vom aufgenommenen Wasser, das zu Anomalien im Verlauf des Elastizit?tsmoduls führt, wird an Hand der vorliegenden Me?resultate und der Ergebnisse anderer Autoren diskutiert. Vorgetragen auf der Arbeitssitzung des Faehausschusses „Physik der IIoehpolymeren“ im Verband der Deutschen Physikalisehen Gesellschaften zur Physikertagung Münehen am 7. September 1956. Die Verfasser der vorliegenden Arbeit sind den Farbenfabriken Bayer AG, Werk Uerdingen, zu Dank verpflichtet für die Genehmigung, die mit den yon ihnen hergestellten Me?proben erzielten Ergebnisse zu ver?ffentlichen. Zu danken haben sie ferner Herrn Dipl.-Ing. W. Weidmann (Farbenfabriken Bayer) ftir anregende Diskussionen und weftvolle Hinweise und den technischen Assistentinnen Fraulein M. L. Angelroth und Frau R. Schmitt für ihre Mitwirkung bei den Messungen und deren Auswertung.     

The canonical Cauchy problem for linear systems of partial difference equations with constant coefficients over the completer-dimensional integral lattice ℕ2r

We canonically define and algorithmically solve the problem of the title. Such algorithms are of great significance for the method of finite differences for the solution of partial differential equations and for many technical applications such as image processing. In contrast to the wide (system theoretic) literature for ordinary difference equations and in spite of the great theoretical and practical significance of this problem, until now, there was no systematic theory of these systems and in particular of the corresponding Cauchy problem, let alone an algorithm. In this paper, we give both. The method consists in a transformation of this problem into a naturally associated problem which is defined over the 2r-dimensional natural number lattice ^2r (the upper quadrant in ^2r ) and for which the canonical initial value or Cauchy problem was defined and constructively solved by the second author.     

Light scalar tetraquarks from a holographic perspective

We discuss how a dominant tetraquark component of the lightest scalar mesons may emerge in AdS/QCD gravity duals. In particular, we show that the exceptionally strong binding required to render the tetraquark ground state lighter than the lowest-lying scalar quark–antiquark nonet can be holographically encoded into bulk-mass corrections for the tetraquark's dual mode. The latter are argued to originate from the anomalous dimension of the corresponding four-quark interpolator. To provide a concrete example, we implement this mechanism into the dilaton soft-wall dual for holographic QCD. Preventing the lowest-lying dual mode from collapsing into the AdS boundary then establishes a rather generic lower bound on the tetraquark mass (which may be overcome in the presence of additional background fields). We further demonstrate that the higher tetraquark excitations can become heavier than their quark–antiquark counterparts and are thus likely to dissolve into the multiparticle continuum.     

Dynamics of alkyl ammonium intercalants within organically modified montmorillonite: Dielectric relaxation and ionic conductivity

The low-frequency (0.01 Hz-10 MHz) dynamic characteristics of alkyl quaternary ammonium exchanged montmorillonite (SC20A) were investigated to determine the correlation between temperature-dependent changes in the interlayer structure and collective mobility of the surfactant. From 25 to 165 degrees C, SC20A exhibits two interlayer transitions, one ascribed to the melting of the intercalated alkyl chains of the surfactant (20-40 degrees C) and another associated with an abrupt decrease in the interlayer's coefficient of thermal expansion (100 degrees C). For this temperature range, the excess surfactant and residual electrolytes present in commercially manufactured SC20A enhance the direct current conductivity and increase low-frequency space-charge polarization, which is believed to occur across percolation paths established by the surfaces of the SC20A crystallites. In contrast, a higher-frequency relaxation, which was less sensitive to process history and impurity content, is ascribed to relaxation within the interlayer at the surfactant-aluminosilicate interface electrostatic couple. The temperature dependence of these dielectric relaxations indicated a drastic increase in mobility as the interlayer organic phase transitions from static and glasslike into molten and mobile. Overall, SC20A displayed features of alternating current universality, including time-temperature superposition, common in other types of disordered ion-conducting media. The presence of long-range transport and its sensitivity to low amounts of impurities imply that from a dynamic perspective the local environment of the surfactants are substantially diverse and a minority fraction, such as at the edge of the crystallite (gallery and aluminosilicate layer), may dominate the lower-frequency dielectric response.