Uniqueness Criteria Research Articles

Comparison ODE theorems related to the method of lines

We prove a comparison theorem for an ODE and DAE system which arises from the method of lines. Under a Perron comparison condition on the functional dependence and a specific Lipschitz and (W+) condition on the classical argument, we obtain strong uniqueness criteria.

Partially Ordered Models

We provide a formal definition and study the basic properties of partially ordered random fields (PORF). These systems were proposed to model textures in image processing and to represent independence relations between random variables in statistics (in the latter case they are known as Bayesian networks). Our random fields are a generalization of probabilistic cellular automata (PCA) and their theory has features intermediate between that of discrete-time processes and the theory of statistical mechanical lattice fields. Its proper definition is based on the notion of partially ordered specification (POS), in close analogy to the theory of Gibbs measures. This paper contains two types of results. First, we present the basic elements of the general theory of PORFs: basic geometrical issues, definition in terms of conditional probability kernels, extremal decomposition, extremality and triviality, reconstruction starting from single-site kernels, relations between POM and Gibbs fields. Second, we prove three uniqueness criteria that correspond to the criteria known as uniform boundedness, Dobrushin uniqueness and disagreement percolation in the theory of Gibbs measures.

On certain non-unique solutions of the Stieltjes moment problem

We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${\rho}_{1}^{(r)}(n)=(2rn)!$ and ${\rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,\dots$, $n=0,1,2,\dots$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = {\rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${\rho}_{1,2}^{(r)}(n)$ give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing ${\rho}_{1,2}^{(r)}(n)$, such as the product ${\rho}_{1}^{(r)}(n)\cdot{\rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,\dots$.

Delay‐dependent asymptotic stability of BAM neural networks with time delay

PurposeThe purpose of this paper is to examine the criteria of uniqueness of the equilibrium point and the new stability criteria for stability of the equilibrium point. The new stability condition is dependent on the size of delays.Design/methodology/approachThe global asymptotic stability of a class of delayed bi‐directional associative memory (BAM) neural networks is studied. Some new sufficient conditions are presented for the unique equilibrium point and the global stability of BAM neural networks with time delays by constructing Lyapunov functions and using the linear matrix inequality. A numerical example is presented to illustrate the effectiveness of the theoretical results.FindingsBased on the mathematical method and matrixes inequality skill, some criteria are obtained which contain the unique equilibrium point and the global stability of BAM neural networks.Research limitations/implicationsThe paper proposes the new Lyapunov function and new skill to compose matrixes inequality.Practical implicationsA very useful method for BAM neural network to judge the uniqueness of the equilibrium point and stability.Originality/valueThe new mathematical model is proposed about the production process, and the new control method is used in the temperature system for a double layers welded pipe in welding process.

Uniqueness of holomorphic Abel functions at a complex fixed point pair

We give a simple uniqueness criterion (and some derived criteria) for holomorphic Abel functions and show that Kneser’s real analytic Abel function of the exponential is subject to this criterion.

The inverse problem for an equation of mixed parabolic-hyperbolic type

We study the inverse problem related to finding the unknown right-hand side of the equation of mixed parabolic-hyperbolic type in a rectangular domain. A uniqueness criterion for the solution is established. The solution is constructed as the sum of the series in eigenfunctions of the corresponding one-dimensional spectral problem.

A method for managing re-identification risk from small geographic areas in Canada

BackgroundA common disclosure control practice for health datasets is to identify small geographic areas and either suppress records from these small areas or aggregate them into larger ones. A recent study provided a method for deciding when an area is too small based on the uniqueness criterion. The uniqueness criterion stipulates that an the area is no longer too small when the proportion of unique individuals on the relevant variables (the quasi-identifiers) approaches zero. However, using a uniqueness value of zero is quite a stringent threshold, and is only suitable when the risks from data disclosure are quite high. Other uniqueness thresholds that have been proposed for health data are 5% and 20%.MethodsWe estimated uniqueness for urban Forward Sortation Areas (FSAs) by using the 2001 long form Canadian census data representing 20% of the population. We then constructed two logistic regression models to predict when the uniqueness is greater than the 5% and 20% thresholds, and validated their predictive accuracy using 10-fold cross-validation. Predictor variables included the population size of the FSA and the maximum number of possible values on the quasi-identifiers (the number of equivalence classes).ResultsAll model parameters were significant and the models had very high prediction accuracy, with specificity above 0.9, and sensitivity at 0.87 and 0.74 for the 5% and 20% threshold models respectively. The application of the models was illustrated with an analysis of the Ontario newborn registry and an emergency department dataset. At the higher thresholds considerably fewer records compared to the 0% threshold would be considered to be in small areas and therefore undergo disclosure control actions. We have also included concrete guidance for data custodians in deciding which one of the three uniqueness thresholds to use (0%, 5%, 20%), depending on the mitigating controls that the data recipients have in place, the potential invasion of privacy if the data is disclosed, and the motives and capacity of the data recipient to re-identify the data.ConclusionThe models we developed can be used to manage the re-identification risk from small geographic areas. Being able to choose among three possible thresholds, a data custodian can adjust the definition of "small geographic area" to the nature of the data and recipient.

Criteria for uniqueness and extrema of moments in continuous end spans with differential settlement

This paper algebraically analyses an end span of a continuous beam which experiences differential settlement under a traveling point load or, separately, a uniformly distributed load (UDL). The beam has two distinct section flexural stiffnesses along the span, with a given ratio between the stiffnesses and with given support stiffnesses. The analysis outputs a novel result that, for the UDL or for any given location of the point load, there is a specific location of stiffness change along the span at which the moment diagram is unique irrespective of the ratio between section stiffnesses. This result punctures the common assumption that stiffening parts of a continuous beam always serves to attract higher moments to those parts. Then, differential calculus is applied to the algebraic analysis to reveal another novel result that, again for the UDL or for any given location of the point load, the moment diagram is an extremum when the point of stiffness change along the span matches a salient point along the moment diagram. For the point load, an important restriction on the location of stiffness change that triggers the moment extremum is identified. Also for the point load, the extremum criterion is further differentiated to arrive at an expression for the specific ratio between the two section stiffnesses which just ensures that any given allowable moment at the inner support is not exceeded as the load travels along the span. Design and research uses of these results are discussed. The paper illustrates how algebra, plots, calculus and physical reasoning may be combined to enhance understanding of mathematically complex systems.

Abstract Hankel Operators in Kreĭn Spaces

Hankel operators and their symbols, as generalized by V. Ptak and P. Vrbova, are considered in the Kreĭn space setting. Under a generic assumption, without which the Krein space case may be untreatable, a necessary and sufficient condition for the existence of Hankel symbols for a given Hankel operator X is given. A parametric labeling of the Hankel symbols of X by means of Schur class functions is obtained. The proof is established by associating to the data of the problem an isometry V acting on a Kreĭn space so that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of V . The result includes uniqueness criteria and a Schur like formula.

Portrait of the four regular super-exponentials to base sqrt(2)

We introduce the concept of regular super-functions at a fixed point. It is derived from the concept of regular iteration. A super-function F of h is a solution of F(z+1)=h(F(z)). We provide a condition for F being entire, we also give two uniqueness criteria for regular super-functions. In the particular case h(x)=b^x we call F super-exponential. h has two real fixed points for b between 1 and e-(1/e). Exemplary we choose the base b=sqrt(2) and portray the four classes of real regular super-exponentials in the complex plane. There are two at fixed point 2 and two at fixed point 4. Each class is given by the translations along the x-axis of a suitable representative. Both super-exponentials at fixed point 4—one strictly increasing and one strictly decreasing-are entire. Both super-exponentials at fixed point 2— one strictly increasing and one strictly decreasing——are holomorphic on a right half-plane. All four super-exponentials are periodic along the imaginary axis. Only the strictly increasing super-exponential at 2 can satisfy F(0)=1 and can hence be called tetrational. We develop numerical algorithms for the precise evaluation of these functions and their inverses in the complex plane. We graph the two corresponding different half-iterates of h(z)=sqrt(2)^z. An apparent symmetry of the tetrational to base sqrt(2) disproved.

Het vonnis van de lezer

After the publication of the story 'Jansen' by L.H. Wiener, someone referred to as L. instituted legal proceedings because of an insulting passage in this story. The court of Amsterdam found that readers probably considered this passage as referring to reality - and to L. in particular - and therefore decided to ban the story. With the help of the so-called criterion of uniqueness, the present article shows how fictional works can refer to reality without nullifying the ontological homogeneity of fictive worlds. It is also shown how readers discover reference to reality in fiction. At the same time, the contribution pays attention to different sorts of readers. Readers with a high level of literary socialization can analyze 'Jansen' more thoroughly and will then see how the suggestion of reality of the passage under discussion is undermined by the story itself. Thus it becomes clear that the ruling of the court largely depends on the choice of the relevant group of readers with respect to 'Jansen'. © 2010 by Spiegel der Letteren.

Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized flow in the sense of partial differential equations, which is proved to have unique jacobian determinant, even though it is itself nonunique.

Novel existence and uniqueness criteria for periodic solutions of a Duffing type p-Laplacian equation

In this study, we investigate a Duffing type p-Laplacian equation. Some new criteria for guaranteeing the existence and uniqueness of periodic solutions of this equation are given by using the Manásevich–Mawhin continuation theorem and some analysis techniques. Our results improve and extend some known results from the literature.

Existence of Solutions of Boundary Value Problems for Impulsive Delay Differential Equations

his paper concerns first-order impulsive delay differential equations with boundary conditions. The existence results are presented by apply a monotone iterative method, and a uniqueness criterion is obtained.

The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain

We study the first boundary-value problem for the following mixed-type equation of the second kind: $$ u_{xx} + yu_{yy} + au_y - b^2 u = 0 $$ in the domain {(x, y) | 0 0, β > 0. Based on the completeness of the system of eigenfunctions of one-dimensional spectral problem we establish a uniqueness criterion. We construct a solution to the problem as the sum of the series in eigenfunctions.

The Neumann problem for a mixed-type equation in a rectangular domain

We establish a uniqueness criterion for the Neumann problem for a mixed-type equation with the Lavrent’ev-Bitsadze operator in a rectangular domain. We construct a solution as the sum of a series of eigenfunctions of the corresponding Sturm-Liouville problem.

A boundary value problem on the half-line for higher-order nonlinear differential equations

This article is devoted to the study of the existence of solutions as well as the existence and uniqueness of solutions to a boundary-value problem on the half-line for higher-order nonlinear ordinary differential equations. An existence result is obtained by the use of the Schauder–Tikhonov theorem. Furthermore, an existence and uniqueness criterion is established using the Banach contraction principle. These two results are applied, in particular, to the specific class of higher-order nonlinear ordinary differential equations of Emden–Fowler type and to the special case of higher-order linear ordinary differential equations, respectively. Moreover, some (general or specific) examples demonstrating the applicability of our results are given.

Tricomi problem for a mixed parabolic-hyperbolic equation in a rectangular domain

For a class of equations of mixed type, we study an analog of the Dirichlet problem satisfying the conjugation conditions on the line of change of the type. We establish a uniqueness and existence criterion for the solution of this problem.

General collision branching processes with two parameters

A new class of branching models, the general collision branching processes with two parameters, is considered in this paper. For such models, it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing states. Regularity and uniqueness criteria are firstly established. Explicit expressions are then obtained for the extinction probability vector, the mean extinction times and the conditional mean extinction times. The explosion behavior of these models is investigated and an explicit expression for mean explosion time is established. The mean global holding time is also obtained. It is revealed that these properties are substantially different between the super-explosive and sub-explosive cases.

Enhanced Gaussian processes and applications

We propose some construction of enhanced Gaussian processes using Karhunen-Loeve expansion. We obtain a characterization and some criterion of existence and uniqueness. Using rough-path theory, we derive some Wong-Zakai Theorem.