We consider the semilinear problem (Formula presented.) where B 1 is the unit ball in (Formula presented.) and assume (Formula presented.) Using a monotonicity formula argument, we prove an optimal regularity result for solutions: (Formula presented.) is a log-Lipschitz function. This problem introduces two main difficulties. The first is the lack of invariance in the scaling and blow-up of the problem. The other (more serious) issue is a term in the Weiss energy which is potentially non-integrable unless one already knows the optimal regularity of the solution: this puts us in a catch-22 situation.
The presented work addresses two-stage stochastic programs (2SPs), a broadly applicable model to capture optimization problems subject to uncertain parameters with adjustable decision variables. In case the adjustable or second-stage variables contain discrete decisions, the corresponding 2SPs are known to be NP-complete. The standard approach of forming a single-stage deterministic equivalent problem can be computationally challenging even for small instances, as the number of variables and constraints scales with the number of scenarios. To avoid forming a potentially huge MILP problem, we build upon an approach of approximating the expected value of the second-stage problem by a neural network (NN) and encoding the resulting NN into the first-stage problem. The proposed algorithm alternates between optimizing the first-stage variables and retraining the NN. We demonstrate the value of our approach with the example of computing operating points in power systems by showing that the alternating approach provides improved first-stage decisions and a tighter approximation between the expected objective and its neural network approximation.
Postage is a big cost for companies and agencies that are sending large volumes of physical mail. This cost can be reduced by getting bulk discounts. Mathematical optimization is used to ensure that the lowest prices possible are found. This thesis will analyze an existing model used by a company and explain how it works. The model will also be expanded to make it applicable to new pricing models.
The model is in the form of a mixed integer linear program and it is used to optimize the cost of a batch of bulk mail. The original model can handle price models where the discounts are reached by sending large volumes of letters. Not all mail delivery companies provide such discounts however. The model is then extended to allow other forms of discounts.
Three extensions of the original model were implemented. The effectiveness of the extensions of the model are measured by comparing the prices attained by using the extended model and the original model. The results are discussed to assess if the implementation of the extension are worthwhile. The first extension of the model was made to allow the mixing of mail with different allowed delivery times. The problem was to ensure that the mail was not delivered slower than what was promised. By mixing the mail with different delivery times larger volumes could be attained and greater discounts were achieved which lead to better results overall.
The second extension that was made was to make the model able to handle discounts based on the mean weight of the letters that are sent. The model is adjusted to work with this price model. This change did not yield a lower price than the original model in all cases. The explanation for this is that the original model is less flexible when different price models are used and therefore the extended model is more suitable to use.
The last extension made was implemented to bypass an existing rule that forbids letters with too great of a weight difference to be sent together. The goal was to make the model more flexible. No conclusive improvement of the results were seen for this extension.
An inverse method for estimating the distributions of the nonlinear elastic properties of inhomogeneous and anisotropic membranes is investigated. The material description of the membrane is based on a versatile constitutive model, including four material parameters: two initial stiffness values pertaining to the principal directions of the material, the angle between these principal directions and a reference coordinate system and a parameter related to the level of nonlinearity of the material. The estimation procedure consists of the following three steps: (i) perform experiments on the membranous structure whose properties are to be determined, (ii) define a corresponding finite-element (FE) model and (iii) minimise an error function (with respect to the unknown parameters) that quantifies the deviation between the numerical predictions and the experimental data. For this finite deformation problem, an FE framework for membranous structures exposed to a pressure boundary loading is outlined: the principle of virtual work, its linearisation and the related spatial discretisation. To achieve a robust parameter estimation, an element partition method is employed. In numerical examples, the proposed procedure is assessed by attempting to reproduce given random reference distributions of material fields in a reference membrane. The deviations between the estimated material parameter distributions and the related reference fields are within a few percent in most cases. The standard deviation for the resulting maximum principal stress was consistently below 1%.
In the present study, a computational framework for studying high-speed crack growth in rubber-like solids under conditions of plane stress and steady-state is proposed. Effects of inertia, viscoelasticity and finite strains are included. The main purpose of the study is to examine the contribution of viscoelastic dissipation to the total work of fracture required to propagate a crack in a rubber-like solid. The computational framework builds upon a previous work by the present author (Kroon in Int J Fract 169:49-60, 2011). The model was fully able to predict experimental results in terms of the local surface energy at the crack tip and the total energy release rate at different crack speeds. The predicted distributions of stress and dissipation around the propagating crack tip are presented. The predicted crack tip profiles also agree qualitatively with experimental findings.
The biomechanical behaviour of biological cells is of great importance in many physiological processes. One such process is the maintenance of fibrous networks, such as collagenous tissues. The activity of the fibre-producing cells in this type of tissue is very important, and a comprehensive material description needs to incorporate the activity of the cells. In biomechanics, continuum mechanics is often employed to describe deforming solids, and modelling can be much simplified if continuum mechanics entities, such as stress and strain, can be correlated with cell activity. To investigate this, a continuum mechanics framework is employed in which remodelling of a collagen gel is modelled. The remodelling is accomplished by fibroblasts, and the activity of the fibroblasts is linked to the continuum mechanics theory. The constitutive model for the collagen fabric is formulated in terms of a strain energy function, which includes a density function describing the distribution of the collagen fibre orientation. This density function evolves according to an evolution law, where fibroblasts reorient fibres towards the direction of increasing Cauchy stress, elastic deformation, or stiffness. The theoretical framework is applied to experimental results from collagen gels, where gels have undergone remodelling under both biaxial and uniaxial constraint. The analyses indicated that criteria 1 and 2 (Cauchy stress and elastic deformations) are able to predict the collagen fibre distribution after remodelling, whereas criterion 3 (current stiffness) is not. This conclusion is, however, tentative and pertains, strictly speaking, only to fibre remodelling processes, and may not be valid for other types of cell activities.
A new plasticity model with a yield criterion that depends on the second and third invariants of the stress deviator is proposed. The model is intended to bridge the gap between von Mises' and Tresca's yield criteria. An associative flow rule is employed. The proposed model contains one new non-dimensional key material parameter, that quantifies the relative difference in yield strength between uniaxial tension and pure shear. The yield surface is smooth and convex. Material strain hardening can be ascertained by a standard uniaxial tensile test, whereas the new material parameter can be determined by a test in pure shear. A fully implicit backward Euler method is developed and presented for the integration of stresses with a tangent operator consistent with the stress updating scheme. The stress updating method utilizes a spectral decomposition of the deviatoric stress tensor, which leads to a stable and robust updating scheme for a yield surface that exhibits strong and rapidly changing curvature in the synoptic plane. The proposed constitutive theory is implemented in a finite element program, and the influence of the new material parameter is demonstrated in two numerical examples.
In 1999 Schramm introduced the one-parameter family of random planar chords known as Schramm-Loewner evolution (SLE(kappa)). More recently,Wang defined a functional on (deterministic) planar chords and loops called Loewner energy. The Loewner energy is the rate function of a large deviation principle on SLE(kappa) as kappa tends to 0. Curves of finite energy are more regular than SLE(kappa) and have several interesting properties. For example, there is a link to Teichmüller theory; the family of finite energy loops coincides with the class of Weil-Petersson quasicircles. In this thesis we study natural generalizations of the chordal Loewner energy. We define a two-sided radial Loewner energy, corresponding to the process of a chordal SLE conditioned to hit a marked interior point. We characterize curves of finite two-sided radial energy and show that there is a unique curve of minimal energy. We then move on to discuss a generalization of the multichordal Loewner energy, introduced by Peltola and Wang, to chords with fused endpoints. First, we construct a multichordal Loewner energy on curves which have not yet reached their respective endpoints, agreeing with the energy defined by Peltola and Wang in the limit. We then generalize this energy to two curves which aim at the same point and define the fused multichordal Loewner energy by taking the limit.
This bachelor thesis in applied mathematics and industrial engineering aims to determine if and how weather affects the consumption of goods at small grocery stores. To study this, we conducted a regression analysis based on sales data from an ICA Nära. We have collected one year’s weather- and sales data and used mathematical statistics to determine how weather affects the sales for different product groups. Our belief is that weather does affect the consumption.
Several large actors in the industry have some sort of consumption of goods forecast. None of these takes weather into account when creating their sale forecast. Hopefully, this thesis will provide information aiding companies in deciding whether or not to use weather forecasts as a prediction parameter.
The results indicate a large effect on sales for some groups of products. The regression reveals how much the sale of a group increase along with an increase of one unit of the different measured weather factors. There is, most likely, not a perfect linear relation between our response variable and the explanatory variables. Therefore, one must interpret the results carefully. In addition, we discuss how a possible implementation affects the supply chain of a large grocery store company and the importance of flexibility in one’s supply chain.
Approximate algorithms for constructing a minimum feasible labeling for a tree are considered based on a greedy approach, combined with a breadth-first search and depth-first search.
We consider a Ricci de Turck flow of spaces with isolated conical singularities, which preserves the conical structure along the flow. We establish that a given initial regularity of Ricci curvature is preserved along the flow. Moreover under additional assumptions, positivity of scalar curvature is preserved under such a flow, mirroring the standard property of Ricci flow on compact manifolds. The analytic difficulty is the a priori low regularity of scalar curvature at the conical tip along the flow, so that the maximum principle does not apply. We view this work as a first step toward studying positivity of the curvature operator along the singular Ricci flow.
We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation—initially derived to integrate over manifolds of codimension one—to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.
We provide a new approach for computing integrals over hypersurfaces in the level set framework. The method is based on the discretization (via simple Riemann sums) of the classical formulation used in the level set framework, with the choice of specific kernels supported on a tubular neighborhood around the interface to approximate the Dirac delta function. The novelty lies in the choice of kernels, specifically its number of vanishing moments, which enables accurate computations of integrals over a class of closed, continuous, piecewise smooth, curves or surfaces; e.g., curves in two dimensions that contain a finite number of corners. We prove that for smooth interfaces, if the kernel has enough vanishing moments (related to the dimension of the embedding space), the analytical integral formulation coincides exactly with the integral one wishes to calculate. For curves with corners and cusps, the formulation is not exact but we provide an analytical result relating the severity of the corner or cusp with the width of the tubular neighborhood. We show numerical examples demonstrating the capability of the approach, especially for integrating over piecewise smooth interfaces and for computing integrals where the integrand is only Lipschitz continuous or has an integrable singularity.
The field of speech recognition has during the last decade left the re- search stage and found its way in to the public market. Most computers and mobile phones sold today support dictation and transcription in a number of chosen languages. Swedish is often not one of them. In this thesis, which is executed on behalf of the Swedish Radio, an Automatic Speech Recognition model for Swedish is trained and the performance evaluated. The model is built using the open source toolkit Kaldi. Two approaches of training the acoustic part of the model is investigated. Firstly, using Hidden Markov Model and Gaussian Mixture Models and secondly, using Hidden Markov Models and Deep Neural Networks. The later approach using deep neural networks is found to achieve a better performance in terms of Word Error Rate.
In this thesis three risk appetite indexes are derived and measured from the beginning of 2006 to the end of the first quarter in 2019. One of the risk appetite indexes relies on annualized returns and volatilities from risky and safe assets while the others relies on subjective and risk neutral probability distributions. The distributions are obtained from historical data on equity indexes and from a wide spectrum of option prices with one month until the options expires. All data is provided by Refinitiv through Öhman Fonder.
The indexes studied throughout the thesis is provided by authors from financial institutions such as Bank of England, Bank of International Settlements and Credit Suisse First Boston.
I conclude in this thesis that the Credit Suisse First Boston index and the Bank of International Settlements index generated the most intuitive result regarding expected response after major financial events. A principal component analysis demonstrated that the Credit Suisse First Boston index held most of the information in terms of explanation of variance. At last, the indexes was used as a trend-following strategy for asset allocation for switching between a safe versus a risky portfolio. A trend in the risk appetite was studied for 2 to 12 months back in time and resulted in that all of the risk appetite indexes studied throughout the thesis can be a helpful tool to asset allocation.
We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang [6].
We show that arithmetic toral point scatterers in dimension three (“Šeba billiards on R3/ Z3”) exhibit strong level repulsion between the set of “new” eigenvalues. More precisely, let Λ : = { λ1< λ2< … } denote the unfolded set of new eigenvalues. Then, given any γ> 0 , |{i≤N:λi+1-λi≤ϵ}|N=Oγ(ϵ4-γ)as N→ ∞ (and ϵ> 0 small.) To the best of our knowledge, this is the first mathematically rigorous demonstration of a level repulsion phenomena for the quantization of a deterministic system.
If f is a non-constant polynomial with integer coefficients and q is an integer, we may regard f as a map from Z/qZ to Z/qZ. We show that the distribution of the (normalized) spacings between consecutive elements in the image of these maps becomes Poissonian as q tends to infinity along any sequence of square free integers such that the mean spacing modulo q tends to infinity.
We prove that any q'>𝑞q‐automatic multiplicative function𝑓:ℕ→ℂ either essentially coincides with a Dirichlet character, or vanishes on all sufficiently large primes. This confirms a strong form of a conjecture of Bell, Bruin and Coons [Trans. Amer. Math. Soc. 364 (2012) 933–959].
This paper studies the maximal size of product-free sets in Z/nZ. These are sets of residues for which there is no solution to ab=c (mod n), with a, b, c being in the set. In a previous paper, we constructed an infinite sequence of integers (n(i))(i >= 1) and product-free sets S-i in Z/n(i)Z such that the density vertical bar S-i vertical bar/n(i) -> 1 as i -> infinity, where vertical bar S-i vertical bar denotes the cardinality of S-i. Here, we obtain matching, up to constants, upper and lower bounds on the maximal attainable density as n -> infinity.
We consider momentum push-forwards of measures arising as quantum limits (semiclassical measures) of eigenfunctions of a point scatterer on the standard flat torus T-2 = R-2/Z(2). Given any probability measure arising by placing delta masses, with equal weights, on Z(2)-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and are completely delocalized in position (i.e., concentration on Lagrangian states). We also show that the mass, in momentum, can fully localize on more exotic measures, for example, singular continuous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals, enabling precise control of the location of perturbed eigenvalues.
In this paper, we show that for almost all primes p there is an integer solution xε [2,p-1] to the congruence xx ≡ x (mod p). The solutions can be interpretated as fixed points of the map x→xx (mod p), and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.
For coprime integers g and n, let l(g) (n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l(g) (n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l(g) (p) as p <= x ranges over primes.
We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori in dimensions . Despite quantum ergodicity holding for the set of "new" eigenfunctions we show that superscars occur-there is phase space localization along families of closed orbits, in the sense that some semiclassical measures contain a finite number of Lagrangian components of the form , for uniformly bounded from below. In particular, for both and , eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues. For , we also show that some semiclassical measures have both strongly localized momentum marginals and non-uniform quantum limits (i.e., the position marginals are non-uniform). For , superscarred eigenstates are quite rare, but for we show that the phenomenon is quite common-with denoting the counting function for the new eigenvalues below x, there are eigenvalues with the property that any semiclassical limit along these eigenvalues exhibits superscarring.
We prove a Polya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for "incomplete intervals" I subset of F-p, provided (p(1/2) log p)/|I| = o(1). Applications include density results for irreducible trinomials in F-p[x], i.e. the number of irreducible polynomials in the set {f(x) = x(d) + a(1)x + a(0) is an element of F-p[x]}a(0) is an element of I-0,I- a(1) is an element of I-1 is similar to |I-0|.|I-1|/d provided |I-0| > p(1/2+is an element of), |I-1| > p(is an element of), or |I-1| > p(1/2+is an element of), |I-0| > p
We prove an analogue of Shnirelman, Zelditch and Colin de VerdiS- re's quantum ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the coupling parameter, in particular the equidistribution holds for both the weak and strong coupling quantizations of the point scatterer.
The Seba billiard, a rectangular torus with a point scatterer, is a popular model to study the transition between integrability and chaos in quantum systems. Whereas such billiards are classically essentially integrable, they may display features such as quantum ergodicity [11], which are usually associated with quantum systems whose classical dynamics is chaotic. Seba proposed that the eigenfunctions of toral point scatterers should also satisfy Berry's random wave conjecture, which implies that the value distribution of the eigenfunctions ought to be Gaussian. However, Keating, Marklof, and Winn formulated a conjecture that suggested that Seba billiards with irrational aspect ratio violate the random wave conjecture, and we show that this is indeed the case. More precisely, for tori having diophantine aspect ratio, we construct a subsequence of the set of new eigenfunctions having even/even symmetry, of essentially full density, and show that its 4th moment is not consistent with a Gaussian value distribution. In fact, given any set Lambda interlacing with the set of unperturbed eigenvalues, we show non-Gaussian value distribution of the Green's functions G(lambda), for lambda in an essentially full density subsequence of Lambda.
We consider the Laplacian with a delta potential (a "point scatterer") on an irrational torus, where the square of the side ratio is diophantine. The eigenfunctions fall into two classes: "old" eigenfunctions (75%) of the Laplacian which vanish at the support of the delta potential, and therefore are not affected, and "new" eigenfunctions (25%) which are affected, and as a result feature a logarithmic singularity at the location of the delta potential. Within a full density subsequence of the new eigenfunctions we determine all semiclassical measures in the weak coupling regime and show that they are localized along four wave vectors in momentum space-we therefore prove the existence of so-called "superscars" as predicted by Bogomolny and Schmit [5]. This result contrasts with the phase space equidistribution which is observed for a full density subset of the new eigenfunctions of a point scatterer on a rational torus [14]. Further, in the strong coupling limit we show that a weaker form of localization holds for an essentially full density subsequence of the new eigenvalues; in particular quantum ergodicity does not hold. We also explain how our results can be modified for rectangles with Dirichlet boundary conditions with a point scatterer in the interior. In this case our results extend previous work of Keating, Marklof andWinn who proved the existence of localized semiclassical measures under a clustering condition on the spectrum of the Laplacian.
A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice (Formula presented.), gives rise to a probability measure on the unit circle in a natural way. Such measures, and their weak limits, are said to be attainable from lattice points on circles. We investigate the set of attainable measures and show that it contains all extreme points, in the sense of convex geometry, of the set of all probability measures that are invariant under some natural symmetries. Further, the set of attainable measures is closed under convolution, yet there exist symmetric probability measures that are not attainable. To show this, we study the geometry of projections onto a finite number of Fourier coefficients and find that the set of attainable measures has many singularities with a “fractal” structure. This complicated structure in some sense arises from prime powers—singularities do not occur for circles of radius (Formula presented.) if n is square free.
We produce a collection of families of curves, whose point count statistics over becomes Gaussian for p fixed. In particular, the average number of points on curves in these families tends to infinity.
We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions.
This is a manuscript containing the full proofs of results announced in [10], together with some recent updates. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components of a random Gaussian field, genuinely depends on the field. We then infer the same for "arithmetic random waves", i.e. random toral Laplace eigenfunctions. (C) 2018 Elsevier Inc. All rights reserved.
We study the defect (or 'signed area') distribution of standard toral Laplace eigenfunctions restricted to shrinking balls of radius above the Planck scale, either for deterministic eigenfunctions averaged w.r.t. the spatial variable, or in a random Gaussian scenario ('arithmetic random waves'). In either case we exploit the associated symmetry of the eigenfunctions to show that the expectation (spatial or Gaussian) vanishes. In the deterministic setting, we prove that the variance of the defect of flat eigenfunctions, restricted to balls shrinking above the Planck scale, vanishes for 'most' energies. Hence the defect of eigenfunctions restricted to most of the said balls is small. We also construct 'esoteric' eigenfunctions with large defect variance, by choosing our eigenfunctions so that to mimic the situation on the hexagonal torus, thus breaking the symmetries associated to the standard torus. In the random Gaussian setting, we establish various upper and lower bounds for the defect variance w.r.t. the Gaussian probability measure. A crucial ingredient in the proof of the lower bound is the use of Schmidt's subspace theorem.
In this paper, we study an approximation of spectral density in terms of the generalized Kullback-Leibler distance minimization. For a given spectral density, we seek a spectral density by minimizing the generalized Kullback-Leibler distance subject to a constraint on the tangential second-order statistics.
A parameterization of the solutions to the positive real residue interpolation with McMillan degree constraint is given. The McMillan degree of the interpolants is equal to the McMillan degree of the maximum entropy interpolant, and the parameterization includes the maximum entropy interpolant.
The main body of this thesis consists of six appended papers.The papers are about the theory of the positive real interpolationwith McMillan degree constraint.In Paper A, a parameterization of the positive real residue interpolantswith McMillan degree constraint is given.For a given interpolation data and for each free parameter,a positive real interpolant, of which McMillan degree isequal to the McMillan degree of the maximum entropy interpolant, is obtained bysolving a nonlinear equation, which is homotopic to a nonlinear equation to determinethe maximum entropy interpolant.In Paper B,the state-space realization of the multivariable rational interpolant with bounded McMillan degreeis given by the block discrete-time Schwarz form.A characterization of the positive realness of the block discrete-time Schwarz form isgiven by a linear matrix inequality.In Paper C,a robust controller synthesis for the mismatch of delay in terms ofthe Nevanlinna-Pick interpolation is presented.In Paper D,a Smith predictor synthesis for unstable and minimum-phaseinput delay system and for a first orderunstable distributed delay system is given in terms of the Nevanlinna-Pick interpolation.In Paper E , we study an approximation of spectral density in termsof the generalized Kullback-Leibler distance minimization.For a given spectral density,we seek a spectraldensity by minimizingthe generalized Kullback-Leibler distance subject to a constraint onthe tangential second-orderstatistics.In Paper F, a property of Schur polynomial of real coefficientsand real Toeplitz matrix is given.Suppose that the vector of coefficients of a Schur polynomial annihilatesa Toeplitz matrix, then the Toeplitz matrix is in facta zero matrix.
For a given partial covariance sequence (C 0,C 1,⋯,C n) and for each MA part of the ARMA modeling filter of degree n, an AR part of the ARMA modeling filter of degree n for the solution to the rational covariance extension problem is obtained by solving a nonlinear equation, which is homotopic to a nonlinear equation determining the maximum entropy AR filter.
In this paper, a robust controller synthesis for the mismatch of delay in terms of the Nevanlinna-Pick interpolation is presented.
The state-space realization of a multivariable rational interpolant with bounded McMillan degree is given by the block discrete-time Schwarz form. A characterization of the positive realness of the block discrete-time Schwarz form is given by a linear matrix inequality.