Proof: Let A and B are s-Hermitian **doubly** **stochastic** matrix if ̅ = V A* V and = V B* V . Since A*and B*are also s-Hermitian **doubly** **stochastic** matrices then A* = V ̅ V and B* = V V To prove A B is s-Hermitian **doubly** **stochastic** matrix we will show that AB = = V (A B )* V Now V (A B )* V= V(B*A*)V = V(V V)( V ̅ V)V where A* = V ̅ V and B* = V V = V 2 V 2 ̅ V 2 = ̅ where V 2 = I

Where each S i is either fully indecomposable or a 1x1 zero matrix. But since S is quaternion **doubly** **stochastic**, the row sums of S 1 are all one and hence the column sums of S 1 being less than or equal to 1 must in fact be equal to 1. Hence S 12 0,...,S K 0 .Repeating this argument we obtain S S 1 S 2 ... S K , where each S i is a quaternion **doubly** **stochastic** and hence fully indecomposable.

A collection of non-empty and non-singular **doubly** **stochastic** matrix R together with two binary operations denoted by “+” and “.” are addition and multiplication which satisfy the following axioms is called a **doubly** **stochastic** Ring.

We know that is a **doubly** **stochastic** matrix, since each row and column of sum to 1. Besides, = , which means that there is at least a **doubly** **stochastic** matrix of order such that its inverse is a **doubly** **stochastic** matrix. In 2015, R. Farhadian (the first author) by a Farsi language article titled “Approximation for stationary distribution of ergodic **stochastic** processes” published in “NEDA: Student Statistical Journal”, showed that there exists some **doubly** **stochastic** matrices except identity matrix, such that their inverses are **doubly** **stochastic** matrices. Consider the following matrices of order 2 and 3:

Although a preliminary version of our method has been presented earlier (Yang and Oja, 2012a), the current paper introduces several significant improvements. First, we show that the proposed clustering objective can be used to learn not only the cluster assignments but also the number of clusters. Second, we show that the proposed structural decomposition is equivalent to the low-rank factorization of a **doubly** **stochastic** matrix. Previously it was known that the former is a subset of the latter and now we prove that the reverse also holds. Third, we have performed much more extensive experiments, where the results further consolidate our conclusions.

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We have derived pricing formulae for stop-loss reinsurance contracts for catastrophic events and for catastrophe insurance derivatives applying **doubly** **stochastic** Poisson process incorporating the shot noise process as its intensity. We have also presented pricing formulae for stop-loss reinsurance contracts for catastrophic events using the Kalman-Bucy filter. For these pricing models, it has been assumed that there are no arbitrage opportunities in the market. This has been achieved by using an equivalent martingale probability measure via the Esscher transform.

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3.1. Background. The idea of equilibrating a matrix such that the 1-norm of the rows and columns are all 1 is not new, dating back to at least the 1930s (see some historical remarks by Knight [?]). Here, we briefly review some of the previous work. Sinkhorn and Knopp [?] studied a method for scaling square nonnegative matrices to **doubly** **stochastic** form, that is a nonnegative matrix with all rows and columns of equal 1-norm. Sinkhorn [?] originally showed that: Any positive square matrix of order n is diagonally equivalent to a unique **doubly** **stochastic** matrix of order n, and the diagonal matrices which take part in the equivalence are unique up to scalar factors. Later, a different proof for the existence part of Sinkhorn’s theorem with some elementary geometric interpretations was given [?].

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An alternative method of generating a random walk on G is to apply a diagonal scaling to both sides of G to form a **doubly** **stochastic** matrix P = DGE. Of course, if we use this approach then the stationary distribution is absolutely useless for rank- ing purposes. However, in §5 we argue that the entries of D and E can be used as alternative measures. We will also see that if we apply the Sinkhorn-Knopp (SK) al- gorithm on an appropriate matrix to find D and E, we can compute our new ranking with a cost comparable to that of finding the PageRank. In order to justify this con- clusion, we need to establish the rate of convergence of the SK algorithm, which we

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An extreme point of a convex set is similar to the idea of maximum and minimum. Geometrically, the extreme points are the vertices of the convex polytope. Note that any point contained within the convex polytope can be written as a linear combination of it’s vertices. Thus, the extreme point matrices in this case are those that represent the vertices of the convex polytope created by the set of **doubly** **stochastic** matrices. This also implies that every **doubly** **stochastic** matrix can be written as a linear combination of the extreme point matrices. Another important property of a matrix, say A, that is an extreme point is that A cannot be written as λA 1 + (1 − λ)A 2 for distinct

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force. A major diﬃculty is related to the very large number of nuclear con- ﬁgurations one has to deal with, when distributing nucleons in the available orbitals of the proton and neutron shells. As an example, in an N=Z nucleus like 44 Ti such a number is of the order of 10 4 , and it increases quickly to 10 10 in 56 Ni and 10 28 in 88 Ru: this makes microscopic calculations computa- tionally challenging in medium mass nuclei and simply impossible in heavy systems. As a consequence, diﬀerent theoretical approaches have been devel- oped to describe the nuclear many body quantum system [1]. In particular, large scale Shell Model calculations can be performed for nuclei up to mass 100 and around closed shells, assuming a frozen core as a truncation scheme. Following this approach, one restricts excitations to valence nucleons within one/two oscillator shells, neglecting almost completely the excitations of the core. This become a relevant limitation particularly in nuclei with one or two particles outside a **doubly**-magic core, since in these systems the lowest structure should be dominated by the couplings between phonon excitations of the core (with a high degree of collectivity) and valence particles, giving rise to series of multiplets [2]. The identiﬁcation of these multiplets can provide precise, quantitative information on the phonon-particle couplings, a phenomen of primary importance, being at the origin of the quenching of spectroscopic factors [3] and of the damping of giant resonances [4, 5].

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instance be found in Borodin and Salminen (1996). The latter approach is called diusion approximation since Brownian motion is a special diusion process. One of the advantages of the diusion approximation is that it is applicable to more general models which derive from the classical risk process. For these more general processes the classical methods from renewal theory usually fail, and the diusion approach is then one of the few tools that work. Brownian motion has been studied for a long time and its usefulness in **stochastic** modeling is well accepted. However, Gaussian processes and variables do not allow for large uctuations and may sometimes be inadequate for modeling high variability. For instance, the above diusion approximation does not apply when the observed data give rise to a heavy{tailed claim size distribution such as Pareto with shape parameter 1 < < 2, implying that the variance 2 does not exist. This phenomenon very often arises in non-life insurance and in

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Abstract. During the last three years strong experimental evidence from B and charm factories has been accumulating for the existence of exotic hadronic quarkonia, narrow resonances which cannot be made from a quark and an antiquark. Their masses and decay modes show that they contain a heavy quark-antiquark pair, but their quantum numbers are such that they must also contain a light quark-antiquark pair. The theoretical challenge has been to determine the nature of these resonances. The main possibilities are that they are either “genuine tetraquarks", i.e. two quarks and two antiquarks within one conﬁnement volume, or “hadronic molecules" of two heavy-light mesons. In the last few months there as been more and more evidence in favor of the latter. I discuss the experimental data and its interpretation and provide fairly precise predictions for masses and quantum numbers of the additional exotic states which are naturally expected in the molecular picture but have yet to be observed. In addition, I provide arguments in favor of the existence of an even more exotic state – a hypothetical deuteron-like bound state of two heavy baryons. I also consider “baryon-like" states QQ q¯ ¯ q , which if found will be direct evidence not just for near-threshold binding of two heavy mesons, but for genuine tetraquarks with novel color networks. I stress the importance of experimental search for **doubly**-heavy baryons in this context.

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Before starting we wish to make precise the various notions of (locally) square-integra- ble martingales used in this paper, since they play a crucial role. As said in the introduc- tion, a process X given on a **stochastic** basis, either with discrete or with continuous time, is called a square-integrable martingale if it is a martingale and if the supremum of X over all time is square-integrable: then the limit X 1 exists and is a square-integrable variable.

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In [2] Grishanov, Meshkov and Omelchenko introduced the idea of representing a fabric with a repeating (**doubly** periodic) pattern by a knot diagram on a torus, having made a choice of a unit cell for the repeat of the pattern. Algebraic invariants of this diagram based on the Jones polynomial were used to associate a polynomial to the fabric which was independent of the choice of unit cell, so long as a minimal choice of repeating cell was made. In this paper we enhance the nature of the diagram used to represent the fabric by including two further auxiliary curves, to produce a link in the 3-dimensional sphere S 3 from which the original fabric can

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Abstract – In this paper, a grid connected wind power generation scheme using a **doubly** fed induction generator (DFIG) is studied. The aims of this paper are: The modelling and simulation of the operating in two quadrants (torque- speed) of a DFIG, the analysis employs a stator flux vector control algorithm to control rotor current using the PI regulator. The simulation calculations were achieved using MATLAB®-SIMULINK® package. The obtained results are presented, illustrating the good control performances of the system.

A great deal of classic works on the relationships of the Schur complements and the diagonal-Schur complements with the original matrices have been contributed, for a com- plete survey of these works we refer to (see e.g., []). As is shown in [, ], the Schur com- plements of positive semideﬁnite matrices are positive semideﬁnite and the Schur comple- ments of strictly diagonally dominant matrices are strictly diagonally dominant, the same is true for M-matrices, H-matrices, inverse M-matrices, strictly **doubly** diagonally dom- inant matrices and generalized strictly diagonally dominant. In addition, Liu and Huang [] proposed that the diagonal-Schur complements of strictly diagonally dominant matri- ces are strictly diagonally dominant, the same is true for strictly γ -diagonally dominant matrices and strictly product γ -diagonally dominant matrices.

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Therefore, it is useful to examine the consistency among heteroscedastic frontier models that have different distributional assumptions. In the present study, we combine the above mentioned four types of distributional assumptions with homoscedastic and **doubly** heteroscedastic **stochastic** production-frontier models, utilizing a sample of 1,221 Japanese water utilities, pooled for two years. Here, the dispersion in the size distribution of utilities suggests that the homogeneity assumption is violated. Thus, we also introduce a **doubly** heteroscedastic variable mean model, and examine the sensitivity of nested models to a more comprehensive heteroscedasticity correction for the one-sided error component.

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In this paper, we study the Carathéodory approximate solution for a class of **doubly** perturbed **stochastic** diﬀerential equations (DPSDEs). Based on the Carathéodory approximation procedure, we prove that DPSDEs have a unique solution and show that the Carathéodory approximate solution converges to the solution of DPSDEs under the global Lipschitz condition. Moreover, we extend the above results to the case of DPSDEs with non-Lipschitz coeﬃcients.

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Abstract. In this paper, we study the existence and uniqueness of solutions to **doubly** perturbed **stochastic** differential equations with jumps under the local Lipschitz condi- tions, and give the p-th exponential estimates of solutions. Finally, we give an example to illustrate our results.

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The origins of web page ranking can be traced to the ranking methods for scientific journals via citations. Kleinberg [71] gives a well-rounded analysis of this fact. One of the major developments in ranking methods for scientific journals was a Markov chains based method proposed by Pinski and Narin [46, 107]. Both of the web ranking models, HITS developed by Kleinberg [71] and PageRank by Brin and Page [103, 18], presented below have elements that resemble the Pinski and Narin method. A thorough analysis of both of these models as well as several others can be found in [94]. The rating vector in the PageRank model is the left eigenvector of an irreducible **stochastic** matrix produced using the “endorsement” approach. The HITS model starts with an adjacency matrix also corresponding to the “endorsement” interpretation of hyperlinks.

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