The plan of the paper is the following. After this preamble in chapter 1, in chapter 2, we develop position of Souriau Symplectic Model of Statistical Physics in historical developments of Thermodynamics concepts. In chapter 3, we develop and revisit Lie Group Thermodynamics model of Jean-Marie Souriau in modern notations. In chapter 4, we make the link between Souriau Riemannian metric and Fisher Metric defined as a Geometric Heat Capacity of Lie Group Thermodynamics. In chapter 5, we elaborate Euler-Lagrange equations of Lie Group Thermodynamics and a Variational model based on Poincaré-Cartan Integral Invariant. In chapter 6, we explore Souriau Affine representation of Lie Group and Lie Algebra (including the notions of: Affine representations and cocycles, Souriau Moment Map and Cocycles, Equivariance of Souriau Moment Map, Action of Lie Group on a Symplectic Manifold and Dual spaces of finite-dimensional Lie Algebras) and we analyse the link and parallelisms with Koszul affine representation, developed in another context (comparison is synthetized in a table). In chapter 7, we illustrate Koszul and Souriau Lie Group models of Information Geometry for Multivariate Gaussian densities. In chapter 8, after identifying the affine group acting for these densities, we compute the Souriau moment map to obtain the Euler-Poincaré equation, solved by geodesic shooting method. In chapter 9, Souriau Riemannian metric defined by cocycle for Multivariate Gaussian Densities is computed. We give a conclusion in chapter 10 with research prospects in the framework of affine Poisson Geometry [112] and Bismut Stochastic Mechanics. We have 3 appendices: Appendix A develops the Clairaut(-Legendre) Equation of Maurice Fréchet associated to “distinguished functions” as seminal equation of Information geometry; Appendix B is about Balian Gauge Model of Thermodynamics and its compliance with Souriau model; Appendix C is devoted to the link of Casalis-Letac works on Affine Group Invariance for Natural **Exponential** **Families** with Souriau works.

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We have assumed a parametric distribution for the matrix variate data to make inferences for the eigenstruc- ture of symmetric random matrix using MLE and LRT, since this assumption has all the convenience of parametric models in terms of analytic expressions and can be efficiently applied to each of tens of thou- sands of voxels in DTI data. A symmetric matrix-variate Gaussian distribution with orthogonally invariant covariance can be considered for a symmetric random matrix Y . Otherwise, we can assume the distribu- tion of a symmetric random matrix as Wishart distribution, since this is a natural model to consider for covariance matrices. Furthermore, we consider an **exponential** family, which includes many common dis- tributions such as matrix-variate Gaussian, **exponential**, matrix gamma, and Wishart distribution (Fisher, 1934; Pitman, 1936; Koopman, 1936). **Exponential** **families** for univariate and multivariate data and infer- ence methods for them are well-known (Mardia, Kent, and Bibby, 1979, page 45; Lehmann and Casella, 1998, page 23; Casella and Berger, 2001, page 217). Analogous to other multivariate **exponential** **families**, we define a matrix-variate version of **exponential** family. Analytical calculations are possible if the distribu- tion is assumed to be equivariant under orthogonal transformations. With this **exponential** family form, we can estimate and test the eigenstructure of M in the general **exponential** family using MLE and LRT. Then we can apply the result to each specific distribution.

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nent norms, whereas the penalty functional in the common smoothing spline is the sum of squared component norms. This difference between the COSSO and the smoothing spline is similar to that between the LASSO and the ridge regres- sion. In fact, as noted in Lin and Zhang (2002), the LASSO can be seen as the COSSO applied to the linear model. Lin and Zhang (2002) showed that this new penalty tends to shrink insignificant functional components to exact zeros, and enjoys excellent model selection and estimation properties. In this paper we shall generalize the COSSO to the more general setting of nonparametric regression in **exponential** **families**, and develop a general algorithm using the iteratively- reweighted least squares (IRLS) estimation procedure. The general framework developed in this paper allows the treatment of many types of responses, such as non-normal responses, binary and polychotomous responses, and the event counts data, with the COSSO-type penalized likelihood methods.

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In general, optimality results for our proposed methods should follow along the lines similar to those established by Moustakides (1986) and related works, but this requires a separate proof. There are other situations of interest in geo- physical studies where an **exponential** family model may not be appropriate. Examples include extremes, cases where the parameter is a boundary point of the support of the random variable, and mixtures of distributions. Our future work will consist of stability detection for such cases.

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being the cumulant generating function, and ( H , h·, ·i H ) a reproducing kernel Hilbert space (RKHS) (Aronszajn, 1950) with k as its reproducing kernel. While various generalizations are possible for different choices of F (e.g., an Orlicz space as in Pistone and Sempi, 1995), the connection of P to the natural **exponential** family in (1) is particularly enlightening when H is an RKHS. This is due to the reproducing property of the kernel, f (x) = hf, k(x, ·)i H , through which k(x, ·) takes the role of the sufficient statistic. In fact, it can be shown (see Section 3 and Example 1 for more details) that every P fin is generated by P induced

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The above equations imply that bootstrap predictive distributions can be evaluated via simple analytical approximations, without the need to use simulation methods. It furthermore reveals that if the function ψ( · ) is convex at µ, as is the case for tail or percentile functions of **exponential** **families**, then the PBE tends to be higher than the MLE. Hence, the use of the bootstrap predictive distributions, e.g in pricing applications, is indeed more conservative than just using the MLE. On the other hand, the following Lemma shows also that using the PBE will approximately double the bias in comparison with the MLE.

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Several researchers have observed relationships between Bregman divergences and exponen- tial **families** (Azoury and Warmuth, 2001; Collins et al., 2001). In this paper, we formally prove an observation made by Forster and Warmuth (2000) that there exists a unique Bregman diver- gence corresponding to every regular **exponential** family. In fact, we show that there is a bijection between regular **exponential** **families** and a class of Bregman divergences, that we call regular Breg- man divergences. We show that, with proper representation, the bijection provides an alternative interpretation of a well known efficient EM scheme (Redner and Walker, 1984) for learning mixture models of **exponential** family distributions. This scheme simplifies the computationally intensive maximization step of the EM algorithm, resulting in a general soft-clustering algorithm for all regu- lar Bregman divergences. We also present an information theoretic analysis of Bregman clustering algorithms in terms of a trade-off between compression and loss in Bregman information.

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There is a very strong relationship between geometry and parameterisation. In particular it is important in a geometrically based theory to distinguish between those properties of the model which are dependent on a particular choice of parametrisation and those which are independent of this choice. Indeed one can define the geometry of a space to be those properties which are invariant to changes in parameterisation, see Dodson and Poston (1991). In Example 2 we noted that the parameters in the structural equations need not be simply related to the natural parameters, !. Structural parameters will often have a direct economet- ric interpretation, which will be context dependent. However there are also sets of parameters for full **exponential** **families** which always play an important role. The natural parameters, ! are one such set. A second form are the expected parameters (. These are defined by

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distance since it is not symmetric (KL ( m : m 0 ) = 6 KL ( m 0 : m ) ), and that it further does not satisfy the triangular inequality [4] of metric distances (KL ( m : m 0 ) + KL ( m 0 : m 00 ) 6≥ KL ( m : m 00 ) ). When the natural base of the logarithm is chosen, we get a differential entropy measure expressed in nat units. Alternatively, we can also use the base-2 logarithm (log 2 x = log log 2 x ) and get the entropy expressed in bit units. Although the KL divergence is available in closed-form for many distributions (in particular as equivalent Bregman divergences for **exponential** **families** [5]), it was proven that the Kullback-Leibler divergence between two (univariate) GMMs is not analytic [6] (the particular case of mixed-Gaussian of two components with same variance was analyzed in [7]). See appendix A for an analysis. Note that the differential entropy may be negative: For example, the differential entropy of a univariate Gaussian distribution is log ( σ √ 2πe ) , and is therefore negative when the standard variance σ < √ 1

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In this section, we perform, as in Minhajuddin et al. (2003), a simulation study of the bivariate **exponential** to examine the properties of estimator of the parameter from the latent distribution, 0 . We focus on 0 because it is an important portion of the correlation structure, and all of the parameters associated with X 1 and X 2 can be easily estimated marginally, since they are the observed diseases or events that occurred after the primary event X 0 . We assess the performance of

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In adaptive cluster sampling estimators, the proposed estimators t GE are more efficient as compare to Dryver and Chao (2007) ratio estimator t 8 . The **exponential** estimators for two auxiliary variables are efficient as compared to other estimator. The **exponential** ratio-cum-ratio estimator t 14 gave maximum percentage relative efficiency which increases with the increase in sample size. The performance of **exponential** ratio estimators remains better than all the other estimators and ratio estimators t 1 and t 8 did not perform in the usual sampling as well as adaptive cluster sampling at lower sample sizes. The adaptive **exponential** estimators perform better and better as initial sample size increases. Thus, it is proposed that for rare and clustered populations, like HIV patients in a locality, a rare type of monkeys in the forest, blind dolphins of river Sindh etc. the **exponential** estimators should be employed. The proposed estimators may be used when there are mild and moderate correlations between study and auxiliary variables and a simulation study is further needed in this respect.

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Abstract. The main purpose of this paper is to investigate the **exponential** stability in mean square of the **exponential** Euler method to linear stochastic delay differential equations (LSDDEs). The classical stability theorem to LSDDEs is given by the Lyapunov functions. However, in this paper we study the **exponential** stability in mean square of the exact solution to LSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to LSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the **exponential** Euler method to LSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

The observed data of COVID-19 progression in Pakistan for first 50 days from the first patient been reported has shown quite an unusual trend which is in opposition to clear **exponential** spread pattern of any infectious disease. The data of positive cases of 50 days of disease progression has been collected from COVID-19 dashboard of Pakistan and analyzed to see the graphical trend and to forecast the behaviour of disease progression for next 30 days. Mathematical equations regarding **exponential** growth are used to analyse the disease progression and different possible trajectories are plotted to understand the approximate trend pattern. The possible projections estimated 20k-456k positive cases within 80 days of disease spread in Pakistan. Although, the disease progression pattern is not perfectly **exponential**, it is still threatening a major fraction of susceptible population and demands effective strategic planning and control.

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In the present article, we study **exponential** attractors of the system (1.1) based on the concept of a non-au- tonomous (pullback) attractor. Thus, in the approach, an **exponential** attractor is also time-dependent. To be more precise, a family t → ( ) t of compact semi-invariant (i.e., U t ( ) ( ) , τ t ⊂ ( ) t ) sets of the dynami- cal process (1.1) is an (non-autonomous) **exponential** attractor if

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level of knowledge and data, the article reflects the status up to 18 April 2020. The prediction was based on the **exponential** growth rate model. For the depiction of the situation, the full length of the epidemic timeline was analyzed (from February 14 th till April 18 th ). The growth rates and their rates of decline during the period from the 22 nd of March till the 18 th of April were calculated and extrapolated in the coming 7 weeks. The predicted hospital needs were assessed against the announced allocated resources.

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total number of 484 cases (404 cases in the Kadiogo province) were reported nationwide. Real-time forecasts of COVID-19 are important to inform decision-making in the country. Here, we propose an approach that tests the performance of four models (**Exponential** Growth model, the Generalized Growth model (GGM), the Generalized Logistic Growth, and Richards Growth model) to select the model that best fit data and to generate short-term forecasting (5-, 10-, and 15-day forecasts from 11 to 25 April 2020) in Kadiogo, the epicenter of the outbreak. Using daily number of confirmed COVID-19 cases, the results suggests that GGM performed the best out of the 4 models. Overall, our GGM predictions suggested an average total number of cumulative cases of 514 (95% CI, 464–559), 629 (95% CI, 559–691), and 750 (95% CI, 661–840) between 11 to 15 April, 16 to 20 April, and 20 to 25 April 2020, respectively. COVID-19 in this province was best approximated by sub **exponential** growth rather than **exponential** or logistic growth. Current data suggest that COVID-19 cases would continue to increase over the next 15-days.

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This paper includes some new investigations and results for post quantum calculus, denoted by (p; q)-calculus. A chain rule for (p; q )-derivative is developed. Also, a new (p; q)-analogue of the **exponential** function is introduced and some its properties including the addition property for (p; q)-**exponential** functions are investigated. Several useful results involving (p; q)-binomial coe¢ cients and (p; q)-antiderivative are discovered. At the …nal part of this paper, (p; q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are an- alyzed extensively.

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u . The **exponential** splines to the base can be presented and evaluated by using **exponential** Euler polynomials, which connect to some famous generating functions (GF’s) in the combinatorics. The details can be found in Lecture 3 of [9-11], and Chapter 2 of [17], and we shall briefly quote them in next section. This approach can be used to derive a strong inter- relationship between two different fields. For instance, in [18], the author presents the equivalence between **exponential** Euler polynomials, Euler-Frobenius polyno- mials, etc. and some frequently used concepts in com- binatorics such as Eulerian fractions, Eulerian poly- nomials, etc., respectively, which will be surveyed as follows.

Stability of **exponential** Lie and **exponential** Strang split- ting methods have won a lot of attention in the recent years [9, 10, 11]. In this section, we look at the linear systems of ODEs and in particular at influence of perturbations at such systems.For doing this, we follow the work in [12]. Consider the initial value problem,

The unifed family is the **Exponential** Dispersion Family (EDF) generated by the uniform distribution (see Chapters 2 and 3 of (Jørgensen 1997) to see how an EDF can be generated from a moment generating function). We created the R package unifed (see (Quijano Xacur 2019b)) that includes functions to work with the unifed. In this section we make references to some functions in the package and we use this font format for those references.

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