Abstract—In this paper, three analytic **closed** **form** **solutions** are introduced for arbitrary Nonuniform Transmission Lines (NTLs). The diﬀerential equations of NTLs are written in three suitable matrix equation forms, ﬁrst. Then the matrix equations are solved to obtain the chain parameter matrix of NTLs. The obtained **solutions** are applicable to arbitrary lossy and dispersive NTLs. The validation of the proposed **solutions** is veriﬁed using some comprehensive examples.

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Abstract—In this study, new **closed**-**form** **solutions** are presented for deriving inductance and capacitance elements of the extended-composite right/left-handed transmission line (E-CRLH TL) unit cell from the cutoﬀ frequencies of right-handed (RH) and left-handed (LH) bands. The characteristics of the E-CRLH TL are investigated for unbalanced, balanced, and mixed cases. The dispersion diagram, Bloch impedance, S -parameters are analyzed by the TL, circuit theories and the Bloch-Floquet theorem. Lastly, the usefulness of our method has been shown in detail by designing the desired characteristics for various cases.

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an underground opening, different methods such as analytical, numerical, and physical modeling approaches are employed. More specifically, **closed**-**form** **solutions** hold for the exact determination of stress and displacement fields, considered for long as the most reliable approach in underground engineering practice. In introducing an analytical solution, the geometry has a profound impact on the equations and, sometimes, complicated geometries have prevented the finding of a **closed**-**form** solution so far. In many geotechnical applications, underground openings in soils and rocks are excavated with more complex geometries such as semi- circular, quadrilateral, oval, horse-shoe shape, rectangular, square, and double-arch cross sections. All shapes mentioned were found to get solved using the complex variable method, as well as conformal mapping. This method is used to transform a noncircular geometry to a circular disc or planes weakened by a circular hole depending on the mapping function as well as the boundary condition of the problem. The advantage of the complex variable method with respect to the method that uses bipolar coordinates [1] is that the complex variable method is of a more general character, enabling the solution of problems for various cross sections and boundary conditions. Another advantage is that it directly gives displacement and stress.

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Abstract. This paper is devoted to **closed**-**form** **solutions** of the partial diﬀerential equa- tion: θ xx + θ yy + δexp(θ) = 0, which arises in the steady state thermal explosion the- ory. We ﬁnd simple exact **solutions** of the **form** θ(x,y) = Φ(F(x)+ G(y)), and θ(x,y) = Φ(f (x + y)+ g(x− y)). Also, we study the corresponding nonlinear wave equation.

the hard-to-obtain graph supp(S opt ), without having to solve the GL. Furthermore, we will show that the GL problem has a simple **closed**-**form** solution that can be easily derived merely based on the thresholded sample covariance matrix, provided that its underlying graph has an acyclic structure. This result will then be generalized to obtain an approxi- mate solution for the GL in the case where the thresholded sample covariance matrix has an arbitrary sparsity structure. This **closed**-**form** solution converges to the exact solution of the GL as the length of the minimum-length cycle in the support graph of the thresholded sample covariance matrix grows. The derived **closed**-**form** solution can be used for two pur- poses: (1) as a surrogate to the exact solution of the computationally heavy GL problem, and (2) as an initial point for common numerical algorithms to numerically solve the GL (see Friedman et al. (2008); Hsieh et al. (2014)). The above results unveil fundamental properties of the GL in terms of sparsification and computational complexity. Although conic optimization problems almost never benefit from an exact or inexact explicit formula for their **solutions** and should be solved numerically, the formula obtained in this paper suggests that sparse GL and related graph-based conic optimization problems may fall into the category of problems with **closed**-**form** **solutions** (similar to least squares problems). 3. Main Results

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Since Fξ is a known function of ξ, the last equation permits us to evaluate ξ with respect to the independent variable t and thus functions wξ and yξ through relations 4.10 and 4.8, respectively. Following the inverse course, one finally defines yx by the way of 4.1. Summarizing, we underline that, in this section, by using the results of 10, we proposed a new mathematical methodology and we succeed in giving new mathematical techniques in constructing analytic or **closed** **form** **solutions** of a class of degenerate Abel’s ODEs in mathematical physics and nonlinear mechanics. The methodology being introduced simplifies that of Section 2, which refers to the solution of an Abel equation of the first kind. For strengthening, we refer to some categories of degenerate Abel’s ODEs of the first kind which are included in 5–7, namely, i ax b 2 y x ax by 3 cy 3 0; ii y

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A way in which one may understand these benefits is to look deeper into the structure of the **closed** **form** solution. The **closed**-**form** solution involves an under- standing of the definitions of the system parameters and the product/s of their interactions, and the combination of this information to reach a desired conclu- sion. This is an integral process in solving physical processes analytically and an effective way in understanding their true nature as it can unveil the limitations and assumptions made. It is through this fundamental structure that it may be said **closed** **form** **solutions** offer an advantage over numerical **solutions**, as the analysis of a system and the interactions of its components is one which can yield great insight. One can examine an equation, alter it, incorporate it; as methods to see the interconnections and intricacies of the processes within it.

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The aim of the present paper is to present the analytical **solutions** for the time-periodic electro-osmotic ﬂow of Newtonian ﬂuids through a microannulus. Analytical **solutions** are rare. Not only do they represent electro-osmotic ﬂows through fundamental cross- sectional shapes but they also serve as standards for asymptotic and fully numerical meth- ods. Most important of all, some new physical and chemical phenomena can be found from the analytical **solutions**.

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The results are compared with the HPM results of.[17].inspite of the obtained results it should be noted that dimensionless variables defined in .[17]are slightly different yet for B=1 (a dimensionless parameter considered in .[17]the relevant equation and the related BCs of .[17] become similar to those in the present study i.e.Eqs.(26) and (28) of .[17] and the present Eqs.(4a)-(4c)are identical to B=1.Moreover ,in .[17] heat transfer effects were considered for the present problem ,while the present work shall exclusively focused on the isothermal situation .Inspiteof the variation, the current contrast occurred while comparing which is substantiated as the works of siddiqui et al. .[17]have assumed that the equation of momentum is unique by itself which is not dependent on temperature and will be handled exclusively. It culminates into having the same velocity profiles whether the heat transfer effects are for? yet it can be observed that the assumption as viewed to consider in .[17]is not correct for higher temperature gradients as the density and viscosity will vary ,and the momentum equations cannot be handled in segregation. Hence, the **solutions** found in.[17] are only ideal to situations where lesser effects of temperature are prevalent .the HPM solution of .[17]for the velocity profile is shown below for B=1.

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In addition, many nonlinear ODEs of the second- as well as of the first-order, gov- erning various problems in Mechanics and Physics, do not accept analytical **solutions** in terms of known functions. As an example, we mention the nonlinear oscillator equations (Rayleigh, Van der Pol, Duﬃng) which have been investigated thoroughly in the literature (also by the present author, see [11]), as well as the most cases of Abel equations of the second kind (the few solvable cases are presented by Polyanin and Zaitsev [12]). Thus, the establishment of methods leading to the construction of new functions, by means of which exact analytical **solutions** can be extracted, would be a desirable advance in the theory of nonlinear di ﬀ erential equations.

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In this paper, we extend Kim (2013) [9] for the optimal single FX risk hedging solution and theory to the multiple FX rates and suggest its application method in the business fields. First, the generalized optimal hedging method of sell- ing/buying of multiple foreign currencies is introduced. Second, the cost of han- dling forward contracts is included. Third, as a criterion of hedging performance evaluation, we consider the Leontief utility (or profit for a firm) function, which represents the risk averseness of a hedger. Fourth, steps are introduced about what is needed to proceed with hedging. There is a computation of the weighting ratios of the optimal combinations of three conventional hedging vehicles, i.e. , call/put currency options, forward contracts, and leaving the position open. As in the standard portfolio theory, the **closed** **form** solution of mathematical opti- mization may achieve a lower level of foreign exchange risk for a specified level of expected return. There is also a suggestion provided for a procedure that may be conducted in the business fields by means of Excel. 4

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The convenience of analytical tractability, paired with maximum stability, make the GEV distribution an attractive choice to model incomplete information structures. Analytical tractability entails the additional benefit in that it permits **closed** **form** expressions for comparative static results, e.g. characterizing the eﬀect of an increase in the number of bidders on the expected revenue in an IPV auction. Such results, next to the well-known analytical expressions for own and cross elasticities in logit demand models, are of great value, e.g. in applied competition analysis. In the microeconometric analysis of incomplete information games, the data typically only capture the value of the winning agents’ optimal strategies, e.g the winning bid or the price reached in the final of a sequence of bargaining episodes. The values of rival agents’ strategies along and oﬀ the equilibrium path, such as losing bids and inferior bargaining matches, typically are not observed. To the extent that agents’ optimal strategies are constrained by, and hence depend on, such values, structural econometric analysis needs to properly account for them. This can be done relatively eﬃciently if they can be replaced - or imputed - by expectations, conditional on observables; see, for example, Beckert, Smith and Takahashi (2015). An analytical E (or expectation) step does not only avoid additional computations necessary in simulation and numerical approximations, but it also improves the precision of resulting estimators relative to their simulation-assisted counterparts 8 .

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Exact **solutions** of the Navier-Stokes equations are rare since these are nonlinear par- tial diﬀerential equations. Exact **solutions** are very important not only because they are **solutions** of some fundamental ﬂows but also because they serve as accuracy checks for experimental, numerical, and asymptotic methods. In an excellent review article, Wang [9] outlines most if not all of the exact **solutions** to the Navier-Stokes equations. Over the past decades, non-Newtonian ﬂuids have become more and more impor- tant industrially. Polymer **solutions** and polymer melts provide the most common examples of non-Newtonian ﬂuids. The equations of motion of such ﬂuids are highly nonlinear and one order higher than the Navier-Stokes equations. In spite of the math- ematical complexity of these nonlinear equations, there exists a few exact **solutions**. Kaloni and Huschilt [3], Siddiqui [8], Rajagopal [6, 7], Benharbit and Siddiqui [1], and Labropulu [4, 5] have given a few such exact **solutions**.

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In this paper, we study an integrable system of coupled KdV equations, derived by Gear and Grimshaw (Stud. Appl. Math. 70(3):235-258, 1984), modeling the strong interaction of two-dimensional, long, internal gravity waves propagating on neighboring pycnoclines in a stratiﬁed ﬂuid. In particular, we present the complete group classiﬁcation of the model and ﬁnd conditions on arbitrary parameters for which the system admits symmetries. Some exact **solutions** of physical relevance are derived.

asymmetry due to inclined sagged configurations have been presented. With the aim of analytically investigating the planar 2:1 resonant, multi-modal, free dynamics of horizontal/inclined cables, approximate **closed**-**form** **solutions** for small sagged cables have been accomplished by means of a multi-dimensional Galerkin discretization and a second-order multiple scales approach. The analytical outcomes highlight the higher-order effects due to system quadratic nonlinearities on the resonantly coupled amplitudes, frequencies, dynamic configurations and velocities associated with the resonant non-linear normal modes. The dependence of cable response on different resonant/non-resonant (modeled/non-modeled) modal contributions has been emphasized. Accuracy of approximate horizontal/inclined cable models has thoroughly been validated by numerically evaluating the associated static as well as non- planar/planar linear and non-linear dynamic results against those of the exact model. Overall qualitative agreement of approximate and exact model results has been found, apart from some quantitative differences, depending on the element kinematics description, system parameters and consideration of planar or non-planar dynamics. Finally, significant insights into the modal coupling role played by system longitudinal dynamics and the effects of disregarding their contributions on non-linear coefficients through kinematic condensation have been obtained for horizontal cables, by also highlighting the influence of cable sag and/or extensibility.

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A mathematical model of Jeffrey fluid flow through an overlapping stenosis is presented. **Closed** **form** **solutions** are obtained for resistance to the flow and wall shear stress. It is analyzed that the resistance to the flow increases with the height and length of the stenosis but decreases with Jeffrey fluid parameter. The pressure drop and wall shear stress increases with stenosis height but decreases with Jeffrey fluid parameter.

Using properties for a prototype box-beam fabricated of pultruded fiber reinforced components and having M10 Unistrut connectors for the method of connection, the **closed** **form** **solutions** for the flange force and vertical deflection are shown to converge to the upper and lower bound limits given by the interaction states of full- and non-interaction. The analytical treatment is found to provide us with predictions of beam performance towards establishing the connection flexibility needed to achieve a given level of interaction. Knowing the shear stiffness from different methods of connections it will be feasible to use the **closed** **form** equations in this paper to aid in the design of ‘flat-pack’ box-beams with higher flexural rigidities (deeper sections) than can be sourced today from the range of single pultruded profiles (see Anon, 2010a; Anon 2010b; Anon 2010c).

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Abstract. We are interested in exploring interacting particle systems that can be seen as microscopic models for a particular structure of coupled transport ﬂux arising when diﬀerent populations are jointly evolving. The scenarios we have in mind are inspired by the dynam- ics of pedestrian ﬂows in open spaces and are intimately connected to cross-diﬀusion and thermo-diﬀusion problems holding a variational structure. The tools we use include a suitable structure of the relative entropy controlling TV-norms, the construction of Lyapunov function- als and particular **closed**-**form** **solutions** to nonlinear transport equa- tions, a hydrodynamics limiting procedure due to Philipowski, as well as the construction of numerical approximates to both the continuum limit problem in 2D and to the original interacting particle systems.

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There are no **closed**-**form** **solutions** for the moments of the mismatch function in (5) and (6). The expectations are multidimensional integrals for which we need to use compu- tationally expensive numerical integrations to calculate the model parameters. With the use of assumption (2) and an additional assumption that the two random variables Y k (l) and N k (l) are uncorrelated, we can reduce the dimensionality of the integration. Using the Gauss-Hermite numerical in- tegral method, we derive the procedures for computing the means and variances of the static features in the log-spectral domain in the next subsections.

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