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# Uncertainty Quantification for Geometry Deformations of Superconducting Cavities using Eigenvalue Tracking

The electromagnetic field distribution as well as the resonating frequency of various modes in superconducting cavities are sensitive to small geometry deformations. The occurring variations are motivated by measurements of an available set of resonators from which we propose to extract a small number of relevant and independent deformations by using a truncated Karhunen-Loève expansion. The random deformations are used in an expressive uncertainty quantification workflow to determine the sensitivity of the eigenmodes. For the propagation of uncertainty, a stochastic collocation method based on sparse grids is employed. It requires the repeated solution of Maxwell's eigenvalue problem at predefined collocation points, i.e., for cavities with perturbed geometry. To ensure consistency of the solution among the various eigenvalue problems and for numerical efficiency reasons, an eigenvalue tracking technique is proposed that is based on homotopies between collocation points and a Newton-based eigenvalue solver. The approach can be efficiently parallelized while tracking the eigenpairs. In this paper, we propose the application of isogeometric analysis since it allows for the exact description of the geometrical domains with respect to common computer-aided design kernels and for a straightforward and convenient way of handling geometrical variations.

## Authors

• 7 publications
• 2 publications
• 8 publications
• 59 publications
• ### Uncertainty Quantification for Maxwell's Eigenproblem using Isogeometric Analysis

The electromagnetic field distribution as well as the resonating frequen...
02/08/2018 ∙ by Niklas Georg, et al. ∙ 0

• ### Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

Stochastic PDE eigenvalue problems often arise in the field of uncertain...
03/05/2021 ∙ by Alexander D. Gilbert, et al. ∙ 0

• ### Uncertainty quantification of microstructure variability and mechanical behaviour of additively manufactured lattice structures

Process-induced defects are the leading cause of discrepancies between a...
03/17/2021 ∙ by Nina Korshunova, et al. ∙ 0

• ### Analytic regularity and stochastic collocation of high dimensional Newton iterates

In this paper we introduce concepts from uncertainty quantification (UQ)...
05/22/2019 ∙ by Julio Enrique Castrillon-Candas, et al. ∙ 0

• ### Isogeometric Analysis Simulation of TESLA Cavities Under Uncertainty

In the design of electromagnetic devices the accurate representation of ...
11/06/2017 ∙ by Jacopo Corno, et al. ∙ 0

• ### High-Dimensional Uncertainty Quantification of Electronic and Photonic IC with Non-Gaussian Correlated Process Variations

Uncertainty quantification based on generalized polynomial chaos has bee...
01/31/2019 ∙ by Chunfeng Cui, et al. ∙ 0

• ### Uncertainty Modeling and Analysis of the European XFEL Cavities Manufacturing Process

This paper reports on comprehensive efforts on uncertainty quantificatio...
06/21/2019 ∙ by Jacopo Corno, et al. ∙ 0

##### This week in AI

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## I Motivation and Introduction

In rf resonators, the electromagnetic fields oscillate at certain frequencies with higher magnitude than at others. The advantageous operation of such devices at selected eigenfrequencies is technically profitable used for example to set up dedicated filter components or to provide field amplitudes, which are higher than the magnitude of the excitation. In the demanding context of charged particle accelerators for example, specifically designed resonators, which are also known as cavities, can efficiently accelerate bunched charged particle beams. Essentially, the acceleration is performed in the longitudinal direction only, although specific devices can also be developed to deflect the beam in a transverse direction. The shape of the resonating structure, as well as the walls materials, determine, to a large extent, the overall performance of the entire system. As a consequence of this, plenty of effort has been put in the development of proper simulation tools to computationally assist scientists and engineers during the complex design process.

Because of the limitations in the available computational resources, in the early days of such simulations the main focus was exclusively put on lossless two-dimensional (2D) eigenanalysis. Meanwhile, even expensive three-dimensional (3D) models including dissipative materials or leaky boundary conditions can be handled routinely. Due to the strong dependence of the solution with respect to the geometry, a huge amount of thought and work has been spent on the precise modeling of the resonator’s surface and the corresponding boundary conditions. Various discretization schemes have been applied to introduce proper computational elements ranging from a flexible irregular triangular mesh [1, 2] to an efficient structured rectangular variant [3, 4]. In both cases, special care has been put on the accurate description of the boundary surface to retain the specific convergence order of the underlying method. A further step in this direction can be found in the application of iga, where the geometric modeling is naturally linked to the utilized cad kernels without further approximations. Thus, no geometry-related modeling error is introduced.

The assembled eigenvalue formulation requires the solution of individual eigenpairs, which are also known as modes. They are characterized by a specific field distribution together with the corresponding eigenfrequency. Whereas the distribution of the electromagnetic field is connected to the determined eigenvector, the extracted eigenvalue specifies the resonating frequency and quality factor if available. In the lossy case, the modal analysis is obviously of complex nature, while in the simplified lossless approximation all calculations can be handled with real arithmetic only. One critical aspect for the manufacturing of the designed cavities is the specification of appropriate tolerances for the shape of the resonator to ensure a proper operability performance. While the sensitivity of the examined mode with respect to unavoidable geometric variations can be obtained with the help of the known sensitivity analysis, e.g.

[9], a profound understanding of the corresponding distributions is obtained using the more expensive uq, see e.g. [10, 11]. Most previous works handle only tiny deformations and represent them rather naively, e.g. using non-curved elements, deforming only one layer of elements or remesh the problem without considering the mesh sensitivity. All these drawbacks are elegantly circumvented by the workflow proposed in this paper.

To study the influence of geometric boundary variations on the distribution of the electromagnetic fields and the related eigenfrequencies, the introduction of proper shape parameters has to be performed. For each cavity, which is related to a specific shape, the respective eigensolutions can be calculated independently of each other. Whereas the assignment of the simulation results with respect to the examined modes is simple for non-overlapping functions, it becomes much worse for touching or crossing graphs. In those cases, a direct evaluation of the derivatives with respect to the shape parameter is not suitable anymore and reliable eigenvalue-tracking methods have to be applied instead. In the following chapters, the necessary steps to establish a uq workflow for electromagnetic fields in superconducting cavities using eigenvalue tracking will be formulated.

In the following section we state Maxwell’s eigenvalue problem and introduce the isogeometric discretization chosen for the approximation of the fields. In section III we present the uncertainty quantification methods used for the simulation and their relation to the geometry parametrization. Given the necessity to properly categorize the eigensolutions for each uncertain configuration, we present the eigenvalue tracking method in section IV. Finally, we show the application of the proposed scheme to the simple case of the pillbox cavity and to a real world example, i.e. the tesla cavity (see Fig. 1 and 2). In the last section we give some concluding remarks.

## Ii Numerical Approximation of Maxwell’s Eigenproblem

### Ii-a Maxwell’s Eigenvalue Problem

Given a bounded and simply connected domain , with Lipschitz continuous boundary , the electromagnetic fields in the source-free, time-harmonic case, are governed by Maxwell’s equations:

 ∇×E =−jωμ0H in Ω (1) ∇×H =jωε0E in Ω ∇⋅ε0E =0 in Ω ∇⋅μ0H =0 in Ω,

where and denote the electric and magnetic field strength, respectively, and we assume the electric permittivity and magnetic permeability of vacuum. We further apply pec boundary conditions

 E×n =0 on ∂Ω (2) H⋅n =0 on ∂Ω.

Starting from Maxwell’s equations, one can derive the classical eigenvalue formulation by eliminating from (1): find the wave number and such that

 ∇×(∇×E) =k2E in Ω (3) ∇⋅E =0 in Ω E×n =0 on ∂Ω.

The standard variational formulation of (3) is: find with such that

 (∇×E,∇×v)=k2(E,v)∀v∈H0(curl;Ω), (4)

where we introduced the function space

of square-integrable vector fields with square-integrable

curl. The subscript in indicates vanishing tangential components on the boundary since we are imposing pec boundary conditions.

In order to be able to numerically solve (4) we deduce its discrete counterpart: following a Galerkin procedure, we introduce a sequence of finite-dimensional spaces and we restrict (4) to it: find with such that

 (∇×Eh,∇×v)=k2h(Eh,v)∀v∈Vh. (5)

Whatever the choice of the functional space , can be expressed in terms of a set of basis functions :

 Eh=Ndof∑j=1ejvj. (6)

The discrete solution of Maxwell’s eigenvalue problem is then obtained by solving the generalized eigenvalue problem

 Ke=k2hMe, (7)

where and are the stiffness and mass matrix for vacuum respectively, given by

 Kij=(∇×vj,∇×vi),Mij=(vj,vi). (8)

There exist multiple choices for the space and its basis although care must be taken to ensure a proper approximation of . For the fem case we refer the interested reader to [12]. In the following section we will follow [13] and introduce a curl-conforming iga discretization.

### Ii-B Isogeometric Analysis

iga is a technique that was introduced by Hughes et al. [5] with the aim of allowing the exact representation of geometries defined via cad software, independently of the level of refinement of the computational grid. It is clear that, since the geometry directly affects the resonant properties of rf cavities, the exact description of the simulation domain is strongly desired, even more so in the case of shape sensitivity analysis and uq.

In order to achieve this goal, the same classes of basis functions that are commonly used for geometry description in cad software, are used for the representation of solution vector fields. Such basis functions are the so called B-splines and nurbs. For the definition and construction of such basis, we refer the interested reader to [14, 15].

The use of iga in cavity simulation has been shown to be beneficial both in terms of accuracy and of overall reduction of the computational cost [6, 16]. For an overview of the applicability of iga to other electromagnetic problems, we refer to [17].

We assume that the domain can be exactly described through a smooth mapping obtained from the cad software, see Fig. 3:

 F0:^Ω→Ω, (9)

where is a parametric domain (e.g. the unit cube). We furthermore assume that is piecewise smoothly invertible.

We define the discretization space in a general way

 Vh :={v∈H(curl;Ω):^v=ι(v)∈^Vh} (10) ≡{v∈H(curl;Ω):v=ι−1(^v),^v∈^Vh},

where is a discrete space on the parametric domain and is a so-called pull-back function defined from the parametrization . Let be a basis for , the set is a basis for that can be substituted in (7)-(8) to obtain a finite-dimensional generalized eigenvalue problem.

We indicate as the standard space of B-Splines of degree along dimension , with regularity , constructed on the reference domain . Following [13] the discrete space is chosen as

 ^Vh:=Sp1−1,p2,p3α1−1,α2,α3×Sp1,p2−1,p3α1,α2−1,α3×Sp1,p2,p3−1α1,α2,α3−1. (11)

It can be proven that basis functions in this space obey to a discrete de Rham sequence, thus being suitable candidates for the discretization of .

The discrete space in the physical domain is obtained through a curl-conforming ‘Piola’ transformation [18]:

 Vh:={v:(dF0)⊤(v∘F−10)∈^Vh}, (12)

where we indicate with the Jacobian of the parametrization.

### Ii-C Domain Parametrization

Given a B-Spline (or nurbs) basis of , the mapping is constructed using a set of control points in the physical space as:

 F=∑iPiBi. (13)

The points create the so-called control mesh.

It is straightforward to show [15] that, given definition (13), cad objects are invariant under affine transformations such as translations, scalings and rotations, i.e. to apply the transformation to the object it is sufficient to apply it to the control mesh. Furthermore, is smooth with respect to the movement of the control points.

Let us now consider the domain and its parametrization to be given in terms of a -vector of parameters , i.e. becomes and the mapping

 Fδ:^Ω→Ω(δ). (14)

The shape variability can be reinterpreted as the variability of the control mesh:

 Fδ=∑iPi(δ)Bi, (15)

and by following the steps described in the previous section in the case of a parametrized domain, one derives the parametrized generalized eigenvalue problem:

 K(δ)e(δ)=k2h(δ)M(δ)e(δ) (16)

where denotes the numerical approximation of the eigenvalue for each parameter value . Suppressing for brevity the dependency, we denote the discrete eigenpairs with where .

## Iii Uncertainty Quantification

In operation, the geometry of a particular TESLA cavity will differ from the design shown in Fig. 2: the manufacturing process, tuning and multiphysical effects as cooling, Lorentz detuning and microphoning will introduce geometry variations [10, 19, 20, 6]. In the simplest case, these variations can be expressed in terms of the design parameters, see Fig. 2, i.e.,

 δdesign=[a1,a2,b1,b2,L,R%iris,Req]⊤ (17)

and lead to an explicit definition of the parametrized geometry mapping . However, in more realistic scenarios the variation of a particular geometry is unknown.

Let us assume that we have measured times the geometric variations and that each observation can be described by an -dimensional vector

 x(m)=[x(m)1,x(m)2,...,x(m)N]⊤,m=1,…,M. (18)

And let us further assume that all components

are realizations of normally distributed random variables

 Xn∼N(μn,σ2n),n=1,…,N, (19)

possibly mutually correlated. Expectation and variance are denoted by

and respectively and can be obtained from the measurement samples.

We propose the discrete Karhunen–Loève expansion (or equivalently principal component analysis

[21]) to convert the observations into a set of linearly uncorrelated variables. uq methods can then be applied to investigate the sensitivity of the eigenmodes to those variables.

### Iii-a Karhunen–Loève

Let us collect all the observations in the matrix

 T=[x(1),x(2),…,x(M)]⊤, (20)

whose columns are the random variables and whose rows are the measured observations. We will denote by its covariance matrix

 C:=1M−1M∑m=1(x(m)−μ)(x(m)−μ)⊤, (21)

where is the sample mean

 μ=1MM∑m=1x(m). (22)

We search for an eigendecomposition of the covariance matrix of the type

 C=VΣV⊤, (23)

where the columns of are the orthonormal right eigenvectors of , and is a diagonal matrix whose elements are the corresponding eigenvalues. Without loss of generality, we will further assume the columns of to be ordered such that .

Now, only the most significant contributions are extracted by approximating

 C≈Ct:=VtΣtV⊤t, (24)

where and is a )-matrix with the corresponding eigenvectors.

Finally, the variations can be approximated by a low-dimensional representation

 ⎡⎢ ⎢⎣X1⋮XN⎤⎥ ⎥⎦=⎡⎢ ⎢⎣μ1⋮μN⎤⎥ ⎥⎦+VtΣ1/2t⎡⎢ ⎢⎣δ1⋮δNt⎤⎥ ⎥⎦. (25)

that is based on normally distributed parameters . With respect to the original vector of parameters , may not have a physical meaning, but has a significantly smaller dimension. Reducing the size of the parameter space has a great impact in the efficiency of a stochastic collocation method.

Let us now consider the case of a nurbs domain specified by the mapping (13). Given , to describe the deformed geometry (14) we will firstly construct the

B-Spline interpolating curve mappings

[14]

 Fμ =∑iPμiBi (26) Fj =∑iPjiBi∀j=1,…,Nt% , (27)

where interpolates the mean deformation vector and is an interpolating curve through the points given by the -th column of the matrix . Unique interpolations can be obtained by requiring additional properties as a certain polynomial degree and minimal curvature [15]. Both and the share the same basis functions as the geometry mapping and differ only in their control mesh. The control points defining the deformed geometry are then given by

 Pi(δ)=Pi+Pμi+Nt∑j=1δjPji. (28)

### Iii-B Stochastic Collocation

Quantities of interest are statistical measures about the eigenmodes with respect to the influence of the vector of random variables. In particular, we aim for expectation and variance of the -th eigenfrequency ,

 E[fj] =∫∞−∞fj(δ)ρ(δ)dδ (29a) var[fj] =∫∞−∞(fj(δ)−E[fj])2ρ(δ)dδ, (29b)

where

denotes the joint probability density of

.

In order to approximate (29) numerically, a stochastic collocation method [22] is applied which can be summarized as follows: based on the assumption that the parametric dependency is smooth [23], we construct a polynomial approximation based on Lagrangian interpolation

 fj(δ)≈Nk∑k=1fh,j(δk)lk(δ), lm(δn)={1if m=n0if m≠n (30)

where are multivariate Lagrange polynomials constructed by tensorization of univariate Lagrange polynomials and is the set of corresponding collocation points (tensor grid). The eigenfrequencies

 fh,j(δk)=kh,j(δk)2π√ε0μ0. (31)

are obtained by solving (16) for each set of discrete parameters . This requires rerunning the simulation times for slightly different geometries. No modifications of the code are necessary and hence the method can be considered as non-intrusive. Finally, inserting (30) into (29) yields, for the -th eigenfrequency:

 E[fj] ≈Nk∑k=1fh,j(δk)wk (32a) var[fj] ≈Nk∑k=1(fh,j(δk)−E[fj])2wk, (32b)

with weights

 wk=∫∞−∞lk(δ)ρ(δ)dδ. (33)

The integrals can be evaluated exactly using Gauss-Hermite quadrature for each independent parameter . The quadrature nodes are also used as collocation points .

The stochastic collocation method (32

) based on tensor grids is known to be very efficient for a small number of random variables but suffers strongly from the curse of dimensionality

[24]. The number of collocation points grows exponentially with the dimension of the parametric space . This problem can be mitigated by the sparse grid approach [25],[26]. Linear combinations of tensor grids with different degrees for different parameters lead to more efficient interpolation and cubature rules for a (moderately) large stochastic space [27]. An illustration of stochastic collocation grids is given in Fig. 4.

In each of the collocation points, eigenvalue problem (7) is solved. The natural ordering of the eigenpairs according to their eigenvalue, however, might not be sufficient to ensure consistency since even small changes in the domain can cause eigenvalue crossing or separation/degeneration of modes. To be able to correctly place the frequencies for each parametric point , an eigenvalue tracking algorithm [28] is applied.

## Iv Eigenvalue Tracking

To be able to track the eigenvalues, we construct an algebraic homotopy between the quadrature points in the parameter space in such a way that:

 Kk,t :=(1−t)K(δ0)+tK(δk) (34a) Mk,t :=(1−t)M(δ0)+tM(δk) (34b)

where and is a given start configuration in the parameter space (e.g. corresponding to an unperturbed geometry). We further assume that at the modes are not degenerate and that there is no bifurcation from to . Other homotopies could be analogously defined, e.g. such that the eigenvalue problems for correspond to actual geometries. However, the algebraically motivated choice (34) allows the straightforward definition of the derivatives,

 K′k,t :=∂∂tKk,t=K(δk)−K(δ0) (35a) M′k,t :=∂∂tMk,t=M(δk)−M(δ0) (35b)

Now, all the quantities depending on the vector , can then be viewed as depending on the scalar quantity which makes the following one-dimensional analysis feasible. For each collocation point and eigenpair , the eigenvalue problem (7) can be rewritten as a linear system of equations with a normalization constraint using an suitable vector

 [Ktet−λtMtetc⊤et−1]=[00], (36)

where and . The subscripts and addressing eigenpair number and collocation point are suppressed for easier readability.

Let us consider a Taylor expansion at along

 [~et~λt]=[e0λ0]+(t−t0)[e′0λ′0] (37)

to obtain a first-order approximation for the eigenpair at . The derivatives of the eigenvalues and eigenvectors are obtained from (36) which is differentiated with respect to the parameter . The resulting linear system

 [K0−λ0M0−M0e0c⊤0][e′0λ′0]=[−K′0e0+λ0M′0e00] (38)

is then solved with respect to the eigenvalue and eigenvector derivatives. The predicted eigenpair can be used as an initial guess

 [e(0)tλ(0)t]:=[~et~λt] (39)

for the Newton-Raphson method in order to obtain the exact eigenpair . In the -th iteration of the Newton-Raphson method, the linear system of equations to be solved is

 [Kt−λ(i−1)tMt−Mte(i−1)tc⊤0][Δe(i)tΔλ(i)t]= −[Kte(i−1)t−λ(i−1)tMte(i−1)tc⊤e(i−1)t−1]. (40)

The increments and improve the prediction by assigning

 [e(i)tλ(i)t]:=[e(i−1)tλ(i−1)t]+[Δe(i)tΔλ(i)t]. (41)

As a termination criterion for the Newton-Raphson method, the norm of the residual of eigenvalue problem (36) can be considered as well as the norm of . The required number of Newton iterations can then be used for a stepsize control in order to increase the efficiency and robustness of the algorithm. The proposed tracking algorithm is illustrated by the flowchart in Fig. 5.

The numerical efficiency of the presented tracking method could be further improved, e.g. by using a simplified Newton method [29], by employing a higher order Taylor approximation in the prediction step (37) or by tracking a subspace of eigenpairs and matching the eigenpairs based on a correlation coefficient [30, 31].

## V Application

In this section, we present the application of the eigenvalue tracking method proposed to cavities with uncertain geometry. As a clear indication of the necessity of the tracking, we first analyze the test case of a cylindrical cavity with uncertain radius. Stochastic collocation is then applied to study the sensitivity of the tesla cavity [8] to eccentric deformation of its 9 cells. In both cases the discretization is carried out with iga using GeoPDEs and NURBS Octave packages [32, 33].

### V-a Pillbox Cavity

As a numerical example with an available closed-form reference solution, we consider a pillbox (i.e. cylindrical) cavity with uncertain radius. We consider a fixed length

and we assume a uniformly distributed radius

. Second degree basis functions are chosen for a total of degrees of freedom. The pillbox cavity as well as the electric and magnetic field lines of the fundamental tm mode are shown in Fig. 6.

#### V-A1 Tracking

We use the proposed eigenvalue tracking technique from section IV to track the eigensolutions over the parameter range . The exact eigenfrequencies are well known [34]. At a radius of approximately a transition of the fundamental mode from te to tm is observed.

To illustrate the relevance of eigenvalue tracking, the eigenvalues obtained by solving (16) at discrete sample points are shown in Fig. 7

. Without tracking, the eigenvalues cannot be associated to specific modes without cumbersome post-processing procedures that typically require some heuristics

[35].

For this first example, the algebraic homotopy (34) is replaced by the physical system matrices such that all eigenvalues computed along the homotopy path correspond to the solutions of a physical problem. This allows for the comparison with the closed-form solution with respect to the radius . In practice, one is probably not interested in the intermediate points. However, in this test case the derivatives cannot be straightforwardly computed and are then approximated by finite differences. Fig. 8 demonstrates the result of the tracked eigenvalues.

#### V-A2 Uq

We determine the uncertainty in the 6 lowest eigenfrequencies at for the uniformly distributed radius using stochastic collocation based on Clenshaw-Curtis quadrature with collocation points. The eigenvalue problem for the first collocation point at is solved based on Arnoldi’s method [36] using the command eigs in Matlab R2017a. For each eigenpair the tracking method is applied for every remaining collocation point. The initial stepsize is chosen such that all points are reached using only one step. For this particular model, the Newton scheme always converges, using on average 2.2 iterations (max. 4) such that the stepsize is never reduced.

The first two statistical moments can be seen in Fig.

9. The relative errors in the numerical results with respect to the analytic reference solutions are below

(for expectations as well as standard deviations).

#### V-A3 Computational Cost

The tracking is necessary to obtain correct results since the eigenmodes need to be matched at each collocation point. However, as shown in this subsection, it is also numerically more efficient than naively solving the repeated eigenvalue problems independently of each other. The computations are performed on a workstation with an Intel(R) Xeon(R) CPU E5-2687W processor and 256 GB RAM. All time measurements are repeated times and averaged. It shall be noted that, for a fair comparison, in all eigenvalue solvers the occurring linear equation systems were solved with the Matlab backslash operator \, i.e. sparse Gaussian elimination. Improvements are to be expected when reusing factorizations or preconditioners with appropriate iterative solvers.

As a reference, we solve the eigenvalue problem at each collocation point using a Matlab implementation of the Jacobi-Davidson algorithm [37, 38] as well as the Matlab command eigs which internally calls the Arpack library [39]. Since we are interested in quantifying uncertainties for 10 eigenmodes and given that the ordering of eigenvalues may change, we calculate 20 eigenvalues to ensure that the corresponding eigenvalues are included at each collocation point. The Jacobi-Davidson algorithm solves the eigenproblems on average in iterations and while eigs takes on average for solving linear systems.

The computational cost of the proposed tracking approach is determined by the time needed for solving (38) in order to compute the derivatives and for solving the linear systems of the Newton scheme (40). Cost for assembly of the system matrices is excluded since the algebraic mapping (35) can be employed for uq which does not require any additional matrix assemblies. On average, the computational time needed for solving linear systems for each collocation point and eigenpair is .

### V-B TESLA Cavity

A more challenging application is the simulation of the so called tesla cavity. This is an rf superconducting cavity for linear accelerators with a design accelerating gradient of . In particular, we focus on the ttf design of the cavities [40] which were installed at desy in Hamburg.

The TESLA cavity is a 9-cell standing wave structure of about in length, whose accelerating mode resonates at . The cavities are produced, one half-cell at a time, by deep drawing of pure niobium sheets [40]. The half cells are then connected together by ebm. The production process inherently introduces geometry imperfections that need to be taken into account in operation. Here, we focus on the misalignment of each of the 9 cells with respect to the ideal axis of the cavity. The misalignment can produce undesired effects on the field quality and on the resonating frequency.

We consider variables, corresponding to the displacement of the nine cell’s centres in the and direction and we collect approximately observations available from DESY measurements [41] in the matrix (see Eq. 20). The deformations are normally distributed, but they also present cross correlations. To obtain uncorrelated variables for the uq process, and to reduce the size of the parameter space, the truncated Karhunen–Loève decomposition introduced in section III-A is applied. This allows for the reduction from to variables according to the relative information criterion

 ∑Nt=7i=1σi∑N=18i=1σi≥0.95. (42)

The collocation points for the numerical quadrature are created with a Smolyak grid of level , with Gaussian points along each direction, for a total of points [24]. For a given collocation point it is possible to use Karhunen–Loève expansion and the curves and to build the deformation curve and thus the deformed geometry . In Fig. 12 the deformed domain is shown.

The measured eccentric deformations, e.g. shown in Fig. 10, cause frequency shifts in the fundamental passband below . These shifts are rather harmless and can be corrected during the initial compulsory tuning procedure [42]. For the following numerical study, the magnitude of the deformations is artificially increased by a factor of 50 to demonstrate that our IGA workflow can cope with arbitrary deformations while linearization or ad-hoc mesh deformation will fail. The increased deformations lead to a frequency shift for the fundamental passband in the order of , a level of accuracy that can be easily achieved by the isogeometric method by choosing second degree basis functions and approximately degrees of freedom.

The results for the standard deviations of the modes in the first TM passband are depicted in Fig. 11. It is noticeable, that the last two modes in the passband are more than times more sensitive than the other ones.

## Vi Conclusion

In this work we investigate the application of uq methods for the evaluation of eigenvalue sensitivity. In particular we are interested in the eigenfrequencies of rf cavities since they are known to be highly sensitive to geometry deformations.

Given the paramount role of geometry parameterization to achieve good accuracy, we discretize Maxwell’s time-harmonic eigenvalue problem with iga. The properties of the isogeometric basis functions also allow for a straightforward description of the domain deformation.

Finally, given that the ordering of the eigenfrequencies can change for different geometry configurations, we apply an eigenvalue tracking method, that exploits the properties of the iga matrices, to ensure a correct classification of the modes in different collocation points. However, the tracking has the positive side effect of reducing the computational costs for solving the repeated eigenvalue problems.

## Acknowledgment

The work of J. Corno, N. Georg and S. Schöps is supported by the Excellence Initiative of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt. J. Corno’s and N. Georg’s work are also partially funded by the DFG grants SCHO1562/3-1 and RO4937/1-1, respectively.

The authors would like to thank M. Kellermann and A. Krimm for their contributions to the first implementation and DESY for providing access to the cavity database.

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# American Institute of Mathematical Sciences

2005, 2005(Special): 216-224. doi: 10.3934/proc.2005.2005.216

## Fitzhugh-Nagumo equations in a nonhomogeneous medium

 1 St. John's University, 300 Howard Av., Staten Island, NY 10301, United States

Received  September 2004 Revised  February 2005 Published  September 2005

In this paper we investigate various propagation phenomena for the FitzHugh-Nagumo system (% (4) with a nonhomogeneous threshold function $a(x)$. It is studied over a range of values $b$, $d,\varepsilon$ and function $a(x).$ Numerical simulations of system show that system (\ref{fh}) exhibits different patterns of behavior and they significantly differ from those in a homogeneous medium.
Citation: Arnold Dikansky. Fitzhugh-Nagumo equations in a nonhomogeneous medium. Conference Publications, 2005, 2005 (Special) : 216-224. doi: 10.3934/proc.2005.2005.216
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## G = C2×Q8.C18order 288 = 25·32

### Direct product of C2 and Q8.C18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C2×Q8.C18
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — C2×Q8⋊C9 — C2×Q8.C18
 Lower central Q8 — C2×Q8.C18
 Upper central C1 — C2×C12

Generators and relations for C2×Q8.C18
G = < a,b,c,d | a2=b4=1, c2=d18=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=bc, dcd-1=b >

Subgroups: 225 in 85 conjugacy classes, 31 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C9, C12, C12, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C18, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×C6, C2×C4○D4, C36, C2×C18, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, Q8⋊C9, C2×C36, C6×C4○D4, C2×Q8⋊C9, Q8.C18, C2×Q8.C18
Quotients: C1, C2, C3, C22, C6, C9, A4, C2×C6, C18, C2×A4, C3.A4, C2×C18, C4.A4, C22×A4, C2×C3.A4, C2×C4.A4, Q8.C18, C22×C3.A4, C2×Q8.C18

Smallest permutation representation of C2×Q8.C18
On 144 points
Generators in S144
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 109)(98 110)(99 111)(100 112)(101 113)(102 114)(103 115)(104 116)(105 117)(106 118)(107 119)(108 120)
(1 138 19 120)(2 56 20 38)(3 83 21 101)(4 141 22 123)(5 59 23 41)(6 86 24 104)(7 144 25 126)(8 62 26 44)(9 89 27 107)(10 111 28 129)(11 65 29 47)(12 92 30 74)(13 114 31 132)(14 68 32 50)(15 95 33 77)(16 117 34 135)(17 71 35 53)(18 98 36 80)(37 99 55 81)(39 140 57 122)(40 102 58 84)(42 143 60 125)(43 105 61 87)(45 110 63 128)(46 108 64 90)(48 113 66 131)(49 75 67 93)(51 116 69 134)(52 78 70 96)(54 119 72 137)(73 130 91 112)(76 133 94 115)(79 136 97 118)(82 139 100 121)(85 142 103 124)(88 109 106 127)
(1 81 19 99)(2 139 20 121)(3 57 21 39)(4 84 22 102)(5 142 23 124)(6 60 24 42)(7 87 25 105)(8 109 26 127)(9 63 27 45)(10 90 28 108)(11 112 29 130)(12 66 30 48)(13 93 31 75)(14 115 32 133)(15 69 33 51)(16 96 34 78)(17 118 35 136)(18 72 36 54)(37 138 55 120)(38 100 56 82)(40 141 58 123)(41 103 59 85)(43 144 61 126)(44 106 62 88)(46 111 64 129)(47 73 65 91)(49 114 67 132)(50 76 68 94)(52 117 70 135)(53 79 71 97)(74 131 92 113)(77 134 95 116)(80 137 98 119)(83 140 101 122)(86 143 104 125)(89 110 107 128)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)

G:=sub<Sym(144)| (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120), (1,138,19,120)(2,56,20,38)(3,83,21,101)(4,141,22,123)(5,59,23,41)(6,86,24,104)(7,144,25,126)(8,62,26,44)(9,89,27,107)(10,111,28,129)(11,65,29,47)(12,92,30,74)(13,114,31,132)(14,68,32,50)(15,95,33,77)(16,117,34,135)(17,71,35,53)(18,98,36,80)(37,99,55,81)(39,140,57,122)(40,102,58,84)(42,143,60,125)(43,105,61,87)(45,110,63,128)(46,108,64,90)(48,113,66,131)(49,75,67,93)(51,116,69,134)(52,78,70,96)(54,119,72,137)(73,130,91,112)(76,133,94,115)(79,136,97,118)(82,139,100,121)(85,142,103,124)(88,109,106,127), (1,81,19,99)(2,139,20,121)(3,57,21,39)(4,84,22,102)(5,142,23,124)(6,60,24,42)(7,87,25,105)(8,109,26,127)(9,63,27,45)(10,90,28,108)(11,112,29,130)(12,66,30,48)(13,93,31,75)(14,115,32,133)(15,69,33,51)(16,96,34,78)(17,118,35,136)(18,72,36,54)(37,138,55,120)(38,100,56,82)(40,141,58,123)(41,103,59,85)(43,144,61,126)(44,106,62,88)(46,111,64,129)(47,73,65,91)(49,114,67,132)(50,76,68,94)(52,117,70,135)(53,79,71,97)(74,131,92,113)(77,134,95,116)(80,137,98,119)(83,140,101,122)(86,143,104,125)(89,110,107,128), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)>;

G:=Group( (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,109)(98,110)(99,111)(100,112)(101,113)(102,114)(103,115)(104,116)(105,117)(106,118)(107,119)(108,120), (1,138,19,120)(2,56,20,38)(3,83,21,101)(4,141,22,123)(5,59,23,41)(6,86,24,104)(7,144,25,126)(8,62,26,44)(9,89,27,107)(10,111,28,129)(11,65,29,47)(12,92,30,74)(13,114,31,132)(14,68,32,50)(15,95,33,77)(16,117,34,135)(17,71,35,53)(18,98,36,80)(37,99,55,81)(39,140,57,122)(40,102,58,84)(42,143,60,125)(43,105,61,87)(45,110,63,128)(46,108,64,90)(48,113,66,131)(49,75,67,93)(51,116,69,134)(52,78,70,96)(54,119,72,137)(73,130,91,112)(76,133,94,115)(79,136,97,118)(82,139,100,121)(85,142,103,124)(88,109,106,127), (1,81,19,99)(2,139,20,121)(3,57,21,39)(4,84,22,102)(5,142,23,124)(6,60,24,42)(7,87,25,105)(8,109,26,127)(9,63,27,45)(10,90,28,108)(11,112,29,130)(12,66,30,48)(13,93,31,75)(14,115,32,133)(15,69,33,51)(16,96,34,78)(17,118,35,136)(18,72,36,54)(37,138,55,120)(38,100,56,82)(40,141,58,123)(41,103,59,85)(43,144,61,126)(44,106,62,88)(46,111,64,129)(47,73,65,91)(49,114,67,132)(50,76,68,94)(52,117,70,135)(53,79,71,97)(74,131,92,113)(77,134,95,116)(80,137,98,119)(83,140,101,122)(86,143,104,125)(89,110,107,128), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144) );

G=PermutationGroup([[(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(97,109),(98,110),(99,111),(100,112),(101,113),(102,114),(103,115),(104,116),(105,117),(106,118),(107,119),(108,120)], [(1,138,19,120),(2,56,20,38),(3,83,21,101),(4,141,22,123),(5,59,23,41),(6,86,24,104),(7,144,25,126),(8,62,26,44),(9,89,27,107),(10,111,28,129),(11,65,29,47),(12,92,30,74),(13,114,31,132),(14,68,32,50),(15,95,33,77),(16,117,34,135),(17,71,35,53),(18,98,36,80),(37,99,55,81),(39,140,57,122),(40,102,58,84),(42,143,60,125),(43,105,61,87),(45,110,63,128),(46,108,64,90),(48,113,66,131),(49,75,67,93),(51,116,69,134),(52,78,70,96),(54,119,72,137),(73,130,91,112),(76,133,94,115),(79,136,97,118),(82,139,100,121),(85,142,103,124),(88,109,106,127)], [(1,81,19,99),(2,139,20,121),(3,57,21,39),(4,84,22,102),(5,142,23,124),(6,60,24,42),(7,87,25,105),(8,109,26,127),(9,63,27,45),(10,90,28,108),(11,112,29,130),(12,66,30,48),(13,93,31,75),(14,115,32,133),(15,69,33,51),(16,96,34,78),(17,118,35,136),(18,72,36,54),(37,138,55,120),(38,100,56,82),(40,141,58,123),(41,103,59,85),(43,144,61,126),(44,106,62,88),(46,111,64,129),(47,73,65,91),(49,114,67,132),(50,76,68,94),(52,117,70,135),(53,79,71,97),(74,131,92,113),(77,134,95,116),(80,137,98,119),(83,140,101,122),(86,143,104,125),(89,110,107,128)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)]])

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 6G 6H 6I 6J 9A ··· 9F 12A ··· 12H 12I 12J 12K 12L 18A ··· 18R 36A ··· 36X order 1 2 2 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 6 6 6 6 9 ··· 9 12 ··· 12 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 6 6 1 1 1 1 1 1 6 6 1 ··· 1 6 6 6 6 4 ··· 4 1 ··· 1 6 6 6 6 4 ··· 4 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 3 3 3 3 3 3 type + + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 C4.A4 Q8.C18 A4 C2×A4 C2×A4 C3.A4 C2×C3.A4 C2×C3.A4 kernel C2×Q8.C18 C2×Q8⋊C9 Q8.C18 C6×C4○D4 C6×Q8 C3×C4○D4 C2×C4○D4 C2×Q8 C4○D4 C6 C2 C2×C12 C12 C2×C6 C2×C4 C4 C22 # reps 1 1 2 2 2 4 6 6 12 12 24 1 2 1 2 4 2

Matrix representation of C2×Q8.C18 in GL3(𝔽37) generated by

 36 0 0 0 36 0 0 0 36
,
 1 0 0 0 1 36 0 2 36
,
 1 0 0 0 6 0 0 12 31
,
 26 0 0 0 25 7 0 23 14
G:=sub<GL(3,GF(37))| [36,0,0,0,36,0,0,0,36],[1,0,0,0,1,2,0,36,36],[1,0,0,0,6,12,0,0,31],[26,0,0,0,25,23,0,7,14] >;

C2×Q8.C18 in GAP, Magma, Sage, TeX

C_2\times Q_8.C_{18}
% in TeX

G:=Group("C2xQ8.C18");
// GroupNames label

G:=SmallGroup(288,347);
// by ID

G=gap.SmallGroup(288,347);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,79,648,172,1153,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^2=d^18=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b>;
// generators/relations

׿
×
𝔽


text:
### Patriots 2022 roster breakdown: Joshuah Bledsoe needs to beat the odds to make the team

by Bake Carlos -
Number of replies: 1

With the offseason workout program in the rear-view mirror, the New England Patriots are already fully on to 2022. The team currently has 87 players under contract, but only 53 of them will be able to survive roster cutdowns in early September and ultimately make the active team. Over the course of spring and summer, just like we have in years past, we will take a look at the men fighting for those spots to find out who has the best chances of helping the Patriots build on their 10-7 record. Today, the series continues with second-year defensive back Joshuah Bledsoe. Hard factsName: Joshuah BledsoePosition: Safety/CornerbackJersey number: 24Opening day age: 23Size: 5-foot-11, 200 poundsContract status: Under contract through 2024 ExperienceWhat is his experience? Bledsoe's experience at the NFL level is rather limited, with the 2021 sixth-round draft choice not taking the field even once during his rookie campaign with the Patriots. Accordingly, his pro level exposure is limited to behind-the-scenes study and a handful of practices both last year and during the current offseason Harvey Langi Jersey. Regardless of how his first year in the NFL went, however, he does have considerable experience on his football rsum from his time at the University of Missouri. During his four years with the Tigers, Bledsoe appeared in a combined 42 games the final 22 of them as one of the team's starting safeties. Along the way, he posted some solid numbers and put himself in a position to get drafted Tre Nixon Jersey. Not only did he lead Mizzou in passes defended in both 2019 and 2020, he also finished his college career with an interception, a sack, a pair of forced fumbles and 131 total tackles. He also scored a touchdown on a fumble recovery during his 2018 sophomore campaign. What did his 2021 season look like? The story of Bledsoe's rookie season in the NFL does not start with him getting selected 188th overall by the Patriots in the sixth round of the draft, but three months earlier in Mobile, Alabama. The draft hopeful, after all, was invited to showcase his talents at the Senior Bowl. Bledsoe participated in practice throughout the week and registered a tackle in the actual game, but apparently also hurt his wrist along the way an injury that would follow him throughout most of 2021. Bledsoe did not only miss his pro day workouts in mid-April after undergoing surgery on his wrist, he also was a limited participant during offseason practices after arriving in New England. As a result, he was sent to the non-football injury list ahead of training camp and would remain there until mid-December Mac Jones Jersey. While the Patriots did activate him ahead of their Week 15 game in Indianapolis, the rookie did not take the field and later suffered a calf injury that kept him out of Week 16's contest versus the Buffalo Bills as well. The calf issue eventually led to Bledsoe being sent to yet another reserve list: on December 30, just two weeks after his activation off NFI, he was sent to injured reserve. The young defender spent the remainder of the Patriots' season there and as a result ended his first year as a pro with no game action whatsoever. Bledsoe did participate in a handful of practices while on the active roster, and also was able to do considerable behind-the-scenes work, but he has yet to compete against another team. 2022 previewWhat is his projected role? While it remains to be seen how New England's coaches plan to use him in their secondary after what was effectively a full year on the sidelines, the expectation is that Bledsoe could fill a Devin McCourty-like role for the team. The youngster, after all, has the athletic profile to play as a deep safety but has also been moved around the formation quite a bit at Missouri. Just like the long-time Patriots captain, he could therefore be trained as a jack-of-all-trades within New England's defense. Does he have positional versatility? There are no NFL games to base this analysis on, but Bledsoe did show some versatility during his time in college https://www.thepatriotsteamstore.com/85-Tre_Nixon_Jersey. He was listed as a safety at Mizzou, but his role was much more flexible than that tag alone would indicate. After becoming a starter in 2019, for example, he saw the majority of his opportunities as a slot cornerback but also aligned as a box safety , free safety , on the defensive edge and split out wide. What is his special teams value? The Patriots like to use their defensive backs in the kicking game, and Bledsoe will likely not be an exception. The second-year man is a realistic candidate to see action on the kickoff and punt return units, the two coverage squads, and occasionally on the field goal/extra point blocking teams as well. Training camp and preseason will paint a clearer picture, but Bledsoe seeing a prominent number of his snaps in the game's third phase would not be a surprise. What is his salary cap situation? Entering the second year of his rookie contract, Bledsoe carries a $755,087 cap hit that is currently not qualifying for Top-51 status https://www.thepatriotsteamstore.com/47-Justin_Bethel_Jersey. That means that only his fully-guaranteed signing bonus proration worth$50,087 is on New England's books at the moment, with his \$705,000 salary only being added in case he survives roster cutdowns and makes the 53-man team. How safe is his roster spot? The Patriots' secondary is in a state of personnel turnover after losing Pro Bowl cornerback J. C. Jackson in free agency, and adding the likes of Jabrill Peppers, Marcus Jones and Jack Jones later during the offseason. Bledsoe falls into the Peppers category in terms of his potential versatility, and he will have to take advantage of it if he wants to make the roster as a cornerback/safety hybrid with special teams potential. After all, New England has plenty of bodies competing for a finite number of spots in the defensive backfield https://www.thepatriotsteamstore.com/13-Trent_Brown_Jersey. One-sentence projection: Bledsoe will have a hard time making the 53-man squad with the numbers stacked against him in the secondary, but he is a realistic candidate to end up on the practice squad.


text:
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Get Full Access to University Physics, Volume 3 - 17 Edition - Chapter 4 - Problem 76
Get Full Access to University Physics, Volume 3 - 17 Edition - Chapter 4 - Problem 76

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ISBN: 9781938168185 2032

## Solution for problem 76 Chapter 4

University Physics, Volume 3 | 17th Edition

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Problem 76

X-Ray Diffraction

On a certain crystal, a first-order X-ray diffraction maximum is observed at an angle of $$27.1^{\circ}$$ relative to its surface, using an X-ray source of unknown wavelength. Additionally, when illuminated with a different, this time of known wavelength 0.137 nm, a second-order maximum is detected at $$37.3^{\circ}$$. Determine (a) the spacing between the reflecting planes, and (b) the unknown wavelength.

Text Transcription:

27.1 degrees

37.3 degrees

Step-by-Step Solution:
Step 1 of 3

Energy Stored in Capacitors U = 1 qV 2 q = CV 1 2 U = CV or 2 1 q2 U = 2 C 37 Energy Density in Capacitors (1)  We define the , u, as the electric potential energy per unit volume  Taking the ideal case of a parallel plate capacitor that has no fringe field, the volume between the plates is the area of each plate times the distance between the plates, Ad  Inserting our formula for the capacitance of a parallel plate capacitor we find 38 Energy Density in Capacitors (2)  Recognizing that V/d is the magnitude of the electric field, E, we obtain an expression for the electric potential energy density for parallel plate capacitor  This result, which we derived for the parallel plate capacitor, is in fact completely general.  This equation holds for all electric fields produced in any way •The formula gives the quantity of electric field energy per unit volume. 39 Example: Isolated Conducting Sphere (1)  An isolated conducting sphere whose radius R is 6.85 cm has a charge of q=1.25 nC. Question 1: How much potential energy is stored in the electric field of the charged conductor Answer: Key Idea: An isolated sphere has a capacitance ofC=4πε R (see prev

Step 2 of 3

Step 3 of 3

##### ISBN: 9781938168185

This full solution covers the following key subjects: . This expansive textbook survival guide covers 11 chapters, and 1200 solutions. University Physics, Volume 3 was written by Aimee Notetaker and is associated to the ISBN: 9781938168185. This textbook survival guide was created for the textbook: University Physics, Volume 3 , edition: 17. The full step-by-step solution to problem: 76 from chapter: 4 was answered by Aimee Notetaker, our top Physics solution expert on 03/18/22, 10:28AM. The answer to “?X-Ray DiffractionOn a certain crystal, a first-order X-ray diffraction maximum is observed at an angle of $$27.1^{\circ}$$ relative to its surface, using an X-ray source of unknown wavelength. Additionally, when illuminated with a different, this time of known wavelength 0.137 nm, a second-order maximum is detected at $$37.3^{\circ}$$. Determine (a) the spacing between the reflecting planes, and (b) the unknown wavelength.Text Transcription:27.1 degrees37.3 degrees” is broken down into a number of easy to follow steps, and 64 words. Since the solution to 76 from 4 chapter was answered, more than 202 students have viewed the full step-by-step answer.

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text:
# Polynomials¶

In this section we illustrate how to create and use polynomials in Sage.

## Univariate Polynomials¶

There are three ways to create polynomial rings.

sage: R = PolynomialRing(QQ, 't')
sage: R
Univariate Polynomial Ring in t over Rational Field

This creates a polynomial ring and tells Sage to use (the string) ‘t’ as the indeterminate when printing to the screen. However, this does not define the symbol t for use in Sage, so you cannot use it to enter a polynomial (such as $$t^2+1$$) belonging to R.

An alternate way is

sage: S = QQ['t']
sage: S == R
True

This has the same issue regarding t.

A third very convenient way is

sage: R.<t> = PolynomialRing(QQ)

or

sage: R.<t> = QQ['t']

or even

sage: R.<t> = QQ[]

This has the additional side effect that it defines the variable t to be the indeterminate of the polynomial ring, so you can easily construct elements of R, as follows. (Note that the third way is very similar to the constructor notation in Magma, and just as in Magma it can be used for a wide range of objects.)

sage: poly = (t+1) * (t+2); poly
t^2 + 3*t + 2
sage: poly in R
True

Whatever method you use to define a polynomial ring, you can recover the indeterminate as the $$0^{th}$$ generator:

sage: R = PolynomialRing(QQ, 't')
sage: t = R.0
sage: t in R
True

Note that a similar construction works with the complex numbers: the complex numbers can be viewed as being generated over the real numbers by the symbol i; thus we have the following:

sage: CC
Complex Field with 53 bits of precision
sage: CC.0  # 0th generator of CC
1.00000000000000*I

For polynomial rings, you can obtain both the ring and its generator, or just the generator, during ring creation as follows:

sage: R, t = QQ['t'].objgen()
sage: t    = QQ['t'].gen()
sage: R, t = objgen(QQ['t'])
sage: t    = gen(QQ['t'])

Finally we do some arithmetic in $$\QQ[t]$$.

sage: R, t = QQ['t'].objgen()
sage: f = 2*t^7 + 3*t^2 - 15/19
sage: f^2
4*t^14 + 12*t^9 - 60/19*t^7 + 9*t^4 - 90/19*t^2 + 225/361
sage: cyclo = R.cyclotomic_polynomial(7); cyclo
t^6 + t^5 + t^4 + t^3 + t^2 + t + 1
sage: g = 7 * cyclo * t^5 * (t^5 + 10*t + 2)
sage: g
7*t^16 + 7*t^15 + 7*t^14 + 7*t^13 + 77*t^12 + 91*t^11 + 91*t^10 + 84*t^9
+ 84*t^8 + 84*t^7 + 84*t^6 + 14*t^5
sage: F = factor(g); F
(7) * t^5 * (t^5 + 10*t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)
sage: F.unit()
7
sage: list(F)
[(t, 5), (t^5 + 10*t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]

Notice that the factorization correctly takes into account and records the unit part.

If you were to use, e.g., the R.cyclotomic_polynomial function a lot for some research project, in addition to citing Sage you should make an attempt to find out what component of Sage is being used to actually compute the cyclotomic polynomial and cite that as well. In this case, if you type R.cyclotomic_polynomial?? to see the source code, you’ll quickly see a line f = pari.polcyclo(n) which means that PARI is being used for computation of the cyclotomic polynomial. Cite PARI in your work as well.

Dividing two polynomials constructs an element of the fraction field (which Sage creates automatically).

sage: x = QQ['x'].0
sage: f = x^3 + 1; g = x^2 - 17
sage: h = f/g;  h
(x^3 + 1)/(x^2 - 17)
sage: h.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field

Using Laurent series, one can compute series expansions in the fraction field of QQ[x]:

sage: R.<x> = LaurentSeriesRing(QQ); R
Laurent Series Ring in x over Rational Field
sage: 1/(1-x) + O(x^10)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)

If we name the variable differently, we obtain a different univariate polynomial ring.

sage: R.<x> = PolynomialRing(QQ)
sage: S.<y> = PolynomialRing(QQ)
sage: x == y
False
sage: R == S
False
sage: R(y)
x
sage: R(y^2 - 17)
x^2 - 17

The ring is determined by the variable. Note that making another ring with variable called x does not return a different ring.

sage: R = PolynomialRing(QQ, "x")
sage: T = PolynomialRing(QQ, "x")
sage: R == T
True
sage: R is T
True
sage: R.0 == T.0
True

Sage also has support for power series and Laurent series rings over any base ring. In the following example, we create an element of $$\GF{7}[[T]]$$ and divide to create an element of $$\GF{7}((T))$$.

sage: R.<T> = PowerSeriesRing(GF(7)); R
Power Series Ring in T over Finite Field of size 7
sage: f = T  + 3*T^2 + T^3 + O(T^4)
sage: f^3
T^3 + 2*T^4 + 2*T^5 + O(T^6)
sage: 1/f
T^-1 + 4 + T + O(T^2)
sage: parent(1/f)
Laurent Series Ring in T over Finite Field of size 7

You can also create power series rings using a double-brackets shorthand:

sage: GF(7)[['T']]
Power Series Ring in T over Finite Field of size 7

## Multivariate Polynomials¶

To work with polynomials of several variables, we declare the polynomial ring and variables first.

sage: R = PolynomialRing(GF(5),3,"z") # here, 3 = number of variables
sage: R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5

Just as for defining univariate polynomial rings, there are alternative ways:

sage: GF(5)['z0, z1, z2']
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
sage: R.<z0,z1,z2> = GF(5)[]; R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5

Also, if you want the variable names to be single letters then you can use the following shorthand:

sage: PolynomialRing(GF(5), 3, 'xyz')
Multivariate Polynomial Ring in x, y, z over Finite Field of size 5

Next let’s do some arithmetic.

sage: z = GF(5)['z0, z1, z2'].gens()
sage: z
(z0, z1, z2)
sage: (z[0]+z[1]+z[2])^2
z0^2 + 2*z0*z1 + z1^2 + 2*z0*z2 + 2*z1*z2 + z2^2

You can also use more mathematical notation to construct a polynomial ring.

sage: R = GF(5)['x,y,z']
sage: x,y,z = R.gens()
sage: QQ['x']
Univariate Polynomial Ring in x over Rational Field
sage: QQ['x,y'].gens()
(x, y)
sage: QQ['x'].objgens()
(Univariate Polynomial Ring in x over Rational Field, (x,))

Multivariate polynomials are implemented in Sage using Python dictionaries and the “distributive representation” of a polynomial. Sage makes some use of Singular [Si], e.g., for computation of gcd’s and Gröbner basis of ideals.

sage: R, (x, y) = PolynomialRing(RationalField(), 2, 'xy').objgens()
sage: f = (x^3 + 2*y^2*x)^2
sage: g = x^2*y^2
sage: f.gcd(g)
x^2

Next we create the ideal $$(f,g)$$ generated by $$f$$ and $$g$$, by simply multiplying (f,g) by R (we could also write ideal([f,g]) or ideal(f,g)).

sage: I = (f, g)*R; I
Ideal (x^6 + 4*x^4*y^2 + 4*x^2*y^4, x^2*y^2) of Multivariate Polynomial
Ring in x, y over Rational Field
sage: B = I.groebner_basis(); B
[x^6, x^2*y^2]
sage: x^2 in I
False

Incidentally, the Gröbner basis above is not a list but an immutable sequence. This means that it has a universe, parent, and cannot be changed (which is good because changing the basis would break other routines that use the Gröbner basis).

sage: B.parent()
Category of sequences in Multivariate Polynomial Ring in x, y over Rational
Field
sage: B.universe()
Multivariate Polynomial Ring in x, y over Rational Field
sage: B[1] = x
Traceback (most recent call last):
...

Some (read: not as much as we would like) commutative algebra is available in Sage, implemented via Singular. For example, we can compute the primary decomposition and associated primes of $$I$$:

sage: I.primary_decomposition()
[Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field]
sage: I.associated_primes()
[Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]


text:
# C_V or C_p for this question

1. Oct 6, 2008

### p3t3r1

1. The problem statement, all variables and given/known data

A quantity of 35.0 of water at 25.0 C (called A) is mixed with 160.0 g of water at 86.0 C (called B). a) calcuate the final temperature of the system, assuming that the mixing is carried out adiabatically and b) calculate the entropy change of A, b, and the entire system

2. Relevant equations
delta S = n x C_V or C_p ln (Tf/Ti)

3. The attempt at a solution

The first part is very easy. Basically, before anything happens, the two gases need to be in thermal equilibrium.

I just did q_A = q_b and figured the temperature at thermal equilbirum, which is 75 C or 348 K.

part b was kind of confusing. I had Gibb's paradox in mind.

I guess I was thinking along the lines of..

if you take 1 L of water and a 2L bucket of water and mix them together, you are going to get three L of water, which is same as before. Thus, no volume change occurs and essentially the only entropy change you are going to get is due to heating.

Which is delta S = n x C_V or C_p ln (Tf/Ti)

I was debating of using C_v or C_p for this question. C_p looked really tempting but I didn't know what pressure it was occuring.. (it didn't say the mixing took place at atmospheric pressure ) so I supposed that it wouldn't be sort of right to using C_p.. (I guess what I missed here is that the pressure would equlibriate between A and B when you mix them and you would get a constant pressure.

Finally I settled for C_v, because I recalled my previous argument that the volume doesn't change so the C_v applies.

However, what I missed here is that the two waters are not at the same temperature. Thus, when the two waters come into thermal equilibrium, the volume of the colder water is going to expand a bit while the volume of the hotter water is going to shrink a bit. Thus C_v doesn't really apply here.

The correct approach would have been using C_p.

but then I ask, if volume do change.. what about adibatic expansion?

$$T2/T1 = (V1/V2) ^ (\gamma -1)$$

If there is a volume change say in A.. wouldn't that mean that the temperature of A is also going to change (drop a bit) .. not only that.. entropy is also given as delta S = nR ln (Vf/Vi)... so wouldn't this contribute to the entropy change of the A too?

but then I was like... if the temperature of A is going to drop a bit.. then temperature of B is going to rise a bit because its volume is going to drop a little since it is in contact with the colder water.. so the two temperature changes would even out and you still would be back at 75 C.

now to address delta S = nR ln (Vf/Vi) for B.. that would be like the opposite of the delta S = nR ln (Vf/Vi) for A so they would really cancel out?

So I guess if instead of using C_p to figure out the entropy change for A..
I would use C_v but I would just have to take the delta S = nR ln (Vf/Vi) into consideration...

so.. that means

nC_pln (Tf/Ti) = n C_vln (Tf/Ti) + nR ln (Vf/Vi)

now.. when you figure out delta S for A and delta S for B, you see that they are just the opposite of each other.. I had Gibb's paradox in mind.. in which I was thinking that since they are all the same material, entropy should be 0 but that wouldn't really be true in this cause due to the temperature difference between the two waters..?

So I guess my question really is.. is my way of thinking correct?

Thanks for the patience!


text:
# Rotolux

## Physics, Python programming, Miscellaneous geekness

### Hailstone pics

Just for reference, I need to link to the recent news reports about the big storm we had last week, which used my photo.

### Pasting syntax-highlighted code

I just found this guide on pasting syntax highlighted code (escaped with this form) into Blogger blogs. It's probably not all that useful, but anyway, let's give it a go with a function I use to read a binary file output by xmds into Python:

def read_file(indat_fname, xsil_fname):   # get datafile name and precisions   if got_lxml:       with open(xsil_fname) as xml:           xsil = objectify.parse(xml).getroot()       ulong_len = xsil.XSIL.Array[1].Stream.Metalink.get("UnsignedLong")       precision = xsil.XSIL.Array[1].Stream.Metalink.get("precision")   else:       xml = open(xsil_fname).read()       ulong_len = str(xml2obj.xml2obj(xml).XSIL[0].Array[1].Stream.Metalink.UnsignedLong)       precision = str(xml2obj.xml2obj(xml).XSIL[0].Array[1].Stream.Metalink.precision)   ulong_type = {'uint32':'<u4','uint64':'<u8'}[ulong_len]   float_type = {'single':'<f4','double':'<f8'}[precision]   with open(indat_fname,'rb') as f:       xlen = fromfile(f, ulong_type, 1)[0]       xs   = fromfile(f, float_type, xlen)       ylen = fromfile(f, ulong_type, 1)[0]       ys   = fromfile(f, float_type, ylen)       zlen = fromfile(f, ulong_type, 1)[0]       zs   = fromfile(f, float_type, zlen)       rlen1  = fromfile(f, ulong_type, 1)[0]       reals1 = fromfile(f, float_type, rlen1)       ilen1  = fromfile(f, ulong_type, 1)[0]       imags1 = fromfile(f, float_type, ilen1)       psi1 = (reals1+1j*imags1).astype(np.complex64).reshape(xlen,ylen,zlen)       del reals1, imags1       rlen2  = fromfile(f, ulong_type, 1)[0]       reals2 = fromfile(f, float_type, rlen2)       ilen2  = fromfile(f, ulong_type, 1)[0]       imags2 = fromfile(f, float_type, ilen2)       psi2 = (reals2+1j*imags2).astype(np.complex64).reshape(xlen,ylen,zlen)       del reals2, imags2   return psi1, psi2

which sort of works, except for the wrapping - maybe I did something wrong with the template. I'll have another go one day (maybe)

### Origami documentary

I just watched an incredible documentary called "Between the Folds" on the art and science of origami. There was something about it that fired my "inspiration" centre. I need to put this aside for now, what with thesis writing, but I must it again later and look into the subjects of that film.

### Changing the eSlick text font

On the Foxit eSlick Forum are various somewhat unclear instructions for changing the text font. To be explicit, the steps are:
1. Create a folder named "Fonts" in the root folder of the eSlick's internal memory.
2. Find a Truetype font, i.e. a *.ttf font - I used the Gentium Book Basic font.
3. Rename "GenBkBasR.ttf" (or whatever) to "fxfont.ttc" and copy it to the "Fonts" folder.
4. Reset the eSlick by depressing the recessed Reset switch on its back whilst it is powered on.

### Chickens and eggs of physics

Whilst listening today to the ABC science show, I mused on the following (poorly thought-out) inequities of science communication.

Astronomers and paleontoligists have an advantage over the rest of us trying to communicate our science in that they can assume a greater level of knowledge amongst the general public. Thus they get to explain more of what's new to the general public within a given amount of time. Is this why Catalyst seem to have an overrepresentation of stories on astronomy and dinosours? Or is it because two of their prominent reporters have backgrounds in these areas? Probably the latter. However, experience of dealing with Monash media and marketing people, would tend to support me, so I'll stand by my suspicion that there is a bias.

Astronomers like to talk about how big things are, how old things are, how far away things are ... in short how far from everyday experience the things they study are. I also do this when talking to people about my research, because this is what "bends their minds". Luckily I at least work in an are where I can explain how small things are, which gives me a "hook". However, I often think that this distorts the intrinsic value of all the other things that I (and other scientists) encounter in research - many of the beautiful parts of my research are simply too hard to explain to non-specialists without also relating a significant number of background concepts.

By the way, Paleontologists like to talk about how big dinosaurs were, how long ago they lived, how easily they could crush a human skull etc.

### Some interesting articles

The latest Edge has an interesting article. I like
this one arguing against Gardner's theory of Multiple Intelligences in favour of a single useful measure "g", which is presumably IQ. The argument is that the curriculum in schools is pandering to Gardner's labels, which aren't orthogonal. I tend to agree that this is a problem, but I wonder whether a roughly-orthogonal set could be found using proper research and data mining or Principle Component Analysis techniques.

### Guide to reducing borders around text in a pdf

OK, here’s a howto on using LaTeX with the pdfpages package to strip whitespace from around the text of an existing pdf document. Say you have a pdf with a large blank marginspace around the text. The following LaTeX file will make a new version. Unfortunately, it will lose any bookmarks, links etc. I usually run it with pages={-8} to process just a few pages, tweaking the offset and scale values until it’s stripping out the bits I want, then change back to pages={-} or maybe a different page range to strip out any guff you don’t want. It’ll also let you glue together multiple pdfs into one file.

\documentclass[a5paper,oneside]{book}\usepackage{pdfpages}\begin{document}\pagestyle{empty}\includepdf[pages={-},offset=0mm -5mm,scale=1.3,pagecommand={}]{”pdf_file_name”}\end{document}


text:
RICAM Events

PhD defence

OOC: Stabilization to trajectories and approximate controllability for the equations of fluid mechanics
Speaker: Duy Phan-Duc
Date: November 23, 2016 13:30
Location: SP2 416-2

In the work, we focus on three problems: Gevrey regularity, approximate controllability and stabilization to trajectories for the equations of fluid mechanics. The Gevrey regularity problem is addressed for Navier--Stokes equations under Lions boundary conditions in case the fluid is assumed to be viscous, incompressible and homogeneous. Sufficient conditions are given to guarantee that the solution of the systems is well-defined in a Gevrey regularity space. Some cases of boundaryless manifolds such as: the 2D Sphere; the 2D and 3D Torus which were mentioned in earlier studies are revisited. Four new results are obtained under Lions boundary conditions namely, the 2D Rectangle, the 2D Cylinder and the 2D Hemisphere; and the 3D Rectangle. The approximate controllability problem is considered for the Navier--Stokes equations and the 1D Burger equations. Following the Agrachev--Sarychev method, the approximate controllability by means of only few actuators is proven for Navier--Stokes system in a 2D infinite channel and in a 3D Rectangle.The actuators are the eigenfunctions of the Stokes operator. By few'' actuators, we mean that the number of them is independent of the viscosity coefficient. For the case of the 1D Burgers equations, a result is obtained for the case where the actuators are constrained in a small subset The exponential stabilization to trajectories problem is addressed for a general class of semilinear parabolic systems. The controls take values in a finite-dimensional space. Both internal and boundary actuators are supported in small region. We prove that under suitable conditions on the family of actuators, they allow us to stabilize the system. Moreover, some estimates on the number of actuators that we need to stabilize the system are established. Using the Dynamical Programming Principle, the stabilizing controls can be taken in feedback form and be computed by solving a suitable differential Riccati equation. Some simulations are presented by using Finite Element Method (FEM) and performed with MATLAB.

To stay informed about new RICAM events subscribe to the RICAM talks mailinglist.

<< back


text:
Tissues

The store got three kinds of tissues - 132 children, 156 women and 204 men. Tissues each species were packed into boxes after the number of pieces the same for all three types (and greatest). Determine the number, if you know that every box has more than 6 tissues.

Result

x =  12

Solution:

$132 = 2^2 \cdot 3 \cdot 11 \\ 156 = 2^2 \cdot 3 \cdot 13 \\ 204 = 2^2 \cdot 3 \cdot 17 \\ \text{GCD}(132, 156, 204) = 2^2 \cdot 3 = 12\\ \ \\ x = 12$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
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text:
Questions on surface area of cone

In this post we will try to solve questions related to surface area cone.

Lateral surface area of cone

The area of the curved part of the cone is called lateral surface area.

Lateral surface area = 𝜋.r.l

Where;
l = slant height of cone

Total surface area of cone

The total surface area includes the area of the base and curved part of the cone.

TSA = 𝜋. r ( l + r)

You need to memorize both the formulas as they would help to solve below questions.

Surface area of cone – Solved examples

Question 01
Find the curved surface area of cone if radius is 7 cm and slant height is 10 cm.

Solution
Slant height (l) = 10 cm

Area of curved surface = 𝜋. r. l

Putting the values in formula;

Area of curved surface = \mathtt{\frac{22}{7} \times 7\times 10\ =220\ cm^{2}}

Hence, 220 sq. cm is the area of curved surface.

Question 02
Consider the cone with radius 3 cm and height 4 cm. Find the curved surface area of the cone.

Solution
Height (h) = 4 cm

To find the curved surface area, we have to first find the slant height.

Applying Pythagoras theorem in triangle OAB.

\mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{l^{2} =4^{2} +3^{2}}\\\ \\ \mathtt{l^{2} =\ 16\ +\ 9=\ 25}\\\ \\ \mathtt{l\ =\ 5\ cm\ }

So, the slant height (l) is 5 cm.

Let’s now calculate the lateral surface area.

Area of curved surface = 𝜋. r. l

Putting the values;

Area = \mathtt{\frac{22}{7} \times 3\times 5\ =47.14\ cm^{2}}

Hence, 47.14 is the curved surface of give cone.

Question 03
The height of the cone is given as 8 cm. If the slant height is 10 cm then find the area of base of the cone.

Solution
Heigh (h) = 8 cm
Slant height (l) = 10 cm

Let’s first calculate the radius of base of cone.

Applying Pythagoras theorem on triangle OAB.

\mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{10^{2} =8^{2} +r^{2}}\\\ \\ \mathtt{r^{2} =100-64=36\ }\\\ \\ \mathtt{r\ =\ 6\ cm}

So, the radius of circular base is 6 cm.

We know that cone is a circular base and its area is calculated by following formula;

Area = \mathtt{\pi .r^{2}}

Area = \mathtt{\frac{22}{7} \times 36\ =\ 113.14}

Hence, 113.14 sq. cm is the area of base of cone.

Question 04
Find the total surface area of cone with radius 5 cm and height 12 cm.

Solution
Height (h) = 12 cm

We have to first calculate the slant height.

Applying Pythagoras theorem in triangle AOB.

\mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{l^{2} =12^{2} +5^{2}}\\\ \\ \mathtt{l^{2} =144+25=169}\\\ \\ \mathtt{l\ =\ 13\ cm\ }

So the slant height is 13 cm.

The formula for total surface area is given as;

TSA = 𝜋. r ( l + r)

\mathtt{T.S.A=\ \frac{22}{7} \times 5\times ( 13+5) \ }\\\ \\ \mathtt{T.S.A\ =\ \frac{22}{7} \times 5\times 18}\\\ \\ \mathtt{T.S.A\ =\ 282.9\ cm^{2}}

Hence, 282.9 sq cm is the total surface area of the cone.

Question 05
Consider a cone with diameter 70 cm and lateral surface area 4070 sq cm. Find the slant height of the cone.

Solution
Diameter = 70 cm
Radius(r) = 70 / 2 = 35 cm

Lateral surface area = 4070 sq cm

The formula for lateral surface is given as;
Lateral surface area = 𝜋. r . l

4070 = (22 / 7) . 35 . l

l = \mathtt{\frac{4070\times 7}{22\times 35} =37\ cm}

Hence, 37 cm is the slant height of the given cone.

Question 06
A tent is in the form of cone of radius 3 meter and height 4 meter. If the cost of canvas is 7$per square meter then calculate the cost of making the tent. Solution In conical tent, the canvas is used to make the curved surface. So, to calculate the cost, we have to find the curved surface area. Let’s calculate the slant height first; Applying Pythagoras theorem in triangle AOB. \mathtt{AB^{2} =OA^{2} +OB^{2}}\\\ \\ \mathtt{l^{2} =4^{2} +3^{2}}\\\ \\ \mathtt{l^{2} =16+9=25}\\\ \\ \mathtt{l\ =\ 5\ m\ } So the slant height of the tent is 5 meter. Calculating the curved surface area of tent. Curved surface area = 𝜋. r. l Curved surface area = \mathtt{\frac{22}{7} \times 3\times 5\ =\frac{330}{7}} Hence, 330/7 sq meter is the curved surface area. Calculating the cost of canvas 1 sq meter cost ⟹ 7$

330/7 sq meter cost ⟹ 330/7 x 7 = 330 $Hence, the total cost of making tent is 330$


text:
# Try This By Hand

Algebra Level 3

How many real roots does the polynomial $P(x)=x^4+2x^2-x+1$ have?

×


text:
1. Math
2. Calculus
3. find the volume of the solid under the surface z...

# Question: find the volume of the solid under the surface z...

###### Question details

Find the volume of the solid under the surface z = 3x + 9y2 and above the region bounded by x = y2 and x = y3.