Discusses the relationship between gender and clothing, first by considering Mary Wollstonecraft’s account of dress, then by considering how we might relate Simone de Beauvoir’s account of objectification to clothing.

5.1 Introduction

The stability analysis of equilibrium points of an autonomous first-order system

is an important topic in the qualitative theory of ordinary differential equations (ODE). Here x = x(t) ∈ ℝn is a vector valued unknown function of the independent variable t ∈ I, an interval in ℝ, and f: ℝn → ℝn is a given vector valued function, which is assumed to be a C1 or more smooth function. This assumption ensures the uniqueness (local or global) of a solution of the system (5.1.1) with a prescribed initial value at an initial time. The positive integer n is referred to as the dimension of the system.

The system (5.1.1) is called an autonomous system because the right-side function f does not depend on t explicitly. When f depends on t explicitly as well, the system is referred to as non-autonomous. For example, the equation x′ = x + t (1D or one-dimensional equation) is non-autonomous.

In some situations, we do assume more smoothness on f, so that global existence of a solution is guaranteed; this means existence for all t. Even when a solution does not exist for all t, we can still do the phase space analysis by considering the maximum interval of existence of the solution in question. However, uniqueness plays a crucial role. In what follows, we introduce many concepts, definitions and list many results that are useful in solving the exercises. For proofs and other details, we refer to Ref.[46] or any other book with similar contents.

On July 3, 2015, four years after the MoRD started data collection for the SEC survey in the state of Tripura, it co-announced the release of provisional data for rural India. Finance minister Arun Jaitley and minister of rural development Chaudhary Birendra Singh—both of the BJP-led government that had come to power in 2014—published a press statement that included a link to SEC survey summary data on the MoRD website. The MoRD had analyzed and made public the data that it required for BPL identification, and envisioned numerous additional possibilities for using the collected data. The press release stated that the MoRD planned “to use the [SEC survey] data in all its programmes,” specifically citing “Housing for all, Education and Skills thrust, MGNREGA [Mahatma Gandhi National Rural Employment Guarantee Act], National Food Security Act, interventions for differently able[d], interventions for women-led households, and targeting of households/ individual entitlements on evidence of deprivation, etc.” It also pointed to other intended uses of the SEC survey data by policy planners across levels of government and stated that combining the SEC survey data with the NPR would allow coordination across programs “to simultaneously address the multi-dimensionality of poverty by addressing the deprivation of households in education, skills, housing, employment, health, nutrition, water, sanitation, social and gender mobilization and entitlement” and tracking the progress of households over time. The concluding paragraph of the press release argued that the “[SEC survey] truly makes evidence based targeted household interventions for poverty reduction possible.” The two summary tables in the press release included details on the percentage of households automatically included on the BPL list based on five possible parameters for automatic inclusion, one of which was “whether any member of the household was from a primitive tribal group” (a question from the household section of the SEC survey questionnaire). This data came from a question that had been field-tested in the 2010 pilot survey—so it was not an addition from when the caste-wise enumeration was combined with the BPL survey. The tables also included the percentage of SC and ST households, which previous BPL surveys had documented.

Examines an apparent tension between clothing’s role as a social artifact and its rationality. Examines how we might think about disagreement when it comes to the fittingness or appropriateness of clothing. Considers how the social role of clothing plays a part in our judgments of individuals belonging to certain groups, and how this might contribute to epistemic injustice.

8.1 Introduction

We briefly recall some of the definitions, formulas and results from the book Ref.[45] which are relevant for solving the exercises of this chapter. A reader can refer the above-cited book or any other book with similar content for more detailed proofs and discussion. The most general form of a second-order linear partial differential equation (PDE) in n variables is given by

where x ∈ Ω, an open set in ℝn, aij = aji. The operator L is said to be uniformly elliptic if there exists an α > 0 such that for all x ∈ Ω and ξ ∈ ℝn. Recall that is the characteristic form, also called the principal symbol associated with the operator L. An important elliptic operator is the Laplace operator. This operator has many interesting properties − mean value property (MVP) and minimum and maximum principles. There is also a notion of a fundamental solution. This is not specific to the Laplace operator Δ, but every constant coefficient differential operator possesses a fundamental solution. A restricted definition is the following. A locally integrable function E is called a fundamental solution of L if for all smooth functions ψ with compact support, where L′ is the adjoint operator of L. Symbolically, this is written as LE = δ, the Dirac delta function. Since the operator Δ is self-adjoint, we have. Interestingly, the fundamental solution is not a solution of the Laplace equation ΔE = 0 in a strict sense but very useful in the construction/representation of solutions to the Laplace and Poisson equations. Note that ΔE = 0 in ℝn\﹛0﹜, and hence E indeed has a singularity at the origin. A fundamental solution is not unique.

6.1 Method of Characteristics

A general first-order partial differential equation (PDE) in n independent variables x = (x1, … , xn) can be written as F (x, u, p) = 0. Here, u = u(x) is the unknown function to be determined and is its gradient. The gradient of a function u = u(x) is also denoted by Du = Dxu = ∇u = ∇xu, to emphasize the variables with respect to which the derivatives are taken; and, F is a given real valued Ck (k ⩾ 2) function defined on Ω × ℝ × ℝn, where Ω is a domain in ℝn. In the two-dimensional case, we also use the notation (x, y) to denote the independent variables and for the first-order derivatives. Sometimes, the variable u is replaced z, as more of a convention. We consider a Cauchy problem or an initial value problem (IVP) associated with the given equation F = 0:

where S is a smooth (n − 1) dimensional surface sitting inside Ω. If S has the parametric representation S = ﹛ℎ(s) = (ℎ1(s), … , ℎn(s)) : s ∈ V ﹜, where V is a region in ℝn−1 and ℎ1, … , ℎn are smooth real valued functions defined on V , then the initial condition is written as u(ℎ(s)) = u0(s) for s ∈ V , where u0 is a given function defined on V. In general, the function F is also required to satisfy a compatibility condition on S.

We attempt to analyse the IVP (6.1.1) through the method of characteristics. The characteristic curves x = x(t) associated with the equation F = 0 are the solutions of the first-order system of ordinary differential equation In general, since F also depends on u and p, this system alone is insufficient to solve for x(t).

11.1 Introduction

The Cauchy problem or initial value problem (IVP) for the homogeneous wave equation in the free space ℝn is given by

Here, n ⩾ 2 is an integer, the (spatial) dimension; c > 0 is a constant, the speed of propagation and u0 and u1 are given smooth functions, the initial values.

Spherical Mean Function: Given a C2 function ℎ defined on ℝn, define its spherical mean function, denoted by Mℎ, by

for x ∈ ℝn and r > 0. The integration is over the sphere of radius r, centred at x and is the surface measure of this sphere with denoting the surface measure of the unit sphere in ℝn; Γ is the Euler gamma function. By a change of variable, equation (11.1.3) can be written as

The form of equation (11.1.4) enables us to define Mℎ for all real r, and it is readily seen that Mℎ(x, −r) = Mℎ(x, r), that is, Mℎ is an even function of r. This property is used repeatedly in the computations below.

A computation using the divergence theorem yields the Darboux equation:

The notation Δx in the above expressions means the Laplacian taken with respect to the x variables. Note that in the above equation, x is a parameter, and the equation (11.1.5) is a second-order ordinary differential equation (ODE) in the variable r.

Using the Darboux equation and some manipulation gives us the solution of IVP (11.1.1) and (11.1.2), for n = 3:

The representation (11.1.6) is known as the Kirchhoff's formula. By carrying out the t differentiation, we can also write the Kirchoff's formula as follows:

Thus, the Kirchoff's formula (11.1.6) is rewritten as

The above formula brings out the essential features of the solution in the case n = 3. Thus, any C2 solution of the Cauchy problems (11.1.1) and (11.1.2) is given by equation (11.1.6) and hence unique.

For n ⩾ 3 odd, we now write down a formula for the solution of the homogeneous wave equation, similar to the Kirchhoff's formula for n = 3; the formula for the solution for n even is obtained by the method of descent from dimension n + 1, which is discussed in the next section.

Examines the relationship between clothing and beauty, especially given the link between clothing and fashion and the importance of function. Considers under which circumstances clothing might be thought of as art.

M. Vijayanunni, who was now retired from the Indian Administrative Service, continued to advocate for a full caste-wise enumeration in the census in the lead-up to Census 2011. As the former head of the ORGI, or the Indian census bureau, Vijayanunni spoke from a unique position when he argued that the ORGI was the only agency with the technical experience to undertake the task. Vijayanunni's certainty stemmed from the technology available for cleaning, analyzing, and tabulating data and his detailed historical knowledge of censuses. If the colonial state could manually process, tabulate, and publish large volumes of caste and religion data, the ORGI was capable of completing a similar task in 2011. Vijayanunni argued that the question was not whether the ORGI had the capacity to do so, but whether it was willing to do so.

Politics was central in deciding the path forward. A coalition of organizations, activists, politicians, and public intellectuals— including Vijayanunni—came together to target a change in census policy on caste. This chapter describes the advocacy efforts to include a caste count in Census 2011, and then details the institutional backlash that followed the campaign's success. In examining the steps that executive bureaucrats and allied groups took to keep a caste-wise enumeration out of Census 2011, this chapter traces the strategies of bureaucratic deflection and how they reinforce castelessness in the census, the state, and beyond.

Securing the Concession: Caste Census to Document Systemic Inequality

Dilip Mandal had worked for several media houses as a writer, editor, and producer for more than fifteen years. His most recent position as managing editor for a national magazine had thrust him into the limelight. Mandal had broken a “glass-ceiling” by being one of a handful of journalists from an oppressed-caste background to enter the editorial ranks, yet his road to becoming a senior journalist had not been easy.