The Tensor Theory of Connections
Daniele Mortari Carl Leake






The Tensor Theory of Connections
    Daniele Mortari[D. Mortari, Professor, Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA. E-mail: mortari@tamu.edu. 
 Work partially presented in Mortari, D. “The Theory of Connections: Connecting Functions,” IAA-AAS-SciTech-072, Forum 2018, Peoples' Friendship University of Russia, Moscow (Russia), November 13-15, 2018.] and Carl Leake[PhD Student, Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA. E-mail: leakec@tamu.edu.]

    December 30, 2023
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    This paper extends the univariate , introduced in (Mortari,2017a), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons,1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g (x, y), satisfy all constraints no matter what the g (x, y) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Tensor Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDEs and stochastic differential equations.





§ INTRODUCTION


The  (ToC), as introduced in <cit.>, consists of a general analytical framework to obtain constrained expressions, f (x), in one-dimension. A constrained expressions is a function expressed in terms of another function, g (x), that is freely chosen and, no matter what the g (x) is, the resulting expression always satisfies a set of n constraints. ToC generalizes the one-dimensional interpolation problem subject to n constraints using the general form,

    f (x) = g (x) + ∑_k = 1^n η_k  p_k (x),

where the p_k (x) are n user-selected linearly independent functions, the η_k are derived by imposing the n constraints, and the g (x) is a freely chosen function subject to be defined and nonsingular where the constraints are specified. Besides this requirement, g (x)  can be any function, including, discontinuous functions, delta functions, and even functions that are undefined in some domains. Once the η_k coefficients have been derived, then Eq. (<ref>) satisfies all the n constraints, no matter what the g (x) function is.

Constrained expressions in the form given in Eq. (<ref>) are provided for a wide class of constraints, including constraints on points and derivatives, linear combinations of constraints, as well as infinite and integral constraints <cit.>. In addition, weighted constraints <cit.> and point constraints on continuous and discontinuous periodic functions with assigned period, can also be obtained <cit.>. How to extend ToC to inequality and nonlinear constraints is currently a work in progress.

The  framework can be considered the generalization of interpolation; rather than providing a class of functions (e.g., monomials) satisfying a set of n constraints, it derives all possible functions satisfying the n constraints by spanning all possible g (x) functions. This has been proved in Ref. <cit.>. A simple example of a constrained expression is,

    f (x) = g (x) + x (2 x_2 - x)2 (x_2 - x_1)[ẏ_1 - ġ (x_1)] + x (x - 2 x_1)2 (x_2 - x_1)[ẏ_2 - ġ (x_2)].

This equation always satisfies . f x|_x_1 = ẏ_1 and . f x|_x_2 = ẏ_2, as long as ġ (x_1) and ġ (x_2) are defined and nonsingular. In other words, the constraints are embedded into the constrained expression.

Constrained expressions can be used to transform constrained optimization problems into unconstrained optimization problems. Using this approach fast least-squares solutions of linear <cit.> and nonlinear <cit.> ODEs have been obtained at machine error accuracy and with low (actually, very low) condition number. Direct comparisons of ToC versus MATLAB's  <cit.> and Chebfun <cit.> have been performed on a small test of ODEs with excellent results <cit.>. In particular, the ToC approach to solve ODEs consists of a unified framework to solve IVP, BVP, and multi-value problems. The extension of s subject to component constraints <cit.> has opened the possibility for ToC to solve in real-time a class of direct optimal control problems <cit.>, where the constraints connect state and costate.

This study first extends the  to two-dimensions by providing, for rectangular domains, all surfaces that are subject to: 1) Dirichlet constraints, 2) Neumann constraints, and 3) any combination of Dirichlet and Neumann constraints. This theory is then generalized to the Tensor   which provide in n-dimensional space all possible manifolds that satisfy boundary constraints on the value and boundary constraints on any-order derivative.

This article is structured as follows. First it shows that the one-dimensional ToC can be used in two dimensions when the constraints (functions or derivatives) are provided along one axis only. This is a particular case, where the original univariate theory <cit.> can be applied with basically no modifications. Then, a two dimensional ToC version is developed for Dirichlet type boundary constraints. This theory is then extended to include Neumann and mixed type boundary constraints. Finally, the theory is extended to n-dimensions and to incorporate arbitrary-order derivative boundary constraints followed by a mathematical proof validating it.



§ MANIFOLD CONSTRAINTS IN ONE AXIS, ONLY


Consider the function, f(x), where f:ℝ^n →ℝ^1, subject to one constraint manifold along the i-th variable, x_i, that is, f (x)|_x_i = v = c (x_i^v). For instance, in 3-D space, this can be the surface constraint, f (x, y, z)|_y = π = c (x, π, z). All manifolds satisfying this constraint can be expressed using the additive form provided in Ref. <cit.>,

    f (x) = g (x) + [c (x_i^v) - g (x_i^v)]

where g (x) is a freely chosen function that must be defined and nonsingular at the constraint coordinates. When m manifold constraints are defined along the x_i-axis, then the 1-D methodology <cit.>, can be applied as it is. For instance, the constrained expression subject to m constraints along the x_i variable evaluated at x_i = w_k, where k ∈ [1, m], that is, f (x)|_x_i = w_k = c (x_i^w_k), is,

    f (x) = g (x) + ∑_k = 1^m {[c (x_i^w_k) - g (x_i^w_k)] ∏_j kx_i - w_jw_k - w_j}.

Note that, this equation coincides with the Waring interpolation form (better known as Lagrangian interpolation form) <cit.> if the free function vanishes, g (x) = 0.



 §.§ Example #1: surface subject to four function constraints


The first example is designed to show how to use Eq. (<ref>) with mixed, continuous, discontinuous, and multiple constraints. Consider the following four constraints,

    c (x, -2) = sin(2 x),     c (x, 0) = 3 cos x   [(x + 1) mod (2)],     c (x, 1) = 9   e^-x^2,   and   c (x, 3) = 1 - x.

This example highlights that the constraints and free-function may be discontinuous by using the modular arithmetic function. The result is a surface that is continuous in x at some coordinates (at y =-2, 1, and 3) and discontinuous at y = 0. The surfaces shown in Fig. <ref> and Fig. <ref> were obtained using two distinct expressions for the free function, g (x, y).




 §.§ Example #2: surface subject to two functions and one derivative constraint


This second example is provided to show how to use the general approach given in Eq. (<ref>) and described in <cit.>, when derivative constraints are involved. Consider the following three constraints,

    c (x, -2) = sin(2 x),      c_y (x, 0) = 0,     and     c (x, 1) = 9   e^-x^2.

Using the functions, p_1 (y) = 1, p_2 (y) = y, and p_3 (y) = y^2, the constrained expression form satisfying these three constraints assumes the form,

    f (x, y) = g (x, y) + η_1 (x) + η_2 (x)  y + η_3 (x)  y^2.

The three constraints imply the constraints,

    sin(2 x)    =    g (x, -2) + η_1 - 2 η_2 + 4 η_3 
    
      0    =    g_y (x, 0) + η_2 
    
      9   e^-x^2   =    g (x, 1) + η_1 + η_2 + η_3,

from which the values of the η_k coefficients,

    η_1    =    2 g_y (x, 0) + 12   e^-x^2 - sin(2 x)3 + 13 g (x,-2) - 43 g (x,1) 
    η_2    =    -g_y (x, 0) 
    η_3    =   sin(2 x)3 - 13 g (x,-2) - g_y (x, 0) - 3   e^-x^2 + 13 g (x, 1),

can be derived. After substituting these coefficients into Eq. (<ref>), the constrained expression which always satisfies the three initial constraints is obtained. Using this expression and two different free functions, g (x, y), we obtain the surfaces shown in Fig. <ref> and Fig. <ref>, respectively. The constraint c_y (x, 0) = 0, difficult to see in both figures, can be verified analytically.





§ CONNECTING FUNCTIONS IN TWO DIRECTIONS


In this section the  is extended to the 2-dimensional case. Note that dealing with constraints in two (or more) directions (functions or derivatives) requires particular attention. In fact, two orthogonal constraint functions cannot be completely distinct as they intersect at one point where they need to match in value. In addition, if the formalism derived for the 1-D case is applied to 2-D case, some complications arise. These complications are highlighted in the following simple clarifying example.

Consider the two boundary constraint functions, f (x, 0) = q (x) and f (0, y) = h (y). Searching the constrained expression as originally done for the 1-dimensional case implies the expression,

    f (x, y) = g (x, y) + η_1   p_1 (x, y) + η_2   p_2 (x, y).

The constraints imply the two constraints,

    {[  q (x) = g (x, 0) + η_1   p_1 (x, 0) + η_2   p_2 (x, 0); h (y) = g (0, y) + η_1   p_1 (0, y) + η_2   p_2 (0, y). ].

To obtain the values of η_1 and η_2, the determinant of the matrix to invert is, p_1 (x, 0)   p_2 (0, y) - p_1 (0, y)   p_2 (x, 0). This determinant is y by selecting, p_1 (x, y) = 1 and p_2 (x, y) = y, or it is x by selecting, p_1 (x, y) = x and p_2 (x, y) = 1. Therefore, to avoid singularities, this approach requires paying particular attention to the domain definition and/or on the user-selected functions, p_k (x, y). To avoid dealing with these issues, a new (equivalent) formalism to derive constrained expressions is devised for the higher dimensional case.

The  extension to the higher dimensional case (with constraints on all axes) can be obtained by re-writing the constrained expression into an equivalent form, highlighting a general and interesting property. Let's show this by an example. Equation (<ref>) can be re-written as,

    f (x) = x (2 x_2 - x)2 (x_2 - x_1)ẏ_1 + x (x - 2 x_1)2 (x_2 - x_1)ẏ_2_A (x) + g (x) - x (2 x_2 - x)2 (x_2 - x_1)ġ_1 - x (x - 2 x_1)2 (x_2 - x_1)ġ_2_B (x).

These two components, A (x) and B (x), of a constrained expression have a specific general meaning. The term, A (x), represents an (any) interpolating function satisfying the constraints while the B (x) term represents all interpolating functions that are vanishing at the constraints. Therefore, the generation of all functions satisfying multiple orthogonal constraints in n-dimensional space can always be expressed by the general form, f (x) = A (x) + B (x), where A (x) is any function satisfying the constraints and B (x) must represent all functions vanishing at the constraints. Equation, f (x) = A (x) + B (x), is actually an alternative general form to write a constrained expression, that is, an alternative way to generalize interpolation: rather than derive a class of functions (e.g., monomials) satisfying a set of constraints, it represents all possible functions satisfying the set of constraints.

To prove that this additive formalism can describe all possible functions satisfying the constraints is immediate. Let f (x) be all functions satisfying the constraints and y (x) = A (x) + B (x) be the sum of a specific function satisfying the constraints, A (x), and a function, B (x), representing all functions that are null at the constraints. Then, y (x) will be equal to f (x) iff B (x) = f (x) - A (x), representing all functions that are null at the constraints.

As shown in Eq. (<ref>), once the A (x) function is obtained, then the B (x) function can be immediately derived. In fact B (x) can be obtained by subtracting the A (x) function, where all the constraints are specified in terms of the g (x) free function, from the free function g (x). For this reason, let us write the general expression of a constrained expression as,

    f (x) = A (x) + g (x) - A (g(x)),

where A (g(x)) indicates the function satisfying the constraints where the constraints are specified in term of g (x).

The previous discussion serves to prove that the problem of extending  to higher dimensional spaces consists of the problem of finding the function, A (x), only. In two dimensions, the function A (x) is provided in literature by the Coons surface <cit.>, f (x, y). This surface satisfies the Dirichlet boundary constraints,

    f (0, y) = c (0, y),    f (1, y) = c (1, y),    f (x, 0) = c (x, 0),   and   f (x, 1) = c (x, 1),

where the surface is contained in the x, y ∈ [0, 1] × [0, 1] domain. This surface is used in computer graphics and in computational mechanics applications to smoothly join other surfaces together, particularly in finite element method and boundary element method, to mesh problem domains into elements. The expression of the Coons surface is,

    f (x, y) =     (1 - x) c (0, y) + x   c (1, y) + (1 - y)   c (x, 0) + y   c (x, 1) - x   y   c (1, 1) 
          - (1 - x) (1 - y)   c (0, 0) - (1 - x)   y   c (0, 1) - x   (1 - y)   c (1, 0),

where the four subtracting terms are there for continuity. Note, the constraint functions at boundary corners must have the same value, c (0, 0), c (0, 1), c (1, 0), and c (1, 1). This equation can be written in matrix form as,

    f (x, y) =  1,     1 - x,     x[r] 0       c (x, 0)     c (x, 1) 
     c (0, y)     -c (0, 0)     -c (0, 1) 
     c (1, y)     -c (1, 0)     -c (1, 1) 1 
     1 - y 
     y,

or, equivalently,

    f (x, y) = v (x)   M (c (x, y))  v (y),

where

    M (c (x, y)) = [r] 0       c (x, 0)     c (x, 1) 
     c (0, y)     -c (0, 0)     -c (0, 1) 
     c (1, y)     -c (1, 0)     -c (1, 1)    and    v (z) =  1 
     1 - z 
     z.


Since the f (x, y) boundaries match the boundaries of the c (x, y) constraint function, then the identity, f (x, y) = v (x)   M (f (x, y))  v (y), holds for any f (x, y) function. Therefore, the B (x) function can be set as,

    B (x) := g (x, y) - v (x)   M (g (x, y))  v (y),

representing all functions that are always zero at the boundary constraints, as g (x, y) is a free function.




§  SURFACE SUBJECT TO DIRICHLET CONSTRAINTS


Equations (<ref>) and (<ref>) can be merged to provide all surfaces with the boundary constraints defined in Eq. (<ref>) in the following compact form,

    f (x, y) = v (x)  M (c (x, y)) v (y)_A (x,y) + g (x, y) - v (x)  M (g (x, y)) v (y)_B (x,y).

where, again, A (x,y) indicates an expression satisfying the boundary function constraints defined by c (x, y) and B (x,y) an expression that is zero at the boundaries. In matrix form Eq. (<ref>) becomes,

    f (x, y) =  1 
     1 - x 
     x[            g (x, y) c (x, 0) - g (x, 0) c (x, 1) - g (x, 1); c (0, y) - g (0, y) g (0, 0) - c (0, 0) g (0, 1) - c (0, 1); c (1, y) - g (1, y) g (1, 0) - c (1, 0) g (1, 1) - c (1, 1) ] 1 
     1 - y 
     y,

where g(x, y) is a freely chosen function. In particular, if g(x, y) = 0, then the ToC surface becomes the Coons surface.

Figure <ref> show the Coons surface (left figure) subject to the constraints,

    c (x, 0)    = sin (3 x - π/4)  cos (π/3) 
    
            c (x, 1)    = sin (3 x - π/4)  cos (4 + π/3) 
    
            c (0, y)    = sin (- π/4)  cos (4 y + π/3) 
    
            c (1, y)    = sin (3 - π/4)  cos (4 y + π/3),

and a ToC surface (right figure) that is obtained using the free function,

    g (x, y) = 13cos(4π x)  sin(6π y) - x^2  cos(2π y).



For generic boundaries defined in the rectangle x,y ∈ [x_i, x_f]×[y_i, y_f], the ToC surface becomes,

    f (x, y) =     g (x, y) + x - x_fx_i - x_f[c (x_i, y) - g (x_i, y)] + x - x_ix_f - x_i[c (x_f, y) - g (x_f, y)] 
         + y - y_fy_i - y_f[c (x, y_i) - g (x, y_i)] + y - y_iy_f - y_i[c (x, y_f) - g (x, y_f)] 
         - (x - x_f)(y - y_f)(x_i - x_f)(y_i - y_f)[c (x_i, y_i) - g (x_i, y_i)] 
         - (x - x_f)(y - y_i)(x_i - x_f)(y_f - y_i)[c (x_i, y_f) - g (x_i, y_f)] 
         - (x - x_i)(y - y_f)(x_f - x_i)(y_i - y_f)[c (x_f, y_i) - g (x_f, y_i)] 
         - (x - x_i)(y - y_i)(x_f - x_i)(y_f - y_i)[c (x_f, y_f) - g (x_f, y_f)].

Equation (<ref>) can also be set in matrix form,

    f (x, y) = v_x (x, x_i, x_f)   M (x, y)  v_y (y, y_i, y_f)

where

    M (x, y) = [                    g (x, y)     c (x, y_i) - g (x, y_i)     c (x, y_f) - g (x, y_f);     c (x_i, y) - g (x_i, y) g (x_i, y_i) - c (x_i, y_i) g (x_i, y_f) - c (x_i, y_f);     c (x_f, y) - g (x_f, y) g (x_f, y_i) - c (x_f, y_i) g (x_f, y_f) - c (x_f, y_f) ]

and

    v_x (x, x_i, x_f) =  1 
    x - x_fx_i - x_f
    x - x_ix_f - x_i    and    v_y (y, y_i, y_f) =  1 
    y - y_fy_i - y_f
    y - y_iy_f - y_i.


Note that all the ToC surfaces provided are linear in g (x, y), and therefore, they can be used to solve, by linear/nonlinear least-squares, 2-dimensional optimization problems subject to boundary function constraints, such as linear/nonlinear partial differential equations.



§ MULTI-FUNCTION CONSTRAINTS AT GENERIC COORDINATES


Equation (<ref>) can be generalized to many function constraints (grid of functions). Assume a set of n_x function constraints c (x_k, y) and a set of n_y function constraints c (x, y_k) intersecting at the n_x   n_y points p_ij = c (x_i, y_j), then all surfaces satisfying the n_x   n_y function constraints can be expressed by,

    f (x, y) = g (x, y)     + ∑_k = 1^n_x[c (x_k, y) - g (x_k, y)] ∏_i  kx - x_ix_k - x_i
         + ∑_k = 1^n_y[c (x, y_k) - g (x, y_k)] ∏_i  ky - y_iy_k - y_i
         -∑_i = 1^n_x{∑_j = 1^n_y(x - x_j)(y - y_i)(x_i - x_j)(y_j - y_i)[c (x_i, y_j) - g (x_i, y_j)]}.

Again, Eq. (<ref>) can be written in compact form,

    f (x, y) = v (x)  ℳ (c (x, y))  v (y) + g (x, y) - v (x)  ℳ (g (x, y))  v (y)

where,

    v (x) =  1 
    ∏_i  1x - x_ix_1 - x_i
    ⋮
    ∏_i  n_xx - x_ix_n_x - x_i    and    v (y) =  1 
    ∏_i  1y - y_iy_1 - y_i
    ⋮
    ∏_i  n_yy - y_iy_n_y - y_i

and

    ℳ (c (x, y)) = [r] 0        c (x, y_1)    …    c (x, y_n_y) 
     c (x_1, y)     - c (x_1, y_1)    …    - c (x_1, y_N_y) 
    ⋮      ⋮      ⋱   ⋮   
     c (x_n_x, y)     - c (x_n_x, y_1)    …    - c (x_n_x, y_n_y)


For example, two function constraints in x and three function constraints in y, can be obtained using the matrix,

    ℳ (c (x, y)) = [r] 0       c (x, y_1)     c (x, y_2)     c (x, y_3) 
     c (x_1, y)     -c (x_1, y_1)     -c (x_1, y_2)     -c (x_1, y_3) 
     c (x_2, y)     -c (x_2, y_1)     -c (x_2, y_2)     -c (x_2, y_3)

and the vectors,

    v (x) =  1 
    x - x_2x_1 - x_2
    x - x_1x_2 - x_1    and    v (y) =  1 
    (y - y_2)(y - y_3)(y_1 - y_2)(y_1 - y_3)
    (y - y_1)(y - y_3)(y_2 - y_1)(y_2 - y_3)
    (y - y_2)(y - y_1)(y_3 - y_2)(y_3 - y_1).


Two examples of ToC surfaces are given in Fig. <ref> in the x,y∈[-2,1]×[1, 3] domain.





§ CONSTRAINTS ON FUNCTION AND DERIVATIVES


Reference <cit.> provides the “Boolean sum formulation” (also called “Hermite-Coons formulation”) of the Coons surface that includes boundary derivatives,

    f (x, y) = v (y) F^x (x) + v (x) F^y (y) - v (x) M^xyv (y)

where

    v (z)    := {2 z^3 - 3 z^2 + 1,    z^3 - 2 z^2 + z,    -2 z^3 + 3 z^2,    z^3 - z^2}
    F^x (x)    := {c (x, 0),    c_y (x, 0),    c (x, 1),    c_y (x, 1)}
    F^y (y)    := {c (0, y),    c_x (0, y),    c (1, y),    c_x (1, y)}

and

    M^xy (x,y) := [r]
        c   (0, 0)     c_y    (0, 0)     c   (0, 1)     c_y    (0, 1) 
    
        c_x (0, 0)     c_xy (0, 0)     c_x (0, 1)     c_xy (0, 1) 
    
        c   (1, 0)     c_y    (1, 0)     c   (1, 1)     c_y    (1, 1) 
    
        c_x (1, 0)     c_xy (1, 0)     c_x (1, 1)     c_xy (1, 1).


The formulation provided in Eq. (<ref>) can be put in the matrix compact form,

    f (x, y) = v (x)   M (c (x,  y))  v (y),

where

    v (z) := {1,    2 z^3 - 3 z^2 + 1,    z^3 - 2 z^2 + z,    -2 z^3 + 3 z^2,    z^3 - z^2}

and the 5× 5 matrix, M (c(x, y)), has the expression,

    M (c(x, y)) := [r]
           0        c   (x, 0)      c_y    (x, 0)      c   (x, 1)      c_y    (x, 1) 
    
        c   (0, y)     -c   (0, 0)     -c_y    (0, 0)     -c   (0, 1)     -c_y    (0, 1) 
    
        c_x (0, y)     -c_x (0, 0)     -c_xy (0, 0)     -c_x (0, 1)     -c_xy (0, 1) 
    
        c   (1, y)     -c   (1, 0)     -c_y    (1, 0)     -c   (1, 1)     -c_y    (1, 1) 
    
        c_x (1, y)     -c_x (1, 0)     -c_xy (1, 0)     -c_x (1, 1)     -c_xy (1, 1).

To verify the boundary derivative constraints the following partial derivatives of Eq. (<ref>) are used,

    f_x (x, y)    = [v_x (x)  M (c (x, y)) + v (x)  M_x (c(x, y))] v (y) 
    
        f_y (x, y)    = v (x) [ M_y (c (x, y)) v (y) +  M (c(x, y)) v_y (y)],

where

    v z =  0 
     6 z (z - 1) 
     3 z^2 - 4 z + 1 
     6 z (1 - z) 
     z (3 z - 2),    M_y = [           0      0_1× 4;  c_y (0, y)      0_1× 4; c_xy (0, y)      0_1× 4;  c_y (1, y)      0_1× 4; c_xy (1, y)      0_1× 4 ],   and   M_x = [           0      0_1× 4;  c_x (x, 0)      0_1× 4; c_xy (x, 0)      0_1× 4;  c_x (x, 1)      0_1× 4; c_xy (x, 1)      0_1× 4 ].


The ToC in 2D with function and derivative boundary constraints is simply,

    f (x, y) = v (x)  M(c (x, y)) v (y)_A(x,y) + g (x, y) - v (x)  M (g(x, y)) v (y)_B (x,y)

where the M matrix and the v vectors are provided by Eq. (<ref>) and Eq. (<ref>), respectively.

Dirichlet/Neumann mixed constraints can be derived as shown in the examples provided in the four subsections. The matrix compact form is simply obtained from the matrix defined in Eq. (<ref>) by removing the rows and the columns associated with the boundary constraints not provided, while the vectors, v (x) and v (y) are derived by specifying the constraints. Note that in general the vectors v (x) and v (y) are not unique.[The reason why the vectors v (x) and v (y) are not unique comes from the fact that the A (x) term in Eq. (<ref>) is not unique.]

In the next subsections four Dirichlet/Neumann mixed constraint examples providing the simplest expressions for v (x) and v (y) are derived. The appendix of this article contains the expressions for the v (x) and v (y) vectors associated with all the combinations of Dirichlet and Neumann constraints.



 §.§ Constraints: c (0, y) and c (x, 0)


In this case, the Coons-type surface satisfying the boundary constraints can be expressed as,

    f (x, y) =  1     p (x)[r] 0       c (x, 0) 
     c (0, y)     -c (0, 0) 1 
     q (y)

where p (x) and q (y) are unknown functions. Expanding we obtain, f (x, y) = c (x, 0) q (y) + p (x) [c (0, y) - c (0, 0) q (y)]. The two constraints are satisfied if,

    c (0, y)    = c (0, 0) q (y) + p (0) [c (0, y) - c (0, 0) q (y)] 
     c (x, 0)    = c (x, 0) q (0) + p (x) [c (0, 0) - c (0, 0) q (0)].

Therefore, the p (x) and q (y) functions must satisfy, p (0) = 1 and q (0) = 1. The simplest expressions satisfying these equations can be obtained by selecting, p (x) = 1 and q (y) = 1. In this case, the associated ToC surface is given by,

    f (x, y) =  1     1[            g (x, y) c (x, 0) - g (x, 0); c (0, y) - g (0, y) g (0, 0) - c (0, 0) ] 1 
     1

Note that any functions satisfying, p (0) = 1 and q (0) = 1, can be adopted to obtain the ToC surface satisfying the constraints f (0, y) = c (0, y) and f (x, 0) = c (x, 0). This is because there are infinite Coons-type surfaces satisfying the constraints. Consequently, the vectors v (x) and v (y) are not unique.




 §.§ Constraints: c (0, y) and c_y (x, 0)


For these boundary constraints, the Coons-type surface is expressed by,

    f (x, y)    =  1     p (x)[r] 0       c_y (x, 0) 
     c (0, y)     -c_y (0, 0) 1 
     q (y)
       = c_y (x, 0) q (y) + p (x) [c (0, y) - c_y (0, 0) q (y)].

The constraints are satisfied if,

    c (0, y)    = c_y (0, 0) q (y) + p (0) [c (0, y) - c_y (0, 0) q (y)], 
     c_y (x, 0)    = c_y (x, 0) q_y (0) + p (x) [c_y (0, 0) - c_y (0, 0) q_y (0).

Therefore, the p (x) and q (y) functions must satisfy p (0) = 1 and q_y (0) = 1. One solution is p (x) = 1 and q (y) = y. Therefore, the associated ToC surface is given by,

    f (x, y) =  1     1[                g (x, y) c_y (x, 0) - g_y (x, 0);     c (0, y) - g (0, y) g_y (0, 0) - c_y (0, 0) ] 1 
     y.




 §.§ Neumann constraints: c_x (0, y), c_x (1, y), c_y (x, 0), and c_y (x, 1)


In this case, the Coons-type surface satisfying the boundary constraints can be expressed as,

    f (x, y) =  1,     p_1 (x),     p_2 (x)[r] 0       c_y (x, 0)     c_y (x, 1) 
     c_x (0, y)     -c_xy (0, 0)     -c_xy (0, 1) 
     c_x (1, y)     -c_xy (1, 0)     -c_xy (1, 1) 1 
     q_1 (y) 
     q_2 (y).

The constraints are satisfied if,

    c_x (0, y)    = q_1 (y) c_xy (0, 0) + q_2 (y) c_xy (0, 1) + 
          + p_1x (0) [c_x (0, y) - q_1 (y) c_xy (0, 0) - q_2 (y) c_xy (0, 1)] + 
          + p_2x (0) [c_x (1, y) - q_1 (y) c_xy (1, 0) - q_2 (y) c_xy (1, 1)] 
    
        c_x (1, y)    = q_1 (y) c_xy (1, 0) + q_2 (y) c_xy (1, 1) + 
          + p_1x (1) [c_x (0, y) - q_1 (y) c_xy (0, 0) - q_2 (y) c_xy (0, 1)] + 
          + p_2x (1) [c_x (1, y) - q_1 (y) c_xy (1, 0) - q_2 (y) c_xy (1, 1)] 
    
        c_y (x, 0)    = q_1y (0) c_y (x, 0) + q_2y (0) c_y (x, 1) + 
          + p_1 (x) [c_xy (0, 0) - q_1y (0) c_xy (0, 0) - q_2y (0) c_xy (0, 1)] + 
          + p_2 (x) [c_xy (1, 0) - q_1y (0) c_xy (1, 0) - q_2y (0) c_xy (1, 1)] 
    
        c_y (x, 1)    = q_1y (1) c_y (x, 0) + q_2y (1) c_y (x, 1) + 
          + p_1 (x) [c_xy (0, 1) - q_1y (1) c_xy (0, 0) - q_2y (1) c_xy (0, 1)] + 
          + p_2 (x) [c_xy (1, 1) - q_1y (1) c_xy (1, 0) - q_2y (1) c_xy (1, 1)].

These equations imply, p_1x (0) = q_1x (0) = 1, p_1x (1) = q_1x (1) = 0, p_2x (0) = q_2x (0) = 0, and p_2x (1) = q_2x (1) = 1. Therefore, the simplest solution is, p_1 (t) = q_1 (t) = t - t^2/2 and p_2 (t) = q_2 (t) = t^2/2. Then, the associated ToC surface satisfying the Neumann constraints is given by,

    f (x, y) = v (x) [                  g (x, y)   c_y (x, 0) - g_y (x, 0)   c_y (x, 1) - g_y (x, 1);   c_x (0, y) - g_x (0, y) g_xy (0, 0) - c_xy (0, 0) g_xy (0, 1) - c_xy (0, 1);   c_x (1, y) - g_x (1, y) g_xy (1, 0) - c_xy (1, 0) g_xy (1, 1) - c_xy (1, 1) ]v (y)

where

    v (x) =  1,     x - x^22,    x^22    and    v (y) =  1,     y - y^22,    y^22.




 §.§ Constraints: c (0, y), c_y (x, 0), and c_y (x, 1)


In this case, the Coons-type surface satisfying the boundary constraints is in the form,

    f (x, y) =  1 
     p (x)[r] 0       c_y (x, 0)     c_y (x, 1) 
    
        c (0, y)     - c_y (0, 0)     - c_y (0, 1) 1 
     q_1 (y) 
     q_2 (y).

The constraints are satisfied if p (0) = 1, p_1y (0) = 1, p_1y (1) = 0, p_2y (0) = 0, and p_2y (1) = 1. Therefore, the associated ToC surface is,

    f (x, y) =  1 
     1[                g (x, y) c_y (x, 0) - g_y (x, 0) c_y (x, 1) - g_y (x, 1);     c (0, y) - g (0, y) g_y (0, 0) - c_y (0, 0) g_y (0, 1) - c_y (0, 1) ] 1 
     y - y^22
    y^22.




 §.§ Generic mixed constraints


Consider the case of mixed constraints,

    [     f (x, y_1) = c (x, y_1); f_x (x, y_2) = c_x (x, y_2);     f (x, y_3) = c (x, y_3) ]    and    [ f_y (x_1, y) = c_y (x_1, y); f_y (x_2, y) = c_y (x_2, y);     f (x_3, y) = c (x_3, y) ].

In this case, the surface satisfying the boundary constraints is built using the matrix,

    M (c (x, y)) = [r] 0       c (x, y_1)     c_x (x, y_2)     c (x, y_3) 
     c_y (x_1, y)     -c_y (x_1, y_1)     -c_xy (x_1, y_2)     -c_y (x_1, y_3) 
     c_y (x_2, y)     -c_y (x_2, y_1)     -c_xy (x_2, y_2)     -c_y (x_2, y_3) 
     c (x_3, y)     -c (x_3, y_1)     -c_x (x_3, y_2)     -c (x_3, y_3)

and all surfaces subject to the constraints defined in Eq. (<ref>) can be obtained by,

    f (x, y) = v (x) M (c (x, y)) v (y) + g (x, y) - v (x) M (g (x, y)) v (y),

where

    v (x) =  1 
     p_1 (x,x_1,x_2,x_3) 
     p_2 (x,x_1,x_2,x_3) 
     p_3 (x,x_1,x_2,x_3)    and    v (y) =  1 
     q_1 (y,y_1,y_2,y_3) 
     q_2 (y,y_1,y_2,y_3) 
     q_3 (y,y_1,y_2,y_3)

are vectors made of the (not unique) function vectors v (x) and v (y) whose expressions can be found by satisfying the constraints (as done in the previous four subsections) along with a methodology similar to that given in section <ref>.



§ EXTENSION TO N DIMENSIONAL SPACES AND ARBITRARY-ORDER DERIVATIVE CONSTRAINTS


This section provides the Tensor , as the generalization to n dimensional rectangular domains with arbitrary-order boundary derivatives of what has been previously presented for 2-dimensional space. Using tensor notation, this generalization is represented in the following compact form,

    F (x) =  M(c (x))_i_1 i_2 … i_n v_i_1 v_i_2…v_i_n_A(x) + g (x) -  M(g (x))_i_1 i_2 … i_n v_i_1 v_i_2…v_i_n_B(x)

where n is the number of orthogonal coordinates defined by the vector x = {x_1, x_2, …, x_n}, v_i_k (x_k) is the i_k-th element of a vector function of the variable x_k, M is an n dimensional tensor that is a function of the boundary constraints defined in c (x), and g (x) is the free-function.

In Eq. (<ref>) the term A (x) represents any function satisfying the boundary constraints defined by c (x) and the term B (x) represents all possible functions that are zero on the boundary constraints. The subsections that follow explain how to construct the M tensor and the v_i_k vectors for assigned boundary constraints, and provides a proof that the tensor formulation of the ToC defined by Eq. (<ref>) satisfies all boundary constraints defined by c (x), independently of the choice of the free function, g (x).

Consider a generic boundary constraint on the x_k = p hyperplane, where k∈[1,n]. This constraint specifies the d-derivative of the constraint function c (x) evaluated at x_k = p and it is indicated by ^k c_p^d := .∂^d c (x)∂ x_k^d|_x_k = p. Consider a set of ℓ_k constraints defined in various x_k hyperplanes. This set of constraints is indicated by ^kc_p^k^d^k, where d^k and p^k are vectors of ℓ_k elements indicating the order of derivatives and the values of x_k where the boundary constraints are defined, respectively. A specific boundary constraint, say the m-th boundary constraint, can then be written as ^kc_p^k_m^d^k_m.

Additionally, let us define an operator, called the boundary constraint operator, whose purpose is to take the d-th derivative with respect to coordinate x_k and then evaluate that function at x_k = p. Equation (<ref>) shows the idea.

    ^kb_p^d[f] ≡∂ ^d f/∂ x_k^d|_(x_1,…,x_k-1,p,x_k+1,…,x_n)

In general, for a function of n variables, the boundary constraint operator identifies an n-1 dimensional manifold. As the boundary constraint operator will be used throughout this proof, it is important to note its properties when acting on sums and products of functions. Equation (<ref>) shows how the boundary constraint operator acts on sums, and Eq. (<ref>) shows how the boundary constraint operator acts on products.

    ^k b^d_p [f_1 + f_2]    =     ^k b^d_p [f_1] +   ^k b^d_p [f_2] 
    
        ^k b^d_p [f_1 f_2]    =   ∑_m = 0^n nm  ^k b^n-m_p [f_1]   ^k b^m_p [f_2]


The remaining subsections show how to build the M tensor and the vectors v given the boundary constraints defined by the boundary constraint operators. In addition, the final subsection contains a proof that, in Eq. (<ref>), the boundary constraints defined by c (x) satisfy the function A (x) and, by extension, the function B(x) projects the free-function g (x) onto the sub-space of functions that are zero on the boundary constraints. Then, it follows that the expression for the ToC surface given in Eq. (<ref>) represents all possible functions that meet the boundary defined by the boundary constraint operators.



 §.§ The M tensor


There is a step-by-step method for constructing the M tensor.

    
  * The element of M for all indices equal to 1 is 0 (i.e. M_11… 1 = 0).
    
  * The first order tensor obtained by keeping the k-th dimension's index and setting all other dimension's indices to 1 can be written as,
        
    M_1, …, 1, i_k, 1, …, 1 =  ^kc_p^k^d^k,      where   i_k ∈[2, ℓ_k+1],

        where the vector ^kc_p^k^d^k contains the ℓ_k boundary constraints specified along the x_k-axis. For example, consider the following ℓ_7 = 3 constraints on the k=7-th axis,
        
    ^7c_p^7^d^7 := {c|_x_7 =-0.3,   .∂^4 c∂ x_7^4|_x_7 = 0.5,   .∂ c∂ x_7|_x_7 = 1.1}     then: d^7 = {0,   4,   1}
    p^7 = {-0.3,   0.5,   1.1}.

    
  * The generic element of the tensor is M_i_1 i_2 … i_n, where at least two indices are different from 1. Let m be the number of indices different from 1. Note that m is also the number of constraint “intersections.” In this case, the generic element of the M tensor is provided by,
        
    M_i_1 i_2 … i_n =  ^1 b^d^1_i_1 - 1_p^1_i_1 - 1[  ^2b^d^2_i_2 - 1_p^2_i_2 - 1[ …[  ^n b^d^n_i_n - 1_p^n_i_n - 1 [c(x)]] …] ] (-1)^m+1.

        If c(x) ∈ C^s, where s = ∑_k=1^n d^k_i_k - 1, then Clairaut's theorem states that the sequence of boundary constraint operators provided in Eq. (<ref>) can be freely permutated. This permutation becomes obvious by multiple applications of the theorem. For example,
        
    f_xyy = (f_xy)_y = (f_yx)_y = (f_y)_xy = (f_y)_yx = f_yyx.


To better clarify how to use Eq. (<ref>), consider the example of the following constraints in three-dimensional space.

    c(x)|_x_1 = 0,    c(x)|_x_1 = 1,    c(x)|_x_2 = 0,   ∂ c(x)∂ x_2|_x_2 = 0,    c(x)|_x_3 = 0,    and  ∂ c(x)∂ x_3|_x_3 = 0


    
  * From step one: M_111 = 0
    
  * From step two:
    
    M_i_1 1 1   =  0,     c (0,x_2,x_3),     c (1,x_2,x_3) =  0,     ^1b_0^0 [c(x)],     ^1b_1^0 [c(x)]
    
            M_1 i_2 1   =  0,     c(x_1,0,x_3),    ∂ c∂ x_2(x_1, 0, x_3) =  0,     ^2b_0^0 [c(x)],     ^2b_0^1 [c(x)]
    
            M_1 1 i_3   =  0,     c(x_1,x_3,0),    ∂ c∂ x_3(x_1, x_2, 0) =  0,     ^3b_0^0 [c(x)],     ^3b_0^1 [c(x)]

    
  * From step three, a single example is provided,
    
    M_323 =  ^1b_1^0[ ^2b_0^0[ ^3c^1_0 (x) ]] (-1)^4 = ∂ c(x)∂ x_3|_x_1 = 1 
     x_2 = 0 
     x_3 = 0

    which, thanks to Clairaut's theorem, can also be written as,
    
    M_323 =  ^2b_0^0[ ^3b_0^1[ ^1c_1^0 ]] (-1)^4 =   ^3b_0^1[ ^1b_1^0[ ^2c_0^0]] (-1)^4.

    Three additional examples are given to help further illustrate the procedure,
    
    M_132 =-.∂ c (x)∂ x_2|_x_2 = 0 
     x_3 = 0,      M_221 =-c (0,0,x_3),     and     M_333 = ∂^2 c (x)∂ x_2 ∂ x_3|_x_1 = 1 
     x_2 = 0 
     x_3 = 0





 §.§ The v vectors


Each vector, v_k, is associated with the ℓ_k constraints that are specified by ^kc_p^k^d^k. The v_k vector is built as follows,

    v_k =  1,    ∑_i=1^ℓ_kα_i1  h_i (x_k),    ∑_i=1^ℓ_kα_i2  h_i (x_k),    …,    ∑_i=1^ℓ_kα_iℓ_k  h_i (x_k),

where the h_i (x_k) are ℓ_k linearly independent functions. The simplest set of linearly independent functions are monomials, that is, h_i (x_k) = x_k^i-1. The ℓ_k ×ℓ_k coefficients, α_ij, can be computed by matrix inversion,

    [       ^kb^d_1_p_1[h_1]       ^kb^d_1_p_1[h_2]                      …     ^kb^d_1_p_1[h_ℓ_k];       ^kb^d_2_p_2[h_1]       ^kb^d_2_p_2[h_2]                      …     ^kb^d_2_p_2[h_ℓ_k];                      ⋮                      ⋮                      ⋱                      ⋮;   ^kb^d_ℓ_k_p_ℓ_k[h_1]   ^kb^d_ℓ_k_p_ℓ_k[h_2]                      … ^kb^d_ℓ_k_p_ℓ_k[h_ℓ_k] ][     α_11     α_12        …   α_1ℓ_k;     α_21     α_22        …   α_2ℓ_k;        ⋮        ⋮        ⋱        ⋮;  α_ℓ_k 1  α_ℓ_k 2        … α_ℓ_kℓ_k ] = [ 1 0 … 0; 0 1 … 0; ⋮ ⋮ ⋱ ⋮; 0 0 … 1 ].

To supplement the above explanation, let's look at the example of Dirichlet boundary conditions on x_1 from the example in section <ref>. There are two boundary conditions, c(x)|_x_1 = 0 and c(x)|_x_1 = 1, and thus two linearly independent functions are needed,

    v_i_1 =  1,    α_11 h_1 (x_1) + α_21 h_2 (x_1),    α_12 h_1 (x_1) + α_22 h_2 (x_1).

Let us consider, h_1 (x_1) = 1 and h_2 (x_1) = x_1. Then, following Eq. (<ref>),

    [ ^1b^0_0 [1] ^1b^0_0 [x];  ^2b^0_1[1]  ^2b^0_1[x] ][ α_11 α_12; α_21 α_22 ] = [ 1 0; 1 1 ][ α_11 α_12; α_21 α_22 ] = [ 1 0; 0 1 ]  →  [ α_11 α_12; α_21 α_22 ] = [r] 1     0 
     -1     1,

and substituting the values of α_ij we obtain v_i_1 =  1,     1 - x_1,     x_1.



 §.§ Proof


This section demonstrates that the term A (x) from Eq. (<ref>) generates a surface satisfying the boundary constraints defined by the function c (x). First, it will be shown that A (x) satisfies boundary constraints on the value, and then that A (x) satisfies boundary constraints on arbitrary-order derivatives.

Equation (<ref>) for d = 0 allows us to write,

    ^kb^0_p_q-1[A(x)] = ^kb^0_p_q-1[ M_i_1i_2… i_k … i_n] v_i_1v_i_2… ^kb^0_p_q-1[v_i_k] …v_i_n.

The boundary constraint operator applied to v_k yields,

    ^kb^0_p_q-1[v_i_k] =  = 1,     i_k = 1, q 
     = 0,     i_k ≠ 1, q.

Since the only nonzero terms are associated with i_k = 1, q we have,

    ^kb^0_p_q-1[A(x)] = ( ^kb^0_p_q-1[ M_i_1i_2… 1 … i_n] +  ^kb^0_p_q-1[ M_i_1i_2… q… i_n]) v_i_1v_i_2…v_i_n.

Applying the boundary constraint operator to the n-1 dimensional M tensor where index i_k = q will have no effect, because all of the functions already have coordinate x_k substituted for the value p_q-1 (see Eq. (<ref>)). Moreover, applying the boundary constraint operator to the M tensor where index i_k=1 will cause all terms in the sum within the parenthesis in Eq. (<ref>) to cancel each other, except when all of the non-i_k indices are equal to one. This leads to Eq. (<ref>).

    ^kb^0_p_q-1[A(x)] = (  M_11… 1 … 1 +  M_11… q… 1) v_1 v_1 …v_1

Since v_j = 1 when j=1 and M_1 1 … 1 = 0 by definition, then,

    ^kb^0_p_q-1 [A (x)] =  M_1 1 … q … 1 = c (x_1, x_2, …, p_q-1, …, x_n),

which proves Eq. (<ref>) works for boundary constraints on the value.

Now we will show that Eq. (<ref>) holds for arbitrary-order derivative type boundary constraints. Equation (<ref>) for d > 0 allows us to write,

    ^kb^d_q-1_p_q-1[A(x)] = ^kb^d_q-1_p_q-1[ M_i_1i_2… i_k … i_n] v_i_1v_i_2…v_i_k…v_i_n +  M_i_1i_2… i_k … i_nv_i_1v_i_2… ^kb^d_q-1_p_q-1[v_i_k] …v_i_n,

because all of the v vectors except v_i_k do not depend on x_k, so applying the boundary constraint operator to them will result in a vector of zeros, and all of the intermediate terms that involve derivatives of order less than d_q-1 result in a term that is equal to zero based on the definitions for M and v_i_k. Applying the boundary constraint operator to v_i_k will yield,

    ^kb^d_q-1_p_q-1 [v_i_k] =  = 1,     i_k = q 
     = 0,     i_k ≠ q,

and applying the boundary constraint operator to M will yield,

    ^k b^d_q-1_p_q-1 [ M_i_1 i_2 … 1 … i_n] =  =   ^kb^d_q-1_p_q-1 [ M_i_1 i_2 … 1 … i_n],     i_k = 1 
     = 0,     i_k ≠ 1.

Substituting these simplifications into Eq. (<ref>) after applying the boundary constraint operator results in Eq. (<ref>).

    ^kb^d_q-1_p_q-1 [A(x)] = ( ^kb^d_q-1_p_q-1[ M_i_1 i_2 … 1 … i_n] +  M_i_1 i_2 … q … i_n) v_i_1v_i_2…v_i_n

Similar to the proof for value-based boundary constraints, based on Eq. (<ref>), all terms in the sum within the parenthesis in Eq. (<ref>) will cancel each other, except when all of the non-i_k indices are equal to one. Thus, Eq. (<ref>) can be simplified to Eq. (<ref>).

    ^kb^d_q-1_p_q-1 [A(x)] = ( ^kb^d_q-1_p_q-1 [ M_1 1 … 1 … 1] +  M_1 1 … q … 1) v_1 v_1 …v_1

Again, all of the vectors v were designed such that their first component is 1, and the value of the element of M for all indices equal to 1 is 0. Therefore, Eq. (<ref>) simplifies to,

    ^kb^d_q-1_p_q-1 [A (x)] =  M_1 1 … q … 1 = ∂^d c (x)∂ x_k^d|_x_k = p_q-1,

which proves Eq. (<ref>) works for arbitrary-order derivative boundary constraints.

In conclusion, the term A (x) from Eq. (<ref>) generates a manifold satisfying the boundary constraints given in terms of arbitrary-order derivative in n-dimensional space. The term B (x) from Eq. (<ref>) projects any free function g (x) onto the space of functions that are vanishing at the specified boundary constraints. As a result, Eq. (<ref>) can be used to produce the family of all possible functions satisfying assigned boundary constraints (functions or derivatives) in rectangular domains in n-dimensional space.



§ CONCLUSIONS


This paper extends to n-dimensional spaces the one-dimensional  (ToC), introduced in Ref. <cit.>. First it provides a mathematical tool to express all possible surfaces subject to constraint functions and arbitrary-order derivatives in a boundary rectangular domain, and then it extends the results to the multivariate case by providing the Tensor Theory of Connections. The Tensor Theory of Connections allows one to obtain n-dimensional manifolds subject to any-order derivative boundary constraints.

In particular, if the constraints are provided along one axis only, then this paper shows that the univariate ToC, as defined in Ref. <cit.>, can be adopted to describe all possible surfaces satisfying the constraints. If the boundary constraints are defined in a rectangular domain, then the constrained expression is found in the form, f (x) = A (x) + B (x), where A (x) can be any function satisfying the constraints and B (x) describes all functions that are vanishing at the constraints. This is obtained by introducing a free function, g (x), into the function B (x) in such a way that B (x) is zero at the constraints no matter what the g (x) is. This way, by spanning all possible g (x) surfaces (even discontinuous, null, or piece-wise defined) the resulting B (x) generates all surfaces that are zero at the constraints and, consequently, f (x) = A(x) + B (x), describes all surfaces satisfying the constraints defined in the rectangular boundary domain. The function A (x) has been selected as a Coons surface <cit.> and, in particular, a Coons surface is obtained if g (x) = 0 is selected. All possible combinations of Dirichlet and Neumann constraints are also provided in the appendix.

The extension to multivariate , called Tensor , is provided in the last section as a mathematical tool to transform n-dimensional constraint optimization problems subject to constraints on the boundary value and  any-order derivative into unconstrained optimization problems. The number of applications of the Tensor  are many, especially in the area of partial and stochastic s: the main subjects of our current research.



 §.§ Acknowledgment


The authors acknowledge Ergun Akleman for pointing out the Coons surface.





§ ALL COMBINATIONS OF DIRICHLET AND NEUMANN CONSTRAINTS


|c|c|c|c|c|c|c|c|c|c|
c_x,0     c_0,y     c_x,1     c_1,y     c^x_0,y     c^x_1,y     c^y_x,0     c^y_x,1     v (x)     v (y) 

                                            1
 1     1
 1 
                                             1
 1     1
y
                                             1
x     1
y
                                           1
 1     1
 1-y^2
 y^2
                                           1
 1     1
 1
 y
                                           1
x     1
 1-y
y
                                           1
x     1
y-y^2
y^2
                                             1 
 1     1 
 y - y^2/2
 y^2/2
                                           1
 x     1
 y - y^2/2
 y^2/2 
                                         1
 1-x
 x     1
 1-y
 y 
                                         1
 1-x
 x     1
 y-y^2
 y^2 
                                         1
 1-x
 x     1
 y-y^2/2
 y^2/2 
                                         1
 x-x^2
 x^2     1
 y-y^2
 y^2 
                                         1
 x-x^2
 x^2     1
 y-y^2/2
 y^2/2 
                                         1
 x-x^2/2
 x^2/2     1
 y-y^2/2
 y^2/2 
                                           1
 1     1
 1
y 
                                           1
 x     1
 1
 y 
                                         1
 1-x
 x     1
 1
 y 
                                         1
 1     1
 1-y^2
 y-y^2
 y^2 
                                         1
 x     1
 1-y^2
 y-y^2
 y^2 
                                         1
 1     1
 1
 y-y^2/2
 y^2/2 
                                         1
 x     1
 1
 y-y^2/2
 y^2/2 
                                         1
 1
 x     1
 1
 y 
                                         1
 x-x^2/2
 x^2/2     1
 1
 y 
                                         1
 1
 x     1
 1
 y 
                                       1
 1
 x     1
 1-y^2
 y-y^2
 y^2 
                                       1
 1
 x     1
 1
 y-y^2/2
 y^2/2 
                                       1
 1-x
 x     1
 1-y^2
 y-y^2
 y^2 
                                       1
 x-x^2
 x^2     1
 1-y^2
 y-y^2
 y^2 
                                       1
 1-x
 x     1
 1
 y-y^2/2
 y^2/2 
                                       1
 x-x^2
 x^2     1
 1
 y-y^2/2
 y^2/2 
                                       1
 x-x^2/2
 x^2/2     1
 1
 y-y^2/2
 y^2/2 
                                     1
 1-x^2
 x-x^2
 x^2     1
 1-y^2
 y-y^2
 y^2 
                                     1
 1-x^2
 x-x^2
 x^2     1
 1
 y-y^2/2
 y^2/2 
                                     1
 1
 x-x^2/2
 x^2/2     1
 1
 y-y^2/2
 y^2/2 
                                     1
 1-x
 x     1
 1-3y^2+2y^3
 y-2y^2+y^3
 3y^2-2y^3
 -y^2+y^3 
                                     1
 -1+x
 1     1
 1-3y^2+2y^3
 y-2y^2+y^3
 3y^2-2y^3
 -y^2+y^3 
                                     1
 x-x^2/2
 x^2/2     1
 1-3y^2+2y^3
 y-2y^2+y^3
 3y^2-2y^3
 -y^2+y^3 
                                     1
 1
 x     1
 1-3y^2+2y^3
 y-2y^2+y^3
 3y^2-2y^3
 -y^2+y^3 
                                   1
 1-x^2
 x-x^2
 x^2     1
 1-3y^2+2y^3
 y-2y^2+y^3
 3y^2-2y^3
 -y^2+y^3 
                                   1
 1
 x-x^2/2
 x^2/2     1
 1-3y^2+2y^3
 y-2y^2+y^3
 3y^2-2y^3
 -y^2+y^3 
                                  1
 1-3x^2+2x^3
 x-2x^2+x^3
 3x^2-2x^3
 -x^2+x^3     1
 1-3y^2+2y^3
 y-2y^2+y^3
 3y^2-2y^3
 -y^2+y^3
 

999
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