These are actual student papers that were not designed to be web pages. They may contain historical, grammatical, mathematical, or formatting errors. These papers were graded using the criterion mentioned in the paper directions, and the writing checklist. The test review sheets and the WebCT tests are good indicators of the mathematics that was discussed in class during and/or after each presentation.

Dorothy Moorefield

Women and Minorities in Mathematics

Dr. Sarah Greenwald

29 March 2001

Ol’ga Aleksandrova Ladyzhenskaya was born on March 7, 1922. The patrons of her hometown, Kologriv, must be proud to claim her as their own.[1] She proves to be a lovely counter example to Hardy’s quote, “Mathematics is a young man’s game.” Her contributions to the fields of Mathematics and Physics are beyond the scope of a single paper.

O. A. Ladyzenskaya received her undergraduate degree from Moscow State University in 1947. In between then and 1949 she furthered her career by earning her PhD from the Leningrad State University.  In 1943 she earned yet another degree from Moscow State University this being a Doctorate of Sciences. Two years later she became a Professor of Mathematics at the Physics Department of St. Petersburg University.[2] She can be reached at the Laboratory of Mathematical Physics in the Steklov Institute of Mathematics at St. Petersburg where she has been presiding as the head of the Laboratory since 1961. [3] [4]

O.A. Ladyzenskaya coauthored the book, Linear and Quasilinear Equations of Parabolic Type, with V. A Solonnikov and N.N. Uranl’ceva.  This book is based on linear, quasi-linear second-order partial differential equations of parabolic type.[5] A partial differential equation contains partial derivatives depending on more than one variable. To solve a partial differential equation one seeks the solution whose derivatives when plugged into the equation are satisfied. The solution is most likely not a constant value but a function of the variables. The order of a differential equation is given by the highest partial derivative within the equation. For example, the order of the wave equation in three dimensions represented by, Utt =Uxx + Uyy + Uzz, is 2. This equation is also linear.[6]

A Linear Partial Differential equation is basically an equation whose dependent variable, which is the solution to the equation, and its derivatives are given in a linear form. Basically this means the unknown dependent variable along and its derivatives, are given to be in a linear combination within the equation. For example here is a second-order linear partial differential equation with two independent variables, A*Uxx + B*Uxy + C*Uyy + D*Ux + E*Uy + F*U = G. A, B, C, D, E, F and G are either constants constant coefficients or they are functions of x and y. In order for the equation to be linear, neither U nor its derivatives taken to any power, multiplied together or plugged to a function of any kind such as sine or cosine.[7]

Equations of the parabolic type are given by the discriminate of the second-order linear partial differential equation. The discriminate is just like the discriminate of the quadratic formula in appearance, B2 – 4*A*C. If the coefficients of the partial differential equation have the property, B2 – 4*A*C=0 then the equation is parabolic. Another type of linear partial differential equations is elliptic. If the discriminate is less than zero, then the equation is elliptic. [8] O.A. Ladyzenskaya wrote another book with N. N. Ural’ceva entitled, Linear and Quasi-Linear Equations of Elliptic Type.[9]

Quasi-linear equations are not to be confused with linear equations. In fact quasi-linear equations are nonlinear. However, quasi-linear means almost or somewhat linear. There are two main interpretations of what “almost linear” means. One interpretation is the coefficients on U and or its derivatives are nonlinear but their value is so small, the nonlinear part of the equation is almost insignificant. The other interpretation is what O.A Ladyzhenskaya appears to follow so we shall focus on equations on this type.[10]

The equations in discussion are a special type of nonlinear equations which take the general form of L U º Ut – aij(x , t, U, Ux)*Uxixj  +a( x, t, U, Ux) = 0.[11]  L is the linear operator on U, which sets up the equation. L U is defined to be Ut – aij(x , t, U, Ux)*Uxixj  +a( x, t, U, Ux) = 0. The coefficients, aij(x , t, U, Ux) and a( x, t, U, Ux) may be dependent on U or its derivatives not with respect to time. However, the partial derivative with respect to time, Ut, is a linear term. This provides the other interpretation of what it means to be quasi-linear.[12]

To understand the notation of this equation one must conceptualize many dimensions. The equation is set into Euclidean space with n dimensions. The variable x stands for n variables where, x = (x1 , …, xn). In two dimensional space x = ( x1 , xn) which is commonly notated by (x , y). Ux stands for the first partial derivatives of U with respect to all variables. So we have, Ux = ( Ux1, … ,Uxn ). Comparing once again to two dimensional space, we have Ux = (Ux1 , Ux2) which is commonly notated as ( Ux , Uy). Uxixj notates the partial derivative with respect to xi, some arbitrary independent variable, and then the partial of that derivative taken with respect to xj, which is another arbitrary independent variable. The variable t usually represents time and is not within our n dimensions of Euclidian space. The terms, a( x, t, U, Ux) and aij(x , t, U, Ux) are the coefficients which my be functions dependent on one to all of the following terms; The term x, denotes a point in n dimensional Euclidian space and can be read as x = (x1 , …, xn). Time, denoted by t, is not in our n dimensions. Ut is the partial derivative of U with respect to t. U is our unknown dependent variable. Ux represents the first derivative with respect to all variables within our n dimensions where, Ux = ( Ux1, … ,Uxn ).[13] If the coefficients are constants or dependent only on x and t, then the equation is now linear.

Quasi-linear equations have properties different than linear equations. The properties that make them hyperbolic, parabolic or elliptic are quite different from the discriminate use to determine these traits in linear equations. Due to time we shall skip these properties and move on to some examples of quasi-linear equations.

Consider the Cauchy problem for systems of quasi-linear equations. Ut = P(x, t, u, Ux)*U + F(x,t), xÎRn , 0#t#T. xÎRn  shows that we are in n dimensions.  P(x, t, u, Ux)=j1v1#m Av(x, t, U)*,v,/ (*x1v1*xnvn). The function, F, is the forcing function and is assumed to be known.  Ut is a linear term and if it is moved across the equality the equation looks quite similar to the general form. This equation is accompanied by an initial the condition, U(x, 0)= f(x) where xÎRn and the initial function, f(x), is assumed to be known.[14]

Another example is the inviscid Burgers’ equation: Ut + (1/2)*(U2)x = 0. Once again the equation is not completely linear but Ut is a linear term thus making the equation quasi-linear. This is a special case of the Navier-Stokes equations.[15]

Navier-Stokes equations are used to model the behavior of fluids. O.A. Ladyzhenskaya’s work on the Navier-Stokes equations has had a profound effect on the field of Fluid Dynamics.[16] She has inspired numerous papers and has received some recognition such as conferences held in her honor[17] and contributions dedicated to her.[18] Her father encouraged her along with many of her colleages.[19] Their encouragement and inspiration were not fruitless. Future generations may very well aknowledge Ol’ga Ladyzhenskaya as one of the most influential people in her field.

Bibliography:

3) Ladyzhenskaya, O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. Linear and Quailinear               Equations of Parabolic Type.  Translated by Smith, S.. American Mathematical Society. Providence, Rhode Island. 1968.

4) Farlow, Stanley J.. Partial Differential Equations for Scientists and Engineers. Dover Publications, Inc.. New York. 1982.

5) Discussion with Dr. Eric S. Marland, Assistant Professor of Mathematical Sciences at Appalachian State University.

6) Kreiss, Heinz-Otto; Lorenz, Jens. Initial-Boundary Value Problems and the Navier-Stokes Equations. Academic press INC.. Boston. 1989.

10) Ladyzhenskaya, O.A.. The Boundary Value Problems of Mathematical Physics. Translated by Jack Lohwater. Springer-Verlag, New York. 1985.

[5] Ladyzhenskaya, O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. Linear and Quailinear Equations of Parabolic Type.  Translated by Smith, S.. American Mathematical Society. Providence, Rhode Island. 1968

[6] Farlow, Stanley J.. Partial Differential Equations for Scientists and Engineers. Dover Publications, Inc.. New York. 1982.

[7] Ibid.

[8] Ibid.

[9] Ladyzhenskaya, O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. Linear and Quailinear Equations of Parabolic Type.  Translated by Smith, S.. American Mathematical Society. Providence, Rhode Island. 1968

[10]  Discussion with Dr. Eric S. Marland, Assistant Professor of Mathematical Sciences at Appalachian State University.

[11] Ladyzhenskaya, O.A.. Solonnikov, V.A.. Ural’ceva, N. N.. Linear and Quailinear Equations of Parabolic Type.  Translated by Smith, S.. American Mathematical Society. Providence, Rhode Island. 1968

[12] Discussion with Dr. Eric S. Marland, Assistant Professor of Mathematical Sciences at Appalachian State University.

[13] Ibid.

[14] Kreiss, Heinz-Otto; Lorenz, Jens. Initial-Boundary Value Problems and the Navier-Stokes Equations. Academic press INC.. Boston. 1989.

[15] Ibid.

`[17] http://pore.csc.fi/math_topics/Mail/NANET98-2/msg00033.html`
`    http://www.mat.uni.torun.pl/~tmna/htmls/archives/vol-9-1.html`

[19] Ladyzhenskaya, O.A.. The Boundary Value Problems of Mathematical Physics. Translated by Jack Lohwater. Springer-Verlag, New York. 1985.