The Riemann problem in gas dynamics
Randolph G.
Smith
1-50
Abstract: We consider the Riemann problem (R.P.) for the $3\, \times \, 3$ system of gas dynamics equations in a single space variable. We assume that the specific internal energy $e = e(v,\,s)$ (s = specific entropy, v = specific volume) satisfies the usual hypotheses, $ {p_v}\, < \,0,\,{p_{vv}}\, > \,0,\,{p_s}\, > \,0\,(p\, = \, - \,{e_v}\, =$ pressure); we also assume some reasonable hypotheses about the asymptotic behavior of e. We call functions e satisfying these hypotheses energy functions Theorem 1. For any initial data $ ({U_l},\,{U_r})\,({U_l}\, = \,({v_l},\,{p_l},\,{u_l})$, ${U_r}\, = \,({v_r},\,{p_r},\,{u_r})$, u = flow velocity), the R. P. has a solution. We introduce two conditions: \begin{displaymath}\begin{array}{*{20}{c}}\tag{$\text{(I)}$} {\frac{\partial } {... ...ant \frac{{{p^2}}} {{2e}}} & {(v,\,e\, > \,0),} \end{array} \end{displaymath} \begin{displaymath}\begin{array}{*{20}{c}}\tag{$\text{(II)}$} {\frac{\partial }{... ... \geqslant - \frac{p} {2}} & {(v,\,p\, > \,0).} \end{array} \end{displaymath} Theorem 2. (I) is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data. (I) is implied by the known condition $\displaystyle {\frac{\partial } {{\partial v}}e(v,p) > 0} \qquad (v,p > 0),$ ($(\ast)$) which holds for all usual gases. (I) implies (II). We construct energy functions e that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*). In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.
Fourier inversion for unipotent invariant integrals
Dan
Barbasch
51-83
Abstract: Consider G a semisimple Lie group and $\Gamma\, \subseteq \,G$ a discrete subgroup such that ${\text{vol(}}G/\Gamma )\, < \,\infty$. An important problem for number theory and representation theory is to find the decomposition of ${L^2}(G/\Gamma )$ into irreducible representations. Some progress in this direction has been made by J. Arthur and G. Warner by using the Selberg trace formula, which expresses the trace of a subrepresentation of $ {L^2}(G/\Gamma )$ in terms of certain invariant distributions. In particular, measures supported on orbits of unipotent elements of G occur. In order to obtain information about representations it is necessary to expand these distributions into Fourier components using characters of irreducible unitary representations of G. This problem is solved in this paper for real rank $ G\, =\, 1$. In particular, a relationship between the semisimple orbits and the nilpotent ones is made explicit generalizing an earlier result of R. Rao.
Degrees of exceptional characters of certain finite groups
Harvey I.
Blau
85-96
Abstract: Let G be a finite group whose order is divisible by a prime p to the first power only. Restrictions beyond the known congruences modulo p are shown to hold for the degrees of the exceptional characters of G, under the assumptions that either all $ p'$-elements centralizing a Sylow p-subgroup are in fact central in G and there are at least three conjugacy classes of elements of order p, or that the characters in question lie in the principal p-block. Results of Feit and the author are thereby generalized.
On the existence of nonregular ultrafilters and the cardinality of ultrapowers
Menachem
Magidor
97-111
Abstract: Assuming the consistency of huge cardinals, we prove that ${\omega _3}$ can carry an ultrafilter D such that $ {\omega _1}^{{\omega _3}}/D$ has cardinality $ {\omega _3}$. (Hence D is not $({\omega _3},\,{\omega _1})$ regular.) Similarly ${\omega _2}$ can carry an ultrafilter D such that $ {\omega ^{{\omega _2}}}/D$ has cardinality $ {\omega _2}$. (Hence D is not $({\omega _2},\,\omega )$ regular.)
The commuting block maps problem
Ethan M.
Coven;
G. A.
Hedlund;
Frank
Rhodes
113-138
Abstract: A block map is a map $f:\,{\{ {\text{0}},\,{\text{1}}\} ^n}\, \to \,\{ 0,\,1\}$ for some $n\, \geqslant \,1$. A block map f induces an endomorphism $ {f_\infty }$ of the full 2-shift $ (X,\,\sigma )$. We define composition of block maps so that $ {(f \circ g)_\infty }\, = \,{f_\infty } \circ {g_\infty }$. The commuting block maps problem (for f) is to determine $\mathcal{C}(f)\, = \,\{ g\vert f \circ g\, = \,g \circ f\}$. We solve the commuting block maps problem for a number of classes of block maps.
Balanced Cohen-Macaulay complexes
Richard P.
Stanley
139-157
Abstract: A balanced complex of type $ ({a_1},\ldots,{a_m})$ is a finite pure simplicial complex $\Delta$ together with an ordered partition $ ({V_1},\ldots,{V_m})$ of the vertices of $\Delta$ such that card $({V_i}\, \cap \,F)\, = \,{a_i}$, for every maximal face F of $\Delta$. If ${\mathbf{b}}\, = \,({b_1},\ldots,{b_m})$, then define ${f_\textbf{b}}(\Delta )$ to be the number of $F\, \in \,\Delta$ satisfying card $({V_i}\, \cap \,F)\, = \,{b_i}$. The formal properties of the numbers ${f_\textbf{b}}(\Delta )$ are investigated in analogy to the f-vector of an arbitrary simplicial complex. For a special class of balanced complexes known as balanced Cohen-Macaulay complexes, simple techniques from commutative algebra lead to very strong conditions on the numbers ${f_\textbf{b}}(\Delta )$. For a certain complex $\Delta (P)$ coming from a poset P, our results are intimately related to properties of the Möbius function of P.
On the zeros of Jacobi polynomials $P\sb{n}\sp{(\alpha \sb{n},\beta \sb{n})}(x)$
D. S.
Moak;
E. B.
Saff;
R. S.
Varga
159-162
Abstract: If ${r_n}$ and ${s_n}$ denote, respectively, the smallest and largest zeros of the Jacobi polynomial $P_n^{({\alpha _n},{\beta _n})}$, where ${\alpha _n}\, > \,1$, ${\beta _n}\, - \,1$, and if ${\lim _{n \to \infty }}\,{\alpha _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,a$ and if $ {\lim _{n \to \infty }}{\beta _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,b$, then the numbers ${r_{a,b}}$ and ${s_{a,b}}$ are determined where $\displaystyle \mathop {\lim }\limits_{n \to \infty } \,{r_{n\,}}\, = \,{r_{a,b}},\mathop {\lim }\limits_{n \to \infty } \,{s_{n\,}}\, = \,{s_{a,b}}$ . Furthermore, the zeros of $\{ P_n^{({\alpha _n},{\beta _n})}(x)\} _{n = 0}^\infty$ are dense in $ [{r_{a,b}},{s_{a,b}}]$.
The sharpness of Lorentz's theorem on incomplete polynomials
E. B.
Saff;
R. S.
Varga
163-186
Abstract: For any fixed $ \theta$ with $0 < \theta < 1$, G. G. Lorentz recently showed that bounded sequences $\{\Sigma_{\theta {n_i} \leqslant k \leqslant {n_i}} {{a_k}(i){{(1 + t)}^k}\} _{i = 1}^\infty }$ of incomplete polynomials on $[ - 1, + 1]$ tend uniformly to zero on closed intervals of $[ - 1,\Delta (\theta ))$, where $2{\theta ^2} - 1 \leqslant \Delta (\theta ) < 2\theta - 1$. In this paper, we show that $\Delta (\theta ) = 2{\theta ^2} - 1$ is best possible, and that the geometric convergence to zero of such sequences on closed intervals $[{t_0},{t_1}]$ can be precisely bounded above as a function of ${t_j}$ and $\theta$. Extensions of these results to the complex plane are also included.
Focal points for a linear differential equation whose coefficients are of constant signs
Uri
Elias
187-202
Abstract: The differential equation considered is ${y^{(n)}} + \Sigma {{p_i}(x){y^{(i)}}} = 0$, where ${\sigma _i}{p_i}(x) \geqslant 0,i = 0,\ldots,n - 1,{\sigma _i} = \pm 1$. The focal point $ \zeta (a)$ is defined as the least value of s, $s > a$, such that there exists a nontrivial solution y which satisfies $ {y^{(i)}}(a) = 0,{\sigma _i}{\sigma _{i + 1}} > 0$ and ${y^{(i)}}(s) = 0$, ${\sigma _i}{\sigma _{i + 1}} < 0$. Our method is based on a characterization of $\zeta (a)$ by solutions which satisfy $ {\sigma _i}{y^{(i)}} > 0,i = 0,\ldots,n - 1$, on $[a,b]$, $ b < \zeta (a)$. We study the behavior of the function $\zeta$ and the dependence of $ \zeta (a)$ on ${p_0},\ldots,{p_{n - 1}}$ when at least a certain ${p_i}(x)$ does not vanish identically near a or near $\zeta (a)$. As an application we prove the existence of an eigenvalue of a related boundary value problem.
Cell-like $0$-dimensional decompositions of $E\sp{3}$
Michael
Starbird
203-215
Abstract: Let G be a cell-like, 0-dimensional upper semicontinuous decomposition of ${E^3}$. It is shown that if $\Gamma$ is a tame 1-complex which is a relatively closed subset of a saturated open set U whose boundary misses the nondegenerate elements of G, then there is a homeomorphism $h:{E^3} \to {E^3}$ so that $h\vert{E^3} - U = {\text{id}}$ and $h(\Gamma )$ misses the nondegenerate elements of G. This theorem implies a disjoint disk type criterion for shrinkability of G. This criterion in turn provides a direct proof of the recent result of Starbird and Woodruff that if G is an u.s.c. decomposition of ${E^3}$ into points and countably many cellular, tamely embedded polyhedra, then ${E^3}/G$ is homeomorphic to ${E^3}$.
Center-by-metabelian groups of prime exponent
Jay I.
Miller
217-224
Abstract: We show that a center-by-metabelian group of prime exponent p is nilpotent of class at most p, and this result is best possible. The proof is based on techniques dealing with varieties of groups.