The conjugacy problem for groups of alternating prime tame links is polynomial-time
Karin
Johnsgard
857-901
Abstract: An alternating projection of a prime link can to used to produce a group presentation (of the link group under free product with the infinite cyclic group) with some useful peculiarities, including small cancellation conditions. In this presentation, conjugacy diagrams are shown to have the form of a tiling of squares in the Euclidean plane in one of a limited number of shapes. An argument based on the shape of the link projection is used to show that the tiling requires no more square tiles than a linear function of word length (with constant multiple based on the number of link crossings). It follows that the computation time for testing conjugacy of two group elements (previously known to be no worse than exponential) is bounded by a cubic polynomial. This bounds complexity in the original link group.
A Characterization of Finitely Decidable Congruence Modular Varieties
Pawel
M.
Idziak
903-934
Abstract: For every finitely generated, congruence modular variety $\mathcal {V}$ of finite type we find a finite family $\cal R$ of finite rings such that the variety $\mathcal {V}$ is finitely decidable if and only if $\mathcal {V}$ is congruence permutable and residually small, all solvable congruences in finite algebras from $\mathcal {V}$ are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra $\mathbf {A}$ from $\mathcal {V}$ is comparable with all congruences of $\mathbf {A}$, each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring $R$ from $\cal R$, the variety of $R$-modules is finitely decidable.
The homology representations of the $k$-equal partition lattice
Sheila
Sundaram;
Michelle
Wachs
935-954
Abstract: We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set $\{1,2,\ldots ,n\}$ obtained by restricting block sizes to the set $\{1,k,k+1,\ldots \}$. A plethystic formula for the generating function of the Frobenius characteristic of the representation is given. We combine techniques from the theory of nonpure shellability, recently developed by Björner and Wachs, with symmetric function techniques, developed by Sundaram, for determining representations on the homology of subposets of the partition lattice.
Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary
Mingxin
Wang;
Yonghui
Wu
955-971
Abstract: This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta \ln(1+u)$ in $\Omega \times (0, +\infty )$, \begin{displaymath}\frac {\partial \ln(1+u)}{\partial n}=\sqrt {1+u}(\ln (1+u))^{\alpha } \quad \text{on} \partial \Omega \times (0, +\infty ),\end{displaymath} and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha \geq 0$ is a parameter, $\Omega \subset\mathbb {R}^N$ is a bounded smooth domain. After pointing out the mistakes in Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), 317-326, by N. Wolanski, which claims that, for $\Omega =B_R$ the ball of $\mathbb {R}^N$, the positive solution exists globally if and only if $\alpha \leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha \leq {1/2}$.
Examples of asymptotic $\ell_1$ Banach spaces
S.
A.
Argyros;
I.
Deliyanni
973-995
Abstract: Two examples of asymptotic $\ell _{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson's.
Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities
T.
Figiel;
P.
Hitczenko;
W.
B.
Johnson;
G.
Schechtman;
J.
Zinn
997-1027
Abstract: The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the $p$ moment of a sum of independent symmetric random variables to that of the $p$ and $2$ moments of the individual variables, are computed in the range $2<p\le 4$. This complements the work of Utev who has done the same for $p>4$. The qualitative nature of the extreme cases turns out to be different for $p<4$ than for $p>4$. The method developed yields results in some more general and other related moment inequalities.
The Mizohata complex
Abdelhamid
Meziani
1029-1062
Abstract: This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ $\omega$ at $0\in \mathbb {R}^{n+1}$ of a $\mathbb {C}$-valued, formally integrable ($\omega \wedge d\omega =0$), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for $(q-1)$-forms $u$, of the equation \begin{equation*}du\wedge \omega =\eta \wedge \omega ,\end{equation*} where $\eta$ is a given $q$-form satisfying $d\eta \wedge \omega =0$ is estimated in terms of the De Rham cohomology relative to $\omega$
Structure of Lorentzian tori with a killing vector field
Miguel
Sánchez
1063-1080
Abstract: All Lorentzian tori with a non-discrete group of isometries are characterized and explicitly obtained. They can lie into three cases: (a) flat, (b) conformally flat but non-flat, and (c) geodesically incomplete. A detailed study of many of their properties (including results on the logical dependence of the three kinds of causal completeness, on geodesic connectedness and on prescribed curvature) is carried out. The incomplete case is specially analyzed, and several known examples and results in the literature are generalized from a unified point of view.
Signed Quasi-Measures
D.
J.
Grubb
1081-1089
Abstract: Let $X$ be a compact Hausdorff space and let $\mathcal A$ denote the subsets of $X$ which are either open or closed. A quasi-linear functional is a map $\rho :C(X)\rightarrow \mathbf R$ which is linear on singly generated subalgebras and such that $|\rho (f)|\leq M\|f\|$ for some $M<\infty$. There is a one-to-one correspondence between the quasi-linear functional on $C(X)$ and the set functions $\mu :\mathcal A \rightarrow \mathbf R$ such that i) $\mu (\emptyset )=0$, ii) If $A,B,A\cup B\in \mathcal A$ with $A$ and $B$ disjoint, then $\mu (A\cup B)=\mu (A)+\mu (B)$, iii) There is an $M<\infty$ such that whenever $\{U_\alpha \}$ are disjoint open sets, $\displaystyle \sum |\mu (U_\alpha )|\leq M$, and iv) if $U$ is open and $\varepsilon >0$, there is a compact $K\subseteq U$ such that whenever $V\subseteq U\setminus K$ is open, then $|\mu (V)|<\varepsilon$. The space of quasi-linear functionals is investigated and quasi-linear maps between two $C(X)$ spaces are studied.
Hilbert-Kunz functions and Frobenius functors
Shou-Te
Chang
1091-1119
Abstract: We study the asymptotic behavior as a function of $e$ of the lengths of the cohomology of certain complexes. These complexes are obtained by applying the $e$-th iterated Frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module. The rings involved here all have positive prime characteristic. For the zeroth homology, these functions also contain the class of Hilbert-Kunz functions that a number of other authors have studied. This asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper: these are defined in terms of the Krull dimensions of the Matlis duals of the local cohomology of the module. There is a more detailed study of this behavior when the given complex is a Koszul complex.
A bound on the geometric genus of projective varieties verifying certain flag conditions
Vincenzo
Di Gennaro
1121-1151
Abstract: Fix integers $n,r,s_{1},...,s_{l}$ and let $\mathcal {S}(n,r;s_{1},...,s_{l})$ be the set of all integral, projective and nondegenerate varieties $V$ of degree $s_{1}$ and dimension $n$ in the projective space $\mathbf {P}^{r}$, such that, for all $i=2,...,l$, $V$ does not lie on any variety of dimension $n+i-1$ and degree $<s_{i}$. We say that a variety $V$ satisfies a flag condition of type $(n,r;s_{1},...,s_{l})$ if $V$ belongs to $\mathcal {S}(n,r;s_{1},...,s_{l})$. In this paper, under the hypotheses $s_{1}>>...>>s_{l}$, we determine an upper bound $G^{h}(n,r;s_{1},...,s_{l})$, depending only on $n,r,s_{1},...,s_{l}$, for the number $G(n,r;s_{1},...,s_{l}):= {max} {\{} p_{g}(V) : V\in \mathcal {S}(n,r;s_{1},...,s_{l}){\}}$, where $p_{g}(V)$ denotes the geometric genus of $V$. In case $n=1$ and $l=2$, the study of an upper bound for the geometric genus has a quite long history and, for $n\geq 1$, $l=2$ and $s_{2}=r-n$, it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data $n,r,s_{1},...,s_{l}$. For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension $n$ in $\mathbf {P}^{n+3}$ as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in $\mathbf {P}^{5}$. Next we discuss how far $G^{h}(n,r;s_{1},...,s_{l})$ is from $G(n,r;s_{1},...,s_{l})$ and show a sort of lifting theorem which states that, at least in certain cases, the varieties $V\in \mathcal {S}(n,r;s_{1},...,s_{l})$ of maximal geometric genus $G(n,r;s_{1},...,s_{l})$ must in fact lie on a flag such as $V=V_{s_{1}}^{n}\subset V_{s_{2}}^{n+1}\subset ...\subset V_{s_{l}}^{n+l-1}\subset {\mathbf {P}^{r}}$, where $V^{j}_{s}$ denotes a subvariety of $\mathbf {P}^{r}$ of degree $s$ and dimension $j$. We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.
Strassen theorems for a class of iterated processes
Endre
Csáki;
Antónia
Földes;
Pál
Révész
1153-1167
Abstract: A general direct Strassen theorem is proved for a class of stochastic processes and applied for iterated processes such as $W(L_t)$, where $W(\cdot )$ is a standard Wiener process and $L_.$ is a local time of a Lévy process independent from $W(\cdot )$.
Mean-boundedness and Littlewood-Paley for separation-preserving operators
Earl
Berkson;
T.
A.
Gillespie
1169-1189
Abstract: Suppose that $(\Omega ,\mathcal {M},\mu )$ is a $\sigma$-finite measure space, $1<p<\infty$, and $T: L^{p}(\mu )\to L^{p}(\mu )$ is a bounded, invertible, separation-preserving linear operator such that the linear modulus of $T$ is mean-bounded. We show that $T$ has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in $L^{p}(\mu )$ which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for $T$ is shown to produce a strongly countably spectral measure on the ``dyadic sigma-algebra'' of $\mathbb {T}$, and to furnish $L^{p}(\mu )$ with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for $\ell ^{p}(\mathbb {Z})$.
Anticanonical Rational Surfaces
Brian
Harbourne
1191-1208
Abstract: A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of $\mathbf {P}^{2}$ defined by complete ideals.
The image of the $BP$ Thom map for Eilenberg-MacLane spaces
Hirotaka
Tamanoi
1209-1237
Abstract: Fundamental classes in $BP$ cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from $BP$ cohomology to mod-$p$ cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod-$p$ cohomology. This subalgebra in mod $p$ cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that $BP$ cohomology determines Morava $K$ cohomology for Eilenberg-MacLane spaces.
Estimation of spectral gap for elliptic operators
Mu-Fa
Chen;
Feng-Yu
Wang
1239-1267
Abstract: A variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time. The main known results are either recovered or improved. A large number of new examples with sharp estimate are illustrated. Moreover, as an application of the march coupling, the Poincaré inequality with respect to the absolute distribution of the process is also studied.
Rings with FZP
P.
R.
Fuchs;
C.
J.
Maxson;
G.
F.
Pilz
1271-1284
Abstract: In this paper we investigate the problem of characterizing those rings \begin{math}R\end{math} such that every nonzero polynomial with coefficients from \begin{math}R\end{math} has a finite number of zeros in \begin{math}R\end{math}. Particular attention is directed to the class of skew polynomial domains.