Dept. of Mathematics
http://hdl.handle.net/2142/16339
Mon, 30 Mar 2015 07:07:16 GMT2015-03-30T07:07:16ZOn the topology of the spaces of coverings
http://hdl.handle.net/2142/72793
On the topology of the spaces of coverings
The spaces of coverings (SoC) arise from various configuration spaces of the covering problems. We study the topology of these spaces for different domains and covering agents. In particular, we study the SoCs for grid domains and metric trees covered by balls. Characterizations of their topology are given for analysis of the topological complexities (the Betti numbers in all dimensions). We also study the spaces of fort coverings, a topological approximation of the SoCs capturing some essential features of the SoCs. The applications of the SoCs abound, in the end we present the feedback control on the SoC for metric tree coverage as a case study.
Covering; Topology; Configuration space; Feedback control
Wed, 21 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2142/727932015-01-21T00:00:00ZIdeals of powers of linear forms
http://hdl.handle.net/2142/72784
Ideals of powers of linear forms
This thesis addresses two closely related problems about ideals of powers of linear forms.
In the first chapter, we analyze a problem from spline theory, namely to compute the dimension of the
vector space of tri-variate splines on a special class of tetrahedral complexes, using ideals of powers of linear forms. By Macaulay's inverse system, this class of ideals is closely related to ideals of fat points.
In the second chapter, we approach a conjecture of Postnikov and Shapiro concerning the minimal free
resolutions of a class of ideals of powers of linear forms in n variables which are constructed from complete graphs on n + 1 vertices. This statement was also conjectured by Schenck in the special case of n = 3. We
provide two different approaches to his conjecture. We prove the conjecture of Postnikov and Shapiro under the additional condition that certain modules are free.
Splines; fat points; free resolutions; powers of linear forms
Wed, 21 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2142/727842015-01-21T00:00:00ZHypergeometric functions, continued fractions for products of gamma functions, and q-analogues
http://hdl.handle.net/2142/72779
Hypergeometric functions, continued fractions for products of gamma functions, and q-analogues
Some of the most interesting of Ramanujan's continued fraction identities are those involving
ratios of Gamma functions in Chapter 12 of his second notebook. This thesis develops
a method for deriving such identities, using hypergeometric functions as the main tool.
We begin by deriving a continued fraction identity, use it to prove Ramanujan's Entry 34,
and then use the method to obtain new identities and relate them to two of Ramanujan's
identities. We next prove Ramanujan's Entries 36 and 39. Finally, we rework the method
for use with basic hypergeometric functions and use it to find q-analogues of the earlier new
results.
hypergeometric functions; continued fractions; gamma function; basic hypergeometric functions; q-analogue; q-series; Ramanujan; Ramanujan's notebooks
Wed, 21 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2142/727792015-01-21T00:00:00ZRamsey theory and its application
http://hdl.handle.net/2142/72752
Ramsey theory and its application
In this dissertation, we study three problems about Ramsey theory. First, we prove a self-dual Ramsey theorem
for parameter systems which is a generalization of the self-dual Ramsey theorem developed by Solecki.
Second, we prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of
linear orders extending the partial order. Third, we study the relations between Ramsey theorems which
have points in common with the classical Ramsey theorem and the dual Ramsey theorem by the concept of
interpretation.
Ramsey theory; abstract Ramsey theorem; Parameter system; Self-Dual; Partial order; Linear order; Interpretation
Wed, 21 Jan 2015 00:00:00 GMThttp://hdl.handle.net/2142/727522015-01-21T00:00:00ZCellular Games
http://hdl.handle.net/2142/72547
Cellular Games
A cellular game is a dynamical system in which cells, placed in some discrete structure, are regarded as playing a game with their immediate neighbors. Individual strategies may be either deterministic or stochastic. Strategy success is measured according to some universal and unchanging criterion. Successful strategies persist and spread; unsuccessful ones disappear.; In this thesis, two cellular game models are formally defined, and are compared to cellular automata. Computer simulations of these models are presented.; Conditions providing maximal average cell success, on one and two-dimensional lattices, are examined. It is shown that these conditions are not necessarily stable; and an example of such instability is analyzed. It is also shown that Nash equilibrium strategies are not necessarily stable.; Finally, a particular kind of zero-depth, two-strategy cellular game is discussed; such a game is called a simple cellular game. It is shown that if a simple cellular game is left/right symmetric, and if there are initially only finitely many cells using one strategy, the zone in which this strategy occurs has probability 0 of expanding arbitrarily far in one direction only. With probability 1, it will either expand in both directions or disappear.; Computer simulations of such games are presented. These experiments suggest the existence of two different kinds of asymptotic behavior.
Mathematics
Sat, 01 Jan 1994 00:00:00 GMThttp://hdl.handle.net/2142/725471994-01-01T00:00:00ZLine Bundles on Projective Homogeneous Spaces
http://hdl.handle.net/2142/72545
Line Bundles on Projective Homogeneous Spaces
The topic of my thesis is the geometry of projective homogeneous spaces G/H for a semisimple algebraic group G in characteristic p $>$ 0, where H is a subgroup scheme containing a Borel subgroup B. In characteristic p $>$ 0 there are an infinite number of subgroup schemes containing B--the reduced ones are the ordinary parabolic subgroups P $\supseteq$ B. Examples of non-reduced parabolic subgroup schemes are extensions of B by Frobenius kernels of P. Using an algebraic analogue of the fixed point formula of Atiyah and Bott, we give a formula for the Euler character of a homogeneous line bundle on G/H generalizing Weyl's character formula. The canonical line bundle on G/H is rarely negative ample. A consequence of this is, that G/H is Frobenius split only when H is an extension of a parabolic subgroup by a Frobenius kernel of G. In an attempt to generalize Kempf's vanishing theorem we discovered, that G/H with H non-reduced can be used to construct new counterexamples to Kodaira's vanishing theorem in characteristic p $>$ 0. For G of type $D\sb5$ and H the extension of B by the first Frobenius kernel of $P\sb\alpha$, where $P\sb\alpha$ is the minimal parabolic subgroup having (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)as its only positive root, we give an example of an ample line bundle $\cal{L}$ on G/H such that ${\cal L}\otimes\omega\sb{G/H}$ has negative Euler characteristic. This also answers an old question of Raynaud.
Mathematics
Fri, 01 Jan 1993 00:00:00 GMThttp://hdl.handle.net/2142/725451993-01-01T00:00:00ZEffective Versions of Ramsey's Theorem
http://hdl.handle.net/2142/72544
Effective Versions of Ramsey's Theorem
Ramsey's Theorem states that if $P = \{C\sb1,\...,C\sb{n}\}$ is a partition of ($\omega\rbrack\sp{k}$ (the set of all unordered k-tuples of natural numbers) into finitely many classes, then there exists an infinite set A which is homogeneous for P; i.e., there exists $j, 1 \le j \le n,$ such that all k-tuples from A are in $C\sb{j}.$ Let H(P) denote the set of all infinite homogeneous sets for a partition P. We consider the degrees of unsolvability and arithmetical definability properties of sets in H(P) for recursive and recursively enumerable partitions P.; We use the notion of effective $\Delta\sbsp{1}{0}$-immunity to show that there exists a recursive partition $P = \{C\sb1, C\sb2\}$ of ($\omega\rbrack\sp2$ such that for every $A \in H(P), A$ is effectively $\emptyset\sp\prime$-immune, and hence every $\Pi\sbsp{2}{0}$ set $A \in H(P)$ is such that $\emptyset\sp\prime\sp\prime \le\sb{T} A \oplus \emptyset\sp\prime.$ From this it follows that every $\Pi\sbsp{2}{0}$ 2-cohesive set is of degree 0$\sp\prime\sp\prime,$ where an infinite set A is 2-cohesive if for each r.e. partition $P = \{C\sb1, C\sb2\}$ of ($\omega\rbrack\sp2,$ there exists a finite set F such that $A - F \in H(P).$; We begin a study of r.e. partitions and show that for every r.e. partition $P = \{C\sb1, C\sb2\}$ of ($\omega\rbrack\sp2,$ there exists $A \in H(P)$ such that $A\sp\prime \le\sb{T} \emptyset\sp\prime\sp\prime.$ In addition, we show that every r.e. stable partition $P = \{C\sb1, C\sb2\}$ of ($\omega\rbrack\sp3$ has a $\Delta\sbsp{4}{0}$ set $A \in H(P),$ while there exists a recursive stable partition $P = \{C\sb1, C\sb2\}$ of ($\omega\rbrack\sp3$ with no $\Delta\sbsp{3}{0}$ set $A \in H(P).$ (A partition $P = \{C\sb1,\...,C\sb{n}\}$ of ($\omega\rbrack\sp{k+1}$ is stable if for all $D \in \lbrack \omega\rbrack\sp{k},$ there exists $i, 1 \le i \le n,$ such that $D \ \cup \{a\} \in C\sb{i}$ for sufficiently large a.); We give Jockusch's proof of Seetapun's recent result that for every recursive partition $P = \{C\sb1, C\sb2\}$ of ($\omega\rbrack\sp2,$ there exists $A \in H(P)$ such that $\emptyset\sp\prime \not\leq\sb{T} A.$ We extend Seetapun's result by establishing arithmetic bounds for such a set A. We discuss applications of these results to Reverse Mathematics and to introreducible sets.
Mathematics
Fri, 01 Jan 1993 00:00:00 GMThttp://hdl.handle.net/2142/725441993-01-01T00:00:00ZExtremal Results and Algorithms for Degree Sequences of Graphs
http://hdl.handle.net/2142/72546
Extremal Results and Algorithms for Degree Sequences of Graphs
A 2-multigraph is a loopless multigraph with maximum multiplicity 2; pairs of vertices induce 0, 1, or 2 edges. A 2-multigraph is parsimonious if it has the minimum number of single edges (multiplicity 1) among all 2-multigraphs with the same degree sequence. In every parsimonious 2-multigraph, the subgraph of single edges consists of isolated stars and possibly one component that is a triangle. We prove the conjecture of Brualdi and Michael that for any fixed degree sequence, either every parsimonious 2-multigraph with those vertex degrees has a triangle of single edges, or no such parsimonious 2-multigraph has a triangle of single edges.; We determine for a planar graph on n vertices the maximum values for the following: (1) The sum of the m largest vertex degrees. (2) For k $\ge$ 12, the number of vertices of degree at least k and the sum of the degrees of vertices with degree at least k. In addition, for 6 $\le$ k $\le$ 11, we determine upper and lower bounds for the latter two values, which match for certain congruence classes of n.; Berge showed that if two graphs G and H have the same degree sequence, then G can be transformed into H by a sequence of elementary degree-preserving transformations. We show that computing the minimum length of such a sequence is an NP-complete problem. In addition we disprove a conjecture of John Gimbel on an analogous result for oriented graphs, and obtain partial results toward a revised conjecture.
Mathematics
Fri, 01 Jan 1993 00:00:00 GMThttp://hdl.handle.net/2142/725461993-01-01T00:00:00ZMaximum Betti Numbers for a Given Hilbert Function
http://hdl.handle.net/2142/72543
Maximum Betti Numbers for a Given Hilbert Function
In his 1927 paper, Macaulay gave a necessary and sufficient condition for a function H to be the Hilbert function of a cyclic module over a polynomial ring $R = k\lbrack x\sb1,\..., x\sb{n}\rbrack$ where k is a field of characteristic 0. He constructed the lex-segment ideal which has a given Hilbert function through degree d, and he showed that this ideal gives a lower bound for H(d + 1). From this, Macaulay showed that the lex-segment ideal has the most generators (that is, the largest first Betti number) of a homogeneous ideal with that Hilbert function. This thesis proves that the lex-segment ideal having a given Hilbert function always has the largest graded Betti numbers of any homogeneous ideal with that Hilbert function. We also construct a submodule of a given free module which has graded Betti numbers at least as large as the graded Betti numbers of any other submodule of the given free module with the same Hilbert function. Moreover, this submodule gives a lower bound on the Hilbert function in degree d + 1 for any submodule with the given Hilbert function through degree d, thus generalizing Macaulay's theorem above.
Mathematics
Fri, 01 Jan 1993 00:00:00 GMThttp://hdl.handle.net/2142/725431993-01-01T00:00:00ZOn P-Radical P-Blocks of Group Algebras
http://hdl.handle.net/2142/72540
On P-Radical P-Blocks of Group Algebras
p-Radical p-blocks of finite group algebras are studied. Much of the p-radical group theory is generalized to p-blocks through R. Knorr's work on simple induction and restriction pairs. In addition, several results concerning such blocks are proved.; The most interesting fact known about p-radical groups is a result of T. Okuyama, who showed that such groups are p-solvable. It turns out that the same conclusion could be reached assuming much less. In fact, it is shown that finite groups, whose principal p-blocks are p-radical, are p-solvable. This is not in general true for non-principal blocks as illustrated by some examples. Nevertheless, in any case, simple modules in p-radical blocks behave somewhat like those in blocks of p-solvable groups.; A p-block, in which every (modular) simple module has vertices contained in the kernel, is characterized. Such a block enjoys a property much stronger than p-radicality. If, in addition, every simple module in this block has vertices conjugate to a defect group of the block, further characterizations are derived. As a consequence, some characterizations of p-length 1 p-solvable groups, in terms of the ordinary irreducible characters of the principal p-blocks, are given.
Mathematics
Fri, 01 Jan 1993 00:00:00 GMThttp://hdl.handle.net/2142/725401993-01-01T00:00:00Z