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9 results on '"Lattice Dimension"'

1. Pure-injective modules over tubular algebras and string algebras

Author

Harland, Richard James, Prest, Michael, and Puninskiy, Gennady

Subjects
510, String Algebras, Tubular Algebras, Lattice Dimension, Pure-Injective Modules, Superdecomposable modules, slope, Wide lattices, Infinite dimensional string modules
Abstract

We show that, for any tubular algebra, the lattice of pp-definable subgroups of the direct sum of all indecomposable pure-injective modules of slope r has m-dimension 2 if r is rational, and undefined breadth if r is irrational- and hence that there are no superdecomposable pure-injectives of rational slope, but there are superdecomposable pure-injectives of irrational slope, if the underlying field is countable.We determine the pure-injective hull of every direct sum string module over a string algebra. If A is a domestic string algebra such that the width of the lattice of pp-formulas has defined breadth, then classify "almost all" of the pure-injective indecomposable A-modules.

Published
2011

3. Photonic Band Gap Structure for Microwave Applications.

Author

Sahoo, Pooja

Subjects
*PHOTONIC band gap structures, *MICROWAVE integrated circuits, *ELECTRIC inductance, *ELECTRIC capacity, *COMPUTER simulation
Abstract

A new Photonic Band Gap (PBG) unit cell is proposed to improve the value of effective components like inductance and capacitance. It makes it easy to control the cutoff frequency by increasing the effective parameters. The proposed periodic PBG structure provides excellent cutoff and stopband characteristics at different center frequencies. PBG structure shows improved effective inductance using six cells structure, was fabricated with identical periodic and different dimensions. Measurements analyzed on the different fabricated PBG circuits show that the cutoff and stopband center frequency characteristics depend on the physical dimension of the proposed PBG unit cell. [ABSTRACT FROM AUTHOR]

Published
2018

6. Weak sense of direction labelings and graph embeddings

Author

Cheng, Christine T. and Suzuki, Ichiro

Subjects
*EMBEDDINGS (Mathematics), *GRAPH theory, *ISOMORPHISM (Mathematics), *CAYLEY graphs, *GRAPH labelings, *CODING theory, *HASSE diagrams, *DISTRIBUTIVE lattices
Abstract

Abstract: An edge-labeling for a directed graph has a weak sense of direction (WSD) if there is a function that satisfies the condition that for any node and for any two label sequences and generated by non-trivial walks on starting at , if and only if the two walks end at the same node. The function is referred to as a coding function of . The weak sense of direction number of , WSD, is the smallest integer so that has a WSD-labeling that uses labels. It is known that WSD , where is the maximum outdegree of . Let us say that a function is an embedding from onto if demonstrates that is isomorphic to a subgraph of . We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when is the coding function that naturally accompanies a Cayley graph and has a node that can reach every other node in the graph, then has a WSD-labeling that has as a coding function if and only if can be embedded onto . Additionally, we show that the problem “Given , does have a WSD-labeling that uses a particular coding function ?” is NP-complete even when and are fairly simple. Second, when is a distributive lattice, is its Hasse diagram and is its cover graph, then WSD, where is the smallest integer so that can be embedded onto the -dimensional mesh. Along the way, we also prove that the isometric dimension of is its diameter, and the lattice dimension of is . Our WSD-labelings are poset-based, making use of Birkhoff’s characterization of distributive lattices and Dilworth’s theorem for posets. [ABSTRACT FROM AUTHOR]

Published
2011
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7. Weak sense of direction labelings and graph embeddings

Author

Ichiro Suzuki and Christine T. Cheng

Subjects
Discrete mathematics, Cayley graph, Applied Mathematics, 010102 general mathematics, Distributive lattice, Hasse diagram, 0102 computer and information sciences, Directed graph, Graph embeddings, 01 natural sciences, Sense of direction, Isometric embeddings, Combinatorics, Lattice dimension, Subgroup, Isometric dimension, Distributive property, 010201 computation theory & mathematics, Lattice (order), Discrete Mathematics and Combinatorics, Embedding, 0101 mathematics, Mathematics
Abstract

An edge-labeling @l for a directed graph G has a weak sense of direction (WSD) if there is a function f that satisfies the condition that for any node u and for any two label sequences @a and @a^' generated by non-trivial walks on G starting at u, f(@a)=f(@a^') if and only if the two walks end at the same node. The function f is referred to as a coding function of @l. The weak sense of direction number of G, WSD(G), is the smallest integer k so that G has a WSD-labeling that uses k labels. It is known that WSD(G)>=@D^+(G), where @D^+(G) is the maximum outdegree of G. Let us say that a function @t:V(G)->V(H) is an embedding from G onto H if @t demonstrates that G is isomorphic to a subgraph of H. We show that there are deep connections between WSD-labelings and graph embeddings. First, we prove that when f"H is the coding function that naturally accompanies a Cayley graph H and G has a node that can reach every other node in the graph, then G has a WSD-labeling that has f"H as a coding function if and only if G can be embedded onto H. Additionally, we show that the problem ''Given G, does G have a WSD-labeling that uses a particular coding function f?'' is NP-complete even when G and f are fairly simple. Second, when D is a distributive lattice, H(D) is its Hasse diagram and G(D) is its cover graph, then WSD(H(D))=@D^+(H(D))=d^*, where d^* is the smallest integer d so that H(D) can be embedded onto the d-dimensional mesh. Along the way, we also prove that the isometric dimension of G(D) is its diameter, and the lattice dimension of G(D) is @D^+(H(D)). Our WSD-labelings are poset-based, making use of Birkhoff's characterization of distributive lattices and Dilworth's theorem for posets.

Published
2011
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