201 research outputs found
Spectral convergence of non-compact quasi-one-dimensional spaces
We consider a family of non-compact manifolds X_\eps (``graph-like
manifolds'') approaching a metric graph and establish convergence results
of the related natural operators, namely the (Neumann) Laplacian \laplacian
{X_\eps} and the generalised Neumann (Kirchhoff) Laplacian \laplacian {X_0}
on the metric graph. In particular, we show the norm convergence of the
resolvents, spectral projections and eigenfunctions. As a consequence, the
essential and the discrete spectrum converge as well. Neither the manifolds nor
the metric graph need to be compact, we only need some natural uniformity
assumptions. We provide examples of manifolds having spectral gaps in the
essential spectrum, discrete eigenvalues in the gaps or even manifolds
approaching a fractal spectrum. The convergence results will be given in a
completely abstract setting dealing with operators acting in different spaces,
applicable also in other geometric situations.Comment: some references added, still 36 pages, 4 figure
Recommended from our members
The life and work of Major Percy Alexander MacMahon
This thesis describes the life and work of the mathematician Major Percy Alexander MacMahon (1854 - 1929). His early life as a soldier in the Royal Artillery and events which led to him embarking on a career in mathematical research and teaching are dealt with in the first two chapters. Succeeding chapters explain the work in invariant theory and partition theory which brought him to the attention of the British mathematical community and eventually resulted in a Fellowship of the Royal Society, the presidency of the London Mathematical Society, and the award of three prestigious mathematical medals and four honorary doctorates. The development and importance of his recreational mathematical work is traced and discussed. MacMahon's career in the Civil Service as Deputy Warden of the Standards at the Board of Trade is also described. Throughout the thesis, his involvement with the British Association for the Advancement of Science and other scientific organisations is highlighted. The thesis also examines possible reasons why MacMahon's work, held in very high regard at the time, did not lead to the lasting fame accorded to some of his contemporaries. Details of his personal and social life are included to give a picture of MacMahon as a real person working hard to succeed in a difficult context
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet
The search for a theory of the S-Matrix has revealed surprising geometric
structures underlying amplitudes ranging from the worldsheet to the
amplituhedron, but these are all geometries in auxiliary spaces as opposed to
kinematic space where amplitudes live. In this paper, we propose a novel
geometric understanding of amplitudes for a large class of theories. The key is
to think of amplitudes as differential forms directly on kinematic space. We
explore this picture for a wide range of massless theories in general spacetime
dimensions. For the bi-adjoint cubic scalar, we establish a direct connection
between its "scattering form" and a classic polytope--the associahedron--known
to mathematicians since the 1960's. We find an associahedron living naturally
in kinematic space, and the tree amplitude is simply the "canonical form"
associated with this "positive geometry". Basic physical properties such as
locality, unitarity and novel "soft" limits are fully determined by the
geometry. Furthermore, the moduli space for the open string worldsheet has also
long been recognized as an associahedron. We show that the scattering equations
act as a diffeomorphism between this old "worldsheet associahedron" and the new
"kinematic associahedron", providing a geometric interpretation and novel
derivation of the bi-adjoint CHY formula. We also find "scattering forms" on
kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual
to the color-dressed amplitudes despite having no explicit color factors. This
is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic
wedge products in the scattering forms satisfy the same Jacobi relations as
color factors. Finally, our scattering forms are well-defined on the
projectivized kinematic space, a property that provides a geometric origin for
color-kinematics duality.Comment: 77 pages, 25 figures; v2, corrected discussion of worldsheet
associahedron canonical for
Kirchhoff Graphs
Kirchhoff\u27s laws are well-studied for electrical networks with voltage and current sources, and edges marked by resistors. Kirchhoff\u27s voltage law states that the sum of voltages around any circuit of the network graph is zero, while Kirchhoff\u27s current law states that the sum of the currents along any cutset of the network graph is zero. Given a network, these requirements may be encoded by the circuit matrix and cutset matrix of the network graph. The columns of these matrices are indexed by the edges of the network graph, and their row spaces are orthogonal complements. For (chemical or electrochemical) reaction networks, one must naturally study the opposite problem, beginning with the stoichiometric matrix rather than the network graph. This leads to the following question: given such a matrix, what is a suitable graphic rendering of a network that properly visualizes the underlying chemical reactions? Although we can not expect uniqueness, the goal is to prove existence of such a graph for any matrix. Specifically, we study Kirchhoff graphs, originally introduced by Fehribach. Mathematically, Kirchhoff graphs represent the orthocomplementarity of the row space and null space of integer-valued matrices. After introducing the definition of Kirchhoff graphs, we will survey Kirchhoff graphs in the context of several diverse branches of mathematics. Beginning with combinatorial group theory, we consider Cayley graphs of the additive group of vector spaces, and resolve the existence problem for matrices over finite fields. Moving to linear algebra, we draw a number of conclusions based on a purely matrix-theoretic definition of Kirchhoff graphs, specifically regarding the number of vector edges. Next we observe a geometric approach, reviewing James Clerk Maxwell\u27s theory of reciprocal figures, and presenting a number of results on Kirchhoff duality. We then turn to algebraic combinatorics, where we study equitable partitions, quotients, and graph automorphisms. In addition to classifying the matrices that are the quotient of an equitable partition, we demonstrate that many Kirchhoff graphs arise from equitable edge-partitions of directed graphs. Finally we study matroids, where we review Tutte\u27s algorithm for determining when a binary matroid is graphic, and extend this method to show that every binary matroid is Kirchhoff. The underlying theme throughout each of these investigations is determining new ways to both recognize and construct Kirchhoff graphs
Solution to the Henckell--Rhodes problem: finite -inverse covers do exist
For a finite connected graph with set of edges , a finite
-generated group is constructed such that the set of relations
satisfied by (with a word over ) is closed under deletion
of generators (i.e.~edges). As a consequence, every element admits a
unique minimal set of edges (the \emph{content} of ) needed
to represent as a word over . The
crucial property of the group is that connectivity in the graph
is encoded in in the following sense: if a word forms a
path in then there exists a -equivalent
word which also forms a path and uses only edges from
their content; in particular, the content of the corresponding group element
spans a connected subgraph of containing the
vertices and . As an application it is shown that every finite inverse
monoid admits a finite -inverse cover. This solves a long-standing problem
of Henckell and Rhodes.Comment: 46 pages, 14 figures, new result Cor. 2.7 included, several
inaccuracies removed, more details include
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