28,995 research outputs found
The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan
We prove several results on the lifespan, regularity, and uniqueness of
solutions of the Cauchy problem for the homogeneous complex and real
Monge-Ampere equations (HCMA/HRMA) under various a priori regularity
conditions. We use methods of characteristics in both the real and complex
settings to bound the lifespan of solutions with prescribed regularity. In the
complex domain, we characterize the C^3 lifespan of the HCMA in terms of
analytic continuation of Hamiltonian mechanics and intersection of complex time
characteristics. We use a conservation law type argument to prove uniqueness of
solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy
problem is ill-posed in C^3, in the sense that there exists a dense set of C^3
Cauchy data for which there exists no C^3 solution even for a short time. In
the real domain we show that the HRMA is equivalent to a Hamilton--Jacobi
equation, and use the equivalence to prove that any differentiable weak
solution is smooth, so that the differentiable lifespan equals the convex
lifespan determined in our previous articles. We further show that the only
obstruction to C^1 solvability is the invertibility of the associated Moser
maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a
positive but generally finite time and cannot be continued even as a weak C^1
solution afterwards. Finally, we introduce the notion of a "leafwise
subsolution" for the HCMA that generalizes that of a solution, and many of our
aforementioned results are proved for this more general object
One-sided Heegaard splittings of RP^3
Using basic properties of one-sided Heegaard splittings, a direct proof that
geometrically compressible one-sided splittings of RP^3 are stabilised is
given. The argument is modelled on that used by Waldhausen to show that
two-sided splittings of S^3 are standard.Comment: This is the version published by Algebraic & Geometric Topology on 20
September 200
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
GIF Today
The revival of the animated GIF marks a point in the history of the web when it finally became sufficiently advanced to take pleasure in its own obsolescence. Like the rusty engines and the leaking pipes of the derelict spaceship in Alien, the lo-fi jitter of the GIF signals a moment when the novelty of technology fades off and becomes the backdrop rather then substance
Digital Image
This paper considers the ontological significance of invisibility in relation to the question ‘what is a digital image?’ Its argument in a nutshell is that the emphasis on visibility comes at the expense of latency and is symptomatic of the style of thinking that dominated Western philosophy since Plato. This privileging of visible content necessarily binds images to linguistic (semiotic and structuralist) paradigms of interpretation which promote representation, subjectivity, identity and negation over multiplicity, indeterminacy and affect. Photography is the case in point because until recently critical approaches to photography had one thing in common: they all shared in the implicit and incontrovertible understanding that photographs are a medium that must be approached visually; they took it as a given that photographs are there to be looked at and they all agreed that it is only through the practices of spectatorship that the secrets of the image can be unlocked. Whatever subsequent interpretations followed, the priori- ty of vision in relation to the image remained unperturbed. This undisputed belief in the visibility of the image has such a strong grasp on theory that it imperceptibly bonded together otherwise dissimilar and sometimes contradictory methodol- ogies, preventing them from noticing that which is the most unexplained about images: the precedence of looking itself. This self-evident truth of visibility casts a long shadow on im- age theory because it blocks the possibility of inquiring after everything that is invisible, latent and hidden
The Grin of Schrödinger's Cat; Quantum Photography and the limits of Representation
The famous quantum physics experiment 'Schrödinger's cat' suggests that some situations are undecidable, i.e. they exist outside of the normative distinctions between 'truth' and 'false' because both states can co-exist under certain conditions. This paper suggests that photography has very close links with this state of affairs, because photography allows one to move from the world of certainty into the quantum dimension of undecidability and indeterminate states
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