147,585 research outputs found
Sum of Squares Lower Bounds from Symmetry and a Good Story
In this paper, we develop machinery which makes it much easier to prove sum of squares lower bounds when the problem is symmetric under permutations of [1,n] and the unsatisfiability of our problem comes from integrality arguments, i.e. arguments that an expression must be an integer. Roughly speaking, to prove SOS lower bounds with our machinery it is sufficient to verify that the answer to the following three questions is yes:
1) Are there natural pseudo-expectation values for the problem?
2) Are these pseudo-expectation values rational functions of the problem parameters?
3) Are there sufficiently many values of the parameters for which these pseudo-expectation values correspond to the actual expected values over a distribution of solutions which is the uniform distribution over permutations of a single solution?
We demonstrate our machinery on three problems, the knapsack problem analyzed by Grigoriev, the MOD 2 principle (which says that the complete graph K_n has no perfect matching when n is odd), and the following Turan type problem: Minimize the number of triangles in a graph G with a given edge density. For knapsack, we recover Grigoriev\u27s lower bound exactly. For the MOD 2 principle, we tighten Grigoriev\u27s linear degree sum of squares lower bound, making it exact. Finally, for the triangle problem, we prove a sum of squares lower bound for finding the minimum triangle density. This lower bound is completely new and gives a simple example where constant degree sum of squares methods have a constant factor error in estimating graph densities
A Solution Set-Based Entropy Principle for Constitutive Modeling in Mechanics
Entropy principles based on thermodynamic consistency requirements are widely
used for constitutive modeling in continuum mechanics, providing physical
constraints on a priori unknown constitutive functions. The well-known
M\"uller-Liu procedure is based on Liu's lemma for linear systems. While the
M\"uller-Liu algorithm works well for basic models with simple constitutive
dependencies, it cannot take into account nonlinear relationships that exist
between higher derivatives of the fields in the cases of more complex
constitutive dependencies.
The current contribution presents a general solution set-based procedure,
which, for a model system of differential equations, respects the geometry of
the solution manifold, and yields a set of constraint equations on the unknown
constitutive functions, which are necessary and sufficient conditions for the
entropy production to stay nonnegative for any solution. Similarly to the
M\"uller-Liu procedure, the solution set approach is algorithmic, its output
being a set of constraint equations and a residual entropy inequality. The
solution set method is applicable to virtually any physical model, allows for
arbitrary initially postulated forms of the constitutive dependencies, and does
not use artificial constructs like Lagrange multipliers. A Maple implementation
makes the solution set method computationally straightforward and useful for
the constitutive modeling of complex systems.
Several computational examples are considered, in particular, models of gas,
anisotropic fluid, and granular flow dynamics. The resulting constitutive
function forms are analyzed, and comparisons are provided. It is shown how the
solution set entropy principle can yield classification problems, leading to
several complementary sets of admissible constitutive functions; such problems
have not previously appeared in the constitutive modeling literature
Two lectures on the arithmetic of K3 surfaces
In these lecture notes we review different aspects of the arithmetic of K3
surfaces. Topics include rational points, Picard number and Tate conjecture,
zeta functions and modularity.Comment: 26 pages; v4: typos corrected, references update
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