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Is there an algorithm that runs faster than O(mn^2) for detecting if there is at least one pair of terms of size m chosen from two sets of size n that are unifiable? You can assume that each variable is used only once across all terms (e.g.: terms are linear).
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The linear canonical transform can be computed efficiently (O(n log n)) in the discrete setting: https://arxiv.org/pdf/0912.1379.pdf …
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In fact, there is a thing called the linear canonical transform that is parameterized by elements of SL₂(C); rotation matrices give the fractional FT, shears give the Fresnel transform, and complex rotation gives the fractional Laplace transform.
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The Fourier transform (FT) has F(F(f)) = -f and F(F(F(F(f)))) = f. Sound familiar? Turns out you can generalize FT to the fractional FT, which does any rotation, not just 90 degree ones.
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Has anyone studied the relationship between classes of linear systems that are easily solved (e.g.: tridiagonal matrices) and the set of graphs that are isomorphic to a graph that has an adjacency matrix in that class?
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The fact that the set of smoothed polynomial-time algorithms coincides with the set of pseudo-polynomial time algorithms is one of the most surprising facts I know.
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Sorry, convexity is f(ax + (1 - a)y) ≤ a f(x) + (1 - a) f(y). And the second derivative is nonnegative everywhere, not positive (that is strict convexity).
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Surprisingly, the polynomial time algorithm for minimizing submodular functions is black-box. it only needs to be able to call the submodular function, despite the fact that minimizing submodular functions is a generalization of many combinatorial optimization problems.
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If f is submodular, then its Lovasz extension is a convex function (i.e.: f(ax + (1 - a)y) = a f(x) + (1 - a) f(y)). Convex functions are easy to minimize, because their local minimum is a global minimum (second derivative positive everywhere)!
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Any submodular function can be thought of as a function defined on vertices of an |X|-dimensional hypercube. The Lovasz extension of a submodular function extends it to a function on the /volume/ of the |X|-dimensional hypercube.
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A function f : P(X) → R is submodular if for all A ⊆ B ⊆ X and x ∉ B, f(A ∪ {x}) - f(A) ≥ f(B ∪ {x}) - f(B). This is a law of diminishing returns -- adding x to B is less effective than adding x to A if A ⊆ B. These functions can be minimized in strongly polynomial time.
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There is a 0.878-approximation algorithm for max cut using semidefinite programming. Much less well known is that there is a 0.614-approximation algorithm using spectral techniques. I wonder how well the latter does in practice; it seems a lot simpler to implement.
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Computing the Fiedler vector of a matrix is like 2 lines of code using the eigSH function. Computing the Laplacian of a graph and partitioning a graph based on the sign of the elements of the Laplacian's Fiedler vector is a few more.
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It is ridiculously easy to implement min-cut via spectral partitioning with hmatrix. Not much reason to bother with the classic min-cut algorithms.
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where X ≤ Y means "there is a substitution s such that s(Y) = X", of course.
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I know unification is the meet in the subsumption lattice, but I'm having difficulty disproving "X unifies with Y iff X ≤ Y or Y ≤ X". anyone know a pair of first-order terms that have a unification but don't satisfy that?
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Remy Goldschmidt proslijedio/la je Tweet
That's a good point, it's actually a homomorphism that preserves two unrelated monoids. Or, if you define degree as going to (N u {-∞}, max, +), where the zero polynomial has degree -∞ (but other constant polynomials have degree 0), then it really is a semiring homomorphism
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it's neat that the function that computes the degree of a polynomial is a semiring homomorphism from the ring of polynomials to the (N, max, +) semiring
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Remy Goldschmidt proslijedio/la je Tweet
The whole country has suffered immensely from the incentive- and culture-shift that caused most schools to remove an array of shop and trade classes
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Remy Goldschmidt proslijedio/la je Tweet
It’d be neat if you could add flag names to WARNING pragmas so users could control groups of them specifically: {-# WARNING ["importing this other module instead is prolly a good bet"] "fused-effects-import-advice" #-} -Wno-fused-effects-import-advice
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