On 2-generated minimal Taylor algebras of size 4
We prove that any 2-generated minimal Taylor algebra on a domain of size 4 is not simple. In addition, we find all such algebras up to isomorphism and term-equivalence.
math.LO↗
arXiv subjects
Zarathustra Brady
Publications and source records attributed to Zarathustra Brady.
The colored edge theory of A. Bulatov and binary absorption in minimal Taylor algebras
We find a new definition of colored edge graphs of finite algebras in the case of minimal Taylor algebras, a definition which includes the graphs invented by A. Bulatov. Next we proceed to reprove the main results of A. Bulatov's theory in the case of minimal Taylor algebras and in our setting, finding several simplifications compared to the more general case of smooth algebras Bulatov considered.
math.LO↗
Notes on CSPs and Polymorphisms
These are notes from a multi-year learning seminar on the algebraic approach to Constraint Satisfaction Problems (CSPs). The main topics covered are the theory of algebraic structures with few subpowers, the theory of absorbing subalgebras and its applications to studying CSP templates which can be solved by local consistency methods, and the dichotomy theorem for conservative CSP templates. Subsections and appendices cover supplementary material.
math.RA↗
Unifying the Three Algebraic Approaches to the CSP via Minimal Taylor Algebras
This paper focuses on the algebraic theory underlying the study of the complexity and the algorithms for the Constraint Satisfaction Problem (CSP). We unify, simplify, and extend parts of the three approaches that have been developed to study the CSP over finite templates -- absorption theory that was used to characterize CSPs solvable by local consistency methods (JACM'14), and Bulatov's and Zhuk's theories that were used for two independent proofs of the CSP Dichotomy Theorem (FOCS'17, JACM'20). As the first contribution we present an elementary theorem about primitive positive definability and use it to obtain the starting points of Bulatov's and Zhuk's proofs as corollaries. As the second contribution we propose and initiate a systematic study of minimal Taylor algebras. This class of algebras is broad enough that it suffices to verify the CSP Dichotomy Theorem on this class only, but still is unusually well behaved. In particular, many concepts from the three approaches coincide in this class, which is in striking contrast with the general setting. We believe that the theory initiated in this paper will eventually result in a simple and more natural proof of the Dichotomy Theorem that employs a simpler and more efficient algorithm, and will help in attacking complexity questions in other CSP-related problems.
cs.CC↗
Coarse classification of binary minimal clones
We classify binary minimal clones into seven categories: affine algebras, rectangular bands, $p$-cyclic groupoids, spirals, non-Taylor partial semilattices, melds, and dispersive algebras. Each category has nice enough properties to guarantee that any clone of one of these types contains a minimal clone of the same type.
math.RA↗
A Semidefinite Framework for the Sieve
We describe a semidefinite programming framework for proving upper bounds on concrete sifting problems, and show that the Large Sieve can be interpreted as a special case of this framework. With a small tweak, the Larger Sieve also falls into this framework. We compare the semidefinite approach to the linear programming approach (i.e., the general framework of the combinatorial sieve and the Selberg sieve), and show that it has a qualitative advantage in a toy case where the primes are completely independent from each other. No new sieve-theoretic bounds are proved.
math.OC↗
Symmetric Operations on Domains of Size at Most 4
To convert a fractional solution to an instance of a constraint satisfaction problem into a solution, a rounding scheme is needed, which can be described by a collection of symmetric operations with one of each arity. An intriguing possibility, raised in a recent paper by Carvalho and Krokhin, would imply that any clone of operations on a set $D$ which contains symmetric operations of arities $1, 2, \ldots, \lvert D \rvert$ contains symmetric operations of all arities in the clone. If true, then it is possible to check whether any given family of constraint satisfaction problems is solved by its linear programming relaxation. We characterize all idempotent clones containing symmetric operations of arities $1, 2, \ldots, \lvert D \rvert$ for all sets $D$ with size at most four and prove that each one contains symmetric operations of every arity, proving the conjecture above for $\lvert D \rvert \leq 4$.
math.RA↗
A hardness of approximation result in metric geometry
We show that it is $\mathsf{NP}$-hard to approximate the hyperspherical radius of a triangulated manifold up to an almost-polynomial factor.
math.DG↗
Examples, counterexamples, and structure in bounded width algebras
We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We classify minimal bounded width algebras which have size at most three, and prove a structure theorem for minimal bounded width algebras which have no majority subalgebra, which form a pseudovariety with a commutative binary operation. As a byproduct of our results, we also classify minimal clones which have a Taylor term.
math.RA↗
Asymptotics of $d$-Dimensional Visibility
We consider the space $[0,n]^3$, imagined as a three dimensional, axis-aligned grid world partitioned into $n^3$ $1\times 1 \times 1$ unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it, or obstructing, in which case no line of sight can pass through it. From a given position, some of these obstructing cubes block one's view of other obstructing cubes, leading to the following extremal problem: What is the largest number of obstructing cubes that can be simultaneously visible from the surface of an observer cube, over all possible choices of which cubes of $[0,n]^3$ are obstructing? We construct an example of a configuration in which $Ω\big(n^\frac{8}{3}\big)$ obstructing cubes are visible, and generalize this to an example with $Ω\big(n^{d-\frac{1}{d}}\big)$ visible obstructing hypercubes for dimension $d>3$. Using Fourier analytic techniques, we prove an $O\big(n^{d-\frac{1}{d}}\log n\big)$ upper bound in a reduced visibility setting.
math.CO↗
Chromatic numbers of directed hypergraphs with no "bad" cycles
Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you then construct a digraph which avoids having any "bad" cycles, but has arbitrarily large chromatic number? We answer this question when the rule is described in terms of a finite state machine. The proof relies on Nesetril and Rodl's structural Ramsey theory of posets with a linear extension. As an application, we give a new proof of the Loop Lemma of Barto, Kozik, and Niven in the special case of bounded width algebras.
math.CO↗
Sums of seven octahedral numbers
We show that for a large class of cubic polynomials $f$, every sufficiently large number can be written as a sum of seven positive values of $f$. As a special case, we show that every number greater than $e^{10^7}$ is a sum of seven positive octahedral numbers, where an octahedral number is a number of the form $\frac{2x^3+x}{3}$, reducing an open problem due to Pollock to a finite computation.
math.NT↗
General divisor function inequalities and the third cumulant
We extend a lower bound of Munshi on sums over divisors of a number $n$ which are less than a fixed power of $n$ from the squarefree case to the general case. In the process we prove a lower bound on the entropy of a geometric distribution with finite support, as well as a lower bound on the probability that a random variable is less than its mean given that it satisfies a natural condition related to its third cumulant.
math.NT↗
Inequalities and higher order convexity
We study the following problem: given n real arguments a1, ..., an and n real weights w1, ..., wn, under what conditions does the inequality w1 f(a1) + w2 f(a2) + ... + wn f(an) >= 0 hold for all functions f with nonnegative kth derivative for some given integer k? Using simple combinatorial techniques, we can prove many generalizations of theorems ranging from the Fuchs inequality to the criterion for Schur convexity.
math.FA↗