THE DIMENSION OF CENTRALISERS OF MATRICES OF ORDER

In this paper, we study the integer sequence$(E_{n})_{n\geq 1}$, where$E_{n}$counts the number of possible dimensions for centralisers of$n\times n$matrices. We give an example to show another combinatorial interpretation of$E_{n}$and present an implicit recurrence formula for$E_{n}$, which may provide a fast algorithm for computing$E_{n}$. Based on the recurrence, we obtain the asymptotic formula$E_{n}=\frac{1}{2}n^{2}-\frac{2}{3}\sqrt{2}n^{3/2}+O(n^{5/4})$.

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ON FREE SPECTRA OF VARIETIES OF LOCALLY THRESHOLD TESTABLE SEMIGROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196712500555 ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250055

Author(s):

IGOR DOLINKA

Keyword(s):

Asymptotic Formula ◽

Regular Language ◽

Finite Alphabet ◽

Combinatorial Interpretation ◽

De Bruijn Graphs ◽

Free Spectrum ◽

Enumeration Problem ◽

De Bruijn ◽

Asymptotic Upper Bound ◽

Free Spectra

A semigroup S is said to be ℓ-threshold k-testable if it satisfies all identities u = v where u, v is an arbitrary pair of words over a finite alphabet Σ such that they simultaneously belong or fail to belong to any ℓ-threshold k-testable (regular) language. We give an asymptotic formula for the free spectrum of the variety [Formula: see text] of all ℓ-threshold k-testable semigroups, thereby providing an asymptotic upper bound on the size of an arbitrary finitely generated locally threshold testable semigroup. The combinatorial interpretation of this task yields an enumeration problem for particular edge labelings of de Bruijn graphs.

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Multiobjective Maximization of Monotone Submodular Functions with Cardinality Constraint

INFORMS Journal on Optimization ◽

10.1287/ijoo.2019.0041 ◽

2020 ◽

pp. ijoo.2019.0041

Author(s):

Rajan Udwani

Keyword(s):

Asymptotic Formula ◽

Fast Algorithm ◽

Asymptotic Approximation ◽

Constant Factor ◽

The Other ◽

Submodular Functions ◽

Cardinality Constraint ◽

Greedy Methods ◽

Deterministic Formula ◽

Single Objective

We consider the problem of multiobjective maximization of monotone submodular functions subject to cardinality constraint, often formulated as [Formula: see text]. Although it is widely known that greedy methods work well for a single objective, the problem becomes much harder with multiple objectives. In fact, it is known that when the number of objectives m grows as the cardinality k, that is, [Formula: see text], the problem is inapproximable (unless P = NP). On the other hand, when m is constant, there exists a a randomized [Formula: see text] approximation with runtime (number of queries to function oracle) the scales as [Formula: see text]. We focus on finding a fast algorithm that has (asymptotic) approximation guarantees even when m is super constant. First, through a continuous greedy based algorithm we give a [Formula: see text] approximation for [Formula: see text]. This demonstrates a steep transition from constant factor approximability to inapproximability around [Formula: see text]. Then using multiplicative-weight-updates (MWUs), we find a much faster [Formula: see text] time asymptotic [Formula: see text] approximation. Although these results are all randomized, we also give a simple deterministic [Formula: see text] approximation with runtime [Formula: see text]. Finally, we run synthetic experiments using Kronecker graphs and find that our MWU inspired heuristic outperforms existing heuristics.

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Heavy Tails of Discounted Aggregate Claims in the Continuous-Time Renewal Model

Journal of Applied Probability ◽

10.1017/s0021900200002965 ◽

2007 ◽

Vol 44 (02) ◽

pp. 285-294 ◽

Cited By ~ 2

Author(s):

Qihe Tang

Keyword(s):

Asymptotic Formula ◽

Continuous Time ◽

Heavy Tails ◽

Tail Behavior ◽

Renewal Model ◽

Discounted Aggregate Claims ◽

Aggregate Claims ◽

Tail Asymptotic

We study the tail behavior of discounted aggregate claims in a continuous-time renewal model. For the case of Pareto-type claims, we establish a tail asymptotic formula, which holds uniformly in time.

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