Related papers: On Chevalley's Extension Theorem

After fixing a canonical ordering (or labeling) of the elements of a finite poset, one can associate each linear extension of the poset with a permutation. Some recent papers consider specific families of posets and ask how many linear…

The notion of minimal complements was introduced by Nathanson in 2011. Since then, the existence or the inexistence of minimal complements of sets have been extensively studied. Recently, the study of inverse problems, i.e., which sets can…

We establish a reversal of Lyapunov's inequality for monotone log-concave sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A strengthened version of the same conjecture is disproved through counter example. We also…

We establish an ideal-theoretic rigidity principle for quadratic distance images over integer residue rings. Specifically, we prove that near-extremal collapse of the distance set in $\mathbb{Z}_n^d$ forces strong algebraic structure…

We revisit classic balancing problems for linear extensions of a partially ordered set $P$, proving results that go far beyond many of the best earlier results on this topic. For example, with $p(x\prec y)$ the probability that $x$ precedes…

We consider a family of finitely presented groups, called Universal Left Invertible Element (or ULIE) groups, that are universal for existence of one--sided invertible elements in a group ring K[G], where K is a field or a division ring. We…

Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices for similar incomplete data sets and no reasonable measure has an associated threshold. This paper…

Extending the discovery by Giles Gardam of a concrete counterexample to Kaplansky's unit conjecture in characteristic 2, a family of counterexamples for every prime characteristic is presented.

We study the distances between the unitary orbits of matrix amplifications of elements in certain C*-algebras. In particular, we show that the distance between unitary orbits of arbitrary elements in unital, separable, UHF-stable…

This paper has two parts. First, we complete the proof of the Kolmogorov extension theorem for unbounded random variables using compactness theorem of integral logic which was proved for bounded case in [8]. Second, we give a proof of the…

In this paper we introduce and develop the concept of expansivity of a tuple whose entries are elements from the polynomial ring $\mathbb{C}[x]$. As an inverse problem, we examine how to recover a tuple from the expanded tuple at any given…

For an irreducible complex reflection group $W$ of rank $n$ containing $N$ reflections, we put $g=2N/n$ and construct a $(g+1)^n$-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the…

We say that a convergence law holds for a sequence of random combinatorial objects if, for any first-order sentence $\varphi$, the proportion of objects satisfying $\varphi$ converges to a limiting value as the size of the objects tends to…

Upon re-examining Arnold's established lemma for explaining his famous limit problem, we have determined that while the lemma itself is correct, there is a defect in the original geometric proof. In this paper, we prove the correctness of…

This paper has been withdrawn by the author due to a sheaf-theoretic error, in the end of the proof of the main theorem.

In this paper, we examine the aptly-named "Lethargy Theorem" of Bernstein and survey its recent extensions. We show that one of these extensions shrinks the interval for best approximation by half while the other gives a surprising…

In Section 6 of the paper ``Fra\"{\i}ss\'{e}'s Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition'', we applied the methods developed in earlier sections to show that a certain reduct of the generic…

Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary…

Unitarity cannot be perserved order by order in ordinary perturbation theory because the constraint $UU^\dagger=\1$ is nonlinear. However, the corresponding constraint for $K=\ln U$, being $K=-K^\dagger$, is linear so it can be maintained…

Let $(R,\fm,k)$ be a commutative noetherian local ring with dualizing complex $\dua R$, normalized by $\Ext^{\depth(R)}_R(k,\dua R)\cong k$. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative)…