Related papers: Variants of {\L}o\'s's Theorem
We present a relatively simple description of binary, definable subsets of models of weakly quasi-o-minimal theories. In particular, we closely describe definable linear orders and prove a weak version of the monotonicity theorem. We also…
We provide a permutation-invariant version of the Koml\'os' theorem for non-negative random variables. The proof is quite elementary in the sense that it did not use the Axiom of Choice, and was based on a recent result in [3].
In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…
We provide a proof of a variant of the Landau-Siegel Zeros conjecture.
We present variations on theorems of Mertens as special cases of Density Hypothesis. Moreover, we study a Serre's estimate concerning Lang-Weil estimate.
We give a new proof of Lucas' Theorem in elementary number theory.
We give a counterexample to a recently conjectured variant of the Penrose inequality.
We study some topics about \L o\'s's theorem without assuming the Axiom of Choice. We prove that \L o\'s's fundamental theorem of ultraproducts is equivalent to a weak form that every ultrapower is elementary equivalent to its source…
A multivariate Gauss-Lucas theorem is proved, sharpening and generalizing previous results on this topic. The theorem is stated in terms of a seemingly new notion of convexity. Applications to multivariate stable polynomials are given.
This note presents a unified theorem of the alternative that explicitly allows for any combination of equality, componentwise inequality, weak dominance, strict dominance, and nonnegativity relations. The theorem nests 60 special cases,…
We discuss the (twisted) weak positivity theorem. We also treat some applications.
We give new proofs of some well-known results from Invariant Theorey using the Kempf-Ness theorem.
We consider variants on the classical Berz sublinearity theorem, using only DC, the Axiom of Dependent Choices, rather than AC, the Axiom of Choice which Berz used. We consider thinned versions, in which conditions are imposed on only part…
We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open.
One of the variants to proof the generalized Ito-Wentzell's formula is introduced and examined in this paper. The relationship between different representations of the generalized Ito-Wentzell's formula/ is considered.
In this paper, we formulate and prove several variants of the Erd\H{o}s-Tur\'{a}n additive bases conjecture.
This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and…
The proofs of K. Oka's Coherence Theorems are based on Weierstrass' Preparation (division) Theorem. Here we formulate and prove a Weak Coherence Theorem without using Weierstrass' Preparation Theorem, but only with power series expansions:…
We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the…
We present a short and self-contained proof of the choosability version of Brooks' theorem.