Related papers: New Counterexamples for Sums-Differences
Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.
By use of a modified Nunokawa's lemma, we obtain some new conditions for univalence. Also, some sharp inequalities concerning univalent functions are presented.
In this short note we report on results on a computational search for a counterexample to the strong coincidence conjecture. In particular, we discuss the method used so that further searches can be conducted.
A new method based on the rejection sampling for finding statistical tests is proposed. This method is conceptually intuitive, easy to implement, and applicable for arbitrary dimension. To illustrate its potential applicability, three…
A new entropy-like measure as well as a new measure of total uncertainty pertaining to the Dempster-Shafer theory are introduced. It is argued that these measures are better justified than any of the previously proposed candidates.
We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.
Starting from a small number of well-motivated axioms, we derive a unique definition of sums with a noninteger number of addends. These "fractional sums" have properties that generalize well-known classical sum identities in a natural way.…
We derive new upper and lower bounds for probabilities that $r$ or at least $r$ from $n$ events occur. These bounds can turn to equalities. The method is discussed as well. It works for measurable space and measures with sign, too. We also…
As counterfactual examples become increasingly popular for explaining decisions of deep learning models, it is essential to understand what properties quantitative evaluation metrics do capture and equally important what they do not…
We give a visually appealing counterexample to the proposition that unbiased estimators are better than biased estimators.
We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood…
The article summarizes some developments about a singular versions of the Sturm Comparison and Separation theorems where the coefficients or the interval of definition may be unbounded.
The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the…
This paper provides new summation inequalities in both single and double forms to be used in stability analysis of discrete-time systems with time-varying delays. The potential capability of the newly derived inequalities is demonstrated by…
We introduce the E-measure: a measure-like generalization of the E-value to a class of hypotheses. Unlike classical measures, E-measures are closed under infimums instead of addition. They arise from a compatibility axiom with logical…
Counterfactual explanations are a common approach to providing recourse to data subjects. However, current methodology can produce counterfactuals that cannot be achieved by the subject, making the use of counterfactuals for recourse…
Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical…
We conjecture new uncertainty relations which restrict correlations between results of measurements performed by two separated parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when…
In [2] the author claims to provide a counterexample to a result in a recent paper [1]. In this note, we prove that the details of his example is false and this example is compatible with our result in [1] and so is not a countreexample.
In this paper, we derive new probability bounds for Chebyshev's inequality if the supremum of the probability density function is known. This result holds for one-dimensional or multivariate continuous probability distributions with finite…