Related papers: Approimate satisfaction of identities

Let $A\subset [1, 2]$ be a $(\delta, \sigma)$-set with measure $|A|=\delta^{1-\sigma}$ in the sense of Katz and Tao. For $\sigma\in (1/2, 1)$ we show that $$ |A+A|+|AA|\gtrapprox \delta^{-c}|A|, $$ for…

In previous work cite{Ha98:Towards} we presented a case-based approach to eliciting and reasoning with preferences. A key issue in this approach is the definition of similarity between user preferences. We introduced the probabilistic…

Let $0<p<\infty$ and $\Psi: [0,1) \to (0,\infty)$, and let $\mu$ be a finite positive Borel measure on the unit disc $\mathbb{D}$ of the complex plane. We define the Lebesgue-Zygmund space $L^p_{\mu,\Psi}$ as the space of all measurable…

We present two measures of distance between quantum processes based on the superfidelity, introduced recently to provide an upper bound for quantum fidelity. We show that the introduced measures partially fulfill the requirements for…

We consider the numbers of positive and negative eigenvalues of matrices of squared distances between randomly sampled i.i.d. points in a given metric measure space. These numbers and their limits, as the number of points grows, in fact…

We consider differential operators $L$ acting on functions on a Riemannian surface, $\Sigma$, of the form $$L = \Delta + V -a K ,$$where $\Delta$ is the Laplacian of $\Sigma$, $K$ is the Gaussian curvature, $a$ is a positive constant and $V…

In this short paper we introduce a new class of performance measures based on certainty equivalents defined via scaled utility functions. We analyse their properties, show that the corresponding portfolio optimization problem is well-posed…

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral…

Metrics and pseudometrics are defined on the group of unitary operators in a Hilbert space. Several explicit formulas are derived. A special feature of the work is investigation of pseudometrics in unitary groups. The rich classes of…

Quantifying numerical data involves addressing two key challenges: first, determining whether the data can be naturally quantified, and second, identifying the numerical intervals or ranges of values that correspond to specific value…

Approximation in measure is employed to solve an asymptotic Dirichlet problem on arbitrary open sets and to show that many functions, including the Riemann zeta-function, are universal in measure. Connections with the Riemann Hypothesis are…

As subjects perceive the sensory world, different stimuli elicit a number of neural representations. Here, a subjective distance between stimuli is defined, measuring the degree of similarity between the underlying representations. As an…

We investigate the sums $\sum_{n\le X, (n,q)=1}\frac{\mu(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this,…

Motivated by rigorous development in the theory of digamma functions, we have first derived some new identities for the digamma function, and then computed the values of digamma function for the fractional orders using these identities…

We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive harmonic measure. Then we…

The category of metric spaces is a subcategory of quasi-metric spaces. In this paper the notion of entropy for the continuous maps of a quasi-metric space is extended via spanning and separated sets. Moreover, two metric spaces that are…

We consider the metric transformation of metric measure spaces/pyramids. We clarify the conditions to obtain the convergence of the sequence of transformed spaces from that of the original sequence, and, conversely, to obtain the…

We show that for any metric probability space $(M,d,\mu)$ with a subgaussian constant $\sigma^2(\mu)$ and any set $A \subset M$ we have $\sigma^2(\mu_A) \leq c \log\left(e/\mu(A)\right)\,\sigma^2(\mu)$, where $\mu_A$ is a restriction of…

In this paper we look at normed spaces of differentiable functions on compact plane sets, including the spaces of infinitely differentiable functions originally considered by Dales and Davie. For many compact plane sets the classical…

Many widely different problems have a common mathematical structure wherein limited knowledge lead to ambiguity that can be captured conveniently using a concept of invisibility that requires the introduction of negative values for…