We construct explicit examples of Frostman-type measures concentrated on arbitrary simple rectifiable curves \(\Gamma\subset\mathbb{R}^{2}\) positive length. Based such constructions we obtain for each \(p \in (1,\infty)\) an exact description the trace space \(W^{1}_{p}(\mathbb{R}^{2})|_{\Gamma}\) first-order Sobolev \(W^{1}_{p}(\mathbb{R}^{2})\) to curve \(\Gamma\) length.

In this work, we present an experimental design and analytical framework to measure the nonparametric joint radius-length (R-L) distribution of an ensemble of parallel, finite cylindrical pores, and more generally, the eccentricity distribution of anisotropic pores. Employing a novel 3D double pulsed-field gradient acquisition scheme, we first obtain both the marginal radius and length distribu...

We derive novel upper and lower finite-length bounds of the error probability in joint source-channel coding when the source obeys the ergodic Markov process and the channel is a Markovian additive channel or a Markovian conditional additive channel. These bounds achieve the tight bounds in the large and moderate deviation regimes. Index Terms Markov chain, joint source-channel coding, finite-l...

We show that the moments of the trace of a random unitary matrix have combinatorial interpretations in terms of longest increasing subsequences of permutations. To be precise, we show that the 2n-th moment of the trace of a random k-dimensional unitary matrix is equal to the number of permutations of length n with no increasing subsequence of length greater than k. We then generalize this to ot...

This paper proposes a joint optimization mechanism for packet length and power in the scenario of covert short D2D communication, so as to effectively improve communication performance between pairs subject constraint. Specifically, we construct short-packet model assisted by further propose notion effective throughput (ECT) quantitatively characterize trade-off covertness reliability IoT state...