47 problems

Let $f$ be a polynomial with no zeros in the closed bidisk $\overline{\mathbb{D}^2}$, and let its orthogonal polynomial approximants (OPAs) be formed in $H^2(\mathbb{D}^2)$. Weak S…

Let $f$ be a polynomial, and let its orthogonal polynomial approximants (OPAs) to $1/f$ in $H^2(\mathbb{D}^2)$ be given. Shanks's conjecture. For every polynomial $f$, the…

Exactness conjecture. The proposed relaxation is tight and yields the exact solution whenever the graph $(V,E)$ has no cycle.

Let $b(n)=\frac{\phi(n)\Lambda(n)}{n}$, where $\phi$ is Euler's totient function and $\Lambda$ is the Mangoldt function. Let $C(h)$ be the Hardy–Littlewood prime-pair constant, def…

Let a signal have bandwidth $2W$ and be observed for a time $T$. Shannon's conjecture. The signal could be represented by $2WT$ parameters. This conjecture is a classical formulati…

Let $f$ be a statistic, let $\mathbf{x}$ be an observation, and let $\kappa(f,\mathbf{x})$ be its structural capacity. Let $C(f)$ denote the single-evaluation variance, and let…

Let $\mathbf{R}$ be a Hermitian positive-definite matrix arising from a broad class of physical signal models, and let $\mathcal{B}$ be a basis. Let…

Let $\mathbf{R}$ be a covariance matrix with spectral eigenvalues $\lambda_k(\mathbf{R})$, and write $M$ for the number of spectral modes. Define the structural entropy by … Let…

Let $L_0$ be the filter length and let $\bm{\pi}_0$ denote a filter pattern. For each pattern, call a filter good when it achieves the $2$-ASK minimum squared Euclidean distance. C…

Let $L_0$ denote the filter length, and call a filter good when it achieves the asymptotic $2$-ASK minimum squared Euclidean distance (MSED). For the filter pattern … consider the…

U-NET optimal trade-off conjecture. A U-NET provides the optimal trade-off between performance and complexity for radio-to-PPG translation, especially given a limited-size dataset.

Let $P(a,z)$ be the self-inversive polynomial defined by … Here $a+r\mathbb{Z}$ is a sampling set for the space $V(Q_m)$, with $m>r$, and $P(a,z)$ arises from the factorization…

Optimal reproducing-filter norm conjecture. Every $s$-dimensional SIS admits a reproducing filter $\varphi\in\mathbb{C}_n(\mathbb{Z})$ such that

Threshold conjecture. This phenomenon holds in general: condition (21) sets a corresponding threshold on the signal-to-thermal-noise ratio. The conjecture predicts that the restric…

Let $\boldsymbol{D}=\boldsymbol{Z}\boldsymbol{A}+\sigma\boldsymbol{W}$ be the signal-plus-noise data matrix, where $\boldsymbol{A}$ is the signal matrix, $\boldsymbol{Z}$ encodes t…

Near second-order Markov conjecture. The MA filtering of the first-order Markov process generates nearly a second-order Markov process.

Consider prewhitening time series using autoregressive models $\operatorname{AR}(p)$ or autoregressive moving-average models $\operatorname{ARMA}(p,q)$, where $p$ and $q$ denote th…

One-bit radar and conventional radar are compared under the same false-alarm-rate settings, with detection performance measured for targets having low signal-to-noise ratios. Lower…

The paper considers Proposition … is not optimal, so its bound can be improved. The authors leave the improvement for future work; the source provides no resolution of this conject…

Superoscillatory signals are band-limited signals containing stretches that oscillate at frequencies exceeding their nominal bandwidth; their dynamic range is the ratio between the…

Let a spectrally sparse signal have spectral components estimated from $2m+1$ uniform temporal measurements, and let the distance between its spectral components be measured on the…

Model-dependent exponent conjecture. There is a function $g(a)$ such that

Minimum-gap scaling conjecture. There exists a constant $c_t$ such that

Normalization-performance conjecture. The normalization is required mostly for the analysis, but has no significant effect on actual performance; in particular, setting…

The discussion concerns signaling methods on the continuous spectrum of a nonlinear optical fiber, with $E$ denoting signal energy and VNT denoting the variance-normalizing transfo…