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Recently an algorithm was given in [Garde & Hyv\"onen, SIAM J. Math. Anal., 2024] for exact direct reconstruction of any $L^2$ perturbation from linearised data in the two-dimensional linearised Calder\'on problem. It was a simple forward substitution method based on a 2D Zernike basis. We now consider the three-dimensional linearised Calder\'on problem in a ball, and use a 3D Zernike basis to obtain a method for exact direct reconstruction of any $L^3$ perturbation from linearised data. The method is likewise a forward substitution, hence making it very efficient to numerically implement. Moreover, the 3D method only makes use of a relatively small subset of boundary measurements for exact reconstruction, compared to a full $L^2$ basis of current densities.

We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.

In this article, we propose a new classification of $\Sigma^0_2$ formulas under the realizability interpretation of many-one reducibility (i.e., Levin reducibility). For example, ${\sf Fin}$, the decision of being eventually zero for sequences, is many-one/Levin complete among $\Sigma^0_2$ formulas of the form $\exists n\forall m\geq n.\varphi(m,x)$, where $\varphi$ is decidable. The decision of boundedness for sequences ${\sf BddSeq}$ and posets ${\sf PO}_{\sf top}$ are many-one/Levin complete among $\Sigma^0_2$ formulas of the form $\exists n\forall m\geq n\forall k.\varphi(m,k,x)$, where $\varphi$ is decidable. However, unlike the classical many-one reducibility, none of the above is $\Sigma^0_2$-complete. The decision of non-density of linear order ${\sf NonDense}$ is truly $\Sigma^0_2$-complete.

We study differentially private (DP) estimation of a rank-$r$ matrix $M \in \RR^{d_1\times d_2}$ under the trace regression model with Gaussian measurement matrices. Theoretically, the sensitivity of non-private spectral initialization is precisely characterized, and the differential-privacy-constrained minimax lower bound for estimating $M$ under the Schatten-$q$ norm is established. Methodologically, the paper introduces a computationally efficient algorithm for DP-initialization with a sample size of $n \geq \wt O (r^2 (d_1\vee d_2))$. Under certain regularity conditions, the DP-initialization falls within a local ball surrounding $M$. We also propose a differentially private algorithm for estimating $M$ based on Riemannian optimization (DP-RGrad), which achieves a near-optimal convergence rate with the DP-initialization and sample size of $n \geq \wt O(r (d_1 + d_2))$. Finally, the paper discusses the non-trivial gap between the minimax lower bound and the upper bound of low-rank matrix estimation under the trace regression model. It is shown that the estimator given by DP-RGrad attains the optimal convergence rate in a weaker notion of differential privacy. Our powerful technique for analyzing the sensitivity of initialization requires no eigengap condition between $r$ non-zero singular values.

In the search for highly efficient decoders for short LDPC codes approaching maximum likelihood performance, a relayed decoding strategy, specifically activating the ordered statistics decoding process upon failure of a neural min-sum decoder, is enhanced by instilling three innovations. Firstly, soft information gathered at each step of the neural min-sum decoder is leveraged to forge a new reliability measure using a convolutional neural network. This measure aids in constructing the most reliable basis of ordered statistics decoding, bolstering the decoding process by excluding error-prone bits or concentrating them in a smaller area. Secondly, an adaptive ordered statistics decoding process is introduced, guided by a derived decoding path comprising prioritized blocks, each containing distinct test error patterns. The priority of these blocks is determined from the statistical data during the query phase. Furthermore, effective complexity management methods are devised by adjusting the decoding path's length or refining constraints on the involved blocks. Thirdly, a simple auxiliary criterion is introduced to reduce computational complexity by minimizing the number of candidate codewords before selecting the optimal estimate. Extensive experimental results and complexity analysis strongly support the proposed framework, demonstrating its advantages in terms of high throughput, low complexity, independence from noise variance, in addition to superior decoding performance.

We consider a class of symmetry hypothesis testing problems including testing isotropy on $\mathbb{R}^d$ and testing rotational symmetry on the hypersphere $\mathcal{S}^{d-1}$. For this class, we study the null and non-null behaviors of Sobolev tests, with emphasis on their consistency rates. Our main results show that: (i) Sobolev tests exhibit a detection threshold (see Bhattacharya, 2019, 2020) that does not only depend on the coefficients defining these tests; and (ii) tests with non-zero coefficients at odd (respectively, even) ranks only are blind to alternatives with angular functions whose $k$th-order derivatives at zero vanish for any $k$ odd (even). Our non-standard asymptotic results are illustrated with Monte Carlo exercises. A case study in astronomy applies the testing toolbox to evaluate the symmetry of orbits of long- and short-period comets.

We study the complexity of constructing an optimal parsing $\varphi$ of a string ${\bf s} = s_1 \dots s_n$ under the constraint that given a position $p$ in the original text, and the LZ76-like (Lempel Ziv 76) encoding of $T$ based on $\varphi$, it is possible to identify/decompress the character $s_p$ by performing at most $c$ accesses to the LZ encoding, for a given integer $c.$ We refer to such a parsing $\varphi$ as a $c$-bounded access LZ parsing or $c$-BLZ parsing of ${\bf s}.$ We show that for any constant $c$ the problem of computing the optimal $c$-BLZ parsing of a string, i.e., the one with the minimum number of phrases, is NP-hard and also APX hard, i.e., no PTAS can exist under the standard complexity assumption $P \neq NP.$ We also study the ratio between the sizes of an optimal $c$-BLZ parsing of a string ${\bf s}$ and an optimal LZ76 parsing of ${\bf s}$ (which can be greedily computed in polynomial time).

In 2017, Aharoni proposed the following generalization of the Caccetta-H\"{a}ggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $c_r$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + c_r$.

Chatterjee's rank correlation coefficient $\xi_n$ is an empirical index for detecting functional dependencies between two variables $X$ and $Y$. It is an estimator for a theoretical quantity $\xi$ that is zero for independence and one if $Y$ is a measurable function of $X$. Based on an equivalent characterization of sorted numbers, we derive an upper bound for $\xi_n$ and suggest a simple normalization aimed at reducing its bias for small sample size $n$. In Monte Carlo simulations of various cases, the normalization reduced the bias in all cases. The mean squared error was reduced, too, for values of $\xi$ greater than about 0.4. Moreover, we observed that non-parametric confidence intervals for $\xi$ based on bootstrapping $\xi_n$ in the usual n-out-of-n way have a coverage probability close to zero. This is remedied by an m-out-of-n bootstrap without replacement in combination with our normalization method.

A graph $G$ is well-covered if all maximal independent sets are of the same cardinality. Let $w:V(G) \longrightarrow\mathbb{R}$ be a weight function. Then $G$ is $w$-well-covered if all maximal independent sets are of the same weight. An edge $xy \in E(G)$ is relating if there exists an independent set $S$ such that both $S \cup \{x\}$ and $S \cup \{y\}$ are maximal independent sets in the graph. If $xy$ is relating then $w(x)=w(y)$ for every weight function $w$ such that $G$ is $w$-well-covered. Relating edges play an important role in investigating $w$-well-covered graphs. The decision problem whether an edge in a graph is relating is NP-complete. We prove that the problem remains NP-complete when the input is restricted to graphs without cycles of length $6$. This is an unexpected result because recognizing relating edges is known to be polynomially solvable for graphs without cycles of lengths $4$ and $6$, graphs without cycles of lengths $5$ and $6$, and graphs without cycles of lengths $6$ and $7$. A graph $G$ belongs to the class $W_2$ if every two pairwise disjoint independent sets in $G$ are included in two pairwise disjoint maximum independent sets. It is known that if $G$ belongs to the class $W_2$, then it is well-covered. A vertex $v \in V(G)$ is shedding if for every independent set $S \subseteq V(G)-N[v]$, there exists a vertex $u \in N(v)$ such that $S \cup \{u\}$ is independent. Shedding vertices play an important role in studying the class $W_2$. Recognizing shedding vertices is co-NP-complete, even when the input is restricted to triangle-free graphs. We prove that the problem is co-NP-complete for graphs without cycles of length $6$.