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We consider the problem of approximating a function from $L^2$ by an element of a given $m$-dimensional space $V_m$, associated with some feature map $\varphi$, using evaluations of the function at random points $x_1,\dots,x_n$. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features $\varphi(x_i)$. We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples $n = O(m\log(m))$, that means that the expected $L^2$ error is bounded by a constant times the best approximation error in $L^2$. Also, further assuming that the function is in some normed vector space $H$ continuously embedded in $L^2$, we further prove that the approximation is almost surely bounded by the best approximation error measured in the $H$-norm. This includes the cases of functions from $L^\infty$ or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

For any positive integer $q\geq 2$ and any real number $\delta\in(0,1)$, let $\alpha_q(n,\delta n)$ denote the maximum size of a subset of $\mathbb{Z}_q^n$ with minimum Hamming distance at least $\delta n$, where $\mathbb{Z}_q=\{0,1,\dotsc,q-1\}$ and $n\in\mathbb{N}$. The asymptotic rate function is defined by $ R_q(\delta) = \limsup_{n\rightarrow\infty}\frac{1}{n}\log_q\alpha_q(n,\delta n).$ The famous $q$-ary asymptotic Gilbert-Varshamov bound, obtained in the 1950s, states that \[ R_q(\delta) \geq 1 - \delta\log_q(q-1)-\delta\log_q\frac{1}{\delta}-(1-\delta)\log_q\frac{1}{1-\delta} \stackrel{\mathrm{def}}{=}R_\mathrm{GV}(\delta,q) \] for all positive integers $q\geq 2$ and $0<\delta<1-q^{-1}$. In the case that $q$ is an even power of a prime with $q\geq 49$, the $q$-ary Gilbert-Varshamov bound was firstly improved by using algebraic geometry codes in the works of Tsfasman, Vladut, and Zink and of Ihara in the 1980s. These algebraic geometry codes have been modified to improve the $q$-ary Gilbert-Varshamov bound $R_\mathrm{GV}(\delta,q)$ at a specific tangent point $\delta=\delta_0\in (0,1)$ of the curve $R_\mathrm{GV}(\delta,q)$ for each given integer $q\geq 46$. However, the $q$-ary Gilbert-Varshamov bound $R_\mathrm{GV}(\delta,q)$ at $\delta=1/2$, i.e., $R_\mathrm{GV}(1/2,q)$, remains the largest known lower bound of $R_q(1/2)$ for infinitely many positive integers $q$ which is a generic prime and which is a generic non-prime-power integer. In this paper, by using codes from geometry of numbers introduced by Lenstra in the 1980s, we prove that the $q$-ary Gilbert-Varshamov bound $R_\mathrm{GV}(\delta,q)$ with $\delta\in(0,1)$ can be improved for all but finitely many positive integers $q$. It is shown that the growth defined by $\eta(\delta)= \liminf_{q\rightarrow\infty}\frac{1}{\log q}\log[1-\delta-R_q(\delta)]^{-1}$ for every $\delta\in(0,1)$ has actually a nontrivial lower bound.

Non-classical generalizations of classical modal logic have been developed in the contexts of constructive mathematics and natural language semantics. In this paper, we discuss a general approach to the semantics of non-classical modal logics via algebraic representation theorems. We begin with complete lattices $L$ equipped with an antitone operation $\neg$ sending $1$ to $0$, a completely multiplicative operation $\Box$, and a completely additive operation $\Diamond$. Such lattice expansions can be represented by means of a set $X$ together with binary relations $\vartriangleleft$, $R$, and $Q$, satisfying some first-order conditions, used to represent $(L,\neg)$, $\Box$, and $\Diamond$, respectively. Indeed, any lattice $L$ equipped with such a $\neg$, a multiplicative $\Box$, and an additive $\Diamond$ embeds into the lattice of propositions of a frame $(X,\vartriangleleft,R,Q)$. Building on our recent study of "fundamental logic", we focus on the case where $\neg$ is dually self-adjoint ($a\leq \neg b$ implies $b\leq\neg a$) and $\Diamond \neg a\leq\neg\Box a$. In this case, the representations can be constrained so that $R=Q$, i.e., we need only add a single relation to $(X,\vartriangleleft)$ to represent both $\Box$ and $\Diamond$. Using these results, we prove that a system of fundamental modal logic is sound and complete with respect to an elementary class of bi-relational structures $(X,\vartriangleleft, R)$.

In a 2005 paper, Casacuberta, Scevenels and Smith construct a homotopy idempotent functor $E$ on the category of simplicial sets with the property that whether it can be expressed as localization with respect to a map $f$ is independent of the ZFC axioms. We show that this construction can be carried out in homotopy type theory. More precisely, we give a general method of associating to a suitable (possibly large) family of maps, a reflective subuniverse of any universe $\mathcal{U}$. When specialized to an appropriate family, this produces a localization which when interpreted in the $\infty$-topos of spaces agrees with the localization corresponding to $E$. Our approach generalizes the approach of [CSS] in two ways. First, by working in homotopy type theory, our construction can be interpreted in any $\infty$-topos. Second, while the local objects produced by [CSS] are always 1-types, our construction can produce $n$-types, for any $n$. This is new, even in the $\infty$-topos of spaces. In addition, by making use of universes, our proof is very direct. Along the way, we prove many results about "small" types that are of independent interest. As an application, we give a new proof that separated localizations exist. We also give results that say when a localization with respect to a family of maps can be presented as localization with respect to a single map, and show that the simplicial model satisfies a strong form of the axiom of choice which implies that sets cover and that the law of excluded middle holds.

We investigate the extremality of stabilizer states to reveal their exceptional role in the space of all $n$-qubit/qudit states. We establish uncertainty principles for the characteristic function and the Wigner function of states, respectively. We find that only stabilizer states achieve saturation in these principles. Furthermore, we prove a general theorem that stabilizer states are extremal for convex information measures invariant under local unitaries. We explore this extremality in the context of various quantum information and correlation measures, including entanglement entropy, conditional entropy and other entanglement measures. Additionally, leveraging the recent discovery that stabilizer states are the limit states under quantum convolution, we establish the monotonicity of the entanglement entropy and conditional entropy under quantum convolution. These results highlight the remarkable information-theoretic properties of stabilizer states. Their extremality provides valuable insights into their ability to capture information content and correlations, paving the way for further exploration of their potential in quantum information processing.

Although the asymptotic properties of the parameter estimator have been derived in the $p_{0}$ model for directed graphs with the differentially private bi-degree sequence, asymptotic theory in general models is still lacking. In this paper, we release the bi-degree sequence of directed graphs via the discrete Laplace mechanism, which satisfies differential privacy. We use the moment method to estimate the unknown model parameter. We establish a unified asymptotic result, in which consistency and asymptotic normality of the differentially private estimator holds. We apply the unified theoretical result to the Probit model. Simulations and a real data demonstrate our theoretical findings.

Distributive laws are important for algebraic reasoning in arithmetic and logic. They are equally important for algebraic reasoning about concurrent programs. In existing theories such as Concurrent Kleene Algebra, only partial correctness is handled, and many of its distributive laws are weak, in the sense that they are only refinements in one direction, rather than equalities. The focus of this paper is on strengthening our theory to support the proof of strong distributive laws that are equalities, and in doing so come up with laws that are quite general. Our concurrent refinement algebra supports total correctness by allowing both finite and infinite behaviours. It supports the rely/guarantee approach of Jones by encoding rely and guarantee conditions as rely and guarantee commands. The strong distributive laws may then be used to distribute rely and guarantee commands over sequential compositions and into (and out of) iterations. For handling data refinement of concurrent programs, strong distributive laws are essential.

We consider a geometric programming problem consisting in minimizing a function given by the supremum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption, we show that a $\varepsilon$-minimizer can be computed in a time that is polynomial in the input size and in $|\log\varepsilon|$. This is obtained by establishing bit-size estimates on approximate minimizers and by applying the ellipsoid method. We also derive polynomial iteration complexity bounds for the interior point method applied to the same class of problems. We deduce that the spectral radius of a partially symmetric, weakly irreducible nonnegative tensor can be approximated within $\varepsilon$ error in poly-time. For strongly irreducible tensors, we also show that the logarithm of the positive eigenvector is poly-time computable. Our results also yield that the the maximum of a nonnegative homogeneous $d$-form in the unit ball with respect to $d$-H\"older norm can be approximated in poly-time. In particular, the spectral radius of uniform weighted hypergraphs and some known upper bounds for the clique number of uniform hypergraphs are poly-time computable.

In the distributional Twenty Questions game, Bob chooses a number $x$ from $1$ to $n$ according to a distribution $\mu$, and Alice (who knows $\mu$) attempts to identify $x$ using Yes/No questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for $\mu$, yet this strategy could potentially uses all $2^n$ possible questions. Dagan et al. constructed a set of $1.25^{n+o(n)}$ questions which suffice to construct an optimal strategy for all $\mu$, and showed that this number is optimal (up to sub-exponential factors) for infinitely many $n$. We determine the optimal size of such a set of questions for all $n$ (up to sub-exponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the $d$-ary setting, obtaining similar results with $1.25$ replaced by $1 + (d-1)/d^{d/(d-1)}$.

Time-fractional parabolic equations with a Caputo time derivative of order $\alpha\in(0,1)$ are discretized in time using continuous collocation methods. For such discretizations, we give sufficient conditions for existence and uniqueness of their solutions. Two approaches are explored: the Lax-Milgram Theorem and the eigenfunction expansion. The resulting sufficient conditions, which involve certain $m\times m$ matrices (where $m$ is the order of the collocation scheme), are verified both analytically, for all $m\ge 1$ and all sets of collocation points, and computationally, for all $ m\le 20$. The semilinear case is also addressed.