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657b4dbba473d04012c5c2108d07d1a0255370769f8fc02ee182c89f2ddb748e
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2026-01-21T00:00:00-05:00
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Squared Bessel processes under nonlinear expectation
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arXiv:2509.24481v2 Announce Type: replace Abstract: In this paper, we define the squared G-Bessel process as the square of the modulus of a class of G-Brownian motions and establish that it is the unique solution to a stochastic differential equation. We then derive several path properties of the squared G-Bessel process, which are more profound in the capacity sense. Furthermore, we provide upper and lower bounds for the Laplace transform of the squared G-Bessel process. Finally, we prove that the time-space transformed squared G-Bessel process is a G'-CIR process.
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https://arxiv.org/abs/2509.24481
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a737ff28ff41918db04328bc956d42d2fc674f69b40ded08c54273729c2ac39e
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2026-01-21T00:00:00-05:00
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Bounds on the propagation radius in power domination
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arXiv:2510.02211v2 Announce Type: replace Abstract: Let $G$ be a graph and let $S \subseteq V(G)$. It is said that $S$ \textit{dominates} $N[S]$. We say that $S$ \textit{monitors} vertices of $G$ as follows. Initially, all dominated vertices are monitored. This step is called the \textit{domination} step. Thereafter, the set of unmonitored vertices of which each is the only unmonitored neighbour of a monitored vertex, is monitored. This step is called a \textit{propagation} step and is repeated until the process terminates. The process terminates when the there are no monitored vertices which have exactly one unmonitored neighbour. This combined process of initial domination and subsequent propagation is called \textit{power domination}. If all vertices of $G$ are monitored at termination, then $S$ is said to be a \textit{power dominating set (PDS) of $G$}. The \textit{power domination number of $G$}, denoted as $\gamma_p(G)$, is the minimum cardinality of a PDS of $G$. The \textit{propagation radius of $G$} is the minimum number of steps it takes a minimum PDS to monitor $V(G)$. In this paper we determine an upper bound on the propagation radius of $G$ with regards to power domination, in terms of $\delta$ and $n$. We show that this bound is only attained when $\gamma_p(G)=1$ and then improve this bound for $\gamma_p(G)\geq 2$. Sharpness examples for these bounds are provided. We also present sharp upper bounds on the propagation radius of split graphs. We present sharpness results for a known lower bound of the propagation radius for all $\Delta\geq 3$.
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https://arxiv.org/abs/2510.02211
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d1f90d77d28f2129e31ae9b10d991c9ad1a0fbb951cba6cb6165c05a6b0b1ddf
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2026-01-21T00:00:00-05:00
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Certain Inequalities for the generalized polar derivative of a polynomial
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arXiv:2510.02775v2 Announce Type: replace Abstract: Recently Rather et al. \cite{NT} considered the generalized derivative and the generalized polar derivative and studied the relative position of zeros of generalized derivative and generalized polar derivative with respect to the zeros of polynomial.\\ \indent In this paper, we establish some inequalities that estimate the maximum modulus of generalized derivative and the generalized polar derivative of the polynomial $P(z)$, which is also the extension of recently known results.
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https://arxiv.org/abs/2510.02775
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ed83fa06b58c114f6d8f4b67db87bfa81ec0e6267069cf54cecb54d96b48ae47
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2026-01-21T00:00:00-05:00
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Perspectives on Stochastic Localization
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arXiv:2510.04460v2 Announce Type: replace Abstract: We survey different perspectives on the stochastic localization process of Eldan, a powerful construction that has had many exciting recent applications in high-dimensional probability and algorithm design. Unlike prior surveys on this topic, our focus is on giving a self-contained presentation of all known alternative constructions of Eldan's stochastic localization, with an emphasis on connections between different constructions. Our hope is that by collecting these perspectives, some of which had primarily arisen within a particular community (e.g., probability theory, theoretical computer science, information theory, or machine learning), we can broaden the accessibility of stochastic localization, and ease its future use.
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https://arxiv.org/abs/2510.04460
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b80c16c139d1cc6dde34e3c08a8c6d37cc0fd9f8c299841e4958d375484f95fc
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2026-01-21T00:00:00-05:00
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Relaxation of quasi-convex functionals with variable exponent growth
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arXiv:2510.04672v3 Announce Type: replace Abstract: We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely continuous and singular parts weighted via a recession function.
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https://arxiv.org/abs/2510.04672
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9cea13cbc18be7f4fd1a14a9cbefd5b78948156b1333e8214d6e3eb6c129de97
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2026-01-21T00:00:00-05:00
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Log-majorizations between quasi-geometric type means for matrices
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arXiv:2510.04691v3 Announce Type: replace Abstract: In this paper, for $\alpha\in(0,\infty)\setminus\{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $\mathcal{M}_{\alpha,p}(A,B):=\mathcal{M}_\alpha(A^p,B^p)^{1/p}$ of several $\alpha$-weighted geometric type matrix means $\mathcal{M}_\alpha(A,B)$ such as the $\alpha$-weighted geometric mean in Kubo--Ando's sense, the R\'enyi mean, etc. The log-majorization $\mathcal{M}_{\alpha,p}(A,B)\prec_{\log}\mathcal{N}_{\alpha,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $\alpha$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}_{\alpha,p}$ is also discussed based on theory of quantum divergences.
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https://arxiv.org/abs/2510.04691
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6067cc598b63aba3f48b97bacd83245aae7a7c1c9c45d2710771eed437a53710
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2026-01-21T00:00:00-05:00
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Locally similar distances and equality of the induced intrinsic distances
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arXiv:2510.05574v2 Announce Type: replace Abstract: Let $X$ be a set and $d_1,d_2$ be two distances on $X$. We say that $d_1$ and $d_2$ are locally similar and write $d_1\cong d_2$ if $d_1$ and $d_2$ are topologically equivalent and, for every $a$ in $X$, \[ \lim_{x\to a} \frac{d_2(x,a)}{d_1(x,a)}=1. \] We prove that if $d_1\cong d_2$, then the intrinsic distances induced by $d_1$ and $d_2$ coincide. We also provide sufficient conditions for $d_1\cong d_2$ and consider several examples related to reproducing kernel Hilbert spaces.
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https://arxiv.org/abs/2510.05574
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3ae4a428702814f42c2c4b519de69de35c300118aa4bf22652d6a85d1f5b4702
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2026-01-21T00:00:00-05:00
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Rearrangements of distributions on integers that minimize variance
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arXiv:2510.07899v2 Announce Type: replace Abstract: Which permutations of a probability distribution on integers minimize variance? Let $X$ be a random variable on a set of integers $\{x_1, \dots, x_N\}$ such that $\mathbb{P}(X_i = x_i) = p_i$, $i \in \{1,\dots,N\}$. Let $(p^{(1)}, \dots, p^{(N)})$ be the sequence $(p_1, \dots, p_N)$ ordered non-increasingly. Let $X^+$ be the random variable defined by $\mathbb{P}(X^+=0)=p^{(1)}$, $\mathbb{P}(X^+=1) = p^{(2)}$, $\mathbb{P}(X^+=-1)=p^{(3)}, \dots, \mathbb{P}(X^+=(-1)^N \lfloor \frac {N} 2 \rfloor)=p^{(N)}$. In this short note we generalize and prove the inequality $\mathrm{Var}\, X^+ \le \mathrm{Var}\, X$.
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https://arxiv.org/abs/2510.07899
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2bad55b1878444f615b3030e7a3d79337985715f89ee4f15acd63d979438795d
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2026-01-21T00:00:00-05:00
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On covering properties of end and ray spaces
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arXiv:2510.10825v3 Announce Type: replace Abstract: We provide new results on combinatorial characterizations of covering properties in end spaces and ray spaces. In particular, we characterize the Lindel\"of degree, the extent, the Rothberger property, $\sigma$-compactness and the Menger property for ray, end and edge-end spaces. We show that $\sigma$-compactness and the Menger property are equivalent for these spaces, and that they are all $D$-spaces. As an application of some of these characterizations, we are able to provide combinatorial characterizations of graphs with countably many ends and edge-ends.
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https://arxiv.org/abs/2510.10825
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1b64264455b1134c2713aea9786d9f6ba35a06cc736bc92cb8134bec398ea787
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2026-01-21T00:00:00-05:00
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Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras
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arXiv:2510.10874v2 Announce Type: replace Abstract: We construct a higher level analogue of Dorey's rule, which describe certain surjective morphisms between Kirillov--Reshetikhin (KR) modules over quantum affine algebras. Building on this, we establish a generalized T-system of short exact sequences and prove the denominator formula between KR modules in all nonexceptional types, except with only mild ambiguities persisting in type $C_n^{(1)}$. As a consequence, we can completely classify when a tensor product of KR modules is simple. These results have further applications to Schur positivity statements, quiver Hecke algebras, and the recently introduced $\mathfrak{d}$-invariants in monoidal categories over quantum affine algebras and quiver Hecke algebras.
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https://arxiv.org/abs/2510.10874
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6e3edd77e4f1fe67ac33917631f3c3a5311c399c73302e3783cf6f7fc520fae5
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2026-01-21T00:00:00-05:00
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A preorder on the set of links with applications to symmetric unions
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arXiv:2510.12372v2 Announce Type: replace Abstract: For a link $L$ in the $3$-sphere, the $\pi$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$, we denote this by $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation.
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https://arxiv.org/abs/2510.12372
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8f808ffd37eeed98ef06fb1a6988269aa02933266008379cfd1a2a91166d3167
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2026-01-21T00:00:00-05:00
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Complete gradient Einstein-type Sasakian manifolds with $\alpha=0$
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arXiv:2510.12441v3 Announce Type: replace Abstract: Catino, Mastrolia, Monticelli, and Rigoli have launched an ambitious program to study known geometric solitons from a unified perspective, which they term Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe solitons, and all of their generalizations simultaneously. Einstein-type manifolds are characterized by four constants $\alpha, \beta, \mu$ and $\rho$. In this paper, we show that when $\alpha = 0$, complete gradient Einstein-type Sasakian manifolds are trivial or isometric to the unit sphere. As a consequence, many geometric solitons on Sasakian manifolds turn out to be trivial or isometric to the unit sphere.
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https://arxiv.org/abs/2510.12441
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4eb063753aebe4c0b800481d6db234deab96a5318645bc2eaebbb72a1130bec5
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2026-01-21T00:00:00-05:00
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Existence of 3 anti-cocircular truncated M\"obius planes and constructions of strength-4 covering arrays
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arXiv:2510.13122v2 Announce Type: replace Abstract: Two projective (affine) planes with the same point sets are orthogoval if the common intersection of any two lines, one from each, has size at most two. The existence of a pair of orthogoval projective planes has been proven and published independently many times. A strength-$t$ covering array, denoted by CA$(N; t, k, v)$, is an $N \times k$ array over a $v$-set such that in any $t$-set of columns, each $t$-tuple occurs at least once in a row. A pair of orthogoval projective planes can be used to construct a strength-$3$ covering array CA$(2q^3-1; 3, q^2 + q + 1, q)$. Our work extends this result to construct arrays of strength $4$. A $k$-cap in a projective geometry is a set of $k$ points no three of which are collinear. In $PG(3,q)$, an ovoid is a maximum-sized $k$-cap with $k =q^2+1$. Its plane sections (circles) are the blocks of a $3-(q^2 + 1, q + 1, 1)$ design, called a M\"obius plane of order $q$. For $q$ an odd prime power, we prove the existence of three truncated M\"obius planes, such that for any choice of these circles, one from each plane, their intersection size is at most three. From this, we construct a strength-$4$ covering array CA$(3q^4-2; 4, \frac{q^2+1}{2}, q)$. For $q \geq 11$, these covering arrays improve the size of the best-known covering arrays with the same parameters by almost 25 percent. The CA$(3q^4 -3; 4, \frac{q^2 +1}{2}, q)$ is used as the main ingredient in a recursive construction to obtain a CA$(5q^4 - 4q^3 - q^2 + 2q; 4, q^2 +1, q)$. Some improvements are obtained in the size of the best-known arrays using these covering arrays.
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https://arxiv.org/abs/2510.13122
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d29aab3c17519f5e4a5edf57a0fdb89b357fafc8134c0bdebc7812ab7e5f7ca7
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2026-01-21T00:00:00-05:00
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A central limit theorem for partitions involving generalised divisor functions
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arXiv:2510.19740v2 Announce Type: replace Abstract: We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ given by the bivariate generating series \begin{align*} Q_f(z,u) \coloneqq 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n =\prod_{k=1}^{\infty}(1+uz^k)^{\Delta_f(k)}, \end{align*} where $\Delta_f(n)=f(n+1)-f(n)$. In this article, we establish a central limit theorem for the number of summands in such partitions when $f(n)=\sigma_r(n)$ denotes the generalised divisor function, defined as $\sigma_r(n)=\sum_{d|n}d^r$ for integer $r\geq 2$. This can be considered as a generalisation of the work of Lipnik, Madritsch, and Tichy, who previously studied this problem for $f(n)=\lfloor{n}^{\alpha}\rfloor$ with $01$. We study this problem employing the identity involving the Ramanujan sum. Furthermore, we analyse the Euler product arising from the above Dirichlet series by adopting the argument of Alkan, Ledoan and Zaharescu.
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https://arxiv.org/abs/2510.19740
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46362fdadaf2d3b08ef20ff3a083c619d636ae34b2589746cb9b0771cb9046c6
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2026-01-21T00:00:00-05:00
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Qualitative Behavior of Solutions to a Forced Nonlocal Thin-Film Equation
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arXiv:2510.20289v2 Announce Type: replace Abstract: We study a one-dimensional nonlocal degenerate fourth-order parabolic equation with inhomogeneous forces relevant to hydraulic fracture modeling. Employing a regularization scheme, modified energy/entropy methods, and novel differential inequality techniques, we establish global existence and long-time behavior results for weak solutions under both time-and space-dependent and time-and space-independent inhomogeneous forces. Specifically, for the time-and space-dependent force $S(t, x)$, we prove that the solution converges to $\bar{u}_0+\frac{1}{|\Omega|}\int_0^\infty \int_\Omega S(r, x)\, dxdr $, where $\bar{u}_0=\frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)\,dx$ is the spatial average of the initial data, and we provide bilateral estimates for the convergence rate. For the time-and space-independent force $S_0$, we show that the solution approaches the linear function $\bar{u}_0 + tS_0$ at an exponential rate.
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https://arxiv.org/abs/2510.20289
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f4110be3dc541af460ee58d419081b66bba325035ff3a2f17e08e28073248dbc
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2026-01-21T00:00:00-05:00
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Binno: A 1st-order method for Bi-level Nonconvex Nonsmooth Optimization for Matrix Factorizations
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arXiv:2510.21390v2 Announce Type: replace Abstract: In this work, we develop a method for nonconvex, nonsmooth bi-level optimization and we introduce Binno, a first order method that leverages proximal constructions together with carefully designed descent conditions and variational analysis. Within this framework, Binno provably enforces a descent property for the overall objective surrogate associated with the bi-level problem. Each iteration performs blockwise proximal-gradient updates for the upper and the lower problems separately and then forms a calibrated, block-diagonal convex combination of the two tentative iterates. A linesearch selects the combination weights to enforce simultaneous descent of both level-wise objectives, and we establish conditions guaranteeing the existence of such weights together with descent directions induced by the associated proximal-gradient maps. We also apply Binno in the context of sparse low-rank factorization, where the upper level uses elementwise $\ell_1$ penalties and the lower level uses nuclear norms, coupled via a Frobenius data term. We test Binno on synthetic matrix and a real traffic-video dataset, attaining lower relative reconstruction error and higher peak signal-to-noise ratio than some standard methods.
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https://arxiv.org/abs/2510.21390
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ab1ebc1deee47e14e8e467902c13eca9d5422860a3c6d348db4c1959f55b9c7a
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2026-01-21T00:00:00-05:00
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Cellular flow control design for mixing based on the least action principle
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arXiv:2510.22703v2 Announce Type: replace Abstract: We consider a novel approach for the enhancement of fluid mixing via pure stirring strategies building upon the Least Action Principle (LAP) for incompressible flows. The LAP is formally analogous to the Benamou--Brenier formulation of optimal transport, but imposes an incompressibility constraint. Our objective is to find a velocity field, generated by Hamiltonian flows, that minimizes the kinetic energy while ensuring that the initial scalar distribution reaches a prescribed degree of mixedness by a finite time. This formulation leads to a ``point to set" type of optimization problem which relaxes the requirement on controllability of the system compared to the classic LAP framework. In particular, we assume that the velocity field is induced by a finite set of cellular flows that can be controlled in time. We justify the feasibility of this constraint set and leverage Benamou--Brenier's results to establish the existence of a global optimal solution. Finally, we derive the corresponding optimality conditions for solving the optimal time control and conduct numerical experiments demonstrating the effectiveness of our control design.
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https://arxiv.org/abs/2510.22703
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11fcd64a205adb77be7e139b648a7ec58345d11d71e4c1f786ead0159ec96ea6
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2026-01-21T00:00:00-05:00
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Tree-Cotree-Based IETI-DP for Eddy Current Problems in Time-Domain
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arXiv:2510.23446v2 Announce Type: replace Abstract: For low-frequency electromagnetic problems, where wave-propagation effects can be neglected, eddy current formulations are commonly used as a simplification of the full Maxwell's equations. In this setup, time-domain simulations, needed to capture transient startup responses or nonlinear behavior, are often computationally expensive. We propose a novel tearing and interconnecting approach for eddy currents in time-domain and investigate its scalability.
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https://arxiv.org/abs/2510.23446
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7569268f07e889a7741a6be4d8bb225bf0ba3811dd6a49bed6662145357e441d
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2026-01-21T00:00:00-05:00
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Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof
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arXiv:2510.23513v2 Announce Type: replace Abstract: The Nesterov accelerated gradient method, introduced in 1983, has been a cornerstone of optimization theory and practice. Yet the question of its point convergence had remained open. In this work, we resolve this longstanding open problem in the affirmative. The discovery of the proof was heavily assisted by ChatGPT, a proprietary large language model, and we describe the process through which its assistance was elicited.
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https://arxiv.org/abs/2510.23513
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d7b0fe0abb02a1cdc1837263cadbb63528152eee1650b2daa2b93db2f6332ae9
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2026-01-21T00:00:00-05:00
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A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces
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arXiv:2510.23865v2 Announce Type: replace Abstract: We calculate the Roger-Yang skein algebra of the annulus with two interior punctures, $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, and show there is a surjective homomorphism from this algebra to the Kauffman bracket skein algebra of the closed torus. Using this homomorphism, we characterize the irreducible, finite-dimensional representations of $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, showing that they can be described by certain complex data and that the correspondence is unique if certain polynomial conditions are satisfied. We also use the relationship with the skein algebra of the torus to compute structural constants for a bracelets basis for $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, giving evidence for positivity.
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https://arxiv.org/abs/2510.23865
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fb38cd24a4a046874b524f6fbdba6681e95cecf9ac947f6a8700f2295a7f2f36
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2026-01-21T00:00:00-05:00
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The multivariate Hermite method for counting real and complex solutions to polynomial systems
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arXiv:2510.23897v2 Announce Type: replace Abstract: This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final section includes an implementation in Macaulay2, a free and open-source computer algebra system.
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https://arxiv.org/abs/2510.23897
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dca86e2c983c2236a274596dfb9323fa09ecc067b968b4b0b52ec4b81b1ccec8
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2026-01-21T00:00:00-05:00
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Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers
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arXiv:2510.25250v2 Announce Type: replace Abstract: Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo powers of $2$ for infinitely many values of $k$. Our approach is truly elementary, relying on generating function manipulations, theta functions and $q$-dissection techniques. We then close by demonstrating an infinite family of congruences modulo 11 which is proven using a result of Ahlgren.
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https://arxiv.org/abs/2510.25250
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16dcd23e0d79a334833c1a6cabd82b070c0c8b67b287613915b301504b670b46
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2026-01-21T00:00:00-05:00
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Cubic Polynomials and Sums of Two Squares
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arXiv:2510.25492v2 Announce Type: replace Abstract: We establish a lower bound for the frequency with which an irreducible monic cubic polynomial can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by Grechuk (2021) concerning the infinitude of such values. Our proof relies on a two-dimensional unit argument and the arithmetic of degree six number fields. For example, we show that if $h \equiv 2 \pmod{4}$, then \begin{align*} \# \{n : n^3+h \in \square_{2}, \ 1 \leq n \leq x \} \gg x^{1/3-o(1)}. \end{align*} These arguments may be generalised to study the representation of irreducible monic cubic polynomials by the quadratic form $x^2+ny^2$, where $n \in \mathbb{N}$.
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https://arxiv.org/abs/2510.25492
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d6af733794f5e2454afaadcb672300b8ae73f71be92cdc873fa211d6e05014ab
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2026-01-21T00:00:00-05:00
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Fine-grained deterministic hardness of the shortest vector problem
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arXiv:2511.01626v2 Announce Type: replace Abstract: Let $\gamma$-$\mathsf{GapSVP}_p$ be the decision version of the shortest vector problem in the $\ell_p$-norm with approximation factor $\gamma$, let $n$ be the lattice dimension and $0<\varepsilon\leq 1$. We prove that the following statements hold for infinitely many values of $p$. $(2-\varepsilon)$-$\mathsf{GapSVP}_p$ is not in $O\left(2^{O(p)}\cdot n^{O(1)}\right)$-time, unless $\text{P}=\text{NP}$. $(2-\varepsilon)$-$\mathsf{GapSVP}_p$ is not in $O\left(2^{2^{o(p)}}\cdot 2^{o(n)}\right)$-time, unless the Strong Exponential Time Hypothesis is false. The proofs are based on a Karp reduction from a variant of the subset-sum problem that imposes restrictions on vectors orthogonal to the vector of its weights. While more extensive hardness results for the shortest vector problem in all $\ell_p$-norms have already been established under randomized reductions, the results in this paper are fully deterministic.
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https://arxiv.org/abs/2511.01626
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c6f25b70d2e4944ceee0ab85a0bce09a6d06b0adedbd81833ec50f6bc733c4c5
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2026-01-21T00:00:00-05:00
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Ergodic Rate Analysis of Two-State Pinching-Antenna Systems
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arXiv:2511.01798v2 Announce Type: replace Abstract: Flexible Antenna Systems (FAS) are a key enabler of next-generation wireless networks, allowing the antenna aperture to be dynamically reconfigured to adapt to channel conditions and service requirements. In this context, pinching-antenna systems (PASs) implemented on software-controllable dielectric waveguides provide the ability to reconfigure both channel characteristics and path loss by selectively exciting discrete radiation points. Existing works, however, typically assume continuously adjustable pinching positions, neglecting the spatial discreteness imposed by practical implementations. This paper develops a closed-form analytical framework for the ergodic rate of two-state PASs, where pinching antennas are fixed and only their activation states are controlled. To quantify the impact of spatial discretization, pinching discretization efficiency is introduced, characterizing the performance gap relative to the ideal continuous case. Finally, numerical results show that near-continuous performance can be achieved with a limited number of pinching points, providing design insights for scalable PASs.
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https://arxiv.org/abs/2511.01798
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d6f9d50680924188b1b8bcc04b5730f09e2eeff454c991631531b018777bbf8b
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2026-01-21T00:00:00-05:00
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The associative-poset point of view on right regular bands
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arXiv:2511.05721v2 Announce Type: replace Abstract: We present two results on the relation between the class of right regular bands (RRBs) and their underlying *associative posets*. The first one is a construction of a left adjoint to the forgetful functor that takes an RRB $(P,\cdot)$ to the corresponding $(P,\leq)$. The construction of such a left adjoint is actually done in general for any class of relational structures $(X,R)$ obtained from a variety, where $R$ is defined by a finite conjunction of identities. The second result generalizes the "inner" representations of direct product decompositions of semilattices studied by the second author to RRBs having at least one commuting element.
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https://arxiv.org/abs/2511.05721
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1aa6e7a412d4adf893bf38bdbc3ca3f1343175ece614c9ec75418be30f95161d
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2026-01-21T00:00:00-05:00
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On the Asymptotic Palindrome Density of Fibonacci Infinite Words
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arXiv:2511.06485v2 Announce Type: replace Abstract: In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0 \to 01$, $1 \to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(\lambda,n)$, $\mathrm{dens}(\alpha,n)$, and $\mathrm{dens}(\beta,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{\varphi_1}$, where $\varphi_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities.
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https://arxiv.org/abs/2511.06485
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ed71adf692748c1875abb2b973eb20fe23fad6b3e2d608b5fb9e67b0ed2a73f2
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2026-01-21T00:00:00-05:00
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Analytical estimations of edge states and extended states in large finite-size lattices
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arXiv:2511.07875v2 Announce Type: replace Abstract: The bulk boundary correspondence, one of the most significant features of topological matter, theoretically connects the existence of edge modes at the boundary with topological invariants of the bulk spectral bands. However, it remains unspecified in realistic examples how large the size of a lattice should be for the correspondence to take effect. In this work, we employ the diatomic chain model to introduce an analytical framework to characterize the dependence of edge states on the lattice size and boundary conditions. In particular, we apply asymptotic estimates to examine the bulk boundary correspondence in long diatomic chains as well as precisely quantify the deviations from the bulk boundary correspondence in finite lattices due to symmetry breaking and finite size effects. Moreover, under our framework the eigenfrequencies near the band edges can be well approximated where two special patterns are detected. These estimates on edge states and eigenfrequencies in linear diatomic chains can be further extended to nonlinear chains to investigate the emergence of new nonlinear edge states and other nonlinear localized states. In addition to one-dimensional diatomic chains, examples of more complicated and higher dimensional lattices are provided to show the universality of our analytical framework.
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https://arxiv.org/abs/2511.07875
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d727506d416c26196efea4bf3ac2ba9902d84986a1a08c97161928f6f820303f
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2026-01-21T00:00:00-05:00
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Nonlinear scalar field equations with a critical Hardy potential
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arXiv:2511.15668v2 Announce Type: replace Abstract: We study the existence of solutions for the nonlinear scalar field equation $$-\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy potential and $N \geq 3$. The nonlinearity $g$ is continuous and satisfies general subcritical growth assumptions of the Berestycki-Lions type. The problem is approached using variational methods within a non-standard functional setting. The natural energy functional associated with the equation is defined on the space $X^1(\mathbb{R}^N)$, which is the completion of $H^1(\mathbb{R}^N)$ with respect to the norm induced by the quadratic part of the functional. We establish the existence of a nontrivial solution $u_0 \in X^1(\mathbb{R}^N)$ that satisfies the Poho\v{z}aev constraint $\mathcal{M}$ and minimizes the energy functional on $\mathcal{M}$. Furthermore, assuming $g$ is odd, we prove the existence of at least one non-radial solution.
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https://arxiv.org/abs/2511.15668
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2026-01-21T00:00:00-05:00
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Normalization of Puiseux Hypersurfaces
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arXiv:2511.15863v2 Announce Type: replace Abstract: It is known that the normalization of a quasi-ordinary complex singularity is a Hirzebruch-Jung, see [Gon00; Pop04; AS05]. We extend this result to Puiseux hypersurfaces. Moreover, we prove that Hirzebruch-Jung singularities are precisely normalizations of Puiseux hypersurfaces. Our result holds over an algebraically closed field whose characteristic does not divide the degree of the polynomial defining the hypersurface. Finally, in the analytic complex case, we conclude that the normalization of an irreducible Puiseux hypersurface is the normalization of a complex analytic quasi-ordinary singularity.
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https://arxiv.org/abs/2511.15863
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73a0700f296e082c47d27b9abfa2e6f83ea6693e7ddf402f286385da6fc37a88
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2026-01-21T00:00:00-05:00
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Subtlety of oscillation indices of oscillatory integrals of real analytic functions
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arXiv:2511.16257v2 Announce Type: replace Abstract: For a locally defined real analytic function, we study the relation between the oscillation index of oscillatory integrals and the real log canonical threshold. The former is always negative, and its absolute value is greater than or equal to the latter. They coincide very often, but there are certain exceptional cases even in the Newton nondegenerate convenient homogeneous case, for instance if the number of variables is even and smaller than the degree. This does not seem compatible with some standard formula in the literature, and there must be some error somewhere, although it does not seem easy to find it inside this paper.
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https://arxiv.org/abs/2511.16257
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aa7bcbf8aaec7980dcdaa3b774b85b3da76f35af836b097489c15dbb997b007f
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2026-01-21T00:00:00-05:00
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Full flexibility of isometric immersions of metrics with low H\"older regularity in Poznyak theorem's dimension
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arXiv:2511.16305v2 Announce Type: replace Abstract: A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $\omega\subset\mathbb{R}^2$, has a smooth isometric immersion into $\mathbb{R}^4$. Using techniques of convex integration, we prove that for any $2$-dimensional $g\in\mathcal{C}^{r,\beta}$, an isometric immersion of regularity $\mathcal{C}^{1,\alpha}(\bar\omega,\mathbb{R}^4)$ for any $\alpha<\min\{\frac{r+\beta}{2},1\}$, may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches $\mathcal{C}^{1,1-}$ for $g\in \mathcal{C}^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity $\mathcal{C}^{1,1/3-}$ achieved by Cao, Hirsch and Inauen for isometric immersions into $\mathbb{R}^{3}$ and the lack of flexibility (rigidity) of such isometric immersions with regularity $\mathcal{C}^{1, 2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity $\mathcal{C}^{1,1-}$ obtained byt K\"allen for isometric immersions into higher codimensional space $\mathbb{R}^{8}$; and (iii) the regularity $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}-}$ proved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Amp\`ere system.
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https://arxiv.org/abs/2511.16305
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ca13af374bfeaf0ebe0da32ab8cb70373f0ee638cd287eca1d83ef8d3a71e755
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2026-01-21T00:00:00-05:00
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Function-Correcting Codes With Data Protection
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arXiv:2511.18420v2 Announce Type: replace Abstract: Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. We first introduce a more general approach and a framework for function-correcting codes that incorporates data protection along with protection of function values. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph called minimum-distance graph to a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. To the best of our knowledge, this is the first instance of FCCs with a linear structure. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection.
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https://arxiv.org/abs/2511.18420
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2026-01-21T00:00:00-05:00
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Elastic scattering by locally rough interfaces
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arXiv:2511.18799v2 Announce Type: replace Abstract: In this paper, we present the first well-posedness result for elastic scattering by locally rough interfaces in both two and three dimensions. Inspired by the Helmholtz decomposition, we discover a fundamental identity for the stress vector, revealing an intrinsic relationship among the generalized stress vector, the Lame constants and certain tangential differential operators. This identity leads to two key limits for surface integrals involving scattered solutions, from which we deduce the first uniqueness result of direct problem for all frequencies. Through a detailed analysis, applying the steepest descent method, subsequently we derive the existence and uniqueness of the corresponding two-layered Green's tensor along with its explicit expression when the transmission coefficient equals 1. Finally, by leveraging properties of the Green's tensor, we establish the existence of solutions via the variational method and the boundary integral equation, thereby achieving the first well-posedness result for elastic scattering by rough interfaces.
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https://arxiv.org/abs/2511.18799
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89bad4c9ddc112b7b8df17be43b8f330bd3e20e03389a42c5e1fccb5ae501412
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2026-01-21T00:00:00-05:00
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Higher property T and below-rank phenomena of lattices
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arXiv:2511.20192v2 Announce Type: replace Abstract: The purpose of this paper is twofold. We explore higher property T as an abstract group-theoretic property. In particular, we provide new operator-algebraic characterizations of higher property T. Then we turn to lattices in semisimple Lie groups. We relate higher property T to other cohomological, rigidity and geometric phenomena below the real rank. The second part outlines a conjectural framework that unifies these aspects and reviews recent advances.
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https://arxiv.org/abs/2511.20192
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e213e40cdc6ca564c1c0f1a35acbdb9b71fbf7d6fa1a0e33ae9f0ccf1a154292
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2026-01-21T00:00:00-05:00
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Uniform inference for kernel instrumental variable regression
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arXiv:2511.21603v2 Announce Type: replace Abstract: Instrumental variable regression is a foundational tool for causal analysis across the social and biomedical sciences. Recent advances use kernel methods to estimate nonparametric causal relationships, with general data types, while retaining a simple closed-form expression. Empirical researchers ultimately need reliable inference on causal estimates; however, uniform confidence sets for the method remain unavailable. To fill this gap, we develop valid and sharp confidence sets for kernel instrumental variable regression, allowing general nonlinearities and data types. Computationally, our bootstrap procedure requires only a single run of the kernel instrumental variable regression estimator. Theoretically, it relies on the same key assumptions. Overall, we provide a practical procedure for inference that substantially increases the value of kernel methods for causal analysis.
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https://arxiv.org/abs/2511.21603
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d411a1ae627509df39863d7d1f1a0412493653243c1851112dbfff8c14ea9175
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2026-01-21T00:00:00-05:00
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Maximal variation of linear systems
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arXiv:2511.22329v4 Announce Type: replace Abstract: Let X be a smooth projective complex variety, and L a line bundle on X . We say that the linear system |L| has maximal variation if its elements have the maximum number dim|L| of moduli. We discuss some cases where this situation is expected: hypersurfaces, double coverings of the projective space, K3 surfaces, hyperkahler manifolds, and abelian varieties.
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https://arxiv.org/abs/2511.22329
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b89a1014729029ef72e1987da47bbda9cb905e1ceab3a197c85ec1f1b835d2db
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2026-01-21T00:00:00-05:00
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Convergence rates of self-repelling diffusions on Riemannian manifolds
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arXiv:2511.23333v2 Announce Type: replace Abstract: We study a class of self-repelling diffusions on compact Riemannian manifolds whose drift is the gradient of a potential accumulated along their trajectory. When the interaction potential admits a suitable spectral decomposition, the dynamics and its environment are equivalent to a finite-dimensional degenerate diffusion. We show that this diffusion is a second-order lift of an Ornstein-Uhlenbeck process whose invariant law corresponds to the Gaussian invariant measure of the environment, and immediately obtain a general upper bound on the rate of convergence to stationarity using the framework of second-order lifts. Furthermore, using a flow Poincar\'e inequality, we develop lower bounds on the convergence rate. We show that, in the periodic case, these lower bounds improve upon those of Bena\"im and Gauthier (Probab. Theory Relat. Fields, 2016), and even match the order of the upper bound in some cases.
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https://arxiv.org/abs/2511.23333
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8360705ce910c93eeeb8cd72e116cb740b7fd5f4d5063dfff0f146b28275cbcb
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2026-01-21T00:00:00-05:00
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The intrinsic subgroup of an elliptic curve and Mazur's torsion theorem
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arXiv:2512.00787v2 Announce Type: replace Abstract: We define and study a biadditive symmetric (not necessarily perfect) pairing on the torsion part $\mathrm{Pic}(X)_{\mathrm{tors}}$ of the Picard group of a smooth projective curve $X$ over a field $k$ with values in $k^\times \otimes \mathbb{Q}/\mathbb{Z}$. We call its kernel the intrinsic subgroup of $X$. It turns out that some information on the reduction type of $X$ can be read off from the intrinsic subgroup. Mazur's torsion theorem says that there are exactly 15 isomorphism classes of abelian groups that appear as the rational torsion points of an elliptic curve $X$ over $\mathbb{Q}$ (identified with $\mathrm{Pic}(X)_{\mathrm{tors}}$). We refine this result by determining which subgroups of those 15 groups appear as the intrinsic subgroups.
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https://arxiv.org/abs/2512.00787
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5678018a9fa04dff1c05e16d362640257a1871a6ca847f53583a384a16d95805
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2026-01-21T00:00:00-05:00
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Spectrally additive maps on the positive cones of the Wiener algebra
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arXiv:2512.01173v2 Announce Type: replace Abstract: We study surjective maps between the positive cones of the Wiener algebra that preserve the spectrum of the sum of every two elements. We show that such maps can be extended to isometric real-linear isomorphisms of the Wiener algebra.
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https://arxiv.org/abs/2512.01173
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fec1700e74ac3130c38391c4f88e852bb5bee30bcbe82eaf9ecfb2672219e8b3
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2026-01-21T00:00:00-05:00
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A discrete approach to Dirichlet L-functions, their special values and zeros
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arXiv:2512.01779v2 Announce Type: replace Abstract: We obtain new infinite families of identities among special values of Dirichlet $L$-functions using finite spectral sums. More precisely, we study Dirichlet $L$-functions via discrete analogues $L_n$ arising from the spectral theory of cyclic graphs as $n\rightarrow \infty$. Applying a refined Euler-Maclaurin asymptotic expansion due to Sidi, together with an independent polynomiality property of these finite spectral sums at integers, we obtain exact special-value formulas, even starting at $n=1$. This yields new expressions for certain trigonometric sums of interest in physics, and recovers, by a different method, the striking formulas of Xie, Zhao, and Zhao. Concerning zeros, using the same asymptotic expansion, we prove that for odd primitive characters, an asymptotic functional equation relating $L_n(1-s,\overline{\chi })$ to $L_n(s,\chi)$ is equivalent to the Generalized Riemann Hypothesis for the corresponding Dirichlet $L$-function $L(s,\chi)$. We also provide some remarks about the non-existence of possible real zeros.
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https://arxiv.org/abs/2512.01779
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d23c7ef095b95ba5d603b914dad6f9da59cf7e3bf81177494f8a091770e488d7
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2026-01-21T00:00:00-05:00
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The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$
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arXiv:2512.03754v2 Announce Type: replace Abstract: This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--L\'evy noises. The equations feature a Caputo time derivative $\partial_t^\alpha$ ($0<\alpha<1$) and a spatial nonlocal operator $\phi(\Delta)$ generated by a subordinate Brownian motion, leading to a doubly nonlocal structure. For the case $p \ge 2$, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of $\phi$-Sobolev spaces $\mathcal{H}_p^{\phi,\gamma+2}(T)$. Our approach combines harmonic analysis techniques (Fefferman--Stein theorem, Littlewood--Paley theory) with stochastic analysis to handle the combined Wiener and L\'evy noise terms. In the special case of cylindrical Wiener noise, a dimensional constraint $d < 2\kappa_0\bigl(2 - (2\sigma_2 - 2/p)_+/\alpha\bigr)$ is obtained.~For the low-regularity case $1 \le p \le 2$, where maximal function estimates fail, we construct unique local mild solutions in $L_p(\mathbb{R}^d)$ for equations driven by pure-jump L\'evy space-time white noise, using stochastic truncation and fixed-point arguments. The results unify and extend previous theories by simultaneously incorporating time-space nonlocality and jump-type randomness.
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https://arxiv.org/abs/2512.03754
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57c81ef919a093229a8d889609a2ae21d1235867442b1e50c2603e454058fde6
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2026-01-21T00:00:00-05:00
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A Second Main Theorem for Entire Curves Intersecting Three Conics
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arXiv:2512.03948v2 Announce Type: replace Abstract: We establish a Second Main Theorem for entire holomorphic curves \( f: \mathbb{C} \to \mathbb{P}^2 \) intersecting a generic configuration of three conics \(\mathcal{C}= \mathcal{C}_1+ \mathcal{C}_2+ \mathcal{C}_3 \) in the complex projective plane $\mathbb{P}^2$. Using invariant logarithmic $2$-jet differentials with negative twists, we prove the estimate \[ T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel, \] where \( T_f(r) \) is the Nevanlinna characteristic function, and \( N_f^{[1]}(r, \mathcal{C}_i) \) is the $1$-truncated counting function. The key innovation of our approach is establishing new vanishing lemmas of the form \[ H^0\bigl(\mathbb{P}^2,\, E_{2,m}T_{\mathbb{P}^2}^*(\log \mathcal{C}) \otimes \mathcal{O}_{\mathbb{P}^2}(-t)\bigr) = 0 \] for specific pairs \((m, t)\), achieved by combining algebro-geometric arguments with computer-assisted computations through a mod-\(p\) reduction technique. This yields a systematic method for proving vanishing results for negatively twisted jet differentials -- a key component in complex hyperbolic geometry.
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https://arxiv.org/abs/2512.03948
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473874c9e0d1965db139a7f9501e3696a2f13786f63b1622fbd761885391abc3
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2026-01-21T00:00:00-05:00
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A Tight-binding Approach for Computing Subwavelength Guided Modes in Crystals with Line Defects
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arXiv:2512.05370v2 Announce Type: replace Abstract: In this paper, we develop an accurate and efficient framework for computing subwavelength guided modes in high-contrast periodic media with line defects, based on a tight-binding approximation. The physical problem is formulated as an eigenvalue problem for the Helmholtz equation with high-contrast parameters. By employing layer potential theory on unbounded domains, we characterize the subwavelength frequencies via the quasi-periodic capacitance matrix. Our main contribution is the proof of exponential decay of the off-diagonal elements of the associated full and quasi-periodic capacitance matrices. These decay properties provide error bounds for the banded approximation of the capacitance matrices, thereby enabling a tight-binding approach for computing the spectral properties of subwavelength resonators with non-compact defects. Various numerical experiments are presented to validate the theoretical results, including applications to topological interface modes.
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https://arxiv.org/abs/2512.05370
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Integrating ethical, societal and environmental issues into algorithm design courses
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arXiv:2512.13216v2 Announce Type: replace Abstract: This document, intended for computer science teachers, describes a case study that puts into practice a questioning of ethical, societal and environmental issues when designing or implementing a decision support system. This study is based on a very popular application, namely road navigation software that informs users of real-time traffic conditions and suggests routes between a starting point and a destination, taking these conditions into account (such as Waze). The approach proposes to intertwine technical considerations (optimal path algorithms, data needed for location, etc.) with a broader view of the ethical, environmental and societal issues raised by the tools studied. Based on the authors' experience conducting sessions with students over several years, this document discusses the context of such a study, suggests teaching resources for implementing it, describes ways to structure discussions, and shares scenarios in different teaching contexts.
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https://arxiv.org/abs/2512.13216
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93e53d1a9725a738ddc486603fb166a80a8ec1ffd74b17d01bd0c37ba77e7c65
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2026-01-21T00:00:00-05:00
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Asymptotics for the number of domino tilings of L-shaped Aztec domains
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arXiv:2512.20388v2 Announce Type: replace Abstract: We obtain precise asymptotics for the weighted number of domino tilings of an L-shaped subset of the Aztec diamond, obtained by removing an approximate rectangle in a corner of the Aztec diamond. By tuning the size of the removed corner, we observe different types of asymptotics. For a small removed corner, the number of tilings is close to that of the full Aztec diamond. Enlarging the removed corner to a critical size, a phase transition described in terms of the Tracy-Widom distribution occurs. Further increasing the size of the removed region, we observe a sharp decrease of the number of tilings, until it is finally approximated by the number of tilings of two smaller disjoint Aztec diamonds. We obtain uniform asymptotics for the number of domino tilings which fully describe these transitions.
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https://arxiv.org/abs/2512.20388
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7bca03b479d43eb32b855b46368474160dfcc7eec69e7dd90450cfd700c32f5d
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2026-01-21T00:00:00-05:00
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Linear varieties and matroids with applications to the Cullis' determinant
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arXiv:2512.21098v2 Announce Type: replace Abstract: Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{\'{e}}'s result on the dimension of vector spaces of square matrices annihilating the ordinary determinant. Furthermore, for certain values of $n$ and $k$, we explicitly describe such vector spaces of maximal dimension. Namely, we establish that if $k$ is odd, $n \ge k + 2$ and $\dim(V) = (n-1)k$, then $V$ is equal to the space of all $n\times k$ matrices $X$ such that alternating row sum of $X$ is equal to zero. Our proofs rely on the following observations from the matroid theory that have an independent interest. First, we provide a notion of matroid corresponding to a given linear variety. Second, we prove that if the linear variety is transformed by projections and restrictions, then the behaviour of the corresponding matroid is expressed in the terms of matroid contraction and restriction. Third, we establish that if $M$ is a matroid, $I^*$ its coindependent set $M|S$ and its restriction on a set $S$, then the union of $I^*\setminus S$ with every cobase of $M|S$ is coindependent set of $M$.
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https://arxiv.org/abs/2512.21098
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acdfa1ccf01726099b184b11e2819b314d454804a5800f1c1592b8e0adad4d51
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$x(1-t(x+x^{-1}))F(x;t) = x-tF(0;t)$
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arXiv:2512.21753v2 Announce Type: replace Abstract: The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their generating functions, and the functional equations they satisfy. We focus on algebraic methods for manipulating and solving these equations. Elementary power series algebra plays a prominent role, computer algebra too, but we repeatedly digress and present ideas and methods of different kind whenever it is appropriate. The exposition is organized around the most simple yet non-trivial problem: the enumeration of simple walks on the half-line. The intention is to illustrate different techniques without getting technical.
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https://arxiv.org/abs/2512.21753
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00314d774dadb40ae10bfc26fee1ba615c6292a8905e6049b60442edd5d06628
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2026-01-21T00:00:00-05:00
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Exchangeability and randomness for infinite and finite sequences
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arXiv:2512.22162v2 Announce Type: replace Abstract: Randomness (in the sense of being generated in an IID fashion) and exchangeability are standard assumptions in nonparametric statistics and machine learning, and relations between them have been a popular topic of research. This short paper draws the reader's attention to the fact that, while for infinite sequences of observations the two assumptions are almost indistinguishable, the difference between them becomes very significant for finite sequences of a given length.
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https://arxiv.org/abs/2512.22162
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a219fcd13551629611b2a67634d61401d2c93a231a3766d4ba593be3ea3fbd3d
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2026-01-21T00:00:00-05:00
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Research projects and Moscow Mathematical Conference for high school students
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arXiv:2512.22191v2 Announce Type: replace Abstract: This paper shares some experience in advanced mathematical education. We show how a high school student can be naturally and gradually introduced to basic steps of scientific research: developing intuition by finding and correcting mistakes through discussions and writing a paper, (transparent) anonymous peer review, recognition and award. We show that most of this can be done in research projects not aiming at scientific novelty. We share the experience (both principles and examples) of the Moscow Mathematical Conference of High School Students.
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https://arxiv.org/abs/2512.22191
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e72cfdfa5b6046bf38918c792597ea141ecb4e783a4cda54c6fbc8ce49f3d23f
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2026-01-21T00:00:00-05:00
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Covering in Hamming and Grassmann Spaces: New Bounds and Reed--Solomon-Based Constructions
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arXiv:2512.22911v2 Announce Type: replace Abstract: We study covering problems in Hamming and Grassmann spaces through a unified coding-theoretic and information-theoretic framework. Viewing covering as a form of quantization in general metric spaces, we introduce the notion of the average covering radius as a natural measure of average distortion, complementing the classical worst-case covering radius. By leveraging tools from one-shot rate-distortion theory, we derive explicit non-asymptotic random-coding bounds on the average covering radius in both spaces, which serve as fundamental performance benchmarks. On the construction side, we develop efficient puncturing-based covering algorithms for generalized Reed--Solomon (GRS) codes in the Hamming space and extend them to a new family of subspace codes, termed character-Reed--Solomon (CRS) codes, for Grassmannian quantization under the chordal distance. Our results reveal that, despite poor worst-case covering guarantees, these structured codes exhibit strong average covering performance. In particular, numerical results in the Hamming space demonstrate that RS-based constructions often outperform random codebooks in terms of average covering radius. In the one-dimensional Grassmann space, we numerically show that CRS codes over prime fields asymptotically achieve average covering radii within a constant factor of the random-coding bound in the high-rate regime. Together, these results provide new insights into the role of algebraic structure in covering problems and high-dimensional quantization.
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https://arxiv.org/abs/2512.22911
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0e67ad8042a1df5ffbd3d7da51ca11a1c82880acda2f7724507ecba8d64423cd
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2026-01-21T00:00:00-05:00
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Limit Computation Over Posets via Minimal Initial Functors
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arXiv:2601.00209v2 Announce Type: replace Abstract: It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation. We systematically study the algorithmic implications of this idea for diagrams indexed by a finite poset. We say an initial functor $F\colon C\to D$ with $C$ small is \emph{minimal} if the sets of objects and morphisms of $C$ each have minimum cardinality, among the sources of all initial functors with target $D$. For $Q$ a finite poset or $Q\subseteq \mathbb N^d$ an interval (i.e., a convex, connected subposet), we describe all minimal initial functors $F\colon P\to Q$ and in particular, show that $F$ is always a subposet inclusion. We give efficient algorithms to compute a choice of minimal initial functor. In the case that $Q\subseteq \mathbb N^d$ is an interval, we give asymptotically optimal bounds on $|P|$, the number of relations in $P$ (including identities), in terms of the number $n$ of minima of $Q$: We show that $|P|=\Theta(n)$ for $d\leq 3$, and $|P|=\Theta(n^2)$ for $d>3$. We apply these results to give new bounds on the cost of computing $\lim G$ for a functor $G \colon Q\to \mathbf{Vec}$ valued in vector spaces. For $Q$ connected, we also give new bounds on the cost of computing the \emph{generalized rank} of $G$ (i.e., the rank of the induced map $\lim G\to \mathop{\mathrm{colim}} G$), which is of interest in topological data analysis.
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https://arxiv.org/abs/2601.00209
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74a1d18b75a896ebb32ec699bce3121600fbacd1f01efbab0148ddf2f479e7ca
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2026-01-21T00:00:00-05:00
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Complexity of deep computations via topology of function spaces
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arXiv:2601.00528v2 Announce Type: replace Abstract: We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically.
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https://arxiv.org/abs/2601.00528
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cb77be29dab5f0d3ce05d824fbd7b38c7b8547360096ca0271fd73c9808dcb63
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2026-01-21T00:00:00-05:00
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On oscillator death in the Winfree model
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arXiv:2601.01203v2 Announce Type: replace Abstract: We show that for the standard sinusoidal Winfree model, a coupling strength exceeding twice the maximal magnitude of the intrinsic frequencies guarantees the convergence of the system for Lebesgue almost every initial data. This is proven by first showing, via an order parameter bootstrapping argument, that the pathwise critical coupling strength is upper bounded by a function of the order parameter, and then showing by a volumetric argument that for Lebesgue almost every data the order parameter cannot stay below and be bounded away from 1 for all time; this is a Winfree model counterpart of the analysis of Ha and the author (2020) performed for the Kuramoto model. Using concentration of measure and the aforementioned volumetric argument, we show that, except possibly on a set of very small measure, oscillator death is observed in finite time; this rigorously demonstrates the existence of the oscillator death regime numerically observed by Ariaratnam and Strogatz (2001). These results are robust under many other choices of interaction functions often considered for the Winfree model. We demonstrate that the asymptotic dynamics described in this paper are sharp by analyzing the equilibria of the Winfree model, and we bound the total number of equilibria using a polynomial description.
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https://arxiv.org/abs/2601.01203
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e86de8bc82a64e1aff84802665e9e8d6b7ad7d11b9ce0a2b0d1a7d20ed9b3a29
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2026-01-21T00:00:00-05:00
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New discretised polynomial expander and incidence estimates
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arXiv:2601.01264v2 Announce Type: replace Abstract: We present two applications of recent developments in incidence geometry. One is a $\delta$-discretised version of a particular `Elekes--R\'onyai' expander problem. The second application is an incidence estimate addressing the scenario when both tubes, squares and their shadings satisfy non-concentration assumptions.
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https://arxiv.org/abs/2601.01264
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ab31cd7d88b51564acd1e01cc584e9dfb97a1dc97f8c6213e08755ce4318b7b2
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2026-01-21T00:00:00-05:00
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Algorithmic Information Theory for Graph Edge Grouping and Substructure Analysis
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arXiv:2601.01760v3 Announce Type: replace Abstract: Understanding natural phenomenon through the interactions of different complex systems has become an increasing focus in scientific inquiry. Defining complexity and actually measuring it is an ongoing debate and no standard framework has been established that is both theoretically sound and computationally practical to use. Currently, one of the fields which attempts to formally define complexity is in the realm of Algorithmic Information Theory. The field has shown advances by studying the complexity values of binary strings and 2-dimensional binary matrices using 1-dimensional and 2-dimensional Turing machines, respectively. Using these complexity values, an algorithm called the Block Decomposition Method developed by Zenil, et al. in 2018, has been created to approximate the complexity of adjacency matrices of graphs which have found relative success in grouping graphs based on their complexity values. We use this method along with another method called edge perturbation to exhaustively determine if an edge can be identified to connect two subgraphs within a graph using the entire symmetric group of its vertices permutation and via unique permutations we call automorphic subsets, which are a special subset of the symmetric group. We also analyze if edges will be grouped closer to their respective subgraphs in terms of the average algorithmic information contribution. This analysis ascertains if Algorithmic Information Theory can serve as a viable theory for understanding graph substructures and as a foundation for frameworks measuring and analyzing complexity. The study found that the connecting edge was successfully identified as having the highest average information contribution in 29 out of 30 graphs, and in 16 of these, the distance to the next edge was greater than log_2(2). Furthermore, the symmetric group outperformed automorphic subsets in edge grouping.
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https://arxiv.org/abs/2601.01760
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8da7096bc397ebb041294bab25957f92c7662074010e7746d25445019f16ddf7
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2026-01-21T00:00:00-05:00
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Hilbert Polynomials of Calabi Yau Hypersurfaces in Toric Varieties and Lattice Points in Polytope Boundaries
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arXiv:2601.02176v2 Announce Type: replace Abstract: We show that the Hilbert polynomial of a Calabi-Yau hypersurface $Z$ in a smooth toric variety $M$ associated to a convex polytope $\Delta$ is given by a lattice point count in the polytope boundary $\partial \Delta,$ just as the Hilbert polynomial of $M$ is known to be given by a lattice point count in the convex polytope $\Delta.$ Our main tool is a computation of the Euler class in $K$-theory of the normal line bundle to the hypersurface $Z,$ in terms of the Euler classes of the divisors corresponding to the facets of the moment polytope. We observe a remarkable parallel between our expression for the Euler class and the inclusion-exclusion principle in combinatorics. To obtain our result we combine these facts with the known relation between lattice point counts in the facets of $\Delta$ and the Hilbert polynomials of the smooth toric varieties corresponding to these facets.
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https://arxiv.org/abs/2601.02176
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52d12bead30a98ede0c18782fc1a50fcc51cc0005c1ca71bfa6185af6a071728
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2026-01-21T00:00:00-05:00
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On difference sets of dense subsets of $\mathbb{Z}^2$
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arXiv:2601.03797v2 Announce Type: replace Abstract: In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc. 146 (2018), 3449-3453] whether, for every such set $E$, there exists a nonzero integer $k$ such that $k \cdot \mathbb{Z} \subseteq \{\, xy : (x,y) \in E - E \,\}.$ Although this question remains open, we establish a relatively weaker form of this conjecture. Specifically, we prove that if $\langle a_j\rangle_{j=1}^m$ is any finite sequence in $\mathbb{N},$ then there exist infinitely many integers $k \in \mathbb{Z}$ and a sequence $\langle x_n \rangle_{n \in \mathbb{N}}$ in $\mathbb{Z}$ such that $k \cdot MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right) \subseteq \{\, xy : (x,y) \in E - E \,\},$ where $MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right)$ denotes the milliken-Taylor configuration generated by the sequences $\langle a_j\rangle_{j=1}^m$ and $\langle x_n \rangle_{n \in \mathbb{N}}$.
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https://arxiv.org/abs/2601.03797
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1c32e8102863034515da093aa83ddf76defc34391cc05de810c2b2e22de60980
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2026-01-21T00:00:00-05:00
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Trade-off between spread and width for tree decompositions
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arXiv:2601.04040v2 Announce Type: replace Abstract: We study the trade-off between (average) spread and width in tree decompositions, answering several questions from Wood [arXiv:2509.01140]. The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. Wood asked for which $c>0$, there exists $c'$ such that each graph $G$ has a tree decomposition of width $c\cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that $c\geq 2$ is necessary and that $c>3$ is sufficient. Moreover, we answer a second question fully by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.
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https://arxiv.org/abs/2601.04040
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eb283c996bff59f25d4c08103fde36c9fad9bec8dd5359415ab4df92b7dcb2ee
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2026-01-21T00:00:00-05:00
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Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms
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arXiv:2601.04965v3 Announce Type: replace Abstract: We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric biquadratic forms. We describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For $m \ge 2$, we present a $m \times 2$ PSD biquadratic form and show that it can be expressed as the sum of $m+1$ squares, but cannot be expressed as the sum of $m$ squares. This provides a lower bound for sos rank of $m \times 2$ biquadratic forms, and shows that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. We also present an $3 \times 3$ SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares.We present a $2 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of three squares, but cannot be expressed as the sum of two squares. Furthermore, we present a $3 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of four squares, but cannot be expressed as the sum of three squares. These show that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. Moreover, we establish a universal upper bound SOS-rank$(P) \le mn-1$ for any SOS biquadratic form, which improves the trivial bound $mn$ and is tight in small dimensions.
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https://arxiv.org/abs/2601.04965
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8155303be1ebe33efa9a28949a0b90d16346fbc649a5eeb414acee12f06637d6
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2026-01-21T00:00:00-05:00
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On Strong Lefschetz Property of 0-dimensional complete intersections
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arXiv:2601.07874v3 Announce Type: replace Abstract: We prove that a homogeneous 0-dimensional complete intersection satisfies the Strong Lefschetz Property (SLP) in degree 1 if and only if its associated form has nonzero Hessian. The result is essentially known in the literature, but our proof is different compared with the previous ones.
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https://arxiv.org/abs/2601.07874
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e37a9fbc82ee88868964991a671b3c07a86e0fda013101369e83809c3b11176e
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2026-01-21T00:00:00-05:00
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Determining the Winner in Alternating-Move Games
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arXiv:2601.08359v2 Announce Type: replace Abstract: We provide a criterion for determining the winner in two-player win-lose alternating-move games on trees, in terms of the Hausdorff dimension of the target set. We focus our study on special cases, including the Gale-Stewart game on the complete binary tree and a family of Schmidt games. Building on the Hausdorff dimension games originally introduced by Das, Fishman, Simmons, and Urba\'nski, which provide a game-theoretic approach for computing Hausdorff dimensions, we employ a generalized family of these games, and show that they are useful for analyzing sets underlying the win-lose games we study.
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https://arxiv.org/abs/2601.08359
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d242c193ba10becfcee59c4376cfe5785a1dd0d6cf741d7c370a60fee885a241
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2026-01-21T00:00:00-05:00
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On the Generalization Error of Differentially Private Algorithms Via Typicality
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arXiv:2601.08386v2 Announce Type: replace Abstract: We study the generalization error of stochastic learning algorithms from an information-theoretic perspective, with a particular emphasis on deriving sharper bounds for differentially private algorithms. It is well known that the generalization error of stochastic learning algorithms can be bounded in terms of mutual information and maximal leakage, yielding in-expectation and high-probability guarantees, respectively. In this work, we further upper bound mutual information and maximal leakage by explicit, easily computable formulas, using typicality-based arguments and exploiting the stability properties of private algorithms. In the first part of the paper, we strictly improve the mutual-information bounds by Rodr\'iguez-G\'alvez et al. (IEEE Trans. Inf. Theory, 2021). In the second part, we derive new upper bounds on the maximal leakage of learning algorithms. In both cases, the resulting bounds on information measures translate directly into generalization error guarantees.
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https://arxiv.org/abs/2601.08386
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e10f8ba9dd1819d4e426ae1a86046f6eb5a3fee4d8fe3698adab2ce4de9031b0
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2026-01-21T00:00:00-05:00
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Fluctuations of the Ising free energy on Erd\H{o}s-R\'enyi graphs
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arXiv:2601.08590v2 Announce Type: replace Abstract: We investigate the ferromagnetic Ising model on the Erd\H{o}s-R\'enyi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(\beta,B)$, where $Z_{\mathbb{G}(n,m)}(\beta,B)$ is the partition function at inverse temperature $\beta>0$ and external field $B\geq0$. If either $B>0$, or $B=0$, $d>1$ and $\beta>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $\Theta(n)$ and is described by a family of stochastic fixed point problems that encode the root magnetisation of two correlated Galton-Watson trees. By contrast, if $B=0$ and either $d\leq1$ or $\beta<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance.
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https://arxiv.org/abs/2601.08590
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620d939a919d661b68dc218ecf7cb537df12d0dfc5b8276cf891e22f061e9102
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2026-01-21T00:00:00-05:00
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Spectral Fusion Deformations for Locally Compact Quantum Groups
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arXiv:2601.08688v2 Announce Type: replace Abstract: We develop a deformation framework for $C^*$-algebras equipped with a coaction of a locally compact quantum group, formulated intrinsically at the level of spectral subspaces determined by the coaction. The construction is defined algebraically on a finite spectral core and extended by continuity to a natural Fr\'echet $*$-algebra completion under mild analytic regularity assumptions. Deformations are governed by scalar fusion data assigning phases to fusion channels of irreducible corepresentations. Associativity and $*$-compatibility are characterized by explicit algebraic identities. The framework recovers a range of known deformation procedures, including Rieffel, Kasprzak, and Drinfeld-type constructions, and also yields genuinely new deformations that do not arise from dual $2$--cocycles or crossed-product methods. At the $C^*$-level, we identify a minimal reduced setting in which the deformed algebra admits a canonical completion, formulated in terms of boundedness of the deformed left regular action on the Haar--GNS space. This separates algebraic coherence from analytic implementability and clarifies the precise role of higher-order fusion data in deformation theory for locally compact quantum groups. In particular, the framework exhibits explicit associator-level deformations governed by fusion $3$--cocycles that cannot arise from any dual $2$--cocycle or crossed-product construction.
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https://arxiv.org/abs/2601.08688
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d02052fd9b14157ec2faf58de0f4239ec391009b30e7bcfca657869c09e86ca2
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2026-01-21T00:00:00-05:00
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Upper and Lower Bounds for The Quantum Dynamics of One-Dimensional Divergence-Type Random Jacobi Operators
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arXiv:2601.08796v2 Announce Type: replace Abstract: We study quantum transport for the discrete one-dimensional random Jacobi operator of divergence-gradient type. For strictly positive and bounded random variables, we analyze the q-moments of the position operator and establish both upper and lower power-law bounds on their growth. Our approach relies on the asymptotic behavior of the integrated density of states and the Lyapunov exponent near the critical energy 0, previously obtained by Pastur and Figotin. A key ingredient in our analysis is the large deviation-type estimates explored via the phase formalism, which play a central role in deriving bounds on the growth of the transfer matrices.
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https://arxiv.org/abs/2601.08796
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62d36b0283c28f79aaa00d7f3eefbc103cca4d9dd3d2567f679fdfe83021f2b9
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2026-01-21T00:00:00-05:00
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Quantum Heegaard diagrams and knot Floer Homology
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arXiv:2601.08805v2 Announce Type: replace Abstract: Given a knot presented as a braid closure, we construct a unified intersection model for the Alexander and Jones polynomials of the knot via what we call quantum Heegaard diagrams. These diagrams are obtained by stabilising the disc model of the first author, which we show are doubly-pointed Heegaard diagrams of the knot together with an additional set of base points. We identify the Alexander grading in the disc model with the Alexander grading in the Heegaard diagram. As the Lagrangian intersection Floer homology of the Heegaard tori in the symmetric power of the Heegaard surface is knot Floer homology, we can view knot Floer homology as a natural categorification of the Alexander polynomial arising from the disc model. The additional base points let us define a new grading on the intersection between the Heegaard tori, which we call quantum Alexander grading. Combining this with the classical Alexander grading, we define a two-variable graded intersection between the Heegaard tori that recovers the Jones and Alexander polynomials as two specialisations of coefficients. The resulting intersection formula for the Jones polynomial opens up a potential avenue to obtaining a new geometric categorification of the Jones polynomial.
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https://arxiv.org/abs/2601.08805
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4bce6aec3db81a5a0b6e737fcdc97e0463f374ccce2a630b9038b00061736087
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2026-01-21T00:00:00-05:00
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Positive Lyapunov Exponents versus Integrability in Random Conservative Dynamics
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arXiv:2601.08814v2 Announce Type: replace Abstract: We study random dynamical systems generated by volume-preserving piecewise $C^{1}$ maps. For this class of systems, we establish an invariance principle stating that if all Lyapunov exponents vanish, then there exists a measurable family of probability measures on the projective bundle that is invariant under the projective cocycle induced by the derivative. We apply this principle to two classes of random systems. First, we consider random additive perturbations of the billiard map associated with a strictly convex planar table on a surface of constant curvature. In this setting, we show that the Lyapunov exponents vanish almost everywhere if and only if the billiard table is a geodesic disk. Second, we study random additive perturbations of a standard map and prove that the Lyapunov exponents vanish almost everywhere if and only if the map is integrable.
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https://arxiv.org/abs/2601.08814
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1c19e6f2d15381ab5ca2b4ec13d3756e66659361f307a245a262e385bf73709d
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2026-01-21T00:00:00-05:00
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Two-dimensional Entanglement-assisted Quantum Quasi-cyclic Low-density Parity-check Codes
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arXiv:2601.08927v2 Announce Type: replace Abstract: For any positive integer $g \ge 2$, we derive general condition for the existence of a $2g$-cycle in the Tanner graph of two-dimensional ($2$-D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Depending on whether $p$ is an odd prime or a composite number, we construct two distinct families of $2$-D classical QC-LDPC codes with girth $>4$ by stacking $p \times p \times p$ tensors. Furthermore, using generalized Behrend sequences, we propose an additional family of $2$-D classical QC-LDPC codes with girth $>6$, constructed via a similar tensor-stacking approach. All the proposed $2\text{-D}$ classical QC-LDPC codes exhibit an erasure correction capability of at least $p \times p$. Based on the constructed $2\text{-D}$ classical QC-LDPC codes, we derive two families of $2\text{-D}$ entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of $2\text{-D}$ EA-QLDPC codes is obtained from a pair of $2\text{-D}$ classical QC-LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of $4$-cycles, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single $2\text{-D}$ classical QC-LDPC code whose Tanner graph is free from $4$-cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of $p \times p$, as the underlying classical codes possess the same erasure correction property.
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https://arxiv.org/abs/2601.08927
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76461704d2d97efbe4754d441e83d013fc1d93043335237c70ad06bd4b445ce5
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2026-01-21T00:00:00-05:00
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Solution to a Problem of Erd\H{o}s Concerning Distances and Points
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arXiv:2601.09102v2 Announce Type: replace Abstract: In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset determines at least 3 distinct pairwise distances. We construct $n$-point sets from an $m\times m$ box of the lattice $L = \{(x,\sqrt{2}y):x,y \in \mathbb{Z}\} \subset \mathbb{R}^2.$ The distinct distance bound follows from applying Bernays' theorem to the number of integers represented by the binary quadratic form $u^2 + 2v^2$. The local 4-point constraint is verified through Perucca's similarity classification of the six similarity types determining exactly two distances.
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https://arxiv.org/abs/2601.09102
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739a4313987d08cdbdc68619f2dc074621699ed95e5d720a99d23bc3dd34d292
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2026-01-21T00:00:00-05:00
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System Availability Optimization: Integrating Quantity Discounts and Delivery Lead Time Considerations
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arXiv:2601.09194v2 Announce Type: replace Abstract: Purpose: The model allocates the system components orders to the suppliers to minimize the parts price and the system construction delay penalties and maximize the system availability during its use. It considers the quantity-based discount and variation of delivery lead time by ordering similar components. The model also reflects the prerequisite relationships between construction activities and calculates the delay penalty resulting from parts delivery lead time. Design/methodology/approach: This research presents a model for selecting suppliers of components of an industrial series-parallel multi-state system. A nonlinear binary mathematical program uses the Markov process results to select system components. It minimizes the total system construction phase costs, including the components' price, and the system construction delay penalty, and the system exploitation phase costs, including the system shutdown and working at half capacity. Findings: The model allocates the optimal orders for a typical industrial system's components, composing four elements. The proposed approach combines the nonlinear binary program and the Markov process results to optimize the system life cycle parameters, including the system construction cost and operational availability. Originality/value: Using the Markov chain results in binary nonlinear mathematical programming, this study attempts to strike the right balance between the construction phase's objectives and an industrial unit's operation phase.
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https://arxiv.org/abs/2601.09194
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846f3fe0214e5f67ad477740e72456aa3df6092bb4d7be3d9a54986c3ca81e38
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2026-01-21T00:00:00-05:00
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On strong law of large numbers for weakly stationary $\varphi$-mixing set-valued random variable sequences
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arXiv:2601.09197v2 Announce Type: replace Abstract: In this paper we extend the notion of $\varphi$-mixing to set-valued random sequences that take values in the family of closed subsets of a Banach space. Several strong laws of large numbers for such $\varphi$-mixing sequences are stated and proved. Illustrative examples show that the hypotheses of the theorems are both natural and sharp.
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https://arxiv.org/abs/2601.09197
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f086ca6579d12b67ba3f8aff0024b3f71951a7cd08388d2c348c3df6dcfdde58
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2026-01-21T00:00:00-05:00
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On Polar Coding with Feedback
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arXiv:2601.09222v2 Announce Type: replace Abstract: In this work, we investigate the performance of polar codes with the assistance of feedback in communication systems. Although it is well known that feedback does not improve the capacity of memoryless channels, we show that the finite length performance of polar codes can be significantly improved as feedback enables genie-aided decoding and allows more flexible thresholds for the polar coding construction. To analyze the performance under the new construction, we then propose an accurate characterization of the distribution of the error event under the genie-aided successive cancellation (SC) decoding. This characterization can be also used to predict the performance of the standard SC decoding of polar codes with rates close to capacity.
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https://arxiv.org/abs/2601.09222
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a69ff316e09cd4ba393797ce868581c9419e47e8cf821dd4522cea7ef603374c
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2026-01-21T00:00:00-05:00
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On generalized Tur\'{a}n problems for expansions
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arXiv:2601.09244v2 Announce Type: replace Abstract: Given a graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Given $r$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{F}$, the generalized Tur\'{a}n number, denoted by $\textrm{ex}_r(n,\mathcal{H},\mathcal{F})$, is the maximum number of copies of $\mathcal{H}$ in an $n$-vertex $r$-uniform hypergraph that does not contain $\mathcal{F}$ as a subhypergraph. In the case where $r=2$ (i.e., the graph case), the study of generalized Tur\'{a}n problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Tur\'{a}n problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Tur\'{a}n number for expansions of complete graphs, and establish the asymptotics of the generalized Tur\'{a}n number for expansions of the vertex-disjoint union of complete graphs. For the degenerate case, we establish the asymptotics of generalized Tur\'{a}n numbers for expansions of several classes of forests, including star forests, linear forests and star-path forests.
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https://arxiv.org/abs/2601.09244
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20c5d24ae6d0d211567b7919ecb9dad9189d0864fdd4f84fdf64a81b6e047d1f
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2026-01-21T00:00:00-05:00
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A Finite-Sample Strong Converse for Binary Hypothesis Testing via (Reverse) R\'enyi Divergence
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arXiv:2601.09550v2 Announce Type: replace Abstract: This work investigates binary hypothesis testing between $H_0\sim P_0$ and $H_1\sim P_1$ in the finite-sample regime under asymmetric error constraints. By employing the ``reverse" R\'enyi divergence, we derive novel non-asymptotic bounds on the Type II error probability which naturally establish a strong converse result. Furthermore, when the Type I error is constrained to decay exponentially with a rate $c$, we show that the Type II error converges to 1 exponentially fast if $c$ exceeds the Kullback-Leibler divergence $D(P_1\|P_0)$, and vanishes exponentially fast if $c$ is smaller. Finally, we present numerical examples demonstrating that the proposed converse bounds strictly improve upon existing finite-sample results in the literature.
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https://arxiv.org/abs/2601.09550
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4a9e6eb6b20afffe3a8497882ce02ebae193a4be34746e792d7a2f8778de51ad
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2026-01-21T00:00:00-05:00
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Counting and Entropy Bounds for Structure-Avoiding Spatially-Coupled LDPC Constructions
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arXiv:2601.09674v2 Announce Type: replace Abstract: Designing large coupling memory quasi-cyclic spatially-coupled LDPC (QC-SC-LDPC) codes with low error floors requires eliminating specific harmful substructures (e.g., short cycles) induced by edge spreading and lifting. Building on our work~\cite{r15} that introduced a Clique Lov\'asz Local Lemma (CLLL)-based design principle and a Moser--Tardos (MT)-type constructive approach, this work quantifies the size and structure of the feasible design space. Using the quantitative CLLL, we derive explicit lower bounds on the number of feasible edge-spreading and lifting assignments satisfying a given family of structure-avoidance constraints, and further obtain bounds on the number of non-equivalent solutions under row/column permutations. Moreover, via R\'enyi entropy bounds for the MT distribution, we provide a computable lower bound on the number of distinct solutions that the MT algorithm can output, giving a concrete diversity guarantee for randomized constructions. Specializations for eliminating 4-cycles yield closed-form bounds as functions of system parameters, offering a principled way to select the memory and lifting degree and to estimate the remaining search space.
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https://arxiv.org/abs/2601.09674
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b3a1d3136a6b1c1c9051d35784f07c0506603e4e7f4f1658651e7987303771b2
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2026-01-21T00:00:00-05:00
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Invariant Algebraic $D$-Modules on Semisimple and General Linear Groups: Classification and Tensor Categories
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arXiv:2601.10934v2 Announce Type: replace Abstract: We study finite-rank left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo algebraic gauge transformations, we recast the classification problem as an explicit moduli problem for constant connections. Our main results treat the semisimple case and the general linear case. For a connected semisimple complex algebraic group, invariant $D$-modules are classified by representations of the finite central kernel of the simply connected cover, yielding an equivalence of tensor categories. For a general linear group, every invariant $D$-module is obtained by pullback along the determinant map, reducing the classification to the one-dimensional torus case and inducing a tensor equivalence with the corresponding invariant category on the torus.
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https://arxiv.org/abs/2601.10934
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256ec63f2cb4e3099f7b7b1fa7858668140c3898d0743cf5349be88e4f1303b8
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2026-01-21T00:00:00-05:00
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Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework
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arXiv:2601.10999v2 Announce Type: replace Abstract: Physics-informed extreme learning machines (PIELMs) typically impose boundary and initial conditions through penalty terms, yielding only approximate satisfaction that is sensitive to user-specified weights and can propagate errors into the interior solution. This work introduces Null-Space Projected PIELM (NP-PIELM), achieving exact constraint enforcement through algebraic projection in coefficient space. The method exploits the geometric structure of the admissible coefficient manifold, recognizing that it admits a decomposition through the null space of the boundary operator. By characterizing this manifold via a translation-invariant representation and projecting onto the kernel component, optimization is restricted to constraint-preserving directions, transforming the constrained problem into unconstrained least-squares where boundary conditions are satisfied exactly at discrete collocation points. This eliminates penalty coefficients, dual variables, and problem-specific constructions while preserving single-shot training efficiency. Numerical experiments on elliptic and parabolic problems including complex geometries and mixed boundary conditions validate the framework.
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https://arxiv.org/abs/2601.10999
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ef4c22ca99a436341e31f0853127cccd50a7dedcdc53474fc73830e13a6be78c
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2026-01-21T00:00:00-05:00
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Countable-Support Symmetric Iterations
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arXiv:2601.11008v2 Announce Type: replace Abstract: We present a framework for iterating symmetric extensions with \emph{countable support}. Assuming the successor-step two-stage symmetric-system construction from the standard finite-support theory, we define the countable-support iteration and its induced automorphism groups, taking $\omega_1$-completions of the successor-stage symmetry filters. At limit stages of uncountable cofinality, countable supports are bounded and we use the resulting direct-limit presentation, and at limits of cofinality $\omega$ we use the inverse-limit presentation induced by restriction maps. We define canonical limit symmetry filters generated by head pullbacks from earlier stages: at limits of cofinality $>\omega$ we take the smallest normal filter containing these generators (and prove it is $\omega_1$--complete by stage-bounding), while at limits of cofinality $\omega$ we take the smallest normal $\omega_1$--complete filter extending the same generators. In either case the resulting limit filter is normal and $\omega_1$--complete. Using this, we define the class of hereditarily symmetric names at each stage and prove closure under the operations required for the $ZF$ axioms. In particular, the resulting symmetric model satisfies $ZF$; for set-length stages over a $ZFC$ ground, the resulting symmetric model satisfies $DC=DC_\omega$. Finally, when the iteration template is first-order definable over a $GBC$ ground with Global Choice and sufficient class recursion (e.g.\ $GBC+\mathsf{ETR}$), the same scheme extends to class-length iterations, and the final symmetric model is obtained by evaluating hereditarily symmetric class-names.
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https://arxiv.org/abs/2601.11008
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95a1f719fd28be29c03039b06220a9001d55d4ff27937ed9d218db0a232aad12
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Constructing Orthogonal Rational Function Vectors with an application in Rational Approximation
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arXiv:2601.11317v2 Announce Type: replace Abstract: We present two algorithms for constructing orthonormal bases of rational function vectors with respect to a discrete inner product, and discuss how to use them for a rational approximation problem. Building on the pencil-based formulation of the inverse generalized eigenvalue problem by Van Buggenhout et al. (2022), we extend it to rational vectors of arbitrary length $k$, where the recurrence relations are represented by a pair of $k$-Hessenberg matrices, i.e., matrices with possibly $k$ nonzero subdiagonals. An updating algorithm based on similarity transformations using rotations and a Krylov-type algorithm related to the rational Arnoldi method are derived. The performance is demonstrated on the rational approximation of $\sqrt{z}$ on $[0,1]$, where the optimal lightning + polynomial convergence rate of Herremans, Huybrechs, and Trefethen (2023) is successfully recovered. This illustrates the robustness of the proposed methods for handling exponentially clustered poles near singularities.
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https://arxiv.org/abs/2601.11317
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d8df04c5f82c00fee0ac4f014b7bc7bd2fe29bdb53bb23463005cd5f8a4a8560
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2026-01-21T00:00:00-05:00
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Efficient Channel Autoencoders for Wideband Communications leveraging Walsh-Hadamard interleaving
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arXiv:2601.11407v2 Announce Type: replace Abstract: This paper investigates how end-to-end (E2E) channel autoencoders (AEs) can achieve energy-efficient wideband communications by leveraging Walsh-Hadamard (WH) interleaved converters. WH interleaving enables high sampling rate analog-digital conversion with reduced power consumption using an analog WH transformation. We demonstrate that E2E-trained neural coded modulation can transparently adapt to the WH-transceiver hardware without requiring algorithmic redesign. Focusing on the short block length regime, we train WH-domain AEs and benchmark them against standard neural and conventional baselines, including 5G Polar codes. We quantify the system-level energy tradeoffs among baseband compute, channel signal-to-noise ratio (SNR), and analog converter power. Our analysis shows that the proposed WH-AE system can approach conventional Polar code SNR performance within 0.14dB while consuming comparable or lower system power. Compared to the best neural baseline, WH-AE achieves, on average, 29% higher energy efficiency (in bit/J) for the same reliability. These findings establish WH-domain learning as a viable path to energy-efficient, high-throughput wideband communications by explicitly balancing compute complexity, SNR, and analog power consumption.
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https://arxiv.org/abs/2601.11407
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ade61aaf873fa096f63dbe507f78897cafef3380a453cf4d15792f27145fc684
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2026-01-21T00:00:00-05:00
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Frame eversion and contextual geometric rigidity
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arXiv:2601.11455v2 Announce Type: replace Abstract: We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental Theorem of Projective Geometry classifies maps between full Grassmannians of two $n$-dimensional Hilbert spaces, $n\ge 3$, preserving dimension and lattice operations for pairs with commuting orthogonal projections, as precisely those induced by semilinear injections unique up to scaling. In a different but related direction, denote the manifolds of ordered orthogonal (linearly-independent) $n$-tuples of lines in an $n$-dimensional Hilbert space $V$ by $\mathbb{F}^{\perp}(V)$ (respectively $\mathbb{F}(V)$) and, for partitions $\pi$ of the set $\{1..n\}$, call two tuples $\pi$-linked if the spans along $\pi$-blocks agree. A Wigner-style rigidity theorem proves that the symmetric maps $\mathbb{F}^{\perp}(\mathbb{C}^n)\to \mathbb{F}(\mathbb{C}^n)$, $n\ge 3$ respecting $\pi$-linkage are precisely those induced by semilinear injections, hence by linear or conjugate-linear maps if also assumed measurable. On the other hand, in the $\mathbb{F}(\mathbb{C}^n)$-defined analogue the only other possibility is a qualitatively new type of purely-contextual-global symmetry transforming a tuple $(\ell_i)_i$ of lines into $\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i$.
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https://arxiv.org/abs/2601.11455
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2dcecd9c72fe96b6454dbc1fc11db11046907298399720357ee68683bfcae6e0
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2026-01-21T00:00:00-05:00
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A New Generation of Brain-Computer Interface Based on Riemannian Geometry
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arXiv:1310.8115v2 Announce Type: replace-cross Abstract: Based on the cumulated experience over the past 25 years in the field of Brain-Computer Interface (BCI) we can now envision a new generation of BCI. Such BCIs will not require training; instead they will be smartly initialized using remote massive databases and will adapt to the user fast and effectively in the first minute of use. They will be reliable, robust and will maintain good performances within and across sessions. A general classification framework based on recent advances in Riemannian geometry and possessing these characteristics is presented. It applies equally well to BCI based on event-related potentials (ERP), sensorimotor (mu) rhythms and steady-state evoked potential (SSEP). The framework is very simple, both algorithmically and computationally. Due to its simplicity, its ability to learn rapidly (with little training data) and its good across-subject and across-session generalization, this strategy a very good candidate for building a new generation of BCIs, thus we hereby propose it as a benchmark method for the field.
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https://arxiv.org/abs/1310.8115
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3ffdb8a5d57c037cb436b74a189541c889ff8bb3232e3789c482da871118a895
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2026-01-21T00:00:00-05:00
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UVIP: Model-Free Approach to Evaluate Reinforcement Learning Algorithms
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arXiv:2105.02135v5 Announce Type: replace-cross Abstract: Policy evaluation is an important instrument for the comparison of different algorithms in Reinforcement Learning (RL). However, even a precise knowledge of the value function $V^{\pi}$ corresponding to a policy $\pi$ does not provide reliable information on how far the policy $\pi$ is from the optimal one. We present a novel model-free upper value iteration procedure ({\sf UVIP}) that allows us to estimate the suboptimality gap $V^{\star}(x) - V^{\pi}(x)$ from above and to construct confidence intervals for \(V^\star\). Our approach relies on upper bounds to the solution of the Bellman optimality equation via the martingale approach. We provide theoretical guarantees for {\sf UVIP} under general assumptions and illustrate its performance on a number of benchmark RL problems.
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https://arxiv.org/abs/2105.02135
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2fb652af0fd74f598759e62ae9f6d238aea237ba3780d94fc90d65b916cc9dd2
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2026-01-21T00:00:00-05:00
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Explicit Non-Abelian Gerbes with Connections
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arXiv:2203.00092v4 Announce Type: replace-cross Abstract: We define the notion of adjustment for strict Lie 2-groups and provide the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is an adjusted 2-group. Most importantly, we depart from the common fake-flat connections and employ adjusted connections. This is an important generalisation that is needed for physical applications especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the spin structure on $S^4$, corresponding to an instanton-anti-instanton pair, to a string structure, a 2-group bundle with connection. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose-Ward transform of string bundles over twistor space.
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https://arxiv.org/abs/2203.00092
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ecd250446a1c60e2c07c3f0dab965af62f51e8cd139a84cc222ffafa0287de7a
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2026-01-21T00:00:00-05:00
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A Note on Comparator-Overdrive-Delay Conditioning for Current-Mode Control
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arXiv:2206.09340v3 Announce Type: replace-cross Abstract: Comparator-overdrive-delay conditioning is a new control conditioning approach for high-frequency current-mode control. No existing literature rigorously studies the effect of the comparator overdrive delay on the current-mode control. The results in this paper provide insights into the mechanism of comparator-overdrive-delay conditioning.
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https://arxiv.org/abs/2206.09340
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cc534d7b13f81522285d7c16c7c320af5ca932d0d84283c45122be8072a14eb1
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2026-01-21T00:00:00-05:00
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Machine Learning Decoder for 5G NR PUCCH Format 0
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arXiv:2209.07861v2 Announce Type: replace-cross Abstract: 5G cellular systems depend on the timely exchange of feedback control information between the user equipment and the base station. Proper decoding of this control information is necessary to set up and sustain high throughput radio links. This paper makes the first attempt at using Machine Learning techniques to improve the decoding performance of the Physical Uplink Control Channel Format 0. We use fully connected neural networks to classify the received samples based on the uplink control information content embedded within them. The trained neural network, tested on real-time wireless captures, shows significant improvement in accuracy over conventional DFT-based decoders, even at low SNR. The obtained accuracy results also demonstrate conformance with 3GPP requirements.
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https://arxiv.org/abs/2209.07861
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eda0fbac666d15e38a8028b2020f5c9e7101bb9c7a5afe4257911ac1f91349d4
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2026-01-21T00:00:00-05:00
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Two convergent NPA-like hierarchies for the quantum bilocal scenario
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arXiv:2210.09065v5 Announce Type: replace-cross Abstract: Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascu\'es et al., New J. Phys. 10, 073013 (2008)], known as the Navascu\'es-Pironio-Ac\'in (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations.
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https://arxiv.org/abs/2210.09065
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888f0b545faa61cf0fa02f804aee387c09458063b2e6a2f64b0248ed7049f036
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2026-01-21T00:00:00-05:00
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On the Global Convergence of Risk-Averse Natural Policy Gradient Methods with Expected Conditional Risk Measures
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arXiv:2301.10932v5 Announce Type: replace-cross Abstract: Risk-sensitive reinforcement learning (RL) has become a popular tool for controlling the risk of uncertain outcomes and ensuring reliable performance in highly stochastic sequential decision-making problems. While it has been shown that policy gradient methods can find globally optimal policies in the risk-neutral setting, it remains unclear if the risk-averse variants enjoy the same global convergence guarantees. In this paper, we consider a class of dynamic time-consistent risk measures, named Expected Conditional Risk Measures (ECRMs), and derive natural policy gradient (NPG) updates for ECRMs-based RL problems. We provide global optimality and iteration complexity of the proposed risk-averse NPG algorithm with softmax parameterization and entropy regularization under both exact and inexact policy evaluation. Furthermore, we test our risk-averse NPG algorithm on a stochastic Cliffwalk environment to demonstrate the efficacy of our method.
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https://arxiv.org/abs/2301.10932
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910d53956637a366f4c0a87a4ec2494c23a2f8edb98cd933d9e920416590a136
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2026-01-21T00:00:00-05:00
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Optimal Conditional Inference in Adaptive Experiments
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arXiv:2309.12162v2 Announce Type: replace-cross Abstract: We study batched bandit experiments and consider the problem of inference conditional on the realized stopping time, assignment probabilities, and target parameter, where all of these may be chosen adaptively using information up to the last batch of the experiment. Absent further restrictions on the experiment, we show that inference using only the results of the last batch is optimal. When the adaptive aspects of the experiment are known to be location-invariant, in the sense that they are unchanged when we shift all batch-arm means by a constant, we show that there is additional information in the data, captured by one additional linear function of the batch-arm means. In the more restrictive case where the stopping time, assignment probabilities, and target parameter are known to depend on the data only through a collection of polyhedral events, we derive computationally tractable and optimal conditional inference procedures.
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https://arxiv.org/abs/2309.12162
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a7f5f062dda286c38966207eb8332565ad357dce4165e00b94dd5c0c37007aa7
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2026-01-21T00:00:00-05:00
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AlphaMapleSAT: An MCTS-based Cube-and-Conquer SAT Solver for Hard Combinatorial Problems
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arXiv:2401.13770v2 Announce Type: replace-cross Abstract: This paper introduces AlphaMapleSAT, a Cube-and-Conquer (CnC) parallel SAT solver that integrates Monte Carlo Tree Search (MCTS) with deductive feedback to efficiently solve challenging combinatorial SAT problems. Traditional lookahead cubing methods, used by solvers such as March, limit their search depth to reduce overhead often resulting in suboptimal partitions. By contrast, AlphaMapleSAT performs a deeper MCTS search guided by deductive rewards from SAT solvers. This approach enables informed exploration of the cubing space while keeping cubing costs low. We demonstrate the efficacy of our technique via extensive evaluations against the widely used and established March cubing solver on three well-known challenging combinatorial benchmarks, including the minimum Kochen-Specker (KS) problem from quantum mechanics, the Murty-Simon Conjecture, and the Ramsey problems from extremal graph theory. We compare AlphaMapleSAT against March using different types of conquering solvers such as SAT Modulo Symmetries (SMS) and SAT+CAS, both built on top of the CaDiCaL SAT solver. We show that in all cases, there is a speedup in elapsed real time (wall clock time) ranging from 1.61x to 7.57x on a 128 core machine for the above-mentioned problems. We also perform cube-level and parallel scaling analysis over 32, 64, and 128 cores, which shows that AlphaMapleSAT outperforms March on all these settings. Our results show that deductively-guided MCTS search technique for cubing in CnC solvers can significantly outperform March on hard combinatorial problems.
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https://arxiv.org/abs/2401.13770
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2026-01-21T00:00:00-05:00
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Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss
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arXiv:2402.00152v4 Announce Type: replace-cross Abstract: Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question. This paper explores a comparison between deeper neural networks (DeNNs) with a flexible number of layers and wider neural networks (WeNNs) with limited hidden layers, focusing on their optimal generalization error in Sobolev losses. Analytical investigations reveal that the architecture of a neural network can be significantly influenced by various factors, including the number of sample points, parameters within the neural networks, and the regularity of the loss function. Specifically, a higher number of parameters tends to favor WeNNs, while an increased number of sample points and greater regularity in the loss function lean towards the adoption of DeNNs. We ultimately apply this theory to address partial differential equations using deep Ritz and physics-informed neural network (PINN) methods, guiding the design of neural networks.
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https://arxiv.org/abs/2402.00152
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ca1710c63d1d6a31faceb7787c4a2ba68102b759fa05257189f9f528df900836
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On Dirac equations on phase spaces
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arXiv:2402.06404v3 Announce Type: replace-cross Abstract: We consider Dirac equations on relativistic phase spaces $T^*{\mathbb R}^{p-1,1}$, where ${\mathbb R}^{p-1,1}$ is Minkowski space with $p=2,4$. We use the geometric quantization approach in which the wave functions are polarized sections of a complex line bundle $L_{\sf{v}}$ over $T^*{\mathbb R}^{p-1,1}$. The covariant derivatives with connection $A_{\sf{vac}}$ in this bundle define canonical commutation relations. Fermions are charged with respect to the field $A_{\sf{vac}}$, so lifting the Dirac equations from space-time ${\mathbb R}^{p-1,1}$ to phase space $T^*{\mathbb R}^{p-1,1}$ results in their solutions being localized in the space ${\mathbb R}^{p-1}$ or in space-time ${\mathbb R}^{p-1,1}$. We describe the explicit form of these solutions.
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https://arxiv.org/abs/2402.06404
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f56124f13a094aac19ef746d557bf58df6db7775ed24047543eb6073506ef710
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2026-01-21T00:00:00-05:00
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Reflexive graph lenses in univalent foundations
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arXiv:2404.07854v2 Announce Type: replace-cross Abstract: Martin-L\"of's identity types provide a generic (albeit opaque) notion of identification or "equality" between any two elements of the same type, embodied in a canonical reflexive graph structure $(=_A, \mathbf{refl})$ on any type $A$. The miracle of Voevodsky's univalence principle is that it ensures, for essentially any naturally occurring structure in mathematics, that this the resultant notion of identification is equivalent to the type of isomorphisms in the category of such structures. Characterisations of this kind are not automatic and must be established one-by-one; to this end, several authors have employed reflexive graphs and displayed reflexive graphs to organise the characterisation of identity types. We contribute reflexive graph lenses, a new family of intermediate abstractions lying between families of reflexive graphs and displayed reflexive graphs that simplifies the characterisation of identity types for complex structures. Every reflexive graph lens gives rise to a (more complicated) displayed reflexive graph, and our experience suggests that many naturally occurring displayed reflexive graphs arise in this way. Evidence for the utility of reflexive graph lenses is given by means of several case studies, including the theory of reflexive graphs itself as well as that of polynomial type operators. Finally, we exhibit an equivalence between the type of reflexive graph fibrations and the type of univalent reflexive graph lenses.
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https://arxiv.org/abs/2404.07854
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2026-01-21T00:00:00-05:00
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Asymmetric canonical correlation analysis of Riemannian and high-dimensional data
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arXiv:2404.11781v3 Announce Type: replace-cross Abstract: In this paper, we introduce a novel statistical model for the integrative analysis of Riemannian-valued functional data and high-dimensional data. We apply this model to explore the dependence structure between each subject's dynamic functional connectivity -- represented by a temporally indexed collection of positive definite covariance matrices -- and high-dimensional data representing lifestyle, demographic, and psychometric measures. Specifically, we employ a reformulation of canonical correlation analysis that enables efficient control of the complexity of the functional canonical directions using tangent space sieve approximations. Additionally, we enforce an interpretable group structure on the high-dimensional canonical directions via a sparsity-promoting penalty. The proposed method shows improved empirical performance over alternative approaches and comes with theoretical guarantees. Its application to data from the Human Connectome Project reveals a dominant mode of covariation between dynamic functional connectivity and lifestyle, demographic, and psychometric measures. This mode aligns with results from static connectivity studies but reveals a unique temporal non-stationary pattern that such studies fail to capture.
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https://arxiv.org/abs/2404.11781
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7c0030c10bab59acbb7f2e6f587b389451880fc4c12aa3d35af1fbb0ae27abcb
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2026-01-21T00:00:00-05:00
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Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models
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arXiv:2405.16377v3 Announce Type: replace-cross Abstract: Using 3D-3D correspondence, we construct 3D dual bulk field theories for general Virasoro minimal models $M(P,Q)$. These theories correspond to Seifert fiber spaces $S^2 ((P,P-R),(Q,S),(3,1))$ with two integers $(R,S)$ satisfying $PS-QR =1$. In the unitary case, where $|P-Q|=1$, the bulk theory has a mass gap and flows to a unitary topological field theory (TQFT) in the IR, which is expected to support the chiral Virasoro minimal model at the boundary under an appropriate boundary condition. For the non-unitary case, where $|P-Q|>1$, the bulk theory flows to a 3D $\mathcal{N}=4$ rank-0 superconformal field theory, whose topologically twisted theory supports the chiral minimal model at the boundary. We also provide a concrete field theory description of the 3D bulk theory using $T[SU(2)]$ theories. Our proposals are supported by various consistency checks using 3D-3D relations and direct computations of various partition functions.
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https://arxiv.org/abs/2405.16377
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2026-01-21T00:00:00-05:00
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Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization
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arXiv:2405.19650v3 Announce Type: replace-cross Abstract: Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.
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https://arxiv.org/abs/2405.19650
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Academic Papers
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cd2d74a2ff9aac3e89398a3b5271ccc6e6ad0d3b040469e309ecf05630899c38
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2026-01-21T00:00:00-05:00
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U-learning for Prediction Inference via Combinatory Multi-Subsampling: With Applications to LASSO and Neural Networks
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arXiv:2407.15301v2 Announce Type: replace-cross Abstract: Epigenetic aging clocks play a pivotal role in estimating an individual's biological age through the examination of DNA methylation patterns at numerous CpG (Cytosine-phosphate-Guanine) sites within their genome. However, making valid inferences on predicted epigenetic ages, or more broadly, on predictions derived from high-dimensional inputs, presents challenges. We introduce a novel U-learning approach via combinatory multi-subsampling for making ensemble predictions and constructing confidence intervals for predictions of continuous outcomes when traditional asymptotic methods are not applicable. More specifically, our approach conceptualizes the ensemble estimators within the framework of generalized U-statistics and invokes the H\'ajek projection for deriving the variances of predictions and constructing confidence intervals with valid conditional coverage probabilities. We apply our approach to two commonly used predictive algorithms, Lasso and deep neural networks (DNNs), and illustrate the validity of inferences with extensive numerical studies. We have applied these methods to predict the DNA methylation age (DNAmAge) of patients with various health conditions, aiming to accurately characterize the aging process and potentially guide anti-aging interventions.
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https://arxiv.org/abs/2407.15301
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Academic Papers
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svg
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cdc674622c852a3d30870f7e3696118da70ce0336e22456eb316ca988ae867af
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2026-01-21T00:00:00-05:00
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Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians
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arXiv:2408.01516v4 Announce Type: replace-cross Abstract: Sampling from Gibbs states -- states corresponding to system in thermal equilibrium -- has recently been shown to be a task for which quantum computers are expected to achieve super-polynomial speed-up compared to classical computers, provided the locality of the Hamiltonian increases with the system size (Bergamaschi et al., arXiv: 2404.14639). We extend these results to show that this quantum advantage still occurs for Gibbs states of Hamiltonians with O(1)-local interactions at constant temperature by showing classical hardness-of-sampling and demonstrating such Gibbs states can be prepared efficiently using a quantum computer. In particular, we show hardness-of-sampling is maintained even for 5-local Hamiltonians on a 3D lattice. We additionally show that the hardness-of-sampling is robust when we are only able to make imperfect measurements.
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https://arxiv.org/abs/2408.01516
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Academic Papers
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svg
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202d3104bbb5ef3b97c4fdac5d36f64cc61fe6fb4e365766a35c770ef2447929
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2026-01-21T00:00:00-05:00
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Fast-forwarding quantum algorithms for linear dissipative differential equations
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arXiv:2410.13189v2 Announce Type: replace-cross Abstract: We establish improved complexity estimates of quantum algorithms for linear dissipative ordinary differential equations (ODEs) and show that the time dependence can be fast-forwarded to be sub-linear. Specifically, we show that a quantum algorithm based on truncated Dyson series can prepare history states of dissipative ODEs up to time $T$ with cost $\widetilde{\mathcal{O}}(\log(T) (\log(1/\epsilon))^2 )$, which is an exponential speedup over the best previous result. For final state preparation at time $T$, we show that its complexity is $\widetilde{\mathcal{O}}(\sqrt{T} (\log(1/\epsilon))^2 )$, achieving a polynomial speedup in $T$. We also analyze the complexity of simpler lower-order quantum algorithms, such as the forward Euler method and the trapezoidal rule, and find that even lower-order methods can still achieve $\widetilde{\mathcal{O}}(\sqrt{T})$ cost with respect to time $T$ for preparing final states of dissipative ODEs. As applications, we show that quantum algorithms can simulate dissipative non-Hermitian quantum dynamics and heat processes with fast-forwarded complexity sub-linear in time.
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https://arxiv.org/abs/2410.13189
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Academic Papers
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