In a $k$-XOR game $\Psi$ over $\mathbb{F}_p$, a verifier samples challenges $(x_1,\ldots,x_k)\sim\mu$ where $\mu$ is a distribution over $\Sigma_1\times\ldots\times \Sigma_k$, sends $x_i$ to player $i$ and receives an answer $f_i(x_i)\in\mathbb{F}_p$, and accepts if and only if $f_1(x_1)+\ldots+f_k(x_k)=t(x_1,\ldots,x_k)$, where $t\colon \Sigma_1\times\ldots\times \Sigma_k\to\mathbb{F}_p$ is a fixed map. We show that if $\Psi$ has value strictly less than $1$, then the value of the $n$-fold repeated game $\Psi^{\otimes n}$ is at most $2^{-cn}$ where $c = c(\Psi)>0$ is a fixed constant. This extends a previous result of [Bhangale-Braverman-Khot-Liu-Minzer] showing exponential decay for some $3$-XOR games. Our proof uses a relaxation of the value of games studied in the quantum setting, called the commuting value of a game. We show a transformation that maps a $k$-player game $\Psi$ with value strictly less than $1$ to a $2k$-player game $\Psi'$ with commuting value less than $1$, such that $\mathsf{val}(\Psi^{\otimes n})\leq \mathsf{val}((\Psi')^{\otimes n})$. We prove that $\mathsf{val}(\Psi')<1$ has a direct proof using nothing more than the Cauchy-Schwarz inequality, which we are able to tensorize to get that $\mathsf{val}((\Psi')^{\otimes n})$ vanishes exponentially fast in $n$.

A time series \(\textbf{x} = (x_1, x_2, \ldots , x_t) \in \mathbb {R}^t\) is a real-valued vector where \(\{x_i\}_{i=1}^t\) typically represent data observations sequentially sampled at equal time intervals from some sensor or system. Time series data is ubiquitous across scientific domains—in this document alone we draw examples from radio frequency (RF) signal analysis 1, biomedical data 2,3, and dynamical systems 4; there are many other examples including seismology, finance, and meteorology. In dynamical systems research, there is a pervasive goal of classifying when and to what extent time series data is chaotic. In 5 it was observed that subsampling of time series from dynamical systems at different time scales resulted in a corresponding “unfolding of the attractor” instinctive phenomenon which could be readily discovered when a system is periodic, partially observed if the system is quasi-periodic, and not at all observed if the system is chaotic. Motivated by the unfolding of the attractor phenomenon, we present novel quantitative methods herein which measure information-theoretic changes in time series. To cast time series analysis in the regime of information theory, we first apply a Takens’ embedding to map the time series to a point cloud. We then approximate the point cloud...

**Authors:** Mark Braverman, Dor Minzer In a $k$-XOR game $Ψ$ over $\mathbb{F}_p$, a verifier samples challenges $(x_1,\ldots,x_k)\simμ$ where $μ$ is a distribution over $Σ_1\times\ldots\times Σ_k$, sends $x_i$ to player $i$ and receives an answer $f_i(x_i)\in\mathbb{F}_p$, and accepts if and only if $f_1(x_1)+\ldots+f_k(x_k)=t(x_1,\ldots,x_k)$, where $t\colon Σ_1\times\ldots\times Σ_k\to\mathbb{F}_p$ is a fixed map. We show that if $Ψ$ has value strictly less than $1$, then the value of the $n$-fold repeated game $Ψ^{\otimes n}$ is at most $2^{-cn}$ where $c = c(Ψ)>0$ is a fixed constant. This extends a previous result of [Bhangale-Braverman-Khot-Liu-Minzer] showing exponential decay for some $3$-XOR games. Our proof uses a relaxation of the value of games studied in the quantum setting, called the commuting value of a game. We show a transformation that maps a $k$-player game $Ψ$ with value strictly less than $1$ to a $2k$-player game $Ψ'$ with commuting value less than $1$, such that $\mathsf{val}(Ψ^{\otimes n})\leq \mathsf{val}((Ψ')^{\otimes n})$. We prove that $\mathsf{val}(Ψ')<1$ has a direct proof using nothing more than the Cauchy-Schwarz inequality, which we are able to tensorize to get that $\mathsf{val}((Ψ')^{\otimes n})$ vanishes exponentially fast in $n$.

In an earlier note (arXiv:2606.11090) I showed that the border determinantal complexity of the power sum x_1^d + ... + x_N^d is ?(Nd). The proof counted the simple solutions of an explicit polynomial system and compared that number with what a small determinant can produce. This left two questions. Does the argument work for other polynomials? And how far can it be pushed? This article answers both. For the first question, the power sum turns out not to be special. What the argument really counts is the smooth points of the zero set whose tangent hyperplanes pass through some fixed generic points. Every smooth form of degree d in N variables has exactly d(d-1)^(N-2) of these. A determinant of size m has at most about (4em/N)^(N-1). So any polynomial with an affine-linear restriction to a smooth form has border determinantal complexity at least (N-1)(d-1)/(4e), and at least (N-1)(d-1)/(2e) in the symmetric model. This lets me move from power sums to more natural families. For the homomorphism polynomial of a connected bipartite graph F with q labels, I get ?(q|E(F)|). When F is a path with q^2 edges, the bound is ?(q^3), and this is tight. These polynomials form a VBP-complete family. For the second question, the answer is that the method does not go much further. I compute the count exactly for generic matrices, and this shows that the constant 4e cannot be improved asymptotically. I then use bounds on the topology of linear sections to show that the comparisons of polar counts in Section 6 cannot prove more than O(dN), even when singular strata are taken into account. For the permanent this means at most O(n^3). For the power sum of degree N it means (1+o(1))N^2/(4e), so the bound from the earlier note is the best this approach can give. The exposition is due to the human author. The mathematical results were obtained in the process of a discussion of lower bounds on the permanent with Claude Fable 5 and later refined and simplified in conversations with Claude Opus 5.5 and ChatGPT Astra.

I give a short proof that the border determinantal complexity of F_n = x_1^n + ... + x_n^n over C is at least (n-1)^2/(4e), and at least (n-1)^2/(2e) in the symmetric model. The same argument shows that the border determinantal complexity of x_1^d + ... + x_N^d is Theta(Nd) for all N >= 3 and d >= 2. The earlier article arXiv:2606.13628, by the same author, introduced the first superlinear lower bounds and includes verification by Lean formalization. However, the AI written exposition was hard to follow. Kumar and Volk (arXiv:2609.34462) give a short proof of the exact ordinary case, but do not address the border or symmetric cases. In this article, I provide a greatly simplified exposition of the border and symmetric cases, using the implicit function theorem and an explicit section of simple solutions in place of the conormal specialization arguments of arXiv:2606.13628. I choose an explicit polynomial system with n(n-1)^(n-2) simple solutions for F_n. I then use the implicit function theorem to ensure that each point persists under small perturbations. Finally, kernel equations and the multihomogeneous Bézout theorem bound the number of these points for a small determinant. The exposition is due to the human author. The mathematical simplification was obtained in discussion with ChatGPT Astra and refined in conversation with Claude Opus 5.5.

**Authors:** Markus Bläser, Radu Curticapean, Sagnik Dutta, Gorav Jindal For a partition $\boldsymbolλ = (λ_1,\ldots,λ_r)$ and $n\ge r$, the monomial symmetric polynomial is defined as $$m_{\boldsymbolλ}(x_1,\ldots,x_n) := \sum x_{i_1}^{λ_1} x_{i_2}^{λ_2} \cdots x_{i_r}^{λ_r},$$ where the sum ranges over all distinct monomials obtained by assigning the parts of $\boldsymbolλ$ to pairwise distinct variables. We establish an almost-complete dichotomy for the complexity of the monomial symmetric polynomials $m_{\boldsymbolλ}$: If a family of partitions contains polynomially many distinct parts, then the corresponding monomial symmetric polynomials (on sufficiently many variables) are VNP-complete. If the number of distinct parts is constant, the family admits polynomial-size circuits. Thus the only remaining gap concerns the intermediate regime where the number of distinct parts grows subpolynomially. As part of our proof, we use combinatorial properties of matching polynomials to establish VNP-hardness of the perfect matching polynomial for blow-up graphs; these are graphs with non-generic edge-weights that are obtained by "blowing up" a smaller graph. We reduce these matching polynomials to monomial-symmetric polynomials by combining classical techniques from algebraic complexity theory with structural properties of Sidon sets.

Computing the approximate quantiles or ranks of a stream is a fundamental task in data monitoring. Given a stream of elements $x_1, x_2, \dots, x_n$ and a query $x$, a relative-error quantile estimati...

概要 この記事では、目的関数が指数1の半陽的微分代数方程式の解を通して初期値やパラメータに依存するとき、その勾配をAdjoint法により数値的に計算する方法を導出します。対象とするのは、目的関数が x_1 \in \mathbb{R}^{N_x}と y_1, \dot y_1 \in \mathbb{R}^{N_y}の関数として J = J(x_1, y_1, \dot y_1) と与えられていて、x_1, y_1, \dot y_1が指数1の半陽的微分代数方程式の初期値問題 \begin{align*} \dot y &= f(t, x, y; \alpha) \\\\...

概要この記事では、目的関数が指数1の半陽的微分代数方程式の解を通して初期値やパラメータに依存するとき、その勾配をAdjoint法により数値的に計算する方法を導出します。対象とするのは、目的関数が x_1 \in \mathbb{R}^{N_x}と y_1, \dot y_1 \in \mathbb{R}^{N_y}の関数としてJ = J(x_1, y_1, \dot y_1)と与えられていて、x

概要この記事では、目的関数が指数1の半陽的微分代数方程式の解を通して初期値やパラメータに依存するとき、その勾配をAdjoint法により数値的に計算する方法を導出します。対象とするのは、目的関数が x_1 \in \mathbb{R}^{N_x}と y_1, \dot y_1 \in \mathbb{R}^{N_y}の関数としてJ = J(x_1, y_1, \dot y_1)と与えられていて、x