math
math
11-26 00:00
arXiv:2511.19439v1 Announce Type: new Abstract: We present an algorithm to solve the Simultaneous Unitary Similarity(S.U.S) problem which is to check if there exists a Similarity transformation determined by a Unitary $U$ s.t $UA_lU^*=B_l$, $l \in \{1,...,p\}$, where $A_l$ and $B_l$ are $nxn$ complex matrices. We observe that the problem is simplest when $U$ is diagonal, where we see that the `paths' in the graph defined by non-zero elements of $A_l$ and $B_l$ determine the solution. Inspired by this we generalize this to the case when $U$ is block-diagonal to identify a form refered to as the `Solution-form' using `paths' determined by non-zero sub-matrices of $A_l,B_l$ which are non-zero multiples of Unitary. When not in Solution form we find an equivalent problem to solve by diagonalizing a Hermitian or a Normal matrix related to the sub-matrices. The problem is solved in a maximum of $n$ steps. The same idea can be extended to solve the Simultaneous Unitary Equivalence(S.U.Eq) problem where we solve for $U,V$ in $UA_lV^*=B_l$, $A_l,B_l$ being $mxn$ Complex rectangular matrices. Here we work with the 'paths' in the related bi-graph to define the Solution-form. The algorithms have a complexity of $O(pn^4)$. This work finds application in Quantum Evolution, Quantum gate design and Simulation.The salient features of each step of the algorithm can be retained as Canonical features to classify a given collection of complex matrices up to Unitary Similarity.
cs.dsquant-phmath.ra
math
math
11-26 00:00
arXiv:2511.19446v1 Announce Type: new Abstract: This paper presents a novel class of information-theoretic strategies for solving the game of Mastermind, achieving state-of-the-art performance among known heuristic methods. The core contribution is the application of a weighted entropy heuristic, based on the Belis-Guias, u framework, which assigns context- dependent utility values to each of the possible feedback types. A genetic algorithm optimization approach discovers interpret-able weight patterns that reflect strategic game dynamics. First, I demonstrate that a single, fixed vector of optimized weights achieves a remarkable 4.3565 average guesses with a maximum of 5. Building upon this, I introduce a stage-weighted heuristic with distinct utility vectors for each turn, achieving 4.3488 average guesses with a maximum of 6, approaching the theoretical optimum of 4.3403 by less than 0.2%. The method retains the computational efficiency of classical one-step-ahead heuristics while significantly improving performance through principled information valuation. A complete implementation and all optimized parameters are provided for full reproducibility.
cs.itcs.gtmath.it
math
math
11-26 00:00
arXiv:2511.19455v1 Announce Type: new Abstract: Linear Geometry describes geometric properties that depend on the fundamental notion of a line. In this paper we survey basic notions and results of Linear Geomery that depend on the flat hulls: flats, exchange, rank, regularity, modularity, and parallelity.
math.homath.mg
math
math
11-26 00:00
arXiv:2511.19461v1 Announce Type: new Abstract: We study the relationship between three combinatorial objects -- a taffy pulling machine, the Calkin-Wilf tree of all fractions, and Conway's rational tangles. After introducing these objects, we develop a taffy analogue for Conway's characterization of rational tangles, and we give a direct geometric connection between rational tangles and taffy pulls.
math.homath.gtmath.co
math
math
11-26 00:00
arXiv:2511.19531v1 Announce Type: new Abstract: We review Euler's work on spherical geometry. After an introduction concerning the general place that trigonometric formulae occupy in geometry, we start by the two memoirs of Euler on spherical trigonometry, in which he establishes the trigonometric formulae using different methods, namely, the calculus of variations in the first memoir, and classical methods of solid geometry in the other. In another memoir, Euler gives several formulae for the area of a spherical triangle in terms of its side lengths (these are ``spherical Heron formulae''). He uses this in the computation of numerical values of the solid angles of the five regular polyhedra, which is his goal in his memoir. We then review memoirs in which Euler systematically starts by establishing a theorem or a construction in Euclidean geometry and then proves an analogue in spherical geometry. We point out relations between Euler's memoirs on spherical trigonometry and works he did in astronomy, on the problem of drawing geographical maps, and in geomagnetism. We also review some other works of Euler involving spheres, including a memoir on the three-dimensional Apollonius problem and others concerning algebraic curves on the sphere. Even though these works are not properly on spherical geometry, they show Euler's interests in various questions related to spheres and we think that they are worth highlighting in such an overview. Beyond spherical geometry, the reader is invited to discover in this article an important facet of the work of the great Leonhard Euler. This article will appear as a chapter in the book ``Spherical geometry in the eighteenth century, I: Euler, Lagrange and Lambert'', Springer, 2026.
math.homath.gt
math
math
11-26 00:00
arXiv:2511.19540v1 Announce Type: new Abstract: In this review article, we provide an overview of recent advances in the numerical approximation of minimizers of the Ginzburg-Landau energy in multiscale spaces. Such minimizers represent the most stable states of type-II superconductors and, for large material parameters $\kappa$, capture the formation of lattices of quantized vortices. As the vortex cores shrink with increasing $\kappa$, while their number grows, it is essential to understand how $\kappa$ should couple to the mesh size in order to correctly resolve the vortex patterns in numerical simulations. We summarize and discuss recent developments based on LOD (Localized Orthogonal Decomposition) multiscale methods and review the corresponding error estimates that explicitly reflect the $\kappa$-dependence and the observed superconvergence. In addition, we include several minor refinements and extensions of existing results by incorporating techniques from recent contributions to the field. Finally, numerical experiments are presented to illustrate and support the theoretical findings.
math.nacs.na
math
math
11-26 00:00
arXiv:2511.19550v1 Announce Type: new Abstract: This paper proposes a novel semiotic framework for analyzing Large Language Models (LLMs), conceptualizing them as stochastic semiotic engines whose outputs demand active, asymmetric human interpretation. We formalize the trade-off between expressive richness (semiotic breadth) and interpretive stability (decipherability) using information-theoretic tools. Breadth is quantified as source entropy, and decipherability as the mutual information between messages and human interpretations. We introduce a generative complexity parameter (lambda) that governs this trade-off, as both breadth and decipherability are functions of lambda. The core trade-off is modeled as an emergent property of their distinct responses to $\lambda$. We define a semiotic channel, parameterized by audience and context, and posit a capacity constraint on meaning transmission, operationally defined as the maximum decipherability by optimizing lambda. This reframing shifts analysis from opaque model internals to observable textual artifacts, enabling empirical measurement of breadth and decipherability. We demonstrate the framework's utility across four key applications: (i) model profiling; (ii) optimizing prompt/context design; (iii) risk analysis based on ambiguity; and (iv) adaptive semiotic systems. We conclude that this capacity-based semiotic approach offers a rigorous, actionable toolkit for understanding, evaluating, and designing LLM-mediated communication.
cs.itcs.aimath.it
math
math
11-26 00:00
arXiv:2511.19556v1 Announce Type: new Abstract: One-shot information theory addresses scenarios in source coding and channel coding where the signal blocklength is assumed to be 1. In this case, each source and channel can be used only once, and the sources and channels are arbitrary and not required to be memoryless or ergodic. We study the achievability part of one-shot information theory, i.e., we consider explicit coding schemes in the oneshot scenario. The objective is to derive one-shot achievability results that can imply existing (first-order and second-order) asymptotic results when applied to memoryless sources and channels, or applied to systems with memory that behave ergodically. Poisson functional representation was first proposed as a one-shot channel simulation technique by Li and El Gamal [118] for proving a strong functional representation lemma. It was later extended to the Poisson matching lemma by Li and Anantharam [117], which provided a unified one-shot coding scheme for a broad class of information-theoretic problems. The main contribution of this thesis is to extend the applicability of Poisson functional representation to various more complicated scenarios, where the original version cannot be applied directly and further extensions must be developed.
cs.itmath.it
math
math
11-26 00:00
arXiv:2511.19560v1 Announce Type: new Abstract: We study the Fourier ratio of a signal $f:\mathbb Z_N\to\mathbb C$, \[ \mathrm{FR}(f)\ :=\ \sqrt{N}\,\frac{\|\widehat f\|_{L^1(\mu)}}{\|\widehat f\|_{L^2(\mu)}} \ =\ \frac{\|\widehat f\|_1}{\|\widehat f\|_2}, \] as a simple scalar parameter governing Fourier-side complexity, structure, and learnability. Using the Bourgain--Talagrand theory of random subsets of orthonormal systems, we show that signals concentrated on generic sparse sets necessarily have large Fourier ratio, while small $\mathrm{FR}(f)$ forces $f$ to be well-approximated in both $L^2$ and $L^\infty$ by low-degree trigonometric polynomials. Quantitatively, the class $\{f:\mathrm{FR}(f)\le r\}$ admits degree $O(r^2)$ $L^2$-approximants, which we use to prove that small Fourier ratio implies small algorithmic rate--distortion, a stable refinement of Kolmogorov complexity.
cs.itmath.camath.it
math
math
11-26 00:00
arXiv:2511.19568v1 Announce Type: new Abstract: Accurate radio propagation and interference modeling is essential for the design and analysis of modern cellular networks. Stochastic geometry offers a rigorous framework by treating base station locations as a Poisson point process and enabling coverage characterization through spatial averaging, but its expressions often involve nested integrals and special functions that limit general applicability. Probabilistic interference models seek closed-form characterizations through moment-based approximations, yet these expressions remain tractable only for restricted parameter choices and become unwieldy when interference moments lack closed-form representations. This work introduces a hybrid approximation framework that addresses these challenges by combining Monte Carlo sampling of a small set of dominant interferers with a Laplace functional representation of the residual far-field interference. The resulting dominant-plus-tail structure provides a modular, numerically stable, and path-loss-agnostic estimator suitable for both noise-limited and interference-limited regimes. We further derive theoretical error bounds that decrease with the number of dominant interferers and validate the approach against established stochastic geometry and probabilistic modeling benchmarks.
cs.iteess.spmath.prmath.it
math
math
11-26 00:00
arXiv:2511.19618v1 Announce Type: new Abstract: We define and study the parabolic K-motivic Hecke category of a (possibly disconnected) Kac-Moody group. Our main result is a combinatorial description via singular K-theory Soergel bimodules which arise from the equivariant algebraic K-theory of parabolic Bott-Samelson resolutions. In the spherical affine case, the K-motivic Hecke category serves as one side of a conjectural quantum K-theoretic derived Satake equivalence, addressing a conjecture of Cautis-Kamnitzer.
math.ktmath.agmath.rt
math
math
11-26 00:00
arXiv:2511.19625v1 Announce Type: new Abstract: Sufficient conditions for a simple graph to be characterized up to isomorphism given its spectrum and the spectrum of its complement graph are known due to Wang and Xu. This note establishes a related sufficient condition in the presence of loops: if the walk matrix has square-free determinant, then the graph is characterized by its generalized spectrum. The proof includes a general result about symmetric integral matrices.
math.ntmath.spmath.co
math
math
11-26 00:00
arXiv:2511.19639v1 Announce Type: new Abstract: Inspired by prior work by Tian and by Cao and Xu, this paper presents an efficient computer-aided framework to characterize the fundamental limits of coded caching systems under the constraint of linear coding. The proposed framework considers non-Shannon-type inequalities which are valid for representable polymatroids (and hence for linear codes), and leverages symmetric structure and problem-specific constraints of coded caching to reduce the complexity of the linear program. The derived converse bounds are tighter compared to previous known analytic methods, and prove the optimality of some achievable memory-load tradeoff points under the constraint of linear coding placement and delivery. These results seem to indicate that small, structured demand subsets combined with minimal common information constructions may be sufficient to characterize optimal tradeoffs under linear coding.
cs.itmath.it
math
math
11-26 00:00
arXiv:2511.19665v1 Announce Type: new Abstract: Persistent homology is a popular technique in topological data analysis that tracks the lifespans of homological features in a nested sequence of spaces. This data is typically presented in a multi-set called a persistence diagram or a barcode. For single parameter filtrations with homology coefficient taken in a principal ideal domain, the persistence diagram/barcode can be computed using the presentation theorem for finitely generated modules over a PID. One way to reconstruct the persistence diagram/barcode is to consider the rank of the pair group at all intervals, as defined by Edelsbrunner and Harer, which counts the number of homology classes whose lifespans are precisely said intervals respectively. In this paper we generalize the rank of the pair group for suitably `tame' filtrations, described as functors from a partially ordered set to a category of chain complexes, and show how it can be captured by a categorical version of the calculus of finite-differences for abelian groups.
math.atmath.ct
math
math
11-26 00:00
arXiv:2511.19675v1 Announce Type: new Abstract: This paper presents the Safe Sequential Quadratically Constrained Quadratic Programming (SS-QCQP) algorithm, a first-order method for smooth inequality-constrained nonconvex optimization that guarantees feasibility at every iteration. The method is derived from a continuous-time dynamical system whose vector field is obtained by solving a convex QCQP that enforces monotonic descent of the objective and forward invariance of the feasible set. The resulting continuous-time dynamics achieve an $O(1/t)$ convergence rate to first-order stationary points under standard constraint qualification conditions. We then propose a safeguarded Euler discretization with adaptive step-size selection that preserves this convergence rate while maintaining both descent and feasibility in discrete time. To enhance scalability, we develop an active-set variant (SS-QCQP-AS) that selectively enforces constraints near the boundary, substantially reducing computational cost without compromising theoretical guarantees. Numerical experiments on a multi-agent nonlinear optimal control problem demonstrate that SS-QCQP and SS-QCQP-AS maintain feasibility, exhibit the predicted convergence behavior, and deliver solution quality comparable to second-order solvers such as SQP and IPOPT.
cs.romath.occs.syeess.sy
math
math
11-26 00:00
arXiv:2511.19679v1 Announce Type: new Abstract: We develop structure-preserving finite volume schemes for the barotropic Euler equations in the low Mach number regime. Our primary focus lies in ensuring both the asymptotic-preserving (AP) property and the discrete entropy stability. We construct an implicit-explicit (IMEX) method with suitable acoustic/advection splitting including implicit numerical diffusion that is independent of the Mach number. We prove the positivity of density, the entropy stability, and the asymptotic consistency of the fully discrete numerical method rigorously. Numerical experiments for benchmark problems validate the structure-preserving properties of the proposed method.
math.nacs.na
math
math
11-26 00:00
arXiv:2511.19502v1 Announce Type: new Abstract: We generalize certain totient functions using elementary symmetric polynomials and derive explicit product forms for the totient functions involving the second elementary symmetric sum. This work follows from the work of Toth [The Ramanujan Journal, 2022] where the totient function was generalized using the first and the kth elementary symmetric polynomial. We also provide some observations on the behavior of the totient function with an arbitrary jth elementary symmetric polynomial. We then outline a method for solving a certain the restricted linear congruence problem with a greatest common divisor constraint on a quadratic form, illustrated by a concrete example. Most importantly, we demonstrate the equivalence between obtaining product forms for generalized totient functions, counting zeros of specific polynomials over finite fields, and resolving a broad class of restricted linear congruence problems .
math.nt
math
math
11-26 00:00
arXiv:2511.19503v1 Announce Type: new Abstract: In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler-Gauss sequences. Noting that every Gauss sequence is an Euler-Gauss sequence, we compare them with certain generalizations of Gauss sequences and provide several counterexamples. In particular, the important Smallest Prime Factor (SPF) and Greatest Prime Factor (GPF) sequences (suitably defined at 1) are Euler-Gauss sequences but not Gauss sequences. We further extend these congruence-based integer sequences to a q-analog setting and establish characteristic properties that reveal their structure and fill gaps in the literature on q-Gauss sequences. In recent works, q-Gauss sequences have been shown to admit interesting combinatorial interpretations and to exhibit the Cyclic Sieving Phenomenon (CSP). Not only do our q-Euler-Gauss sequences satisfy the standard CSP with some restriction, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature.
math.nt
math
math
11-26 00:00
arXiv:2511.19552v1 Announce Type: new Abstract: We construct the five-variable $p$-adic $L$-function attached to Hida families on $\mathrm U(2,1)\times\mathrm U(1,1)$, interpolating the square-root of Rankin-Selberg $L$-values in the \emph{shifted piano} range. Our construction relies on a new theta operator and its $p$-adic variation which plays a role analogous to the classical Ramanujan-Serre theta operator in Hida's $p$-adic Rankin-Selberg method. The interpolation formula, including the modified Euler factors at $p$ and at the real place, is consistent with the conjectural shape of $p$-adic $L$-functions predicted by Coates and Perrin-Riou.
math.nt
math
math
11-26 00:00
arXiv:2511.19579v1 Announce Type: new Abstract: The present note contains a new proof of Y. Hashizume's 1958 theorem that every non-split link in $S^3$ admits a unique factorization into prime links. While the new proof does not go far beyond standard techniques, it is considerably shorter than the original proof and avoids most of its case exhaustion. We apply this proof to obtain a string link version (and also an alternative proof) of a 1972 theorem of D. Rolfsen: two PL links in $S^3$ are ambient isotopic if and only if they are PL isotopic and their respective components are ambient isotopic. It is tempting to dismiss this string link version as obvious by deriving it directly either from Rolfsen's or Hashizume's theorem. But this does not seem to be possible, as it turns out that there exists a string link that has no local knots, while its closure has a local knot.
math.gt
math
math
11-26 00:00
arXiv:2511.19637v1 Announce Type: new Abstract: The Douglas-Rachford splitting method is a classical and widely used algorithm for solving monotone inclusions involving the sum of two maximally monotone operators. It was recently shown to be the unique frugal, no-lifting resolvent-splitting method that is unconditionally convergent in the general two-operator setting. In this work, we show that this uniqueness does not hold in the convex optimization case: when the operators are subdifferentials of proper, closed, convex functions, a strictly larger class of frugal, no-lifting resolvent-splitting methods is unconditionally convergent. We provide a complete characterization of all such methods in the convex optimization setting and prove that this characterization is sharp: unconditional convergence holds exactly on the identified parameter regions. These results immediately yield new families of convergent ADMM-type and Chambolle-Pock-type methods obtained through their Douglas-Rachford reformulations.
math.oc
math
math
11-26 00:00
arXiv:2511.19643v1 Announce Type: new Abstract: An isotopy between two diffeomorphisms means the existence of an arc connecting them in the space of diffeomorphisms. Among such arcs there are so-called stable arcs, which do not qualitatively change under small perturbations. In the present paper we consider a set of gradient-like diffeomorphisms f of 2-torus whose induced isomorphism given by a matrix $\begin{pmatrix} -1 & -1\cr 1& 0\end{pmatrix}$. We prove that the set of such diffeomorphisms is decomposed into four stable components. Moreover, we establish that two diffeomorphisms under consideration are stably connected if and only if they have the same number of fixed sinks.
math.ds
math
math
11-26 00:00
arXiv:2511.19666v1 Announce Type: new Abstract: Carbon matching aims to improve corporate carbon accounting by tracking emissions rather than energy consumption and production. We present a mathematical derivation of carbon matching using marginal emission rates, where the unit of matching is tons of carbon emitted. We present analysis and open source notebooks showing how marginal emissions can be calculated on simulated electric bus networks. Importantly, we prove mathematically that distinct emissions rates can be assigned to all aspects of the electric grid - including transmission, storage, generation, and consumption - completely allocating electric grid emissions. We show that carbon matching is an accurate carbon accounting framework that can inspire ambitious and impactful action. This research fills a gap by blending carbon accounting expertise and power systems modeling to consider the effectiveness of alternative methodologies for allocating electric system emissions.
math.oc
math
math
11-26 00:00
arXiv:2511.19678v1 Announce Type: new Abstract: Given a word $w$, what is the maximum possible number of appearances of $w$ reading contiguously along any of the directions in $\{-1, 0, 1\}^d \setminus \{\mathbf{0}\}$ in a large $d$-dimensional grid (as in a word search)? Patchell and Spiro first posed a version of this question, which Alon and Kravitz completely answered for a large class of ``well-behaved" words, including those with no repeated letters. We study the general case, which exhibits greater variety and is often more complicated (even for $d=1$). We also discuss some connections to other problems in combinatorics, including the storied $n$-queens problem.
math.co