math.GM

Based on the growth patterns of 166 CHO monoclones observed over a 15 day period, we show that the standard population growth in a confined space equation, i.e. the sigmoid/logistic function, is alone does not capture the complex behaviour of the cell growth in a confined space. Thus, combining the sigmoid function and the exponential of the sigmoid function, we present a more accurate model for modelling cell growth in a confined space. We also present a working algorithm to obtain population growth variables (growth capacity, growth time and growth rate), model the growth patterns of the CHO monoclones, and we include subset of the dataset, along with a sample python script for the reader to replicate the results. Furthermore, we derive a model for cell confluence growth in a confined space, numerically model the confluence and present the reader with a working algorithm. With Kolmogorov-Smirnov analysis conducted on the area of the CHO monoclones, we show that the cell area of the incipient population is normally distributed, the sparse cell population is gamma distributed and the dense colony population is log-normally distributed. Thus, we further derive models for the mean, the standard deviation, the coefficient of variation and the inverse coefficient of variation for the log cell area growth in a confined space, numerically model them and present the reader with working algorithms. Finally, based on the growth patterns of another 48 CHO monoclones observed over a 16 day period, and their titer and viability measurements, we find the correlation coefficients with our calculated growth variables, and titer and viability measurements, and show that our derived growth variables can be used to predict the productivity and the health of a cell. Thus, we conclude our study by demonstrating that the productivity and the health of a cell (also the overall population) are interdependent.

In the Josephus problem with stepsize four, the participants in a circle are eliminated one by one, every fourth person leaving, until a single survivor remains. A fixed point occurs when the survivor turns out to be the person who began in the last seat. The circle sizes with this property form the sequence 1; 21; 38; 51; 122; 163; 689; 919; 2{,}906; and so on, whose gaps fluctuate erratically. This paper explains the fluctuation and turns it into a recurrence. Between consecutive fixed points, the circle sizes at which the survivor falls exactly one or two seats short of the last one, the near-misses, group into alternating blocks of the two kinds, and the length of every block is the number of times three divides a simple quantity built from the circle size that precedes the block. Iterating these divisibility counts carries each fixed point to the next. Stepsize four is the first case in which two kinds of near-miss coexist, and the alternation they force is what separates it from the solved cases of stepsizes two and three. As a byproduct, the survivor's position for an arbitrary circle size can be computed by walking the near-misses of a single interval, in a number of steps proportional to their count, rather than stepping through every smaller circle as the defining recursion does.

The sequence of Mersenne numbers $\{M_n\}_{n\geq 0}$ is defined as $M_n = 2^n-1.$ In this study we introduce the Mersenne-Bernoulli and Mersenne-Euler polynomials. Using the generating functions and $M$-calculus we find some identities associated with them. Moreover, we define the corresponding matrices with these polynomials, factorise them and find their inverses.

In this work, we study a normalized remainder $T_{n,\lambda}[\e_\lambda]$ for the degenerate exponential $\e_\lambda(u)=(1+\lambda u)^{1/\lambda}$ ($\lambda>0$). We establish an integral representation, an exact monotonicity threshold at $\lambda=1/(n+1)$, and rigorous conditions for the local failure of logarithmic convexity at the origin. We then prove a sharp asymptotic result: for every $\lambda$ in the increasing regime $(0,1/(n+1))$, the second logarithmic derivative satisfies $u^2L(u)\to -\alpha<0$ as $u\to\infty$, showing that global logarithmic convexity on $(0,\infty)$ fails throughout this regime. We further give a necessary and sufficient condition for absolute monotonicity, showing it holds only on a countable, measure-zero set of parameters, and we derive explicit two-sided truncation-error bounds that are pointwise sharp at the origin.

We classify a reciprocal degree-five quadrinomial family over the quadratic extension F_{q^2}, where q is an odd prime power. The family has four terms, coefficients in F_q, and a coefficient constraint that makes the induced rational function on the unit circle highly structured. The classification has two sharply different branches. When q is congruent to 1 modulo 4, infinite families occur and are governed by two quadratic-character conditions on the parameter b. When q is congruent to 3 modulo 4, a square-class obstruction converts the problem into a character-sum problem on a conic. A Weil-bound argument eliminates all large fields in this branch, and finite verification leaves only the sporadic fields q = 7, 19, 23. The result is a complete classification of the nondegenerate members of the family for all odd prime powers q.

We introduce a new mathematical constant $\lambda_\infty = 0.674036183193696139936660007576508455780\ldots$ (OEIS A396695), defined as the unique solution in $(1/4,+\infty)$ of $h(x) := \sum_{p \text{ prime}} 1/(xp^2-p+1) = 1$. This equation arises as the $n\to\infty$ limit of the secular equation of rank-$n$ truncations of the infinite prime LCM matrix $\mathcal{L}[p_i,p_j]=1/\mathrm{lcm}(p_i,p_j)$, where $\mathcal{L}=D+vv^T$. Viewed as a compact self-adjoint operator on $\ell^2(\mathcal{P})$, $\mathcal{L}$ has spectral radius $\rho(\mathcal{L})=\lambda_\infty$ (Theorem 5.4). The corresponding integer LCM matrix satisfies $\lambda_{\max}(W_N^*)/N\to\lambda_\infty$ (Theorem 5.8), a prime-indexed counterpart to $\lambda_{\max}(M_N)/N\to\zeta(2)=\pi^2/6$ for the integer divisor matrix. We prove that $h$ is real-analytic, strictly decreasing and strictly convex on $(1/4,+\infty)$, ensuring existence and uniqueness of $\lambda_\infty$. We compute 500 decimal digits of $\lambda_\infty$, certified by rigorous error bounds and independently verified by six computational runs, including a fully independent recomputation (Run E, PARI-GP) and a machine-verifiable Arb interval certificate (Run F) covering 505 digits. Extensive PSLQ and LLL searches find no minimal polynomial of degree $\le 8$ satisfied by $\lambda_\infty$ (at 560 decimal digits), and no integer relation against catalogs of up to 31 classical constants. The arithmetic nature of $\lambda_\infty$ remains open.

This paper investigates the asymptotic behavior of the tail of the singular product arising in the Hardy Littlewood and Bateman Horn conjectures for one dimensional systems of polynomials. A universal estimate is proved, showing that the contribution of large primes decays like the reciprocal of the logarithm, regardless of the structure of the system. For linear systems (trivial Galois group) superfast convergence is obtained. For nonlinear systems a coefficient is defined that is expressed via the average over the Galois group; in the abelian case and under the Riemann Hypothesis for Dirichlet L functions a more precise error estimate is obtained. Mixed systems containing both linear and nonlinear polynomials are also considered. Numerical experiments, presented as summary tables, confirm the theoretical conclusions. The results provide a rigorous theoretical foundation for computing singular series and refine the Bateman Horn formula.

We calculate the first derivative of the Jacobi polynomials with respect to their order in explicit form. This derivative is not an elementary function, but contains elementary special cases. As an application, we use our result with a recently devised method for resolving a new class of indefinite integrals containing Jacobi polynomials.

The ranking of fuzzy numbers has become a challenging task in fuzzy set theory due to their complex, multi-dimensional nature. While the Klir-Yuan partial order provides a natural term-wise comparison of $\alpha$-cuts, it often leaves many fuzzy numbers incomparable. To address this, various ranking methods have been developed to construct total preorders between them. However, many classical approaches suffer from significant information loss as they imply a defuzzification process. On the other hand, approaches such as admissible orders allow defining total orders, but at the expense of imposing strict algebraic rules that may contradict human intuition. In this study, we introduce a generalized sequential ordering framework to overcome these limitations. By establishing a sequence space over a totally preordered base space, we construct a flexible lexicographical structure that sequentially resolves ties. We prove that this framework yields total preorders and, under injectivity conditions, total orders. Furthermore, we analyze the compatibility of these sequential orders with the notion of admissibility. We also show that our proposed framework provides a unified mathematical umbrella that encompasses and generalizes existing ranking techniques, offering highly discriminative ordering relations for fuzzy numbers and beyond.

We describe an artistic project consisting of fabricating the 3532 different soccer balls that can be obtained by randomly assembling the 32 pieces of a classic Telstar soccer ball.

In the aerial pickup and delivery of goods in a distribution network, transport aviation faces risks of load imbalance due to the urgency required for loading, immediate take-off, and mission accomplishment. Transport planners deal with trip itineraries, prioritisation of items, building up pallets, and balanced loading, but there are no commercially available systems that can integrally assist in all these requirements. This enables other risks, such as improper delivery, excessive fuel burn, and possible safety issues due to cargo imbalance, as well as a longer than necessary turn-around time. This NP-hard problem, named "Air Cargo Load Planning with Routing, Pickup, and Delivery Problem" (ACLP+RPDP), is mathematically modelled using standardised pallets in fixed positions. We developed a strategy to solve this problem, considering historical transport data from some Brazilian hub networks, and performed several experiments with a commercial solver, five known meta-heuristics, and a new heuristic designed specifically for this problem. By using a portable computer, our strategy quickly found practical solutions to a wide range of real problems in much less than operationally acceptable time.

In this paper, we introduce the Apostol-type Mersenne-Bernoulli and Mersenne-Euler polynomials of order $\alpha$. By employing the $M$-calculus, based on the Mersenne numbers, we establish explicit series representations, addition theorems, difference equations and convolution identities.

This paper presents a detailed application of the Unified Transform Method (Fokas method) to the one-dimensional heat equation on $[0,1]$ with Dirichlet boundary conditions. The analysis formulates the Initial-Boundary Value Problem and derives an integral representation of the solution via a generalised spatial Fourier transform with complex spectral parameter $\lambda \in \mathbb{C}$, yielding the Global Relation -- an algebraic identity coupling the initial datum, prescribed boundary values, and unknown Neumann data. The unknowns are eliminated by exploiting the symmetry $\lambda \mapsto -\lambda$, reducing the solution to a contour integral over $\partial D^+$. An explicit evaluation is carried out for exponential initial datum $u_0(x)=e^{-x}$ and Dirichlet conditions $g_0(t)=\cos(t)$, $h_0(t)=e^{-1}\cos(t)$. The integral representation is analysed in the complex plane, with emphasis on exponential decay and analyticity, providing rigorous justification for contour deformation via Cauchy's Theorem and Jordan's Lemma. Numerical implementation in Maple uses a trapezoidal contour parametrisation ensuring exponential decay along each segment; the solution over $x\in[0,1]$, $t\in[0,2\pi]$ matches prescribed data to machine precision. The results confirm the analytical and numerical efficacy of the Unified Transform for classical parabolic problems and illustrate how rigorous contour analysis yields stable, accurate solutions.

In this paper, we propose a matrix-based polyalphabetic data encoding and decoding scheme using Fibonacci, Leonardo, Jacobsthal, and Lucas sequences. The method employs three sequence-based alphabets for character substitution and a Lucas-based auxiliary alphabet for word separators. A position-dependent selector, \[ \sigma=\bigl(v^2+(i-1)+(j-1)\bigr)\pmod 3, \] distributes repeated plaintext symbols among different numerical alphabets, thereby reducing frequency concentration. The resulting numerical matrix is divided into $3\times 3$ blocks and transformed using powers of the Leonardo $Q$-matrix with block-dependent keys generated from pre-shared parameters $(s,p)$. A collision-free public prime $P$ is used to keep ciphertext entries bounded while preserving unique decoding. A worked example and preliminary statistical, entropy, avalanche, and timing results indicate that the proposed modular construction is computationally efficient and provides improved distributional behavior compared with standard monoalphabetic substitution.

In this paper, we introduce and study the notion of fuzzy normed BCK-algebras and fuzzy normed BCI-algebras as a natural extension of clas sical normed algebraic structures into the fuzzy setting. A fuzzy norm on a BCK/BCI-algebra is defined as a mapping from the algebra and a positive real parameter into the unit interval satisfying suitable axioms analogous to those of fuzzy normed linear spaces. Several examples are presented to illustrate the validity of the axioms. Fundamental properties of fuzzy normed BCK/BCI algebras are established, including monotonicity, chained triangle inequalities, and order-related behaviors. It is shown that every BCK/BCI-algebra admits a fuzzy norm, and the behavior of fuzzy norms under algebra homomorphisms is investigated. Necessary and sufficient conditions are obtained for the transfer of fuzzy norms via injective, surjective, and bijective homomorphisms. A charac terization theorem is proved showing that the main fuzzy norm inequality can be reduced to a simpler condition. These results generalize known concepts in fuzzy algebra and provide a new analytical framework for studying BCK/BCI algebras.

We develop a finite-type framework for studying the Jones polynomial and its ability to distinguish the unknot. The main difficulty is to propagate the vanishing of its finite-type coefficient layers from the natural upper degree bound down to the first potentially informative orders. To overcome this, we introduce a local clasped-twist construction whose diagrammatic reductions remain uniformly controlled even when the twist is made arbitrarily long. This separates the role of the twist length from the degree bound and permits a descending vanishing argument. A single construction transfers high-order finite-type vanishing to a long residual twist, where local smoothing relations and an anchor reproducing the original knot force vanishing at all relevant lower orders. Two constructions placed in disjoint regions then generate a controlled two-crossing family. On this family, the resulting low-order relations impose a component count after successive smoothings that contradicts the topology of a suitably chosen pair of crossings. As a consequence, every nontrivial knot has a nonzero Jones finite-type coefficient at an order bounded above by three times its crossing number. In particular, a knot whose Jones polynomial is equal to that of the unknot must itself be the unknot.

We examine a Riccati--Gamma approach to the logarithmic derivative of the completed Riemann zeta function. The first part proves, in full local detail, that a naive two-sided vertical concavity criterion for $\Xi'/\Xi$ cannot be a proof of the Riemann Hypothesis, because every zero produces opposite vertical curvatures on the two horizontal sides of the pole of the logarithmic derivative. The second part replaces this obstruction by a rigorously formulated finite spectral averaging framework. We prove cancellation at the critical line, positivity of the off-critical paired contribution on the left of the critical line under a concrete low-frequency kernel condition, a conditional zero-density consequence, and a precise conditional theorem showing which additional localisation hypotheses would imply the Riemann Hypothesis. The results are therefore not presented as an unconditional proof of RH. They give a partial resolution of the Riccati--Gamma question: one natural route is ruled out unconditionally, a second symmetric mechanism is proved at the finite spectral level, and the remaining step is isolated as explicit analytic hypotheses. Reproducible Python routines and numerical figures accompany the analytic discussion.

Isohedral or face-transitive polyhedra are polyhedra with identical faces, such that any face can be mapped onto any other via a symmetry of the polyhedron. Examples include the Platonic solids, Catalan solids, bipyramids, and trapezohedra. We use the orbit-counting theorem to count the number of ways of tiling the faces of such polyhedra up to their isometries given any arbitrary set of tile designs. This approach also counts tilings fixed under each isometry of a given polyhedron and provides explicit enumeration formulas. We use this framework to recover sixteen sequences from the On-Line Encyclopedia of Integer Sequences and fill in the gaps by contributing twelve new sequences.

In a recent paper, Chatterjee, the author and Mohan posed the problem of determining all solutions of the Diophantine equation $F_n=\Phi(m)$, where $F_n$ is the $n$-th Fibonacci number and $\Phi(m)$ counts the number of nonempty sets $A \subseteq \{1, 2, \dots, m\}$ for which $\gcd(A)$ is relatively prime to $m$. In this paper, we prove that the Diophantine equation has the only solutions $(n,m)=(1,1),(2,1),(3,2)$. The main tools used in this paper are lower bounds for linear forms in logarithms due to Matveev and Dujella-Peth{\H{o}} version of the Baker-Davenport reduction method in diophantine approximation.

A complete theory of null Cartan normal helices in Minkowski space-time $\mathbb{E}^4_1$ is developed. Two algebraic conditions, obtained by successive differentiation of the helix invariant along a unit $C$-constant normal field, fully characterize null Cartan helices; the quadratic condition yields two mutually orthogonal helix axes in the Lorentzian metric. Special field types are analyzed and null Cartan cubics are shown to be normal helices. On a timelike hypersurface, a Darboux frame with six curvature functions is constructed from first principles, the normal isophotic condition is shown to reduce to a linear first-order ODE, and the existence of normal silhouettes in $\mathbb{E}^4_1$ is established.

We approach Fa\`a di Bruno as a composition theorem for Taylor polynomials. For $C^k$ maps $\phi: E \to F$ and $\psi: F \to G$ between Banach spaces, let $T^k_\ast(\phi; x)$ denote the reduced Taylor polynomial of $\phi$ at $x$, obtained by removing the constant term. We show that $$T^k_\ast(\psi \circ \phi; x) = \pi_{\le k}\bigl(T^k_\ast(\psi; \phi(x)) \circ T^k_\ast(\phi; x)\bigr).$$ The proof is an elementary estimate of the Peano remainder and does not use partitions or combinatorial enumeration. Expanding this composition identity recovers the classical Fa\`a di Bruno formulas. Polarization gives the multivariate partition formula (L\'evy 2006), while coefficient extraction gives the multi-index formula (Constantine and Savits 1996). Our approach separates the functorial nature of Taylor approximation from the combinatorial bookkeeping of polarization and coefficient extraction. As an application, we give a general higher-order product rule.

We describe a certified arbitrary-precision framework for evaluating a family of generalized multiple zeta functions. The family includes strict and weak-star chain sums, ordinary and colored multiple zeta values, affine-base and polynomial-base variants, and composite levels containing several affine or polynomial letters with complex coefficients. The numerical strategy combines finite-prefix recurrences with two complementary analytic-tail mechanisms: recursive Euler-Maclaurin expansion of one-variable tails and direct absolute tail majorants. The Euler-Maclaurin branch is fast when the relevant suffix expansions are regular, while the direct-tail branch gives robust certificates for multi-letter, weak-star, complex-coefficient, and branch-sensitive inputs. A computation is called certified only when its reported radius is obtained from a proved analytic bound for the omitted infinite tail. Strict-disk colored sums and boundary-color cases with summable absolute majorants are therefore within the certified scope; conditionally convergent colored cases whose convergence relies only on non-one unit-modulus oscillation are kept separate and reported as explicitly non-certified diagnostic outputs unless an independent analytic remainder bound is available.

We study the parametric subfamily $p = 3m(m+1) + 1$ with $m = 2^a 3^b - 1$, $a,b \in \mathbb{N}^*$, a 3-smooth slice of the centred hexagonal numbers $3m^2 + 3m + 1 = (m+1)^3 - m^3,$ from the point of view of unconditional primality certification via the Pocklington-Lehmer criterion. The 3-smoothness of $m+1 = 2^a 3^b$ yields, for every $(a,b)$, a fully factored divisor $F = 2^a 3^(b+1)$ of $p-1$ satisfying $F > \sqrt(p)$ unconditionally, reducing the certificate to two witnesses, for $q = 2$ and $q = 3$. Our main new contribution is a complete, deterministic characterisation of the two canonical witnesses. We prove that $w_2 = 5$ is a valid witness if and only if $a - b$ = 1, 2 (mod 4), by quadratic reciprocity; and that $w_3 = 7$ is a valid witness if and only if $m$ is not congruent to 2 (mod 7), by cubic reciprocity in $\mathbb{Z}[omega]$ using the explicit Eisenstein factorisation $p = ((1+m) - m \omega)((1+m) - m \omega^2)$. These two results turn the heuristic "5 and 7 always work" (which is in fact false) into exact congruence conditions, and yield a deterministic witness-selection rule. Alongside, three elementary arithmetic filters (mod 6, a (-3) quadratic-residue sieve, and a mod-7 forbidden-class test) remove about 87% of candidates at negligible cost. As a demonstration, a multi-core implementation produced four unconditional certificates on consumer hardware, the largest a prime of 29998 decimal digits.

This article deals with the upper and lower bounds of triangular subnorms generated by continuous, strictly decreasing additive generators. It first establishes necessary and sufficient conditions for the comparability of such triangular subnorms. It then explores the existence of strict (resp. nilpotent) bounds of a finite family of strict (resp. nilpotent) triangular subnorms generated by continuous, strictly decreasing additive generators. By duality, completely analogous results are derived for triangular superconorms generated by continuous, strictly increasing additive generators. In particular, it supplies the corresponding algorithms for computing those bounds, which are illustrated by several examples.

In many real-world domains, knowledge is inherently vague or imprecise - features that classical ontology languages, based on crisp Description Logics (DLs), are unable to capture. This shortcoming poses particular challenges for applications in the Semantic Web and Explainable Artificial Intelligence (XAI), where robust reasoning over graded information is essential. Fuzzy ontologies address this limitation by enriching DLs with fuzzy logic, enabling the expression of partial truth and supporting more nuanced modelling of real-world knowledge. We present fuzzy-dl-owl2, a complete re-engineering in Python of the fuzzyDL reasoner and the Fuzzy OWL 2 framework. The former is an expressive fuzzy DL reasoner, while the latter allows for defining fuzzy ontologies within OWL 2. Our contribution addresses several shortcomings of the original software, including semantic inconsistencies, rigid architectural design, and limited solver integration. The re-implementation features a modular class hierarchy tailored for extensibility, supports a broader range of Mixed-Integer Linear Programming (MILP) solvers (including open-source alternatives), and corrects IRI ambiguities arising from overlapping ontological elements. Furthermore, a dedicated Python library (pyowl2) has also been developed to handle OWL 2 annotations in a standards-compliant manner, improving interoperability with existing Semantic Web tooling and resolving IRI ambiguities. The resulting framework offers a portable, extensible, and theoretically grounded platform for reasoning with fuzzy ontologies, suitable for both research and deployment in vague-aware systems. Performance tests have also been conducted that show improved execution times w.r.t. the original Java implementation. The source code and full documentation are publicly available to facilitate community adoption and further development.

The irrationality of e+pi remains open, despite the separate transcendence of e and pi. This paper studies the problem from the viewpoint of finite irrationality certificates and gives a bounded no-go audit for low-complexity Ap\'ery-type proof mechanisms. First, we prove exact equivalences between the hypothesis e+pi in Q and eventual factorial-arithmetic phenomena: a ceiling recurrence, a factorial-Cantor digit condition, and a divisibility criterion. These criteria identify what rationality would force, while showing why tail conditions are not finite obstructions. Second, we formulate an Ap\'ery-type certificate framework based on integer linear forms L_n = A_n(e+pi)+B_n with A_n,B_n in Z, L_n nonzero, and |L_n| tending to zero. A mixed integration-by-parts identity produces such forms from integer polynomials. We then audit several low-complexity constructions, including mixed Pad\'e approximation, crossed separate approximations to e and pi, simple J-fractions, holonomic ansatzes, Rodrigues-type families, and an integer kernel-lattice search. The main contribution is a rigid boundary probe: no-go filters marking a tested zone where analytic smallness is destroyed by denominator clearing, coefficient growth, primitive reduction, or continued-fraction shadows. In the final kernel-lattice audit, 145 raw candidates reduce to 133 primitive records; the best signals are dominated by continued-fraction shadows, while non-CF candidates do not form a degree-continuing family. Thus, within the tested low-complexity families, no non-circular Ap\'ery-type mechanism for e+pi is found.

The essence of the density topology lies in the family of Lebesgue measurable sets where each point of a set is a density point of that set. The motivation of this work is to investigate the family of measurable sets for which, at every point within a set belonging to this family, the upper density of that set is positive. We obtain a strong generalized topology, and its essential properties are demonstrated in comparison with those of the classical density topology.

We study the parametric family $p_{k,m,\varepsilon,q}=km(m+1)+\varepsilon +2kq$ with $k,m\in\mathbb{N}^{\ast}$, $\varepsilon \in \{\pm1\}$, $q\in\mathbb{Z}$, which extends the elementary observation $p\equiv\pm1\pmod 6$ for every prime $p>3$. Each prime $p>k+1$ has a canonical triple $(m,\varepsilon,q)$ with $|q|$ minimal, mapping it to a generalised hexagonal base $H^{(k)}_m:=km(m+1)+1$ and a signed minimal offset $q_{\rm min}(k,m,\varepsilon)$. Every quantitative assertion is labelled rigorous, conditional, or heuristic. \emph{(Rigorous, structural.)} For every prime $\ell\mid 2k$ the family satisfies $p \equiv \varepsilon \pmod\ell$: the degenerate modular axis is fixed exactly, independently of $q$. \emph{(Rigorous, constructive.)} For the $3$-smooth subfamily $p=3m(m+1)+1$ with $m=2^a3^b-1$, Pocklington--Lehmer certificates are unconditional; two a-priori arithmetic filters remove $\approx 87\%$ of candidates before any large-integer arithmetic, and the algorithm produced an unconditionally certified prime of $29\,998$ decimal digits, independently re-verified in a separate computational environment. \emph{(Rigorous, negative.)} The earlier-reported spectral correlation between the cumulative functional $Q(r)=\sum q_n$ and the zeros of $\zeta$ is a spurious-regression artefact. Beyond three permutation tests and a bias-free test on $10^8$ primes ($R^2=1.16\times10^{-7}$), we prove \emph{unconditionally} that the regression amplitude $A_N(\gamma)\to 0$ for every fixed $\gamma$, by reduction to square-root-phase prime exponential sums. The Riemann zeros are not spectral frequencies of $Q(r)$. \emph{(Conditional, GRH / Bateman-Horn.)} $|q_{\rm min}|=O_k(m\log^2 m)$; $\mathbb{E}[|q|\mid m]=m/4+O((\log m)^2/\sqrt m)$; a modular distribution law. \emph{(Conditional, RH.)} The per-prime $\zeta$-footprint on the layer occupancy is $\ll p^{-1/4}(\log p)^2$ (Selberg variance) -- the mechanism behind the negative result. \emph{(Heuristic.)} $\mathbb{E}[|q_{\rm min}|]\sim\log m/C_k$ ($R^2=0.984$); the geometric constant $C_0(k)=\langle|q|\rangle/\sqrt p=1/(4\sqrt k)$, stable to ${<}0.02\%$ across $3\le k\le 29$; and, for $\ell\mid 2k$, $q_{\rm min}$ is empirically equidistributed modulo $\ell$ (\emph{not} a consequence of Bombieri--Vinogradov). This is experimental mathematics with rigorously tracked hypotheses: two genuinely unconditional pillars -- constructive (the certified prime) and analytic-negative (the vanishing $\zeta$-signal) -- together with a precise law on each axis. No classical question is settled.

Machine-assisted mathematical discovery has been a long-standing challenge in machine learning and artificial intelligence. In recent years, we have seen tremendous progress with generative AI, yet its contribution to automated discovery in advanced mathematical research has been limited. One of the most difficult benchmarks in this context is the Birch test, which asks whether a machine can discover truly novel and non-trivial mathematical structures without human intervention. In this work, we particularly focus on the branch of automated conjecture discovery. We use HypothesiX, an automated conjecture mining agent and analyse its generated conjectures related to the distribution of twin primes to verify the conditions of the Birch test. Furthermore, note that automated discovery is now operating at scale, but verifying its non-triviality still depends on human evaluation. We propose a benchmark to quantify the non-triviality of machine-generated conjectures using the Mahalanobis distance within an embedding cluster of selected known mathematical conjectures. We also note that this quantified benchmark can be used as an error indication signal to localise the incorrectness of a new mathematical statement, which autoformalisers fail to verify due to their limitations in proof discovery capability.

Borel and symmetric Borel derivatives are generalized derivatives defined through local averages of difference quotients. Distributional point values, in the sense of {\L}ojasiewicz and its symmetric variants, are a classical way of describing the local value of a distribution. This paper connects these two ideas. Writing $T_f$ for the regular distribution generated by $f$, we prove that finite first and second symmetric Borel derivatives give symmetric distributional point values of $T_f'$ and $T_f''$, respectively. For the first symmetric derivative, Borel smoothness is used as a sufficient condition to pass from the symmetric point value to the full {\L}ojasiewicz point value. We also prove that the one-sided Borel derivatives determine the right and left distributional point values of $T_f'$, and that the ordinary Borel derivative gives the full point value when the two one-sided averages agree. Examples show why the second-order symmetric result cannot be strengthened automatically.

In fuzzy-set theory, the notion of convexity has often been formulated through scalar reductions, typically via scores and aggregation functions. Although useful, such reductions may obscure relevant order-theoretic information in the codomain, especially in complex set-valued settings. This article develops a lattice-theoretic framework for convexity on lattice-valued mappings over point-convex segment-generated abstract convexity spaces. The framework separates the segment structure of the domain from the lattice structure of the codomain, distinguishing intrinsic convexity, defined through the lattice meet, from relational and observable convexities induced by preorders or scalar maps. The article characterizes when scalar observables preserve or reconstruct intrinsic convexity, and which codomain operators preserve it through their pointwise extensions. It then specializes the framework to hesitant fuzzy sets endowed with the symmetric lattice, recovering classical fuzzy and interval-valued convexities as natural restrictions. The structural result shows that the symmetric order cannot be represented by any finite family of scalar observables. This obstruction appears even on two-point typical hesitant fuzzy elements. Consequently, symmetric hesitant convexity cannot, in general, be reconstructed by any finite family of scalar observables. Moreover, every finite family of monotone scalar observables admits a three-point hesitant profile that is scalar-convex for all selected descriptions but not symmetrically convex.

In 1916, Hausdorff proved that the Baire hierarchy on $\mathbb{R}$, starting with the continuous functions, generates all Borel functions on $\mathbb{R}$. It remained open whether, starting with the a.e. continuous functions, the corresponding hierarchy generates all Lebesgue measurable functions on $\mathbb{R}$. We prove that, assuming the Axiom of Choice, the answer is negative.

We consider the enumeration of $N$-ary convex structures on finite sets. Our main result shows that the total number of convexities is expressed as the binomial transform of the sequence of numbers of grounded convexities. We present the exact numbers of all $N$-ary and grounded $N$-ary convexities for $|X|\leqslant 5$. The obtained integer sequences have been cross-referenced with the OEIS. As a result, we identified both known sequences (such as A000798) and new ones not previously listed.

Using a sequence of high-school level mathematics questions that were not available on the Internet, we compare the performance of the popular AI software Claude with that of my friends and fellow human beings Pierre and Catherine. Pierre had solid scientific training as a young man, while Catherine studied literature. All three were subjected to a simulated pre-calculus oral exam with main questions and follow-up questions. Their performances are compared and the ones with the best and worst performances are identified. The outcome is that the current version of Claude, even though it is an extremely useful tool that has probably recorded the solution to nearly all calculus questions that are available on the Internet, {\em exhibits only a very limited understanding of the subject} and {\em does not exhibit the ability to make intelligent connections} between different features of a pre-calculus mathematics problem that it has never seen before.

The $abc$-conjecture (Masser and Oesterle) has remained open for decades. By measuring $abc$-triples using a particular quality metric, the conjecture may be framed as seeking the asymptotic distribution of triples of sufficient quality. We create new classes of quality metrics to develop variants on the $abc$-conjecture, with each metric based upon the doubly geometric mean of the prime factors of triples. We investigate the behavior of the resulting class of quality metrics; by determining families of triples that yield high quality, we establish several asymptotic results that are analogous to the $abc$-conjecture for our metrics. We also develop sharp phase transitions for the behavior of families of such quality metrics within specified parametrizations for smoothness of primes in $abc$-triples, using heuristics from the Szpiro ratio for associated Frey curves. Finally, we implement algorithms to determine triples with high qualities with sub-linear runtime, an asymptotic speedup over na\"ive approaches. Our analysis offers robust variations of, and connections to, the $abc$-conjecture that offer independent questions of analytical interest.

This paper bridges combinatorial geometry and music theory to solve the fundamental challenge of embedding classical Western harmony into the microtonal 19-tone equal temperament (19-TET). Inspired by Roger Penrose's observations on the mathematical elegance of 19-TET, we provide the theoretical foundation for a physical 19-TET acoustic piano currently under construction. However, playing classical 12-TET music on such an instrument poses a topological problem: emvedding the classical Euler-Riemann Tonnetz into the 19-TET universe inevitably distorts structural chords, creating dissonant ``wolves.'' By formalizing these harmonic spaces as incidence configurations (the 12_3 and 19_3 graphs) and utilizing integer cuts in our optimization model, we exhaustively prove that exactly 32 out of 36 Neo-Riemannian harmonic connections can be preserved. We demonstrate a strict 5-fold degeneracy of this optimum: there exist exactly 5 mathematically equivalent local packings for the wolf chords. Among these, we identify a unique canonical realization in which the 14 excised vertices form a perfectly contiguous geometric void along the primary Hamiltonian cycle. We reveal that the 4 inevitably broken edges represent the exact topological scars of the historical enharmonic diesis, and we formulate the Vicentino Hypothesis regarding 16th-century microtonal composition. Finally, to make this theoretical geometry physically playable, we design a novel 19-TET split-key keyboard, formalized through a biomechanical cost function that optimizes the performer's hand span. This work provides the complete theoretical, historical, and ergonomic blueprint for the next generation of microtonal acoustic instruments.

In this paper, we investigate uncertainty principles for random signals associated with the linear canonical transform (LCT). First, the LCT of random signals is formulated on the probability space. Based on this representation, the Heisenberg uncertainty principle is established to characterize the relationship between the expectations in the time and frequency domains. Furthermore, the Donoho-Stark uncertainty principle, developed from a measure theoretic perspective, reveals that a random signal cannot be simultaneously concentrated on arbitrarily small sets in both the time and frequency domains. The bounds obtained in these two uncertainty principles explicitly depend on the LCT parameters, indicating that the LCT offers greater flexibility than the Fourier transform (FT). The corresponding results in the fractional Fourier transform and FT domains are also given as special cases.

We prove that if $j>0$ is not an integer, then there is a choice of signs $\varepsilon_k\in\{\pm1\}$ such that the partial sums $ \sum_{k=1}^{N}\varepsilon_k k^j $ are dense in $\mathbb R$. The proof groups consecutive terms into Thue--Morse blocks, whose Prouhet--Tarry--Escott cancellation produces nonzero block sums tending to zero but with divergent total variation. A standard steering argument then chooses block signs so that the resulting partial sums visit arbitrarily small neighbourhoods of every real number.

We develop a spectral theory of scale-invariant operators on the multiplicative half-line $(\mathbb{R}_+, dx/x)$. A symmetric kernel $M(x, y)$ satisfying $M(kx, ky) = k^{-a}M(x, y)$ necessarily factorizes as $(xy)^{-a/2}F(x/y)$, where the shape function $F$ depends only on the ratio of its arguments. The Mellin transform diagonalizes such operators: the generalized eigenfunctions are $\psi_\omega(x) = x^{-a/2+i\omega}$, and the eigenvalues are the Mellin multiplier $\tilde{F}(\omega)$. This structure reveals a fundamental decoupling of two exponents. The geometric exponent $a$, carried by the power-law envelope $(xy)^{-a/2}$, governs the matrix scaling under dilation. The spectral exponent $b$, measured from the eigenvalue decay of the finite-dimensional truncation, is an effective quantity determined by the shape of $\tilde{F}(\omega)$. For the explicit kernel $F(t) = c \rho^{|\ln t|}$, the Mellin multiplier is a Lorentzian of width $\sigma = -\ln \rho$, not a power law -- so $b$ is generically distinct from $a$. This decoupling provides a precise mathematical characterization of multicriticality: the equality $a = b$ corresponds to a simple critical fixed point of the Renormalization Group, while $a \neq b$ signals the presence of multiple independent scaling dimensions. We prove that the discrete self-similarity condition forces eigenvector collapse on the lattice, motivating the continuum formulation. Finite-size corrections from lattice sampling are quantified numerically.

The classical theorems of Kolmogorov--Nagumo--de Finetti and of Aczel--Maksa characterize quasi-arithmetic means from two complementary directions: the former for compatible families of means satisfying the replacement axiom, and the latter for bisymmetric means of fixed arity. We refine both representation results by showing that the required continuity follows automatically. Our main result states that every reflexive, symmetric, bisymmetric and partially strictly increasing $n$-variable operation on a real interval is continuous and hence quasi-arithmetic. The proof is based on a recursive construction on $n$-adic rationals given by bisymmetry, and a dense-domain continuity argument. The same method also yields the regularity-free Kolmogorov--Nagumo--de Finetti theorem for compatible families of strictly increasing symmetric means.

The Analytic Hierarchy Process (AHP) and other classic Multi-Criteria Decision Making (MCDM) techniques excel when decision makers can provide consistent pair wise judgments. Real world problems, however, often involve inconsistent, n-wise, or non-linear preference structures that render traditional methods inadequate. The {\alpha}-Discounting MCDM ({\alpha}-D MCDM) extends AHP by embedding a global discounting parameter {\alpha} that transforms an inconsistent system of preference equations into a solvable algebraic system.

A multiset is a 'set' in which elements may occur multiple times. These structures are ideal for expressing the outcome of an election, for instance of the form 60 'yes' and 40 'no'. Moreover, multisets are a useful datatype in vote counting algorithms. This will be illustrated in three different forms of vote counting, known as: 'instant-runoff', 'De Borda', and 'single transferrable vote'. The relevant abstract properties of multisets are: (1) they form a (free) commutative monoid, and (2) they form a functor, and (3) also a monad. This paper illustrates how such categorical properties can be put to good use in deriving and expressing election outcomes. The emphasis is not on the (elementary) category theory involved, but on its application in voting systems.

We develop an analytic umbral approach to Bessel moments, using them as a concrete testbed justifying the passage from formal indicial umbral calculus to Mellin--Barnes umbral transmutation theory. [...] While the formal procedure reproduces the correct results in suitable convergence chambers, it may lead to non-admissible hypergeometric expansions at physically relevant parameter values. The cubic moment provides the basic example [...] We show that this obstruction is removed by replacing the purely formal expansion with an analytic umbral transmutation. In this setting, exponential umbral pairings are interpreted through Mellin--Barnes integrals, and Ramanujan's Master Theorem acts as an inverse selection principle for the spectral ground state, or clock, associated with a given Bessel product. The factorisation \(J_0^3=J_0J_0^2\) produces two distinct clocks and reduces the cubic full-line moment to a one-dimensional Barnes integral, equivalently to a Meijer \(G\)-function. This gives the classical value of the cubic Bessel moment and clarifies why the divergent Appell realisation is only a local representation of a globally meaningful umbral identity. The same mechanism is then applied to scaled cubic products and to the fourth Bessel moment. [...] The fifth moment marks the first genuinely higher-rank case: the natural umbral grouping leads to a bivariate Barnes transmutation rather than to an ordinary Meijer \(G\)-function. Finally, we discuss real fractional powers \(J_0^\alpha\), \(\alpha>2\), showing that the same interpretation persists beyond integer moments. [...] The resulting picture identifies Bessel moments as values of effective umbral transmutations and separates the global analytic meaning of the umbral representation from the local convergence properties of its hypergeometric residue expansions.

We propose a systematic way to construct trigonometric-like functions beyond the classical sine--cosine pair by factorising rational umbral operators. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an $n$-fold decomposition of rational umbral expressions. For each integer $n\geq 2$, the construction produces $n$ functions which play the role of higher-order trigonometric components: their sum reconstructs the corresponding umbral function, while the individual components isolate the different cyclic sectors of its expansion. The construction is developed first in the formal umbral setting. The quadratic case $n=2$ gives the Gaussian trigonometric functions, in which the cosine-like component is a Gaussian and the sine-like component is its natural umbral companion. The cubic case $n=3$ yields a three-component cyclic system and shows how the same idea extends beyond the usual even--odd decomposition. These examples suggest that trigonometric factorisation is not restricted to ordinary rotations, but belongs to a broader cyclic principle in umbral calculus. We then reinterpret the same formal identities through the recently developed analytic umbral framework. In this second step, the cyclic components are realised by Mellin--Barnes pairings, and the root-of-unity decomposition is related to the splitting of the corresponding spectral kernel. This analytic formulation provides contour representations, local expansions, and sectorial asymptotics for the functions obtained formally. Finally, we indicate how the same cyclic kernels act on Fourier transforms. The resulting framework presents higher-order umbral trigonometric functions as natural cyclic components of factorised rational or exponential umbral operators.

The Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) spacetime, which was first proposed by Friedmann (1922--1924) and Lema\^{i}tre (1927) and subsequently developed by Robertson and Walker (1935), is an isotropic and homogeneous cosmological model of the universe. This paper addresses a significant gap in the differential geometry literature by providing a comprehensive examination of the curvature properties of the FLRW spacetime. It is demonstrated that the FLRW spacetime satisfies the curvature condition R \cdot R - Q(S, R)=L_C Q(g, C) alongside several pseudosymmetric-type conditions related to the conformal and conharmonic curvature tensors. Furthermore, the Tachibana tensors Q(g,C) and Q(S, C) are found to exhibit a linear dependence on the tensor $(C \cdot R + R \cdot C)$. Additionally, the spacetime is shown to be a 2-quasi-Einstein manifold, generalized Roter type and Ein(3). The Ricci tensor is shown to be neither cyclic parallel nor of Codazzi type, yet it satisfies several compatibility requirements concerning the R, C, P, K and W curvature tensors. A thorough analysis of Ricci solitons and curvature inheritance properties reveals that the spacetime admits almost Ricci soliton and $\eta$-Ricci Yamabe soliton structures with respect to the non-Killing soliton vector fields $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial r}$. Moreover, the spacetime admits generalized curvature inheritance symmetry properties for the Riemann curvature tensor, as well as for the Weyl conformal, concircular, and conharmonic curvature tensors with respect to the coordinate vector field $\frac{\partial}{\partial t}$ and the gradient of $t$. Later, a comparison of the FLRW and Lema\^{i}tre--Tolman--Bondi (LTB) spacetimes is provided in terms of various curvature-related geometric properties and physical characteristics. Finally, a noteworthy conclusion of the entire study is presented.

The graph Fourier transform (GFT) is a fundamental tool in graph signal processing and has recently been extended to the graph fractional Fourier transform (GFRFT). Existing sampling methods in the GFRFT domain are primarily designed to minimize error, whereas a wider range of alternative sampling strategies should be admitted. In this paper, a unified and efficient GFRFT sampling theory is proposed. First, a new definition of graph fractional bandlimited signals is introduced, with the corresponding graph fractional sampling and perfect reconstruction theorem, as well as the associated graph fractional localization operator. Next, several GFRFT sampling strategies are developed based on different criteria, including maximum cutoff frequency, minimum error, and maximum localized basis, along with the corresponding representations of their localization operators. Then, by exploiting a localization operator that jointly considers vertex and spectral localization, a fast sampling set selection method in the GFRFT domain is proposed. Finally, numerical experiments investigate the reconstruction errors and execution time of the proposed sampling methods and evaluate their performance in applications, demonstrating the effectiveness of the unified GFRFT sampling theory and its advantages over GFT methods.

For $p,q\in\mathbb{N}$ and $\alpha,\beta\in\mathbb{R}$, we investigate the family of improper integrals \[\int_0^\infty\frac{(\cos\alpha x-\cos\beta x)^p}{x^q}dx.\] We establish a complete classification of the parameter ranges $(p, q; \alpha, \beta)$ for which the integrals converge or diverge, and we derive explicit closed-form evaluations in all convergent cases. The analysis also reveals a family of combinatorial identities arising naturally from coefficients in the trigonometric power expansions. As a further application of the same method, we study an analogous class of integrals involving powers of sine differences. This extends the work of Laoharenoo and Boonklurb in 2022.

This is an incomplete attempt to show that the upper bound of $\lesssim n^\frac{4}{3}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$ points and $n$ lines that attain the sharp bound of the Szemer\'edi-Trotter point-line incidence bound.

Majorization is a fundamental tool for comparing vectors, with connections to convexity, doubly stochastic matrices, eigenvalues, singular values, and zeros of polynomials. In matrix analysis, it plays a central role in the study of eigenvalue inequalities, particularly those arising from classical determinantal inequalities such as those attributed to Hadamard and Fischer in the context of positive semidefinite matrices. A result of Fischer and Holbrook shows that equality in the Hardy--Littlewood--P\'olya theorem for non-affine convex functions is closely linked to block structure in the associated doubly stochastic transformations. Motivated by this, we introduce $*$-majorization, a structured extension of majorization that respects prescribed block decompositions of vectors. This framework naturally corresponds to block diagonal doubly stochastic matrices and provides a refinement of the classical Hardy--Littlewood--P\'olya and Rado theorem. We show that such transformations are precisely the linear operators that preserve $*$-majorization, and we extend fundamental constructions such as $T$-transforms and convex combinations to this setting. In an application, we study the eigenvalue relations associated with the principal submatrices of positive definite matrices. Classical majorization does not, in general, capture determinantal inequalities such as those of Koteljanskii, whereas $*$-majorization provides a natural framework for structured comparisons of eigenvalue vectors. This leads to new insights into the interplay between majorization theory, determinantal inequalities, and spectral properties of matrices.

We present a graph-theoretic reformulation of pitch-class set theory in which each set in $\mathbb{Z}_n$ is represented as a complete weighted graph whose edge weights are interval classes. We show that this construction is invariant under the dihedral group $D_n$, and that the full interval structure is encoded by a cyclic step composition, from which all interval data are recovered via an additivity principle. This framework yields a direct correspondence between T/I equivalence and graph isomorphism, and reinterprets Z-relation as non-isomorphic graphs with identical edge-weight multisets. We extend the model to weighted clique complexes, linking higher-order homometry to simplex-weight structure, and introduce a cent-weighted formulation enabling comparisons across different equal temperaments. Finally, we define a polynomial invariant derived from antipodal step pairings for algebraic analysis of pitch class space.

We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual stuffle, or quasi-shuffle, algebra on the associated nested sums. This viewpoint places classical multiple harmonic numbers, colored harmonic sums, and several generalized Euler sums under a common structural mechanism. We focus on three fundamental families of monoidal alphabets: the ordinary power alphabet generated by $n$, the affine alphabet generated by linear factors $an+b$, and the polynomial-base alphabet generated by polynomial factors $P(n)$. The resulting classes of multiple harmonic numbers, multiple affine harmonic numbers, and multiple polynomial-base harmonic numbers provide systematic containers for a wide range of finite and infinite Euler-type sums. We prove closure and lifting results showing that nested sums whose summands are built from these alphabets, possibly multiplied by harmonic-number factors, reduce to the corresponding finite harmonic-number objects. As consequences, the framework recovers many known Euler-sum identities and produces many new identities in a uniform way. While reduction to simpler functions remains a separate and often difficult problem, the monoidal-alphabet perspective provides a unified algebraic language for organizing, transforming, and extending harmonic-sum identities.

This note studies a concrete bitwise and triangular coordinate model for the central product of n copies of the quaternion group Q8. The positive basis elements are words of length n in the alphabet {1, 2, 4, 7}, identified with i, j, k, and the identity element e. The signed basis group Fn is the corresponding central product of n copies of Q8, and the real algebra generated by the basis words is H^{\otimes n}. The contribution is the coordinate model: in this basis, Boolean multiplication, recursive triangular tilings, digitwise S3-actions, reflection anti-automorphisms, parity cancellation, and centralizer tile sets can be expressed in a single language. A local XNOR/AND rule recovers quaternionic basis multiplication and gives a table-free digitwise multiplication rule in every order. The associated centroid map to a recursive triangular tiling is equivariant for the digitwise S3-action and the dihedral action on the triangle. Odd digit permutations reverse multiplication order, yielding ordinary or twisted commutation criteria for products of elements symmetric about triangular axes. Synchronized cyclic changes of selected noncentral digits give equilateral triangles of centroids. Finally, for every non-unit basis word, the centralizer in the signed group has cardinality 4^n and its positive tile set occupies exactly one half of the order-n tiling.

We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of descents. Our approach relies on a known bijection that associates each integer composition $\sigma$ with a pair $(\pi,\lambda)$, where $\pi$ is a permutation and $\lambda$ is an integer partition. We show that the distribution of inversions and the distribution of descents over $k$-compositions are related, respectively, to the distribution of (maj,inv) and to the distribution of (inv,des) over permutations of $\{1,2,\ldots,k\}$, where maj, inv, and des denote the classical permutation statistics major index, inversion number, and descent number, respectively.

In this paper, we introduce a Clifford algebra framework for Julia-type dynamics driven by the geometric product. The nonlinear iteration \[ f(\vec{x}) = (\vec{x}\diamond \vec{n})^p \diamond \vec{n} + \vec{c}, \qquad p \ge 2, \] is studied in a real $n$-dimensional inner-product space $V$, where $\vec{x}, \vec{n}, \vec{c} \in V$ and $\vec{n}$ is a unit vector. The main result reveals a previously unreported invariance phenomenon: although the geometric product generates higher-grade multivector components at intermediate stages, a built-in grade-reduction mechanism ensures complete collapse back to the vector subspace. Consequently, the Clifford Julia operator is shown to be closed on $V$, and the iteration defines a well-posed nonlinear dynamical system in arbitrary dimensions. This invariance is established through a structural decomposition of the Clifford product and an inductive closure argument, supported by explicit verification in low-dimensional cases and a general proof in $\mathbb{R}^n$. The results demonstrate that classical Julia dynamics can be consistently extended beyond the complex plane into higher-dimensional geometric algebra without loss of geometric interpretability. The framework opens a new direction for fractal-type dynamics in Clifford algebras, providing a unified algebraic setting for higher-dimensional invariant-preserving iterative systems.

We develop a unified analytical and dynamical framework for the qualitative study of the one-parameter family of generalized Dirichlet eta functions $\eta_{a}(t)=\sum_{m\ge0}(-1)^{m}(am+1)^{-t}$, $a>0$, $t>0$, which includes the classical Dirichlet eta and beta functions. Using a Mellin--Laplace representation of $\eta_{a}$ as $\mathbb{E}[f_{a}(X_{t})]$, where $f_{a}$ is a scaled logistic function and $(X_{t})$ a standard Gamma process, we show that the logarithmic derivative $\varphi_{a}(t)=\eta_{a}'(t)/\eta_{a}(t)$ satisfies a non-homogeneous Riccati equation with strictly negative forcing. This single inequality yields strict concavity and strict log-concavity of $\eta_{a}$, positivity and monotonicity of $\varphi_{a}$, and the precise asymptotic law $\varphi_{a}(t)=\log(a+1)(a+1)^{-t}+O((a+2)^{-t})$. We further prove that $\varphi_{a}(t)/\varphi_{a,e}(t)\to 2/\log(a+1)$ as $t\to\infty$, where $\varphi_{a,e}(t)=-\eta_{a}''(t)/(2\eta_{a}(t))$, obtaining in particular the trapping inequality $0<\varphi_{a,e}(t)<\varphi_{a}(t)$ for all sufficiently large $t$ when $a<e^{2}-1$. We also present a self-contained geometric-rate algorithm (rate $1/3$) for computing $\eta_{a}^{(k)}(t)$ together with a sharp error bound. High-precision numerical experiments confirm all results. As an application, we show that the Riccati--Gamma dynamics of $\eta_{a}$ and $\varphi_{a}$ provide a principled mechanism for musical synthesis, generating a complete melody whose pitch and rhythm are governed by these functions.

A unified algebraic framework is developed to study nilpotency as a structural mechanism for exactness in operational, combinatorial, and computational problems. The central object is the Nilpotent Operational System (SON), formalized as a tuple (R, N, m, M_R), where R is a C-algebra, N satisfies N^{m+1}=0, and M_R(m) is the arithmetic cost of a product in R. The basic result is the Exact Termination Lemma: every formal series evaluated at N collapses to an exact finite sum of m+1 terms. Three complexity regimes are obtained: truncated series (quasi-linear bound via Newton iteration), nilpotent operators (linear bound via Horner evaluation), and incidence algebras (quasi-quadratic bound). Applications include classical and free cumulants, Appell sequences, orthogonal polynomials, Stirling numbers, M\"obius inversion, Witt vectors, and local holonomic functions. In all cases except Stirling numbers, strict complexity improvements over classical algorithms are obtained. For classical cumulants, equivalence with polynomial multiplication is achieved: L(C_n)=Theta(M(n)). For free cumulants, the complete equivalence T_{FC}(n)=Theta(M(n)) is established via the Voiculescu functional equation and compositional reversal.

We formulate a Maxwell version of the codimension-three Riesz/Gaussian quadratic-form representation for perfectly conducting parallel plates. This paper is the Maxwell follow-up to the scalar codimension-three Riesz/Gaussian representation theorem presented earlier in arXiv:2605.06693(2026): the same transverse Riesz reduction and prescribed-covariance quadratic-form mechanism are carried over here to the physical parallel-plate Maxwell operator. The construction is carried out in finite lateral volume $\Omega_{L,a}=T_L^2\times[0,a]$, using the physical electric-field Hilbert space of divergence-free fields satisfying the perfect-conductor tangential condition $n\times E=0$, with the static normal zero mode removed. The Maxwell curl-curl operator is defined by its closed quadratic form, and an explicit Fourier-domain analysis proves the finite-volume spectral gap, compact resolvent, and heat-trace admissibility needed for the stochastic construction. For this reduced Maxwell operator $\mathcal L_{\mathrm{Mx}}$, the codimension-three Riesz integral gives the transversely reduced Riesz mediator $g\mathcal L_{\mathrm{Mx}}^{-1}$. A prescribed heat-regularized Gaussian source with covariance $(\hbar c/g)\mathcal L_{\mathrm{Mx}}^{3/2}e^{-\tau\mathcal L_{\mathrm{Mx}}}$ then has expected quadratic Green energy equal to the heat-regularized physical Maxwell trace. The finite-volume trace is shown to be spectrally equivalent to a scalar Dirichlet channel plus a scalar Neumann channel with its constant zero mode removed. Under the standard parallel-plate interaction finite-part prescription, the large-area energy density is $$ -\frac{\pi^2\hbar c}{720a^3}. $$ The result is a representation theorem for the Maxwell parallel-plate trace under a prescribed covariance in the flat-plate geometry considered here.

On the n x n chessboard, the move totals of distinct pieces satisfy a small number of striking arithmetic identities. The total diagonal mobility of the bishop and the total 8-neighbor mobility of the king are exactly proportional, with constant n/12, valid for every n. Among nontrivial boards, the strengths of two distinct pieces drawn from a natural thirteen-piece alphabet coincide only for n in {6, 8, 12}. We define the strength of a piece P on the n x n board as the probability that a uniformly random ordered pair of distinct squares forms a legal P-move on the empty board, and prove four main results. (1) An asymptotic dichotomy classifies pieces into riders (Theta(1/n) strength) and leapers (Theta(1/n^2) strength), with explicit rational leading constants. (2) A stable-ordering theorem identifies the threshold n* = 24 beyond which the strength order becomes fixed, with a complete tabulation of every transition for 4 <= n <= 24. (3) A complete classification of strength coincidences shows they occur only at the three magic boards n in {6, 8, 12}, accompanied by the closed-form identity str(K) - str(N) = 12/(n^2(n+1)), the unique near-coincidence between bishop and knight at n = 10 (gap 0.0606%), and the bishop-king proportionality str(B)/str(K) = n/12. (4) A Strength Algebra Theorem expresses the strength of any compound army as a linear functional of a four-dimensional atomic vector, and confines strength coincidences between distinct single pieces to the three magic boards. As immediate consequences we obtain explicit strength-preserving single-piece substitution rules on each magic board, and a characterization of the 8 x 8 board as the unique nontrivial board on which the rook attains a strength matched by another piece in the alphabet (the archbishop).

This paper continues the author's previous work on a limit-free algebraic-geometric construction of the derivative in the class of polynomial functions and extends the proposed framework to elementary functions. Derivatives of rational power, exponential, logarithmic, trigonometric, and inverse trigonometric functions are constructed through the geometric interpretation of the tangent line, inverse symmetry, and local linear structure, without treating the limit as the initial defining mechanism. Within the proposed approach, the derivative is introduced from the outset as a functional correspondence assigning to each point the slope coefficient of the tangent line. The paper demonstrates that the classical differentiation formulas arise naturally from interconnected geometric and algebraic structures and are subsequently consistent with standard limit-based analysis. From a methodological perspective, the study proposes the logical sequence: Tangent, Local Linear Structure, Limit formalisation. Thus, the paper presents a conceptual bridge between geometric intuition, algebraic construction, and classical differential calculus.

Soft set theory is an important and emerging area within soft computing, owing to its attribute-oriented mathematical framework and its wide applicability in diverse domains, including science and social sciences. The theoretical constraints associated with the selection of subsets of the sets of attributes in soft set theory have further motivated the development of hybrid and extended theoretical models. In this paper, we introduce two distinct roughness measures and six entropy measures for soft sets and systematically investigate their properties using both theoretical analysis and computational techniques. The proposed roughness measures are defined within two distinct conceptual frameworks. Throughout the development of these measures and the corresponding results, the foundational principles of soft set theory, as established by Molodtsov, are strictly preserved. Furthermore, the proposed framework is shown to be novel with respect to roughness characterization, and a comparative analysis with classical rough set theory is presented to highlight the theoretical distinctions and contributions of this work.

In this paper, we investigate fractional B splines and their connections with Fourier analysis, and establish connections with generalized Stirling-type numbers and distribution theory. Employing a generating function approach inspired by recent results of Simsek [24], we derive a novel Fourier type expansion for fractional B splines that involves generalized Stirling type numbers. Our main contribution is the derivation of a Fourier-type expansion of fractional B splines in terms of generalized Stirling-type numbers. This representation allows us to express fractional B splines as infinite linear combinations of derivatives of the Dirac delta in the distributional sense. Furthermore, we establish an explicit shifted distributional representation and obtain shifted distributional representations that characterize the action of fractional B-splines on test functions. In addition, we introduce a new class of fractional spline polynomials and derive their generating function in terms of the Mittag Leffler function. These results provide a unified framework that connects spline theory, fractional calculus, and combinatorial structures.

Traditional graph analysis focuses on nodes and edges, that is, pairwise relationships. Yet many real-world networks, including biological, social, and communication networks, involve higher-order relationships in which multiple nodes interact simultaneously. This has led many to develop network topology analysis methods based on higher-order structures and higher-order connectivity, seeking to reveal complex interactions beyond node pairs. Many of the latter address only undirected networks. To overcome this, we lay out a mathematical formalism resting on directed clique complexes constructed from directed graphs (their "higher-order structures" or "simplicial structures''), stressing the interrelations between directed cliques (their "directed higher-order connectivities''), leading towards a more complete directed Q-analysis that allows quantifying, characterizing, and comparing similarities involving simplicial structures.

Omar Khayyam's treatment of cubic equations by intersections of conic sections has often been read as an anticipation of analytic or coordinate geometry. This paper argues that such a reading obscures the conceptual structure of Khayyam's own method. Working within the geometric framework of Euclid and Apollonius, it reconstructs Khayyam's thirteen cubic species through the local conic relations generated by his proportional arguments. In each case, the construction yields not merely the two conics Khayyam uses, but a third algebraically available conic relation that remains geometrically unused. This hidden conic reveals the extent to which Khayyam's algebra and geometry cooperate without yet merging into a global coordinate system. From this perspective, Khayyam is not an incomplete analytic geometer, but a complete geometric algebraist working within a different conceptual world.

In this note, we derive an elementary version of the coarea formula by considering the mass of a solid body with density $g (x)$. Then we present an rigorous proof using the changing variable formula. To this end we construct the diffeomorphism $\Phi$ via the gradient flow and compute its Jacobian determinant via geometric method.

When one inserts a number of identical bars in between blocks of an ordered set partition, they get a barred preferential arrangement. In this study we define a new generalization of barred preferential arrangements, by considering barred preferential arrangements with no fixed blocks, and ones where the first r elements of a set are singletons. We derive several combinatorial identities. Combinatorially these numbers are a kind of generalized barred preferential arrangements. We also provide some asymptotic results for these numbers.

Automated crack inspection is increasingly recognized as a critical component of infrastructure monitoring; however, cracks continue to be reported primarily as binary segmentation masks by many current vision-based systems. While localization is facilitated by such masks, limited structural information is provided for robust engineering interpretation. For practical crack assessment, measurable morphological features -- including centerline geometry, branching behavior, junction locations, topology, and severity-related indicators -- are required. In this work, \textit{CrackMorph-XAI-Net}, an explainable morphology-aware framework for image-based crack analysis, is presented. Crack image and region-mask data are converted into a sequence of interpretable structural outputs through four distinct stages: topology-preserving skeleton extraction, junction detection via Gaussian heatmap regression, morphology descriptor computation, and severity-oriented screening. To support rigorous stage-wise evaluation, the standard \textit{CRACK500} benchmark is extended with aligned skeleton maps, junction heatmaps, and topology labels. Experimental validation demonstrates that a mean Dice coefficient of 0.991 is achieved by the learned skeleton extraction stage, with topology preserved in 98.5\% of test images. Furthermore, a recall of 0.964 and an F1-score of 0.887 are obtained in the junction detection stage, highlighting the efficacy of heatmap regression for sparse structural targets. Strong agreement between predicted and reference morphology values is revealed by descriptor-level evaluation, with correlations exceeding 0.95 for length, width, orientation, junction count, and tortuosity.

We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the discrete Fourier transform, and we identify general conditions for it to be invertible when applied to any real or complex sequence. Applications to the mathematical sciences are also presented.

Using Cartan equivalence method, invariant coframes are constructed for two branches of rank one and zero, which characterize linearizable third-order ODEs under contact transformations with four- and five-dimensional Lie symmetry algebras, respectively. A procedure for deriving the corresponding contact transformations is also presented, along with illustrative examples.

In this paper, we present a fixed point method for the arctangent based on sine and cosine. Let $t\in \mathbb{R}^{+}$ and $P\in \mathbb{N}$. We define: \[T\left(x\right)=x-\sum_{k=1}^{P}\,\frac{\left(-1\right)^{k-1}}{2\,k-1} \left(\frac {\sin\!\left(x\right)-t\cos\!\left(x\right)} {\cos\!\left(x\right)+t\sin\!\left(x\right)} \right)^{2\,k-1}.\] For every initial value $x_0$ sufficiently close to $\arctan\left(t\right)$, the sequence \[x_{n+1}=T\left(x_{n}\right)\;;\,n=0,1,\ldots\] is converging to $\arctan\left(t\right)$ with order of convergence exactly $\left(2\,P+1\right)$. The computational test we performed demonstrates the efficiency of the method. \selectlanguage{ngerman} \[\] \[\textbf{Zusammenfassung}\] In dieser Abhandlung stellen wir ein Fixpunktverfahren zur Berechnung des arcustangens auf Basis von sinus und cosinus vor. Es sei $t\in \mathbb{R}^{+}$ und $P\in\mathbb{N}$. Wir definieren: \[T\left(x\right)=x-\sum_{k=1}^{P}\,\frac{\left(-1\right)^{k-1}}{2\,k-1} \left(\frac {\sin\!\left(x\right)-t\cos\!\left(x\right)} {\cos\!\left(x\right)+t\sin\!\left(x\right)}\right) ^{2\,k-1}.\] F\"ur jeden Startwert $x_0$ hinreichend nahe bei $\arctan\left(t\right)$ konvergiert die Folge \[x_{n+1}=T\left(x_{n}\right)\;;\,n=0,1,\ldots\] gegen $\arctan\left(t\right)$ mit Konvergenzordnung genau $\left(2\,P+1\right)$. Anhand einer praktischen Berechnung von $\frac{\pi}{4}$ zeigen wir die Effizienz des Verfahrens. \[\text{Deutsche Version ab Seite 17}\]

The fuzzy Sombor index applies the classical Sombor index to fuzzy graphs, incorporating both edge membership values and fuzzy vertex degrees. For $\alpha>1$, the general fuzzy Sombor index it is defined as \[ \mathrm{SO}^{\mu}_{\alpha}(\Gamma)=\sum_{uv\in V(\Gamma)} \left( \mu(u,v)\, \sqrt{\mu_u^2+\mu_v^2} \right)^{\alpha}. \] This paper analyses extremal features of $\mathrm{SO}^{\mu}$ across different types of fuzzy graphs. We determine the maximum value (resp. minimum value) of $\mathrm{SO}^{\mu}$ characterise in regular fuzzy graph. We established significant inequality between the fuzzy Sombor index and other well-known fuzzy topological indices.

The aim of this paper is to generalize some fixed point theorems in the class of convex contraction of order $m$ on a complete suprametric space. Then, we will prove that the class of convex contraction of order m is strong enough to generate a fixed point on a complete suprametric spaces but do not force the mapping to be continuous at the fixed point, and it can be replaced by relatively weaker conditions of $k$-continuity or $T$-orbitally lower semi-continuous. On this way a new and distinct solution to the open problem of Rhoades (Contemp Math 72:233-245,1988) is found. In sequel, we will prove some fixed point results in the setting suprametric spaces which are generalizations of the results regarding Sehgal, \'Ciri\'c and Fisher's quasi-contraction. Some examples and application will be approved our results.

Recently, Stull [18], [17] resolved a long-standing open problem posed by Lutz, on whether the set of effective Hausdorff dimensions of points on a straight line in $\mathbb{R}^2$ -- the effective dimension spectrum of the line -- contains a unit interval. This question is related to problems in classical fractal geometry like the Kakeya conjecture and Furstenberg sets. Stull posed an open question on the dimension spectra of polynomial curves. For the first result, with new techniques which adapt the theory of classical real root-finding of polynomials to the current setting, we show that the dimension spectra of every polynomial curve contains at least two points. This answers an open question posed by Stull [18], [17]. We use the main result to construct a class of polynomials which have width strictly greater than 1, answering a second problem stated in [18],[17]. Stull [18] resolved the dimension spectrum conjecture for planar lines, showing that it contains a unit interval. For the second result, we resolve the conjecture for a subfamily of polynomials whose coefficients form a "low" dimension point in $\mathbb{R}^{d+1}$.

Under a prescribed heat-regularized Gaussian source covariance, we give a quadratic-form representation of the scalar Casimir trace associated with a codimension-three Riesz reduction. For a product operator $L_M=L_B-\Delta_\perp$, with $L_B$ positive self-adjoint and bounded below, transverse reduction of the ambient Riesz operator $L_M^{-s}$ produces the brane multiplier $L_B^{m/2-s}$, up to an explicit Gamma-function constant. The exponent $s=1+m/2$ is therefore the critical Riesz exponent for obtaining the ordinary brane Green operator $L_B^{-1}$; in codimension three this gives $s=5/2$. Using this induced Green kernel, we prescribe a Gaussian generalized scalar source with covariance proportional to $L_B^{3/2}e^{-\tau L_B}$. The expectation of its quadratic Green-kernel energy is then exactly the heat-regularized scalar Casimir trace \[ \frac{\hbar c}{2} \operatorname{Tr}\!\left(L_B^{1/2}e^{-\tau L_B}\right). \] With the same finite-part prescription, the identity specializes in the Dirichlet parallel-plate geometry to the standard scalar finite part. We also record a deterministic flat Green-energy calibration at the plate scale. Within the plate-compatible rectangular aspect-ratio family, the cubical cell is selected by spectral, heat-trace, and Green-energy extremal criteria, and the associated comparison coefficient is the corresponding extremal calibration value. The construction is a scalar spectral representation theorem; no electromagnetic, gravitational, brane-dynamical, or fundamental-constant identification is asserted.

This paper introduces a class of extended central factorial numbers generated by a parity-dependent recurrence relation, termed the "flickering operator". We demonstrate that the resulting triangular structure, now indexed as OEIS A395021, provides a unified recursive framework for alternating bit sequences (A000975) and normalized tangent-secant coefficients (A036969). This study provides an alternative integer-based expansion for power sums. While similar to the central factorial methods explored by Knuth (1993), our flickering basis offers an integrated computational scheme that avoids fractional Bernoulli numbers by construction. We provide explicit closed-form expressions, discuss its geometric derivation from finite difference tables, and present a full Python implementation. Structural Synthesis. A key contribution of this work is the unification of previously disparate combinatorial sequences into a single coherent framework. While certain columns of the flickering triangle T(n, k) (such as A008957) could be partially retrieved from the diagonals of existing central factorial arrays, our structure provides a complete representation including previously unindexed even-positioned terms. Furthermore, the row-wise analysis reveals that the flickering operator generates full integer sequences where previously only the odd-indexed elements (e.g., A002451) were identified. This synthesis bridges the gap between these sequences, positioning A395021 as the underlying master structure.

The resummation of superfactorially divergent series represents a significant computational challenge in mathematical physics. In the present paper the resummation of a specific class of Stieltjes series characterized by a moment sequence growing as $(2n)!$ will be addressed. Despite the fact that Carleman's condition is satisfied for these series, the convergence rate of Pad\'e approximants is severely hindered by the logarithmic divergence of the associated Carleman series. Weniger's $\delta$ transformation is proposed as a highly efficient alternative resummation tool. By employing recently established results on the converging factors of superfactorially divergent Stieltjes series, an exact integral representation for the truncation error is obtained. This representation enables the rigorous derivation of the leading-order asymptotic behavior of the transformation error, as well as the estimation of the related convergence rate, for real positive arguments. Numerical experiments strongly support the theoretical findings, suggesting that the $\delta$ transformation offers a robust and computationally efficient framework for decoding this class of wildly divergent expansions

Given that the restricted equivalence functions (REFs) can serve to measure the similarity of two fuzzy sets, this motivates the integration of REFs with similarity-based approximate reasoning systems to enhance inference capabilities. Therefore, this work primarily constructs hierarchical similarity-based approximate reasoning (SBAR) using REFs. Specifically, we first characterize REFs with a given aggregation function, then discuss the approximation equality of SBAR method proposed by Raha et al. with REFs. Finally, we suggest two REF-based hierarchical Raha's SBAR methods which efficiently restrain the explosion of fuzzy rules.

In the paper [E. Jim\'enez-Fern\'andez, J. Rodr\'{\i}guez-L\'opez, E. A. S\'anchez-P\'erez, Fuzzy Sets and Systems 406 (2021),66-81], a McShane-Whitney extension theorem is presented for real-valued fuzzy Lipschitz maps between fuzzy metric spaces. Specifically, the codomain space is considered as a so-called Euclidean fuzzy metric space $(\mathbb{R},M_{\phi,g},\ast).$ However, while the function $\phi$ is only required to be increasing, some results of the paper implicitly assume that $\phi$ is invertible, even though this is not explicitly stated. We propose here an alternative possibility that only requires $\phi$ to be also left-continuous.

We show that every family of quasi-arithmetic means generated by (a subset of) $\mathcal{C}^1$ functions with nonvanishing derivative which is bounded (from below or from above) by a quasi-arithmetic mean, possesses the best (lower or upper) bound which is a quasi-arithmetic mean generated by a function belonging to the same family.

The transitivity of fuzzy relations plays an important role in fuzzy set theory, artificial intelligence, clustering and decision-making. However, it is often difficult for fuzzy relations to satisfy the transitivity property in many practical applications. This has motivated researchers to investigate the degree to which a fuzzy relation is transitive. Therefore, this work first investigates two different measures of T-transitivity for fuzzy relations using some well-known fuzzy implications. And then, the relationship between two different degrees of transitivity is investigated. Further, the concept of an {\epsilon}-T-transitive fuzzy relation is introduced, and the aggregation functions that preserve the {\epsilon}-T-transitivity of fuzzy relations are characterized. Finally, the {\epsilon}-T-transitive fuzzy relation is utilized to make inferences and cluster objects. Compared to finding the T-transitive closure, it is reasonable to cluster objects using the {\epsilon}-T-transitive fuzzy relation under the permissible error.