Computer Science and Mathematics

Abstract: The Yang-Baxter equations are important tools in mathematics, physics and computer science. Yang-Baxter equations are recognised as the origin of the theory of quantum groups or quasitriangular Hopf algebras. We propose new solutions for Yang-Baxter equations. Connecting celebrated topics, the current paper also deals with a many circles version of the Johnson–Tzitzeica theorem. Taking as a model some results from the Yang-Baxter theory, a self-dual theorem for cones is stated and proved.

Abstract: This work presents a practical implementation of the Zero Interaction Spectral Framework for financial market forecasting and economic analysis. The framework introduces five powerful analytical techniques: Spectral Mirrors for asset correlation analysis, Orbital Dynamics for trend prediction, Fractal Time Warping for pattern recognition, Quantum Spectral Resonance for turning point detection, and Predictive Coherence Collapse Detection for market crash prediction. These techniques utilize the mathematical structure of the Riemann zeta function zeros to identify hidden patterns in market behavior. The framework provides a practical tool for investors, financial analysts, and policymakers, with applications in algorithmic trading, risk management, and economic forecasting. Numerical simulations using the first 10,000 non-trivial zeros confirm the convergence and stability of the model with 91.3% prediction accuracy. The framework incorporates 17 distinct Zero Behavior Motions that characterize market dynamics.

Abstract: A recent conjecture states that a finite-dimensional complex evolution algebra is solvable if and only if it has no non-zero idempotents. We exhibit a three-dimensional counterexample over C whose isomorphism class already appears in the classification of three-dimensional complex evolution algebras. Since the conjecture is known in dimensions one and two, this counterexample has the smallest possible dimension. The algebra is defined over every field of characteristic different from 2 and is the exceptional member of a one-parameter family of pairwise non-isomorphic non-solvable evolution algebras whose idempotents are determined explicitly. For the exceptional parameter, the derived series stabilises at a non-zero two-dimensional subalgebra whose only idempotent is zero. Direct sums with zero algebras give complex counterexamples in every dimension at least three. We also prove the conjectured equivalence whenever the stable term of the derived series is an evolution algebra.

Abstract: Let p > 3, and let A be a finite pre-Lie ring whose additive group has p-power order and which satisfies Property 3 of Smoktunowicz. We give a two-part answer to Kourovka Notebook Problem 21.123, distinguishing the typographically literal formulas from an explicit factorial-compatible corrected prescription.First, the formulas displayed around the proposed modified group-of-flows construction do not, when read typographically literally, define a total operation. Several local indices do not align, an arbitrary set-theoretic right inverse of multiplication by p need not be iterable, and the movement identity printed in Remark 63 of the source need not hold with its stated shielding exponent. Second, after the factorial-compatible local adjustments and replacement of that identity by the block-shielding law proved here, the corrected partial prescription has a unique conservative totalization. Here conservative means that the totalized value agrees with every corrected literal section iterate at every point where that iterate is defined. The result covers both natural readings of the pullback notation: a direct set-theoretic \( p^{s_j} \)-pullback and a repeated p-section wherever the latter is defined.We construct this totalization through protected divided operator words. Every p-pullback is separated from the next pullback and from both endpoints by enough operator factors for Property 3 to annihilate all root-choice and additivity defects. This gives a choice-independent noncommutative functional calculus on finite abelian p-groups, stable under products and substitution and supporting protected exponential, logarithm, and Baker-Campbell-Hausdorff series.A faithful affine representation of the pre-Lie ring yields the decisive closure statement: the protected BCH series returns to the affine image, so the protected exponentials are closed under actual composition. Their orbit map is bijective, and transport of composition defines a group law on A whose \( \lambda \)-maps are additive automorphisms. Thus the conservative totalization is a left brace. If one fixed section has all required literal iterates defined, its literal construction is exactly this group law. Consequently, the typographically literal reading does not yield a total operation, whereas the explicitly corrected reading formalized in this paper gives an affirmative answer to the mathematical question underlying Problem 21.123.

Abstract: Suppose that R is a commutative multiplicative hyperring with identity and \( S\subseteq R \) is a multiplicatively closed subset. In this paper, we introduce and study the notions of weakly 2- prime hyperideals and weakly S-2 prime hyperideals, which extend the concepts of prime, 2- prime , weakly prime, weakly S - prime, and S-2 prime hyperideals in multiplicative hyperrings. A proper I of R is called a weakly 2- prime hyperideal if whenever \( \left\{0\right\}\neq x\circ y\subseteq I \) for \( x,y\in R \), then either \( x^2\subseteq I \) or \( y^2\subseteq I \). Furhermore, a proper hyperideal I with \( I\cap S=\emptyset \) is said to be a weakly S-2 prime hyperideal if there exists an element \( s\in S \) such that for all \( x,y\in R \) with \( \left\{0\right\}\neq x\circ y\subseteq I \), we have \( s\circ x^2\subseteq I \) or \( s\circ y^2\subseteq I \). We examine various basic properties of these hyperideals and investigate their characterizations under good hyperring homomorphisms, quotient hyperrings, and fraction hyperrings \( S^{-1}R \). In addition, we generalize the classical 2- prime avodience theorem to weakly 2-prime and weakly S-2 prime hyperideals, and we delineate the structural relationships among prime, 2- prime, weakly 2- prime, weakly S- prime, S-2 prime, weakly \( S-2 \) prime hyperideals.

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We develop an exact "mantissa" formalism for the binary expansions of natural numbers, in which the gap structure between consecutive ones is encoded by a sequence of fractional parts \(\sigma_j\in(0,1]\). Within this formalism we prove a sharp threshold dichotomy: the next binary gap equals 1 precisely when \(\sigma_j\le \kappa\), where \(\kappa=2-\log_2 3\approx 0.41504\). For \(M=3^n\) we obtain rigorous consequences: an exact characterization of the leading run of ones in terms of \(\{n\log_2 3\}\), whence, by Weyl equidistribution, the asymptotic density of \(n\) whose expansion begins with \(m\) ones equals \(-\log_2(1 - 2^{-m})\), and, by effective bounds for linear forms in logarithms, the leading run has length \(O(\log n)\); and a proof that the trailing run of ones always has length 1 or 2. On the trajectory side, we give an exact decomposition of Collatz iterations, a lemma describing precisely how a block of trailing ones is consumed, and a conditional contraction criterion: if the total 2-adic valuation accumulated along \(M\) odd steps satisfies \(Q_M\ge\beta M\) with \(\beta>\log_2 3\), the trajectory contracts at an explicit exponential rate; conversely, any divergent trajectory must keep \(Q_M\) below \(M\log_2 3+\log_2 X_0\) for every \(M\). The heuristic self-correcting dynamics of the mantissas, together with computations up to \(n=5000\) and valuation statistics over 2000 random trajectories, supports the conjecture that the density of ones in the binary expansion of \(3^n\) tends to \(1/2\); we state this precisely as a conjecture and delimit exactly which steps remain open. We then prove the average-case form of the density statement: for every fixed \(j\ge3\) the \(j\)-th low-order bit of \(3^n\) equals 1 for exactly half of the exponents \(n\) in each period \(2^{j-1}\) (the exceptional positions \(j=0,1,2\) balance to the same mean, so the lowest \(J\) bits carry exactly \(J/2\) ones on average), and the bit at fixed offset \(j\) from the top carries ones with density \(\mu_j\to1/2\) at rate \(2^{-j}\), with \(\mu_1=\kappa\). Thus the outer \(O(\log n)\) digits provably obey the density-\(1/2\) law, and the conjecture is reduced to the middle range of the expansion. We further determine the dispersion and the full asymptotic law of the digit counts in these windows: in the low window the numbers of zeros and of ones each follow an exactly binomial law \(1+\mathrm{Bin}(J-2,\tfrac12)\) over every period—mean \(J/2\), variance \((J-2)/4\), Hoeffding tails, Gaussian local limit—while in the top window the variance equals \(W/4-c_*+O(W2^{-W})\) with \(c_*=0.2399227044\ldots\), and the normalized count of zeros satisfies a central limit theorem. Finally, we identify the natural invariant measure of the gap dynamics itself: in tail coordinates the dynamics is an induced binary shift preserving Lebesgue measure (invariant \(\sigma\)-density \(2^{1-\sigma}\ln2\)), under which the gaps are i.i.d. geometric \(\tfrac12\); for almost every mantissa the density of zeros is \(1/2\) with Gaussian fluctuations of variance \(D/4\), and the density conjecture becomes precisely the assertion that the mantissas of \(3^n\) are typical for this measure—a statement strongly supported by the data (Kolmogorov–Smirnov distance 0.0116 at \(n=2000\)). Moreover, we prove that at every fixed depth \(j\) the interior mantissa \(\sigma_j\) is equidistributed over \(n\), with its law converging to the invariant law at the exponential rate \(2^{-j}/\ln2\); that the gaps decorrelate over \(n\) \((|\operatorname{Cov}(\delta_i,\delta_j)|\le36\cdot2^{-|j-i|})\); and that, consequently, for all \(n\) outside a set of density \(O(1/(\eta^2K))\) the leading segment spanned by the first \(K\) gaps has density of ones within \(\eta\) of \(1/2\). This yields the density conjecture as an asymptotic law on the outer ranges: with windows \(J(n),K(n)\to\infty\) of logarithmic length at both ends of the expansion, the set of \(n\) whose outer segments have ones-density \(\tfrac12+O(\eta)\) has natural density 1. The genuinely open regime is thereby isolated as depths growing linearly with \(n\)—a barrier of the same nature as the classical \(\times2,\times3\) rigidity problems.

Abstract: This work presents a unified spectral theory of light based on the non-trivial zeros of the Riemann zeta function. Light emerges as a manifestation of spectral coherence in a fundamental frequency space, where information is encoded through the alignment and interaction of spectral modes. A Superluminal Coherence Coefficient is introduced to quantify information transfer in spectral space. Numerical simulations using the first one thousand zeta zeros yield a spectral coherence approaching unity. The model provides a mathematical foundation for wave-particle duality, quantum entanglement, and the holographic principle, consistent with the theory of relativity. The framework suggests that the Riemann zeta zeros may constitute a fundamental frequency spectrum encoding information across multiple scales of reality.

Abstract: Graphical discreteness is not a quasi-isometry invariant in general [16, Corollary 1.3]. For every finite thick generalized m-gon Γ with m ≥ 3, however, it is constant on the class of finitely generated groups quasi-isometric to the right-angled Coxeter group W_Γ. Such a group Λ is graphically discrete if and only if Γ is nonflexible, where nonflexible means that no nontrivial graph automorphism fixes a closed star pointwise [20, p. 1]. The proof combines Fuchsian-building quasi-isometric rigidity with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. The standard Coxeter Cayley-graph criterion is also strengthened from discreteness to compact-by-discreteness. Further results derive obstructions from one-vertex doublings, classify compact-by-discreteness for standard chamber graphs of graph products of finite groups, and give applications to projective planes and to right-angled Coxeter groups with Menger curve boundary. The arguments do not determine whether any nonflexible finite thick generalized polygon exists.

Abstract: In this paper, we develop a higher-order theory for the numbers \( y_{9,n}^{(\alpha)}(\lambda;a) \) and the associated polynomials \( y_{9,n}^{(\alpha)}(x,\lambda;a) \), extending several identities and interpolation formulas previously obtained for the case \( \alpha=1 \). Using the corresponding generating functions, we derive explicit representations, recurrence relations, and structural identities, and we establish connections with a broad range of classical and modern special numbers, including the Apostol--Bernoulli, Apostol--Euler, Euler--Frobenius, Fubini, and Stirling numbers. We further construct an interpolation function for the higher-order numbers and prove that its special values at negative integers reproduce these numbers up to an explicit normalization factor. A residue-class decomposition of this interpolation function is then obtained, leading naturally to a generalized hypergeometric Hurwitz--Lerch-type zeta function. Finally, we study several special cases and analytic properties of this zeta-type function, including its reduction to the classical Lerch transcendent and a differential identity.

Abstract: Let pk denote the k-th prime and let θ(x) = Σ_p≤x log p be Chebyshev’s function. For S(k) :=^k−1Σ_i=1(log θ(p_i+1)/θ(p_i)−log p_i/p_i, we show that S(k) telescopes exactly to log θ(p_k)−log log 2− M(p_k−1), where M(x) := Σ_p≤x log p/p, and that consequently lim _k→∞S(k) = −log log 2 − E = 1.699095 . . . , where E ≈ −1.332582 is the constant of Mertens’ first theorem – a consequence of the Prime Number Theorem alone. Using the Vinogradov–Korobov zero-free region, we upgrade this to the rate statement S(k) < 1.699095 . . . + C exp(−c₄(log p_k)^3/5(log log p_k)^−1/5) for absolute, effectively computable but not explicitly evaluated constants C, c₄ > 0, the best known unconditional shape for the convergence of S(k) to its limit. We also show that a natural candidate combination of S(k) with Σ_p<pk log p/p and e^−γ Π_p≤pk p/(p−1) provides strong structural evidence that Nicolas’ 1983 inequality is never violated, thus unconditionally supporting the truth of the Riemann Hypothesis. We also record an explicit bound of Axler for the product Π_p≤x(1 − 1/p), an explicit Dusart-type estimate for the Mertens sum, and Nicolas’ criterion itself, which recasts RH as the statement that X_k := Π_p≤pk p/(p − 1) − e^γ log θ(p_k) is positive for every k; if RH is false, X_k is known to change sign infinitely often.

Abstract: We present novel explicit representations for the core inverse of block anti-triangular matrices involving core invertible product subblocks. Specifically, the corresponding EP property in each of these cases is examined.

Abstract: The notion of a nodal filter has recently gained attention. A filter is called nodal (of type 1) if it is a node in the poset of all filters ordered by inclusion. This seemingly simple order- theoretic condition imposes a strong rigidity on the filter lattice and makes nodal filters natural candidates for serving as reference points or “skeletons” within the filter structure. The present work performs a systematic study of nodal filters in equality algebras. We investigate the relationship between different types of equality algebras (dense, pure, pre-linear, bounded) and various properties of nodal filters, and establish several key characterization theorems. Furthermore, by defining several operators, we show that the set of all nodal filters of type1 is not only a lattice but also a Kleen algebra and even a Stonean equality algebra. The examples provided make the concepts presented more intuitive, and in particular, counterexamples are given that delineate the precise boundaries between these concepts. Our results therefore contribute to a deeper understanding of filters in equality algebras and pave the way for further applications in algebraic logic and related fields.

Abstract: We classify arbitrary, not necessarily additive, maps Φ:R→R satisfying the right bi-skew commuting identity Φ(a)b*+ηbΦ(a)*=aΦ(b)*+ηΦ(b)a*. For unital algebras with a second-kind scalar involution, all scalar parameters η are treated; solutions are symmetric right multipliers except at η=1, when a term from the left annihilator of the commutator ideal may occur. For first-kind involutions and η=±1, non-PI prime rings and prime PI-rings of central degree at least three admit only symmetric right multipliers. Orthogonal and symplectic degree-two exceptions on M2(F) for both signs show that the degree bound is sharp. Using central closure and orthogonal completion, we extend the classification to 2-torsion-free semiprime rings without assuming an identity. At η=1 every solution is additive; at η=−1 nonadditivity may occur only on the commutative central summand.

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Let \(C=ap_k^\#\) and consider symmetric offsets \(\{C-d,C+d\}\) under the sieve of Eratosthenes. Small primes generate a primorial wheel, while each later prime forbids one or two lift residues. The full Chinese remainder theorem (CRT) pattern has positive density, but a fixed center supplies only a translated fragment. We characterize terminal survivors as the disjoint union of a conservative avoiding set and explicit endpoint-prime exceptions, obtaining an exact unequal-prime-pair count and \(R_G(2C)\geq\max\{0,|U(C)|-1\}\). We derive arbitrary-order finite-phase CRT intersection formulas and odd Bonferroni lower bounds. The third-order bound gives \(|U(86)|\geq3\) and \(|U(128)|\geq2\), whereas first-moment, spanning-tree, and complete-block bounds are nonpositive; the latter certificate guarantees a prime-pair representation. We also express the survivor count as a shift-aware cyclic Fourier correlation, factor the pattern transform locally, and obtain computable spectral bounds. For a fixed wheel, \(\log P\sim\sqrt{2ap_k^\#}\), so the complete-block condition is asymptotically too restrictive. Exact-integer computations reproduce all finite cases. The results isolate the finite-window obstruction but prove neither an unconditional Goldbach theorem nor a new infinite prime-pair family.

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Abstract: We investigate the existence and representations of generalized core-EP inverses for a class of anti-triangular matrices over Banach *-algebras. These newly established representations naturally yield innovative formulations for the core and core-EP inverses, providing fresh insights into their algebraic structures.

Abstract: To a Collatz-type map T_{a,b,q}(n) = (an+b)/q^{v_q(an+b)} one may attach two numerical invariants of opposite character: an archimedean one, the mean logarithmic drift delta = log a - E[v] log q, which governs whether orbits grow or shrink; and a non-archimedean one, the similarity dimension dim_S = H(nu_q)/log a of the invariant measure of an associated iterated function system on the a-adic integers. Each reaches its critical value at a particular multiplier a. We prove that these two critical multipliers coincide precisely when q = 2, and that in general they differ by the exact factor q-1: a_drift = (q-1) a_dim. The proof rests on the pointwise identity -log nu_q(m) = m log q - log(q-1), valid for every m >= 1, which is not a normalisation but a coincidence between the information content of a valuation and its archimedean cost. We show that the condition q = 2 is equivalent to three further properties of the family - the triviality of (Z/qZ)^*, the vanishing of an associated free energy, and the map's being everywhere defined on the units - so that the arithmetic distinguishing the classical 3x+1 map is one condition in four guises. Three complements are proved: the full Renyi spectrum of the invariant measure admits a closed form, of which the similarity dimension is the value at t=1; for q >= 3 there is a nonempty band of multipliers on which the map contracts on average while its invariant measure is singular; and the transfer operator, though quasi-compact on Holder spaces, has no eigenvalue other than 1, its entire remaining spectrum being essential spectrum at the contraction rate. Two further results give the family a sharper shape. First, a purity theorem: mu_{a,b,q} is either purely absolutely continuous or purely singular with respect to Haar measure, never a mixture. Its proof is a direct consequence of the uniqueness of the invariant measure, and we know of no route to it from the probabilistic description of the underlying random variable. Second, a complete classification: among all integer pairs (a,q) with a >= 2, q prime and gcd(a,q) = 1, exactly two - namely (3,2) and (2,3) - lie in the supercritical regime a < a_dim, and for every other pair the invariant measure is unconditionally singular. The first of these is the Collatz map. We are explicit that none of this bears on the Collatz conjecture, and we prove why it cannot.

Abstract: In this article, we offer a complete, self-contained, and entirely elementary proof of the mean-square estimate for the Chebyshev function. From this we deduce the convergence of the integral is valid, thus proving the validity of the Riemann hypothesis. The proof primarily employs elementary estimates of the Chebyshev function, the Cauchy-Schwarz inequality, a dyadic decomposition, and Abel summation, in which the argument results of this article are already optimal within the elementary framework and sufficient to derive the convergence of the required integral that it is a suffficient condition for the Riemann hypothesis. In particular, we provide a theoretical complement linking integral convergence, pointwise bounds and analyticity, and concludes that the well-known \(o\)-bound is valid, thereby reconfirming the validity of the Riemann hypothesis. In other words, we give a self-contained elementary proof for the mean-square estimate that \(\displaystyle\int_2^{X} \bigl(\psi(t)-t\bigr)^{2}\,dt = O(X^{2}\log^{2} X),\) where \(\psi(x)\) is the Chebyshev function. From this we deduce that \(\displaystyle\int_{1}^{\infty}\frac{|\psi(x)-x|}{x^{\frac{3}{2}+\varepsilon}}\,dx < \infty\) holds for every \(\varepsilon>0,\) thus concluding the integral \(\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) converges absolutely for every \(\varepsilon>0,\) so that the integral \(\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) converges conditionally for every \(\varepsilon>0,\) whereas the integral converges conditionally \(\iff \text{RH},\) so then the Riemann hypothesis is true. In particular, we conclude that the property of absolute convergence of the integral \(\displaystyle\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) for every \(\varepsilon>0\) is equivalent to the property of conditional convergence of the integral \(\displaystyle\int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx\) for every \(\varepsilon>0,\) either of which is equivalent to the property of the \(o\)-bound: $|\psi(x)-x| = o(x^{\frac{1}{2}+\varepsilon})$ for every \(\varepsilon>0,\) and all of them imply the \(O\)-bound: \(\psi(x)-x= O(x^{\frac{1}{2}+\varepsilon})\) is also valid for every \(\varepsilon>0,\) which is the well-known equivalent form of the Riemann hypothesis, thus reconfirming the validity of the Riemann hypothesis.

Abstract: Let R be a commutative ring. We introduce and systematically study a new class of ideals, called r-sdf-absorbing ideals, which unifies and extends the classical notions of r-ideals and sdf-absorbing ideals. A proper ideal I of R is r-sdf-absorbing if whenever a²-b²∈I for a,b∈R, then either a+b is a zero divisor or a-b∈I. We fully characterize these ideals in some certain types of rings, determine when the ideal (x) in polynomial rings over characteristic 2 rings is r-sdf-absorbing, and classify rings in which every proper ideal is r-sdf-absorbing. We further investigate their transfer under quotients, localizations, products, idealizations, and amalgamations, providing sharp criteria in each case. A wealth of examples and counterexamples illustrates that this novel class lies strictly between several known ideal families and possesses a robust and nuanced structure.

Abstract: We pioneered a new class of generalized inverses, termed generalized weighted EP elements, unifying the concepts of weighted EP and *-DMP elements within Banach *-algebras. This framework allows us to establish a novel characterization of Banach elements expressible as the sum of a weighted EP element and a quasinilpotent. Furthermore, we derive distinct characterizations of weighted *-DMP elements, uncovering previously unknown properties of DMP elements.