ALT In this work we present an extension of the classical trigonometric functions which include, amongst others, the Lindqvist p-trigonometric functions. Additionally, we show their relevance in the study of certain versions of the nonlinear harmonic oscillator equation. Furthermore, we study their more important mathematical properties and obtain their corresponding Pythagorean-like relations. We also define a new extension of the Beta and Ateb sine functions using these new χ-trigonometric functions. Indeed, these new Beta and Ateb functions are involved in the closed form expressions obtained for the solutions of the nonlinear oscillator under certain constraints.

ALT We define packaged structures through carriers, internal units, incidence, candidate spaces, and selected families. A realized packaging system additionally retains a context and formation datum related to its output by a possibly partial, multivalued realization relation. We establish comparison-relative recoverability and non-collapse criteria and an intrinsic refinement-completion theory. For atlas specializations, exact extension becomes a specialization-relative packaging completion problem. Under additional recovery and elimination axioms, we establish general globalization sufficiency criteria and sharp classifications for three classes of compatible finite atlases on a fixed cover ℋ. A geometric obstruction rank ν(ℋ) yields thresholds 2k+1<ν(ℋ) for empty-preserving closures of local Carathéodory number at most k≥1; d<ν(ℋ) for complete functional-dependency theories of local minimum Armstrong size at most d≥3; and 3<ν(ℋ) for transitive relations, including equivalence relations.

ALT In this paper, we establish several identities involving Bernoulli numbers, harmonic numbers, and Stirling numbers. Our approach extends the method developed by Kargin and Mutluer.

ALT We study two families of coefficients associated with powers of the tangent function and with products of the secant and tangent functions, and develop a unified treatment of the corresponding higher-order tangent and secant numbers. The first part establishes a common identity for even derivatives of tanh x and odd derivatives of sechx. Combined with integral formulae for these derivatives, this yields finite power-integral representations for the normalized values ζ(2n+1) and β(2n) and naturally introduces two polynomial families Pₙ and Qₙ. We determine several arithmetic and structural properties of these polynomials. In particular, we identify the exact content of Pₙ, prove that Qₙ is primitive, establish strict log-concavity and unimodality of their coefficient sequences, and describe their real zeros. For even indices each family has a unique negative real zero, and the two sequences of real zeros have the same limit. We also derive differential recurrences and generating functio

ALT The perimeter of an ellipse is expressed exactly through the complete elliptic integral of the second kind, motivating accurate closed-form approximations. We introduce ELIP (Ellipse Linear Interpolated Perimeter), a family built by correcting the straight line joining the two exact normalized endpoint values. Two branches are developed: logarithmic-power corrections with optional deformation and exponential modulators, and rational-logarithmic corrections incorporating the exact asymptotic behavior of highly elongated ellipses. The framework distinguishes coefficients fixed analytically by Maclaurin matching from parameters optimized numerically by minimax fitting. Compact rationalized formulas achieve maximum relative errors of approximately 6.84, 3.57, 0.904, and 1.596 ppm. An ELIP R5,2 + 3 log formula with three fitted degrees of freedom has a certified maximum relative error below 0.001623 ppm. For ELIP R5,3 + 3 log, all rational-core coefficients are determined by exact Maclaurin

ALT The large number of calculus formulas makes learning calculus heavily memory-dependent and can scare away many students. Based on Issac Newton's idea of prediction, this paper shows that calculus students need to master only one critical formula: his proposition that, given boundary data, one can predict the interior and vice versa. We can treat other formulas as consequences of the formula for this proposition. In 1-, 2-, and 3-dimensional spaces, this formula corresponds to the fundamental theorem of calculus, Green's theorem, and the divergence theorem. This approach is summarized as OFFER (one-formula-facilitates-every-relationship) for calculus education. This paper demonstrates the method and advantages of the OFFER pedagogy. The OFFER approach can accelerate the pace of calculus teaching, provide deeper insight into calculus, inspire questions from students, and naturally facilitate the inclusion of real-world data, artificial intelligence (AI), and computer coding in the calcul

ALT In this paper, we discuss two related Diophantine tuple problems involving star numbers Sₙ=6n(n-1)+1. First, we investigate classical D(k)-tuples whose elements are star numbers. We construct infinitely many D(k)-pairs for several positive and negative admissible values of k, and we prove several nonexistence results for D(k)-triples and quadruples. Second, we introduce a modified notion of a star number D(k)-tuple, requiring aᵢaⱼ+k itself to be a star number. We construct several infinite families of D(1)-triples, and study their extensions to quadruples. In particular, we prove that {4,108,165} has the unique extension {4,108,165,360} to a star number D(1)-quadruple. The main tools used in this paper are generalized Pell equations, Nagell's theorem, lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethő version of the Baker-Davenport reduction method in diophantine approximation.

ALT We develop a common two-by-two matrix framework for the four classical Hopf fibrations associated with the real, complex, quaternionic and octonionic numbers. In all four cases the Hopf map is the image of the north pole of a sphere of trace-free Hermitian matrices under conjugation by a bi-unitary matrix, and it depends only on the first column of that matrix, without any bracketing convention, even for the non-associative octonions. Our main results concern general octonionic bi-unitary matrices, beyond the compatible matrices, with entries in a single complex subalgebra, on which earlier work on octonionic Lorentz transformations relies. We prove that every octonionic bi-unitary matrix is in fact quaternionic: its entries lie in a quaternion subalgebra depending on the matrix. We show that requiring Hermitian output leaves exactly four inequivalent ways of bracketing the octonionic triple products, among them an interlaced rule called RLLR/LRRL, and that each of them splits into an

ALT For the Lucas numbers L₀=2, L₁=1, and L_n=L_n-1+L_n-2, we obtain \sum_{substackn,m≥1\n,m)=1}1/L₂nL₂mL₂(n+m)=frac145. This follows from the absolutely convergent identity \sum_{substackn,m≥1\n,m)=1}frac{q^n+m} {(1+q^n)(1+q^m)(1+q^n+m)}=fracq²(1-q²)², |q|<1. We give a self-contained proof by local cancellation on the Calkin–Wilf tree, as a specialization of Hata's Farey-interval telescoping principle. Specializations yield a hyperbolic-cosine identity \sum_{substackn,m≥1\n,m)=1} frac1cosh(nx)cosh(mx)cosh((n+m)x)=frac2sinh²(2x), x>0 and weighted sums for canonical second-order Lucas recurrences. Decomposing arbitrary positive pairs according to their greatest common divisor gives an unrestricted sum expressed by a Lambert series and the quasimodular Eisenstein series E₂. Conversely, Möbius inversion recovers the primitive identity from a double Lambert-series result of Cui, Kumar, and Singh.

ALT Let Qp,q(t)∈ℤ[t] be Sharipov's even monic degree-10 second cuboid polynomial, with coprime positive integers p≠ q. We prove that Qp,q(t) is irreducible over ℚ, and hence over ℤ. More generally, the normalized associated quintic Pₛ(x) and its composition Pₛ(t²) are irreducible over ℚ for every s∈ℚ∖{0,±1}. The proof reduces linear and quadratic factors to rational points and horizontal conjugate pairs on one fixed genus-2 curve. A certified rank upper bound for its Jacobian, together with an exact divisor calculation, gives a rank-one annihilating differential at the prime 3. For the quadratic-factor case, the symmetric coordinates of a horizontal pair lie on an explicit double cover of an elliptic quartic. All rational affine points on this model are 3-adically integral and reduce to six smooth residue classes. The pulled-back differential is a unit in each class, proving that six displayed points form the complete list. Their coordinates exclude every admissible parameter. A negative r