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

ALT In the present paper, we investigate geometric properties of a Riemannian submersion for which the total manifold is endowed with hyperbolic and almost hyperbolic Ricci solitons. We establish conditions under which the fibers of the submersion admit hyperbolic and almost hyperbolic Ricci solitons. Further, we derive necessary conditions for the base manifold to be an Einstein and a hyperbolic Ricci soliton. Also, we determine the scalar curvature associated with the fibers and the base manifold. Moreover, when the total space admits an almost hyperbolic Ricci soliton with a torse-forming potential vector field, we obtain conditions ensuring that the extrinsic vertical and horizontal distributions are almost generalized quasi-Einstein. We also obtain necessary conditions for the fibers and base manifold to be an Einstein and almost generalized quasi-Einstein manifold. Finally, examples are included to illustrate the existence of hyperbolic and almost hyperbolic Ricci solitons in Riemann

ALT We study integers of the form N=(d+1)(3d+1)(4d+1). Using Korselt's criterion, we prove that, provided the three factors are prime, N is a Carmichael number if and only if d≡4±od12. We also show that the special case d=2^m produces only 1105 under the prime-factor hypothesis.

ALT The secondary zeta function is defined by Z(s)=∑γ>₀¹⁄γₛ, where tfrac12+iγ runs over the non-trivial zeros of the Riemann zeta function with positive ordinate, and the Riemann hypothesis is assumed throughout. In this article, we derive convergent integral representations of Z(s) at positive even and odd integer argument, expressed through a differential operator acting on the logarithm of the Riemann ξ-function. Define L(x)=log(ξ(x+tfrac12)/ξ(tfrac12)), ℰ=xᵈ⁄dₓ, Dₖ=ℰ∏ⱼ₌₁ᵏ(ℰ-2j). Then, for k≥1, Z(2k)=(-1)ᵏ/2ᵏ(k-1)!∫₀∞Dₖ L(x)/x²ᵏ⁺¹ dx and Z(2k+1)=(-1)ᵏ/π (2k-1)!!∫₀∞Dₖ L(x)/x²ᵏ⁺² dx. In the last part, we write these formulas out explicitly for Z(k), 2≤ k≤ 7, both in terms of L(x) and in terms of ξ(x+tfrac12) for reference. We also derive a limit formula for γ₁, the ordinate of the first nontrivial zero, and demonstrate its convergence numerically.

ALT Generalized Pellet's theorem is extended to analytic matrix-valued functions using Descartes' rule of signs for infinite series. Cauchy's estimate is incorporated to allow numerical computation of the bounds. To address the issue that Pellet's theorem and its generalizations may not always guarantee the existence of bounds, for the case of matrix size one, we employ the Dandelin- Lobachevsky-Graeffe method, which ensures the existence of the bounds for the roots and further improves them as the order of the transform increases. Without the need for any a priori information on the function, these theorems allow us to compute mathematically verified bounds for the roots in a certain region using an interval arithmetic implementation of the function. Numerical examples and an implementation are provided to illustrate the effectiveness of the proposed method.

ALT We present an elementary but powerful method for proving that a two-variable equation has no integer solutions. The method produces quadratic identities which every integer solution must satisfy. Once these identities are established, a standard quadratic reciprocity argument finishes the proof. We illustrate the power of the method by resolving 25 out of 30 two-variable equations of size up to 34 for which the problem of integer solubility remained open. We also resolve the remaining 5 equations by deeper methods, and further determine the integer solubility of all two-variable equations of size up to 35 and of length up to 12, with two exceptions. Before this work, there were 211 equations in this category that resisted the previously known methods, and we proved that 209 of them have no integer solutions, with all proofs formalized in Lean. In addition, we show that our method can be used to solve certain two-variable equations whose integer solution set is non-empty.

ALT A continuous shape parameter enters both a radial density and the normalization of its transform coefficients. This coupling allows changes in shape to be compensated by changes in scale and exponential tilt. We study the coefficients bₖ=eᵃ E[Yᵏ ]/[k!(d)ₖ ] for densities proportional to y⁽d-1) exp(-αyq+e̲tay), with known integer q>=2, d,α>0, and unrestricted real a,e̲ta. For q=2, the first four coefficients determine all parameters globally, whereas three coefficients leave an analytic curve of solutions. We characterize the exact image of the two shape ratios and prove that two alternating corners decide whether an entire coefficient box is realizable and give its exact shape range. For every fixed q, the first q(q+1) coefficients give global identification and an everywhere full-rank map. A finite-difference criterion yields generic identification from 2q+1 coefficients, including every zero-tilt model against the full tilted family. For q=3, an exact rational certificate constructs

ALT A set of m distinct positive integers is called a Diophantine m-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple {a,b,c,d} with a<b<c<d is called regular if d=a+b+c+2abc+2√((ab+1)(ac+1)(bc+1)). We prove that every irregular Diophantine quadruple satisfies c<10³³⁸ and d<10¹⁰²⁴.

ALT The 2026 measles resurgence in Bangladesh motivates a spatial modelling framework that accounts for heterogeneous local transmission, vaccination activity, and interdivisional connectivity. We formulate a multipatch susceptible-vaccinated-infectious-recovered model for the eight administrative divisions of Bangladesh. Division-specific parameters are estimated from reported measles cases and smoothed vaccination data using bounded nonlinear least squares, with uncertainty quantified through 1000 parametric bootstrap replicates. The fitted divisional systems are then connected through compartment-specific movement and cross-patch infection coefficients adjusted for connectivity and the source division's relative reproduction number. The fitted model captures the main division-specific epidemic patterns. Estimated local reproduction numbers range from 1.1386 in Khulna (95% CI:0.5703–1.3834) to 4.3491 in Dhaka (95% CI: 1.4449–5.6307), while Chattogram also exhibits high transmission poten

ALT Let K be an imaginary quadratic field with fundamental discriminant D=-5n, where n>0 is odd, and cyclic ideal class group Cl(K)=langle granglesimeq C₄M. Assume that the ramified prime class above 5 is the unique involution g²M. For the Kronecker eta invariant F(A)=|η(z_A)|²/√a_A, where z_A and a_A arise from a positive definite quadratic form representing the ideal class A, we show that the logarithms of the mirror ratios R_j=F(g²M-j)/F(g^j), 0≤ j<M, are related by an explicit discrete cosine transform to the odd class-character Hecke L(1)-values. A single level-5 generalized-Weber orbit realizes all of these mirrors. Combining Fourier nonvanishing, lower-power Weber class invariants and a ray-class cancellation argument, we prove mathbbQ(R_j)=H^langlecrangle, R_j^{σ²M}=R_j^-1, mathbbQ(R_j+R_j^-1)=H^{langlec,σ²Mrangle}. Here H is the Hilbert class field, c denotes complex conjugation, and σ corresponds to g under the Artin map. Consequently [mathbbQ(R_j):mathbbQ]=4M, [mathbbQ(R_j+R_j^-

ALT For an odd prime p, we study the change in matrix order under lifting from Z/pᵏ Z to Z/p⁽k+1) Z. The first variation of the order defect is an affine translate of a cyclic adjoint transfer on sl₂(Fₚ). We classify the full-order transfer for every q in GL₂(Fₚ): its rank is 3 for scalar elements, 1 for nonscalar semisimple elements, 1 for nonscalar Jordan elements when p = 3, and 0 for nonscalar Jordan elements when p > 3. For a Jordan element q = lambda(I + U), U² = 0 and U!=0, the vanishing first-order transfer for p > 3 admits an exact higher-order refinement. Writing d = (p - 1)/2, the full-order perturbation has depth d when tr(UX)!=0, depth p - 1 when tr(UX) = 0 and X is not in the span of U, and vanishes identically for X in the span of U. For p = 3, among the 27 lifts of a nonscalar order-preserving parent with determinant fixed to +1 or -1, exactly 9 preserve its order and 18 triple it; after the first increase, every further lift triples the order. We also give a scalar obstruc

ALT We give a self-contained derivation of the closed form of the power sums Sₘ(n)=∑ᵢ₌₁ⁿ(a₀+(i-1)d)ᵐ in which the Bernoulli numbers enter as constants of integration rather than as prescribed coefficients. The derivation rests on an elementary fact: a sum whose terms are polynomial in the index is the restriction to the integers of a unique polynomial, which satisfies the defining difference equation identically and may therefore be differentiated and integrated exactly. Differentiating lowers the power by one; integrating back introduces a single constant, equal to the slope Sₘ'(0) of the polynomial at the origin and fixed by the value Sₘ(1)=a₀ᵐ. This yields a closed form for Sₘ(n) in terms of the constants and a recurrence that determines them, obtained without induction on the exponent and without generating functions. With the Bernoulli numbers defined by their classical recurrence, we identify the constants as dᵐBₘ(a₀/d); the progression starting at 0 reproduces that recurrence exactl