Mathematics Papers

ALT We consider special cases of the discontinuous Weber–Schafheitlin integral in which the orders μ and ν of the Bessel functions differ by an integer and the exponent λ equals 0 or ±1, with μ-ν+λ odd. Explicit closed-form results are obtained in terms of Jacobi polynomials.

ALT Let p(n) be the minimum cardinality of a finite planar point set in which each point has exactly n unit-distance neighbours. Harborth proved p(5)=18 in work presented in 1985 and published in 1987, using a detailed case analysis. We revisit the proof using convex-hull arguments, incidence counts, and elementary properties of intersections of unit circles. These tools give an alternative treatment of configurations with 14 and 16 points and organize the final case distinction by a common vector-sum structure. We also treat a configuration not explicitly covered by the case analysis in the online PDF copy of Harborth's paper available to us, and give an explicit justification of the six-point classification used in the argument. No result subsequent to Harborth's proof is required.

ALT We show that (1/(g(x))), with g(x)=x-1/xln x, is a complete Bernstein function and give its Lévy measure explicitly. For h(x)=(x²-1)ln(x+1)/x²ln x, we prove that h is not a Stieltjes function, so (1/h) is not a complete Bernstein function, but (1/h) is a Bernstein function. The latter rests on the positivity of a function built from the convolution of the exponential integrals E₁ and E₂. We also give a related completely monotonic function that is not a Stieltjes function.

ALT Power series expansions in the polar distance about the center are derived for the normal direction and signed distance from an ellipse to points near its center. After extracting constant terms and common factors, the series coefficients are polynomials in the axis ratio and the absolute value of the sine of the polar angle, with rational coefficients. The region of convergence is shown to be the region bounded by the ellipse evolute. The expansions complement previously derived expansions valid for points far from the center.

ALT In this note, we address a traditional pedagogical asymmetry between algorithmic differentiation and heuristic integration by introducing an extra parametric-order differential framework. By treating the order of differentiation as a parametric axis (discrete or continuous), classical integration is reframed as a local differential evaluation at the non-positive boundary level. This analytical shift alleviates some traditional routines of integration by parts, by replacing the quadrature of a product of elementary functions with a direct differential series that either truncates into a finite algebraic sum or resolves through cyclic symmetries. This parametric-order approach also bridges the gap with fractional calculus, showing that non-integer derivatives emerge as a seamless extension of the same operational dimension. By reducing global quadrature to local differentiation, this note offers an efficient, conceptual tool for undergraduate analysis.

ALT We prove that the largest absolute determinant of a 15×15 matrix with entries in {1,ω,ω²}, where ω³=1 and ω≠1, is 120932352√(19). A matrix attaining this value was already known; the issue is maximality. The proof follows the Gram-matrix search method used in the real maximal-determinant problem. A sharp determinant bound in terms of the off-diagonal Gram energy, combined with Eisenstein congruences and a published classification of difference matrices, reduces a strict improvement to 23 possible energies. Componentwise Schur bounds and exact support catalogues then leave a small number of cases, excluded by orthogonality, rank, and spectral-intertwining conditions. We present the finite tests in a common framework and supply their replay programs. For the displayed extremizer we give a dephased representative, its Gram matrix and spectrum, and an exact computation of its monomial automorphism group, which has order 1008. The proof determines the value of the maximum, but does not clas

ALT We define a binary operation T₁⋆ T₂ on the fuzzy unit interval \I by extending the natural action of T₁ and T₂ on step functions. When both T₁ and T₂ are left-continuous, this operation coincides with the tensor product of the corresponding quantales and therefore yields a t-norm on \I. Without left-continuity, the situation becomes more subtle. We prove that if at least one of the two t-norms is continuous, then T₁⋆ T₂ is still a t-norm on \I. We also show by counterexample that continuity cannot simply be dropped. The operation DN fails to be associative. On the other hand, T_D⋆ T_D is a t-norm despite T_D being neither continuous nor left-continuous, indicating that continuity is not a necessary condition.

ALT We study the cyclic inequality ∑ᵢ₌₁ⁿ xᵢ ∑ᵢ₌₁ⁿ xᵢᵃ xᵢ₊₁ᵇ ≤ (∑ᵢ₌₁ⁿ xᵢ⁽¹⁺ᵃ⁺ᵇ/2))², where x₁,…,xₙ≥ 0 and indices are taken modulo n. Let N(a,b) be the largest n for which the inequality holds for all nonnegative xᵢ. We show that N(a,b) is bounded above by an explicit quantity N⁽a,b) obtained from the second variation at the constant vector, and we conjecture that N(a,b)=N⁽a,b). For a=b=1 we prove that the inequality holds if and only if n≤ 8, which settles the case left open in the original version of this paper. For n=4 the inequality holds whenever 1,a,b are the sides of a triangle, with an explicit remainder term. We show that N is monotone along rays emanating from the line a+b=1; this corrects and extends an earlier monotonicity statement. Finally, any n nonnegative numbers can be arranged so that the inequality holds; for n≠ 6 this is achieved by one of n-1 prescribed arrangements, which yields a volume bound for every such n.

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.