General Mathematics
Mathematical material of general interest, topics not covered elsewhere
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For any positive integer N and offset p, the infinite Bessel series Σ(±1)^m J_{Nm+p}(x) is expressed in closed form using incomplete…
· “On one-sided series of Bessel functions of the first kind: sum_(m=0)^(infty) (pm 1)^m J_(Nm+p)(x)”
The Legendre symbol (6/p) = -1 rules out every case except 13 + 6^2 = 7^2.
· “The Diophantine equation p^(x)+6^(y)=z^(2n): a complete solution for half of the primes”
Set n=0 to recover the original Bauer-Ramanujan series and its harmonic-number siblings; every member is an exact 1/pi identity.
A universal tiling of 2x4 bricks connects every piece, and no smaller factor works for all shapes.
· “Fourfold Is Enough: Connected LEGO Models from One Brick Type”
A bound that holds for every prime p and numerator m, forcing Type II solutions to cluster at x/p = 1/m.
The conjectured bound reduces to a correlation statement, and two shortcut proofs are closed.
· “Odd square roots and the sum of element orders on Sn and An”
Scale-invariant contrast formulation avoids oblique projection and makes the spectral limit a well-posed perturbation problem.
· “Construction of Finite Hilbert--P\'olya Matrices from Weil's Explicit Formula”
The Itô correction and its cancellation for conformal martingales come from the same quadratic remainder term.
The pair count's mean and variance now have exact closed forms for every deck geometry and deal length.
· “Analysis of pairs in a generalized deck of playing cards”
As constraints match variables in number, only sporadic small-n solutions exist, suggesting a sharp satisfiability threshold.
For Mersenne triples, the abc radical excess splits exactly into a Wieferich-prime part and an exponent-divisor part, so infinite…
· “Radical defects, Wieferich primes, and the abc conjecture”
Coefficient extraction yields strict sign alternation linked to orthogonal polynomials.
Proofs of Wirsching's Conjectures 1 and 3 reduce the five-step program to a single open implication.
· “Wirsching's Positive-Predecessor-Density Program: Proofs of Conjectures 1 and 3”
From vector additivity alone, the Pythagorean sum $a^2+b^2$ and the quantum rule $|z|^2$ become the only possible magnitudes.
A perturbation of at most ε/2 restores exact convexity; convex sequences of length 2^n−1 embed into perfect binary trees.
First edge theorem derives information geometry as the infinite-rigidity limit of a fiber-bundle statistical space
Splitting the first derivative into equal steps and demanding addition reproduces the standard Gamma formula without integrals.
An explicit 6x6 integer matrix joins the settled orders; order 7 is now the smallest open case.
The familiar sum of cubes equals the square of the first n integers; no other power does.
A new logic keeps valuation, context, and caution separate until a scalar choice is made.
Near Goldbach graphs count prime-pair sums exactly; a single dependence bound would settle the large case.
A segment-counting proof closes the last two-ply gap: no legal game reaches 17,698 half-moves, and 17,697 is attained.
· “The maximum length of a chess game under the 2023 FIDE Laws”
New graph proof gives zero-retuning station switches, with colocated spacecraft needing only 3+c channels.
· “Resilient Spectrum Scheduling for Colocated Space Networks”
Half-integer Bessel, Struve, and incomplete-gamma forms with Gaussian weights and power denominators now have explicit antiderivatives.
Generalizing Zumkeller numbers: for each k≥0, infinitely many odd numbers split their divisors with sum difference k.
It shows degree-2 buffers dominate extremal trees and the near-optimal gap stays bounded as n grows.
Weights proportional to log p extend smoothly to the reals; all others blow up.
Coefficients crossing a threshold create an extra solution, so boundary value problems need extra conditions.
For incomplete comparisons, the priority vector is the mean over all spanning trees, reached by sequential sampling.
Uniform for all n at least m, the log-count series carries explicit arithmetic corrections; sparse cases reveal a 1/4 constant.
A 314-trillion-digit Monte Carlo run finds no detectable bias in pi's decimal expansion.
· “Can π generate itself? A Monte Carlo analysis of 314 trillion digits”
A graded Lie algebra captures their deformations: infinitesimals are 1-cocycles, obstructions are 2-cocycles.
A pointwise explicit formula shows the signed count D_k(x) stays negative for all large x, assuming RH.
Four inequalities bound sums by pairwise differences, replacing monotonicity with the ultrametric max rule.
· “Non-Archimedean Cauchy-Schwarz Angle-Length and Chebyshev Arithmetic Mean Inequalities”
The result settles a 2023 conjecture and opens closed forms for hypergeometric and beta functions.
For people timed 1 to n, the capacity-3 optimum is a quadratic plus a small periodic correction.
A positive Markov identity puts one zero in each gap (n,n+1) and ties the zeros to Euler's constant.
· “The Gregory function and its completed Gregory transform”
The paper's matrix condition writes out Green's functions for delayed, impulsive, and nonlocal problems in one formula.
For primes below 500 at conductor 5, each zero line yields a certified degree-p unramified extension.
· “Character Fourier Spectra of Circular Units and Twisted Bernoulli Class Components”
Previously limited to symmetric means, the formulas also flag the two singular pairs where coefficients stay undetermined.
· “Asymptotic Analysis of Nonsymmetric Gaussian and Archimedean Compound Means”
Functions satisfying the one-way Jensen inequality at 1/3 on R turn out to satisfy it both ways, closing the question.
A power-saving error bound for any ℓ-totient would prove the Riemann hypothesis.
Cut a triangle by any line; the paper classifies which centers always give perspective triangles.
· “Perspective Central Triangles Formed from a Triangle and a Transversal”
A new proof shows that when divisor sums get large enough, a number must be a sum of its own distinct proper divisors.
Sign tests on the variables pick the sharpest product upper bound and ratio lower bound
· “On some inequalities for weighted products and ratios of the sine function”
Burnside's lemma reduces the 3^20 orientation choices to an exact count of rotationally distinct balls.
Complete characterizations and exact identities replace empirical guesses in the online encyclopedia.
First numerics of a candidate zeta operator give a conductor-only ladder; zeros live in the symbol's error term.
A column-partition triple replaces the old reduction cascade, giving the 44 a structural meaning.
· “Counting, Symmetries and Equivalence Classes of Sudoku Grids”
All other even exponents are 2, and that one exponent is at least 23,000,000,000.
· “On odd perfect numbers with exactly one even exponent greater than 2”
No integrals or limits; double factorials emerge, and matching the classical fractional rule fixes the scale at 1/√π.
If true, membership-based fuzzy vector spaces reduce to nested filtrations that keep an order structure membership alone discards.
· “A Parametric Theory of Vector Spaces with Applications to Fuzzy Vector Spaces”
The pair of equations encodes each factorization and yields a second decomposition, powering a complete sieve.
For fixed n, S_N×S_X-orbits of relations follow a degree-2^n−1 polynomial with periodic coefficients; the n=2 formula is explicit.
Fourteen sharp bounds tie the index to degree, Zagreb, Randic, and more, each with exact equality cases.
The integer matrix has 25 distinct entry squares and equal square sums on both diagonals, ending the smallest open case.
F(a,b) is precisely σ_a(F(b,a)); frequencies, return words, and slope follow from one morphism.
· “Duality in Biperiodic Fibonacci Words Substitution Frequencies and Combinatorial Invariants”
Cartan's method gives exact point-transformation criteria for two ODEs to reach the eight-symmetry linear form.
· “Linearization Problem for a System of Two Second-Order ODEs via Cartan's Method: Branch I”
A single summation-by-parts trick unifies and extends three-, four-, and five-factor identities.
Embedding the system as a doubled block matrix produces generalized-inverse least-squares solutions.
· “Direct Methods for Singular Fuzzy Linear Systems Using Generalized Inverses and Its Applications”
All dependence on the cutting rule collapses into C(c•), yielding series for π and log 2.
· “Area of the Limit Curve for Symmetric Nonstationary Corner-Cutting Schemes”
A cubic-Gauss-sum recursion evaluates the Hardy function in (t/eps)^0.25–0.3 (log t)^2 operations for t up to 10^35.
Balanced gamma quotients: derivative sums become exact rational MZV combinations, certified without numerics.
A single windowed square wave, evaluated twice, yields tanh, ζ(2), ζ(2k), and Dirichlet L-values.
· “From a Windowed Square Wave to the Hyperbolic Tangent, Even Zeta Values, and Dirichlet L-Values”
Weighted gamma-quotient sums reduce to zeta and log-sine values; Sun's conjecture follows at a special value.
· “Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations”
Ensemble variance flags branching junctions and bad pixels at 90% precision, far cheaper than FEM.
A recurrence with coefficient q^{n-2} yields generating functions, negative indices, and a matrix Cassini identity
Lengths, not ratios, yield the identities and the law of cosines from a single diagram
· “OFFER (One-Figure-Facilitates-Every-Relationship)for Trigonometric Functions”
Hyers–Ulam stability recovers monotonicity, subadditivity or convexity without large weight changes
· “Approximate monotonicity, subadditivity, and convexity in weighted topologies”
A number is correct only if it appears that many times; the resulting tests become ordinary partitions.
Classical uniqueness of Haar measure is extended to fuzzy analogues of Lie groups
At squarefree level one finite Fourier transform diagonalizes cusp orders, Fricke, Atkin–Lehner and Eisenstein constants at once
· “Boolean Walsh Eta Units and Eisenstein Bases For Squarefree Levels”
A six-cube identity plus the Pell unit 8+√65 forces each error to be exactly +1 or -1, and points to a K3 surface.
· “A Ramanujan Pell Elliptic K3 Surface and Primitive Five-Cube Near Misses”
Even modest puzzles have far more assembly routes that go through separate intermediate pieces
· “Counting Connected and Disconnected Ways to Assemble a Jigsaw Puzzle”
An explicit formula exposes the mechanism: x^{-1/2} decay forces every nontrivial zero onto the critical line.
That settles the shortest open Diophantine equation and finishes all length < 9 cases.
Eigenvalue sum and product are fixed by the trace and determinant of two coefficient matrices
Quadratic generators never give a bi-UFS; the only positive rational bi-HFS is N0.
· “The Bi-UF Positive Conjecture for quadratic monogenic semirings and related progress”
Five minimal-prime branches closed by valuation balance and machine-checked certificates; only q=3 and q≥19 remain open
· “Certified Minimal-Prime Branch Closures for Odd Perfect Numbers”
Six inequalities force every level set of a multigroup to be a classical subgroup