Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
We give an explicit curvature argument for three-point positivity of weighted products of the Riemann theta density. For every alpha at least 5/8 and every nonnegative Borel center measure with a finite positive integral of Phi^(2alpha), the averaged product Phi(s+t/2)^alpha Phi(s-t/2)^alpha has positive definite three-point translation matrices at distinct real positions. Repeated positions give positive semidefinite matrices. A general curvature criterion combines with positive-Laplace estimates for the complete theta-series tail; all infinite-tail bounds and rational interval estimates are explicit. Positive definiteness of arbitrary order is not established.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
For every nonnegative integer radius, we prove joint log-concavity of the raw three-dimensional truncated simplex lattice sum at arbitrary real endpoints when each independent real power is at least two. The coordinatewise threshold two is sharp. Exhaustive cell coverage and exact positive-coefficient polynomial identities certify the quadratic endpoint over unbounded parameter orthants, rather than numerical samples or finite parameter cutoffs. In every dimension, the same endpoint holds with up to two unrestricted occupancies and arbitrarily many binary matroid companions with activities in [0,1]. Degree-elevated homogenization, classical complete-log-concavity and compensated composition supply that extension. The unrestricted quadratic simplex problem in dimensions four and higher remains open.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis · Probability and Statistics
- Paper type
- Research Article
- Manuscript stage
- Preprint
For a nonnegative coordinatewise nondecreasing integrable weight on a finite d-dimensional box, joint Gamma-normalized mixed moments are log-concave when every exponent is at least d/4, for d at least two. No input log-concavity, independence or positive association is required. Finite product grids with ordered log-concave coordinate values and index-increasing weights admit the sufficient bound d/4+4. Continuous densities and finite-grid examples show that both thresholds have optimal leading constant one quarter; additive constants remain open. Conditional logarithmic-deficit estimates, a signed-block covariance argument and trigamma bounds also give spatial product-power smoothing for arbitrary decreasing inputs at powers d/4+4 in the continuous case and d/4+7 for arrays. A classical matroid input yields a dimension-free binary endpoint, while unweighted hereditary binary supports require dimension-linear powers.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis · Algebra
- Paper type
- Research Article
- Manuscript stage
- Preprint
Let h be a real polynomial with h(0)>0 whose zeros lie in the closed sector of half-angle pi/4 around the negative real axis. For every nonnegative real step and geometric bias at least one, its geometrically weighted integer prefix sums admit a finite concave logarithmic extension to the nonnegative quadrant. This implies weighted logarithmic Jensen inequalities for arbitrary probability laws with finite first moments and integral mean prefix length. The universal sector angle and bias range are sharp, including integral-height counterexamples. Joint log-concavity of a two-variable truncated-square lattice sum follows from complete geometry coverage and exact rational certificates; classical Prekopa marginalization removes the degree bound. Applications cover split projective-line bundles over projective space in every dimension. The extension agrees with integer-prefix rows rather than asserting ordinary polynomial continuation in the prefix variable.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
For measurable nonincreasing relative widths w, this paper identifies a square-integral threshold for exact eventual subadditivity on the integer corridor n ≤ m ≤ n + nw(log n). Divergence of ∫w² forces normalized-slope convergence for nondecreasing nonnegative linearly bounded sequences, without continuity or an adjacent-pair assumption. If adjacent integer pairs are eventually admissible, a prefix-envelope and dyadic-record argument removes the monotonicity, sign and growth hypotheses: every real sequence obeying the corridor inequality has a normalized limit in ℝ ∪ {−∞}. In particular, n/√log n is a sufficient sublinear corridor for arbitrary real sequences. For a finite square integral, a shifted-tail correction produces strictly increasing integer submultiplicative and supermultiplicative counterexamples with logarithmic growth endpoints 1.99 and 2.01. The resulting dichotomy is exact in the monotone counting class and, with the adjacent-pair condition, for arbitrary real sequences. Divergent widths with missing adjacent pairs remain unresolved in the unrestricted real-sequence setting.