Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory · Number Theory and Special Functions · Linear Algebra and Matrix Inequalities
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study the contrast between finite-order positivity and all-order failure for averaged Riemann theta products. Our main new theorem proves that, at the exact upper-center cutoff A=23/2, each polynomial parity through degree N has negative inertia at least 3 log(N)/(8 pi)-O(log log N). A fully explicit conservative bound constructs common negative subspaces, with every theta term and approximation error retained. Consequently a correction making the full form positive semidefinite must have rank at least 3 log(N)/(4 pi)-O(log log N), even if the correction changes with N or couples parities. The proof combines classical sinc concentration with gamma-reference cancellation and uniform Meixner-Pollaczek projection. We consolidate, with explicit attribution, previously established positivity through degree eight, degree-dependent terminal positivity, exact rational negative witnesses of degrees 4096 and 4097, and eventual witness degree at most ceil(A^(31/4)) in both parities. A classical lognormal counterexample explains why finite moments and a negative density region alone are insufficient. The physical all-order problem remains open; no zero theorem or optimal inertia asymptotic is claimed.
Edition 2
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory · Differential Equations
- Paper type
- Research Article
- Manuscript stage
- Preprint
We construct bounded connected nonconvex planar pentagons for which the affine 2-Laplacian has a strictly positive Dirichlet eigenfunction at an eigenvalue strictly above its first variational eigenvalue. The same domain therefore supports positive affine eigenfunctions at two distinct values. The construction gives a negative answer, for n = p = 2, to the general-domain positive-eigenfunction question of Haddad, Jiménez and Montenegro. The mechanism is an affine-isotropic saddle formed by pinching a pentagon into two right isosceles triangles. A symmetry-preserving diagonal normalization has positive curvature, while a shear couples the ground state to a nearly coincident odd mode. We prove the sharp limit (λ₂ − λ₁)∂²λ₁/∂q² → −32768/81 at the isotropic positions. An area-preserving apex slide has the corresponding limit −50π⁴. The saddles persist under small vertex perturbations, including examples with nonparallel convex-hull sides. Consequences include distinct isotropic positions on one linear orbit and failure of positive-definite-deformation log-convexity outside convexity. We further preserve the saddle under smooth inner exhaustion and use a symmetry-balanced product construction to obtain positive affine 2-eigenfunctions at two distinct eigenvalues on smooth connected nonconvex domains in every dimension n ≥ 2.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
Odd logarithmic twisting modes in a fixed polar-strip tube are invisible to the leading Weyl profile but detectable in the next heat profile. This paper gives a quantitative continuation of that complete-spectrum result. An estimate on the actual curved normal cap yields a heat remainder O(T¹⁷⁄⁶) and normalized bias O(T⁻¹⁄⁶). A complete finite spectral prefix, certified uncertainties, a counting envelope and a certified complex multiplier enclosure give a deterministic recovery disk. With exact inputs, the fixed-mode cutoff rate is O((log Λ/Λ)¹⁄¹²); the relative-noise balance is O(σ¹⁄⁷) before heat-multiplier inversion. A long-straight-core construction shows why a uniform geometric prior is necessary, and two-sided multiplier bounds quantify exponential conditioning. The geometric remainder constants are not fully evaluated numerically in this manuscript. The concrete fundamental-mode interval execution is valid but noninformative; practical numerical reconstruction is not claimed.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
For each fixed parameter 0 < λ ≤ 64/27, this paper determines an anisotropic corner limit of the inverse Laplace distribution of a stable Vámos quartic raised to the power −1/2. The limit is an explicit positive density with arcsine slope sections. Convergence holds jointly in all scaled variables, including the variable where the exact partial inverse has a logarithmic singularity. The proof uses a modified Bessel partial inverse, differentiation to remove logarithmic counterterms and quantitative estimates for inversion on growing frequency regions. It also determines the first correction in the same joint distribution topology. Positivity of the full inverse distribution is not assumed. The result does not establish positivity at a positive scale or justify restriction of the converging distributions to a graph.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
The inverse square root of the specialized four-variable Vámos quartic is shown not to be completely monotone. More generally, the obstruction holds for every interior stable member hλ = 4e₁e₃ + λa²b² with 0 < λ < 64/27. The proof computes a regular correction to a singular positive leading inverse, establishes distributional Laplace-inversion convergence through uniform complex estimates, and identifies a negative density outside the leading support. A smooth-section argument transfers this sign to the unrestricted inverse. For the original quartic an explicit interval and uniform negative bound are obtained. The result gives a qualitative strict exponent gap above 1/2, uniform on compact interior parameter sets. At λ = 64/27 the first correction is nonnegative outside the leading support; complete monotonicity at that endpoint remains open.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
A fixed rapidly twisted Euclidean tube has an infinite lattice of logarithmic inverse-speed harmonics invisible to its leading Weyl profile. This paper proves that every odd harmonic on that lattice is detected by the next heat profile. The connected-versus-cut relative heat trace of the step aperture is represented as a Laplace transform of a positive measure through a finite positive-root matrix lemma, threshold atoms and controlled aperture limits. A noncommuting trace estimate gives an analytic mass bound, and Euler-product domination yields uniform noncancellation at every original odd frequency. Combined with the preceding actual-tube heat expansion, this recovers each admissible pure amplitude and gives injectivity on admissible finite odd Fourier profiles. The argument requires no sampled aperture calculation or floating-point PDE enclosure. Frequency-dependent lower bounds do not establish recovery stable under finite spectral data.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
A complex Mellin invariant of a fixed connected step junction is certified on a ray with negative real part. Positive-axis exponential flatness is propagated through an exterior sector using a classical Phragmén–Lindelöf multiplier with explicit constants. A left-sector contour suppresses reciprocal-Gamma conditioning and detects the original fifth inverse-speed harmonic, which is invisible to the leading Weyl profile of the same Euclidean tube. Complete residual bounds retain the noncommuting aperture inverse and infinite input tails. The certificate includes 67 complex nodes, the endpoint calculation, infinite quadrature tails and stated arithmetic errors. The junction Mellin modulus is below 3.198 × 10⁻⁹, while cancellation requires a modulus above 1.18 × 10⁻⁸; the fifth heat multiplier therefore has modulus greater than 2.16 × 10⁻⁷. Higher modes and finite-data stability are outside this certificate.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
The leading Weyl profile of a rapidly twisted tube with a fixed connected step section annihilates every odd logarithmic harmonic of inverse twist speed. A preceding heat-trace result detects the fundamental harmonic. This paper gives a computer-assisted resolution for the original third harmonic without changing the section or adding a lower-order perturbation. A half-strip resolvent formula and complete residual estimates enclose a complex Mellin invariant of the connected junction. Sectorial trace bounds reduce the certified quadrature to 27 complex nodes while retaining infinite tails, arithmetic error, node conversion and aliasing. The resulting Mellin modulus is below 5.653 × 10⁻⁶, whereas cancellation requires a modulus above 1.13 × 10⁻⁵; the third heat coefficient therefore has modulus greater than 5.08 × 10⁻⁵. The source certificate states its arithmetic assumptions and regenerates its inputs. Higher odd modes and finite-spectrum stability are outside this paper’s scope.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
A fixed connected polar section can hide every odd logarithmic harmonic of inverse twisting speed from the leading Weyl profile. We show that its original pure cosine perturbation is nevertheless audible in the next heat-trace term. The second coefficient contains the relative heat trace of a connected step strip; replacing the exposed radial face by an independent angular channel gives the wrong model. We derive a uniform momentum-integrated radial expansion and use an exact angular clock, a small real shear and weighted Dirichlet–Neumann bracketing to obtain an actual two-term heat expansion for the Euclidean tube. A positive corner-loop remainder is nonincreasing in heat time. Its Stieltjes representation controls the complex Mellin transform sharply enough to prove noncancellation at the original fundamental frequency. Thus a parameter invisible in the leading Weyl law is determined by the lower-order heat profile, without adding any lower-homogeneity perturbation. Higher odd frequencies and a two-term unsmoothed counting law are left open.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We determine the leading Dirichlet eigenvalue count for rapidly twisted Euclidean tubes whose inverse twisting speed is a power times a smooth positive logarithmically periodic function. The actual operator count is an explicit periodic profile, without a regular-variation hypothesis. The Fourier multiplier factors into a nonvanishing universal factor and a complex angular-width moment. This gives a necessary and sufficient criterion for recovery of the inverse-speed modulation from the complete leading profile. A fixed connected two-width cross-section hides every odd logarithmic harmonic. In particular, smooth strictly increasing non-regularly-varying speeds on this same geometry have the pure-power law N(Λ) = Λ²/(6π) + o(Λ²) for two identical ends. The ambiguity is infinite dimensional, whereas visible modes admit conditional Sobolev recovery. For constant-step sections and inverse speeds with derivative bounded above and away from zero, we also prove an O(Λ⁷⁄⁴) counting remainder. A mixed-homogeneity inverse speed then has a genuine second periodic spectral profile: every mode of its lower-order modulation is visible on the example geometry. Whether lower terms reveal the original pure blind modulation remains open, as does equality of complete spectra.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We determine exact eigenvalue-counting asymptotics for infinite-volume Dirichlet tubes confined by rotation. Continuous regularly varying reference speeds of positive index require neither monotonicity nor derivative bounds; a second class includes smooth rapidly varying speeds such as exponential and double-exponential growth. The two axial ends may have different types. Finite-band cross-sections allow continuous angular endpoints, pinching widths, multiple components and finitely many topology transitions. The law remains unchanged under angle perturbations with finite limits, without control of their derivatives, and therefore permits infinitely many actual speed reversals and zeros. In the positive-index regime the coefficient is a nonlinear angular-width moment; in the rapid regime it depends only on area. We give full box and localization proofs, logarithmic trace thresholds, width-distribution recovery from a family of coefficients and a finite-data obstruction.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We determine the leading eigenvalue-counting asymptotic for a class of infinite-volume Dirichlet tubes confined by diverging twisting. The planar cross-section is a compact annular polar strip with positive C¹ angular width α(r) < 2π, and the twisting speed is c(1 + s²)^(p/2), where c, p > 0. We prove N(Λ) ∼ CΛ^(3/2 + 1/(2p)) and compute C explicitly in terms of ζ(1 + 1/p) and ∫ rα(r)^(1 + 1/p) dr. The proof retains the angular Dirichlet modes while treating the radial and longitudinal variables semiclassically. A uniform Dirichlet-box comparison obtained from quasiperiodic phase averaging and IMS localization controls the artificial longitudinal boundaries. A separate tail estimate and absolute-coefficient freezing justify the global limit. The result gives a concrete-class answer to the Weyl-asymptotics question for tubes with asymptotically diverging twisting.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study nonnegative initial temperatures whose Dirichlet heat evolution has one common signed outward normal derivative on both boundary components of a fixed spherical shell. In every dimension at least two, the admissible observation times form an interval (0, τ) with a finite, unattained endpoint characterized by a boundary heat-kernel ratio. Finite schedules satisfy an exact interior convex-hull criterion, and a datum-independent terminal interval contains at most one observation.
With unit ground-state projection, every weighted minimum-Lq preparation, 1 < q < ∞, is unique at each admissible time and is determined by a monotone real scalar moment equation. We obtain sharp cost constants throughout the fixed norm scale, including separate endpoint arguments. For every finite strictly convex power, quantitative ground-probability stability prevents scaled second-moment loss, and physical near-minimizers have a universal strong rescaled profile with angular symmetry. Exactly feasible counterexamples exclude universal higher-moment convergence.
At the L1 endpoint the ground-weighted probabilities concentrate at √6 in quadratic Wasserstein distance. At the cap endpoint the preparations approach a filled layer in every finite Lr, while an exact counterexample excludes general strong L∞ convergence. All parameters are fixed in the stated asymptotic limits; classical positivity, convex-moment and coupling methods are explicitly credited.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
For every dimension d >= 2 and every measurable spatial weight w >= 1 essentially bounded on bounded sets, we construct a connected tower of enlarging cubes whose ordinary Dirichlet Laplacian has a complete orthonormal eigenbasis and essential spectrum [0,infinity). Given any finite nondecreasing r(E) >= 1 tending to infinity, the tower can be chosen so that weighted local-source evolution below energy E is bounded uniformly in time by C_K min(r(E),sqrt(E+1)). More generally, any prescribed Borel spectral filter 0 < h <= 1 uniformly vanishing at high energy admits a uniform weighted local-source bound. Local Neumann approximation targets and a summable spectral-projection budget supply the uniform source estimate. With a rational initial room, computable local weight upper bounds and a supplied computable filter-tail modulus, the geometry and its finite spectral certificates can be produced by a procedure terminating at every stage. Crack-aware primal and flux approximation certify whole-resolvent compressions and strict whole-cluster spectral tests. Explicit sufficient aperture bounds are logarithmic in the plane and linear in higher dimensions. For uniformly diverging spatial weights, a uniformly bounded all-energy factor is impossible; the geometry depends on the prescribed rate.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study the intrinsic Wasserstein metric on affine Fourier-density models on the unit circle. For three distinct frequencies, with one arbitrary fixed phase per frequency, every sectional curvature at the uniform density is strictly negative unless the frequencies have ratio 1:2:3. In that exceptional family we determine the exact attained curvature range and a phase-dependent sign criterion: the same frequency magnitudes can have mixed curvature or strictly negative curvature solely through relative phase. For the aligned four-frequency family {m, 2m, n, n+m}, n > 5m, we determine the exact attained maximum sectional curvature at every frequency ratio. Two sharp algebraic thresholds separate mixed signs from strict negativity and then mark saturation of the maximum at −18π²m². Finally, exact rational Fourier integration with a bounded infinite reciprocal-series remainder certifies nonuniform closed parameter cubes: the {m, 2m, 3m} model has both signs throughout [−.01, .01]³, whereas every sectional curvature of the {m, 2m, 4m} model is less than −.98π²m² there. The phase and four-frequency classifications are analytic; the explicit neighborhood bounds are computer-assisted and accompanied by reproducible exact-arithmetic certificates.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
For every dimension d ≥ 2, we construct a connected quasi-conical open set whose Dirichlet Laplacian has a complete orthonormal basis of eigenfunctions and spectrum [0, ∞). The domain is a tower of adjacent enlarging cubes joined through positive windows tending to zero. Cube sizes and windows are chosen together. The construction gives an affirmative existence answer to the singular-continuous-spectrum question raised for this geometry by Krejčiřík and Lotoreichik and repeated in a recent spectral-geometry survey. At each finite attachment we continue a nested family of whole spectral clusters, including their multiplicities. A summable projection budget and a refreshed dense family of test vectors make the limiting finite-dimensional reducing subspaces complete. We also obtain a room-index L² tail bound and retain the possibility of placing one prescribed positive embedded eigenvalue by dilation.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
A sharp integrated concentration comparison is proved for heat flow on finite compact connected metric graphs with a nonempty Dirichlet vertex set and standard Kirchhoff conditions elsewhere. The comparison with the mixed interval of equal total length applies to every nonnegative square-integrable initial datum. For initial datum one, it yields an all-times heat-content bound and characterizes equality at a single positive time. An exact positive-kernel deficit representation and a separate convolution formula for an asymmetric three-edge pitchfork accompany the proofs.