Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Data of the Hamilton-Jacobi Equation of the Heat Equation with Square-Integrable White Noise on the Torus: the Noise, the Nonlinearities and the Drift
lemmalem:white-noise-heat-data-torus-2026aAnalysisPDEOn the Sobolev triple of order s of the torus, the Fourier coefficient families of the trigonometric classes, enumerated along the lattice, form a sequence that is square-summable in the order -s space with an explicit bound on the sum; the ambient space is infinite-dimensional;…The Laplacian of the Trigonometric System, and the Fourier Coefficients of a Laplacian on the Torus
lemmalem:laplacian-trigonometric-system-torus-2026aAnalysisEach member of the trigonometric system is twice continuously differentiable and is an eigenfunction of the Laplacian with eigenvalue minus four pi squared times the squared length of its frequency; consequently the Fourier coefficients of the Laplacian of a twice continuously di…Summability of the Negative Powers of the Fourier Weights of the Torus
lemmalem:fourier-weights-summable-torus-2026aAnalysisFor an exponent at least the dimension, the reciprocals of the powers of the Fourier weights of the torus, taken along any injective indexing of the integer lattice by the natural numbers, form a convergent series with a bound independent of the indexing.Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares
lemmalem:reciprocal-squares-bounded-2026aAnalysisReciprocals reverse the order on the positive reals; the partial sums of the reciprocals of the squares are bounded by two, so that series converges; and a quadratic sum taken symmetrically about its centre is bounded independently of its length.Real Sums over a Finite Index Set: Comparison, Nonnegativity, Monotonicity, Term Bounds, Absolute Values, Counting and Limits
lemmalem:finite-set-indexed-sum-nonnegative-2026bAnalysisFor real-valued maps on a nonempty finite set, sums over the set preserve order and nonnegativity, grow with the index set, dominate each nonnegative term, satisfy the triangle inequality, count the elements of the set, and commute with limits.Well-Posedness of the Viscous Allen-Cahn Hamilton-Jacobi Equation on the Torus
corollarycor:viscous-allen-cahn-hamilton-jacobi-well-posed-torus-2026aAnalysisPDEWith a finite noise built from the trigonometric system, the Allen-Cahn Hamilton-Jacobi equation on the torus carries a second-order trace term and a Hamiltonian coefficient between zero and one, and still has a unique bounded continuous viscosity solution on the square-integrabl…The Trigonometric System is Continuously Differentiable and Orthogonal in the Sobolev Space of the Torus
lemmalem:trigonometric-system-sobolev-torus-2026aAnalysisPDEEach member of the trigonometric system is continuously differentiable and periodic, with partial derivatives again members of the system up to a factor; its classes therefore lie in the Sobolev space of the torus and are orthogonal there, with squared norm one plus four pi squar…The Lebesgue Space of Square-Integrable Functions on the Torus is Not Finite-Dimensional
corollarycor:l2-torus-not-finite-dimensional-2026aAnalysisThe classes of the trigonometric system indexed along a single coordinate axis form an orthonormal sequence, so the Lebesgue space of square-integrable functions on the torus is not finite-dimensional.The Trigonometric System on the Torus is Orthonormal
theoremthm:trigonometric-system-orthonormal-torus-2026aAnalysisEach member of the trigonometric system is continuous and periodic, so defines a class in the Lebesgue space of square-integrable functions on the torus; these classes form an orthonormal family indexed by the integer lattice.- The real trigonometric system on the flat torus, indexed by the integer lattice: a normalised cosine or sine in each coordinate according to the sign of the index, multiplied together.
- The functions and for integer : their derivatives and bounds, and the values of the integrals over the half-open unit interval of each of them and of each pairwise product.
Zero Extension of a Real-Valued Function, and the Unit-Cell Integral of a Continuous Function
lemmalem:zero-extension-cell-integral-2026aAnalysisA real-valued function on a measurable subset is measurable and integrable exactly when its zero extension is, with the same integral; consequently the integral over the half-open unit interval of a continuous function is its Riemann integral over the closed one.The Integral over the Unit Cell of a Product of One-Variable Functions
lemmalem:product-function-integral-cell-2026aAnalysisMultivariable CalculusFor bounded Borel functions of one variable, the integral over the torus of the product of their coordinate evaluations is the product of their integrals over the unit interval.Quarter-Turn Identities and Periodicity of Sine and Cosine
theoremthm:sine-cosine-periodicity-2026aAnalysisShifting the argument by exchanges the sine and the cosine up to sign; iterating gives , and then invariance under every integer multiple of . The sine is positive on .- is twice the least positive zero of the cosine.
- There is exactly one positive real number at which the cosine vanishes and before which it is positive; it is at most , and the sine is positive up to it and equals there.
- Each of the products , and is half a sum or difference of cosines or sines of and .
- The cosine and sine of a sum expand as and .
- The sum of the squares of the cosine and the sine is everywhere; consequently both functions are bounded in absolute value by .
Uniform Convergence, Continuity, Parity and Derivatives of Sine and Cosine
theoremthm:sine-cosine-calculus-2026aAnalysisThe defining series for cosine and sine converge uniformly on every interval , with an explicit tail bound; both functions are continuous on the real line, satisfy and , are respectively even and odd, and are differentiable everywhere with derivatives…