On the fundamental theorem of surface theory under weak regularity assumptions

We consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that together satisfy the Gauss and Codazzi–Mainardi equations in a connected and simply connected open subset of $\text{R}{}^2$. If these fields are of class C² and C¹ respectively, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with the given matrix fields as its first and second fundamental forms. The purpose of this Note is to prove that this theorem still holds true under the weaker regularity assumptions that these fields are of class W^1,∞_loc and L^∞_loc respectively, the Gauss and Codazzi–Mainardi equations being then understood in a distributional sense. To cite this article: S. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Geometric mechanics: Foundations of the theory and fundamental equations

Geometric mechanics, which develops traditional geometric methods of mechanics, enables one to construct theories of complex coupled systems on the basis of Newtonian axiomatics without recourse to the methods of Lagrangian analytical mechanics, Euler's methods for the dynamics of a rigid body and other theories and principles. Geometric methods simplify the general theory of complex mechanical systems and bring it closer to computerized computational technologies and to engineering practice.     

A generalization of the fundamental theorem of spherical harmonic theory

We consider a boundary-value problem of the first kind for a self-adjoint differential operator with constant coefficients on a domain in ℝⁿ bounded by an ellipsoid; boundary conditions are defined by an arbitrary polynomial of degree N. It is proved that the solution of the problem is again a polynomial of degree ≤N. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

The fundamental theorem of mutual insurance

The essence of mutual insurance is the notion that re-distributing risk in a pool of risks is more beneficial than taking the risk alone. Interpreting ‘more beneficial’ as an increase in utility and considering sequences of exchangeable risks, we are able to formalize this notion from the policyholder’s perspective and demonstrate its validity for various alternative preference functionals (e.g., expected utility, Choquet expected utility, and distortion risk measures). To obtain this result, we exploit that for a sequence of exchangeable risks the corresponding sequence of arithmetical averages is a reversed martingale.We conclude that pooling risks is fundamental for understanding the mechanisms of insurance because it favourably affects the utility of policyholders, and we refer to this phenomenon as the ‘utility-improving effect of risk pooling’. Moreover, we demonstrate that the utility of the policyholder is (strictly) increasing with the size of the risk pool.     

On the fundamental theorem of algebra for polynomial equations over real composition algebras

A topological proof is given that real compositions algebras of finite dimension greater than one are algebraically closed under polynomial equations with a tame tail.     

A rigid analytic version of M. Artin's theorem on analytic equations

Mathematische Annalen -     

A fundamental theorem on lambda-matrices with applications —II. Difference equations with constant coefficients

The fundamental theorem of the title refers to a spectral resolution for the inverse of a lambda-matrix $\text{L(λ) =}\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{λ}{}^{\mathrm{i}}$ where the A_i are n×n complex matrices and detA_l ≠ 0. In this paper general solutions are formulated for difference equations of the form $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{u}{}_{\mathrm{r\; +\; i}}\text{= ?}{}_{\mathrm{\gamma }}\text{, r = 1, 2,…}$. The use of these solutions is illustrated i new proof of Franklin's results describing the sums of powers of the eigenvalues of L(λ) (the generalized Newton identities), and in obtaining convergence proofs for the application of Bernoulli's method to the solution of $\text{∑}\text{i=0}\text{l}\text{A}{}_{\mathrm{i}}\text{S}{}^{\mathrm{i}}\text{= 0}$ for matrix S.     

Euclid and the fundamental theorem of arithmetic

Slight changes or benevolent interpretations of certain theorems and proofs in Euclid's Elements make his demonstration of the fundamental theorem of arithmetic satisfactory for square-free numbers, but Euclid's methods cannot be adapted to prove the uniqueness for numbers containing square factors.     

The F. and M. Riesz theorem revisited

A result (Lemma 1) is obtained concerning the absolute continuity properties of operators implementing the unitary equivalence of two self-adjoint operators having a positive difference. A proof of the F. and M. Riesz theorem is given as a consequence.This work was supported by a National Science Foundation research grant.     

The fundamental theorem on the relation between spin and statistics

The fundamental theorem on the connection of spin and statistics is deduced from the basic symmetry between emission and absorption processes in quantum field theory. The new theorem which prescribes the physical connection of Bose fields with integral spin and Fermi fields with half-integral spin, is valid for all field theories; and it contains Pauli’s theorem as a special case.