For any n-end catenoids, conformal invariants called relative weights are defined by using the flux formula. In this paper, we observe some obstructions for the existence of n-end catenoids with prescribed flux, and describe what happens near the data of such obstructions by means of the relative weights.

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Regularities and higher order regularities of ground states of quantum field models are investigated through the fact that asymptotic annihilation operators vanish ground states. Moreover, a sufficient condition for the absence of a ground state is given.

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The Calabi-Yau conjecture is one of the main problems in the global theory of complete minimal surfaces in R³. Francisco Martin and Santiago Morales have constructed complete proper minimal surfaces in convex bodies of R³. In this paper, we modify their technique in the cylindrical case, and construct a complete minimal cylinder properly immersed in the unit ball.

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A generic smooth map of a closed 2k-manifold into (3k - 1)-space has a finite number of cusps (Σ^1,1-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities (Σ^1,0-singularities). Two fold maps are left-right fold bordant if there are cobordisms between their source and target manifolds with a fold map extending the two maps between the boundaries. If the two targets agree and the target cobordism can be taken as a product with a unit interval, then the maps are fold cobordant. Cobordism classes of fold maps are known to form groups. We compute these groups for fold maps of (2k - 1)-manifolds into (3k - 2)-space. Analogous cobordism semigroups for arbitrary closed (3k - 2)-dimensional target manifolds are endowed with Abelian group structures and described. Left-right fold bordism groups in the same dimensions are also described.

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An evaluation code is a generalization of a one-point algebraic geometry code. The aim of this article is to present a method of constructing evaluation codes from the Hermitian curves, and compute the Feng-Rao bounds for their minimum distances. It is proved that some of such evaluation codes have a better property than one-point Hermitian codes.

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First, it is pointed out that the uniform distribution of points in [0, 1]^d is not always a necessary condition for every function in a proper subset of the class of all Riemann integrable functions to have the arithmetic mean of function values at the points converging to its integral over [0, 1]^d as the number of points goes to infinity. We introduce a formal definition of the d-dimensional high-discrepancy sequences, which are not uniformly distributed in [0, 1]^d, and present motivation for the application of these sequences to high-dimensional numerical integration. Then, we prove that there exist non-uniform (∞, d)-sequences which provide the convergence rate O(N^-1) for the integration of a certain class of d-dimensional Walsh function series, where N is the number of points.

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We consider representations of Cuntz algebras on self-similar fractal sets for proper/improper systems of contractions. Natural representations, called Hausdorff representations, are associated with self-similar sets and Hausdorff measures in the case of similitudes in Rⁿ. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list(λ₁^D, . . . ,λ_N^D), where λ_j is the Lipschitz constant of the j th contraction and D is the Hausdorff dimension of the fractal set. Any non-trivial list can be realized by similitudes on the unit interval. There exists an improper system of contractions such that its representation of a Cuntz algebra on the self-similar fractal set is not unitarily equivalent to any Hausdorff representation for a proper system of similitudes in Rⁿ.

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The distribution function for the first eigenvalue spacing in the Laguerre unitary ensemble of finite rank random matrices is found in terms of a Painlevé V system, and the solution of its associated linear isomonodromic system. In particular, it is characterized by the polynomial solutions to the isomonodromic equations which are also orthogonal with respect to a deformation of the Laguerre weight. In the scaling to the hard edge regime we find an analogous situation where a certain Painlevé III' system and its associated linear isomonodromic system characterize the scaled distribution. We undertake extensive analytical studies of this system and use this knowledge to accurately compute the distribution and its moments for various values of the parameter a. In particular, choosing a = ±1/2 allows the first eigenvalue spacing distribution for random real orthogonal matrices to be computed.

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We locate almost all of the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of Rankin and Swinnerton-Dyer (On the zeros of Eisenstein series. Bull. London Math. Soc. 2 (1970), 169-170). We also use the arguments of some terms of the Eisenstein series in order to improve existing error bounds.

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This note studies a q-analogue of Mellin's transform of a modular form via Jackson's integral.

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