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The singly periodic Scherk surfaces with higher dihedral symmetry have -ends that come together based upon the value of . These surfaces are embedded provided that . Previously, this inequality has been proved by turning the problem into a Plateau problem and solving, and by using the Jenkins–Serrin solution and Krust’s theorem. In this paper we provide a proof of the embeddedness of these surfaces by using some results about univalent planar harmonic mappings from geometric function theory. This approach is more direct and explicit, and it may provide an alternate way to prove embeddedness for some complicated minimal surfaces.
Let be a degree polynomial with zeros . The total distance from the zeros of to the unit circle is defined as . We show that up to scalar multiples, sits between and . This leads to an equivalent statement of Lehmer’s problem in terms of . The proof is elementary.
We examine a one-dimensional reaction diffusion model with a weak Allee growth rate that appears in population dynamics. We combine grazing with a certain nonlinear boundary condition that models negative density dependent dispersal on the boundary and analyze the effects on the steady states. In particular, we study the bifurcation curve of positive steady states as the grazing parameter is varied. Our results are acquired through the adaptation of a quadrature method and Mathematica computations. Specifically, we computationally ascertain the existence of -shaped bifurcation curves with several positive steady states for a certain range of the grazing parameter.
Extending the work of Deborah L. Massari and Kimberly L. Patti, this paper makes progress toward finding the probability of elements randomly chosen without repetition generating a finite abelian group, where is the minimum number of elements required to generate the group. A proof of the formula for finding such probabilities of groups of the form , where and is prime, is given, and the result is extended to groups of the form , where and is prime. Examples demonstrating applications of these formulas are given, and aspects of further generalization to finding the probabilities of randomly generating any finite abelian group are investigated.
This paper studies the existence of free and very free curves on the degree Fermat hypersurface in over an algebraically closed field of characteristic . We explicitly compute a free curve in degree , and a very free curve in degree . We also prove that free and very free curves cannot exist in lower degrees.
For an integral domain , the irreducible divisor graph of a nonunit gives a visual representation of the factorizations of . Here we consider a higher-dimensional generalization of this notion, the irreducible divisor simplicialcomplex . We show how this new structure is a true generalization of , and show that it often carries more information about the element and the domain than its two-dimensional counterpart.
It is well known that there exist arbitrarily long sequences of consecutive happy numbers. In this paper we find the smallest numbers beginning sequences of fourteen and fifteen consecutive happy numbers.
We study the inertia space of with the standard action of the special orthogonal group . In particular, we indicate a decomposition of the inertia space that induces the orbit Cartan type stratification of the inertia space recently defined by C. Farsi, M. Pflaum, and the first author for an arbitrary smooth -manifold where is a compact Lie group.
We propose a generalization of Greenberg’s unrelated-question randomized response model allowing subjects the option of giving a correct response if they find the survey question nonsensitive, and to give a scrambled response if they find the question sensitive. Models are provided for both the binary response and the quantitative response situations. Mathematical properties of the proposed models are examined and validated with computer simulations.
Let be the sum of the divisors of . Although much attention has been paid to the possible values of (the sum of proper divisors), comparatively little work has been done on the possible values of . Here we present some theoretical and computational results on these values. In particular, we exhibit some infinite and possibly infinite families of integers that appear in the image of . We also find computationally all values of for which is odd, and we present some data from our computations. At the end of this paper, we present some conjectures suggested by our computational work.