Draining a polygon—or—rolling a ball out of a polygon

We introduce the problem of draining water (or balls representing water drops) out of a punctured polygon (or a polyhedron) by rotating the shape. For 2D polygons, we obtain combinatorial bounds on the number of holes needed, both for arbitrary polygons and for special classes of polygons. We detail an $O(n^2\log n)$ algorithm that finds the minimum number of holes needed for a given polygon, and argue that the complexity remains polynomial for polyhedra in 3D. We make a start at characterizing the 1-drainable shapes, those that only need one hole.     

Maintenance of diversity in a hierarchical host–parasite model with balancing selection and reinfection

Inspired by DNA data of the human cytomegalovirus we propose a model of a two-type parasite population distributed over its hosts. The parasite is capable to persist in its host till the host dies, and to reinfect other hosts. To maintain type diversity within a host, balancing selection is assumed.For a suitable parameter regime we show that in the limit of large host and parasite populations the host state frequencies follow a dynamical system with a globally stable equilibrium, guaranteeing that both types are maintained in the parasite population for a long time on the host time scale.     

Some Properties of a Class Self—homomorphism of BCI—algebras

ＢｙａＢＣＩ－ａｌｇｅｂｒａｗｅｍｅａｎａｎａｌｇｅｂｒａ（Ｘ；，０）ｏｆｔｙｐｅ（２，０）ｓａｔｉｓｆｙｉｎｇｔｈｅａｘｉｏｍｓ：（１）（（ｘｙ）（ｘｚ））（ｚｙ）＝０；（２）（ｘ（ｘｙ））ｙ＝０；（３）ｘｘ＝０；（４）ｘｙ＝ｙｘ＝０ｘ＝ｙｆｏｒａｎｙｘ，ｙａｎｄｚｉｎＸ．ＦｏｒａｎｙＢＣＩ－ａｌｇｅｂｒａＸ，ｔｈｅｒｅｌａｔｉｏｎ≤ｄｅｆｉｎｅｄｂｙｘ≤ｙｉｆａｎｄｏｎｌｙｉｆｘｙ＝０ｉｓａｐａｒｔｉａｌｏｒｄｅｒｏｎＸ［１］．ＩｎａｎｙＢＣＩ－ａｌｇｅｂｒａＸ，…     

Global Asymptotics of Krawtchouk Polynomials——a Riemann-Hilbert Approach

In this paper, we study the asymptotics of the Krawtchouk polynomials KnN(z;p,q) as the degree n becomes large. Asymptotic expansions are obtained when the ratio of the parameters n/N tends to a limit c∈(0,1) as n→∞. The results are globally valid in one or two regions in the complex z-plane depending on the values of c and p; in particular, they are valid in regions containing the interval on which these polynomials are orthogonal. Our method is based on the Riemann-Hilbert approach introduced by Deift and Zhou.     

Sum Complexes—a New Family of Hypertrees

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k−1)-dimensional skeleton and \binomn-1k\binom{n-1}{k} facets such that H _k (X;ℚ)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group ℤ _n . The sum complex X _A is the pure k-dimensional complex on the vertex set ℤ _n whose facets are σ⊂ℤ _n such that |σ|=k+1 and ∑ _x∈σ x∈A. It is shown that if n is prime, then the complex X _A is a k-hypertree for every choice of A. On the other hand, for n prime, X _A is k-collapsible iff A is an arithmetic progression in ℤ _n .     

On a Comparison Property of M—matrices

IntheresearchonthetheoryofM-matrices,wefrequentlyneedthefollowingtheoremasatool,whichiscalledthecomparisontheoremofanonsingularM-matrix(forexamplesee[5,Theorem2.2.5j).TheoremALetAbeanonsingularM-matrix,andletBbeaZ-matrixwithB2A.Then(1)BisanonsingularM-matrixI'(2)detB2detAj(3)A-'2B-'.obviously,theabovetheoremisnottrueforAtobeasingularM-matrix.InthispaperweshallextendTheoremAtOanyM-matrix,herethegeneralizedinverseisconsideredforgeneralizingassertion(3)ofTheoremA.AllnOtationanddefiniti…     

Subgroups of odd depth—a necessary condition

This paper gives necessary and sufficient conditions for subgroups with trivial core to be of odd depth. We show that a subgroup with trivial core is an odd depth subgroup if and only if certain induced modules from it are faithful. Algebraically this gives a combinatorial condition that has to be satisfied by the subgroups with trivial core in order to be subgroups of a given odd depth. The condition can be expressed as a certain matrix with {0, 1}-entries to have maximal rank. The entries of the matrix correspond to the sizes of the intersections of the subgroup with any of its conjugate.     

Stability of a Chem ostatModelfor a Single Specieswith Delayed NutrientRecycling——Case ofWeak KernelFunction

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｃｏｍｍｏｎｃｈｅｍｏｓｔａｔｍｏｄｅｌｈａｓｂｅｅｎｃｏｎｓｉｄｅｒｅｄｂｙｍａｎｙｍａｔｈｅｍａｔｉａｎｓ（ｓｅｅ［１］）．Ｃｈｅｍｏｓｔａｔｍｏｄｅｌｓｉｎｃｏｒｐｏｒａｔｉｎｇｄｉｓｃｒｅｔｅｔｉｍｅｄｅｌａｙｈａｖｅｂｅｅｎｉｎｖｅｓｔｉｇａｔｅｄｉｎ［２］ａｎｄ［３］．Ｉｎ［４］ａｎｄ［５］，ｃｈｅｍｏｓｔａｔｍｏｄｅｌｓｗｉｔｈｃｏｎｔｉｎｕｏｕｓｔｉｍｅｄｅｌａｙｉｎｎｕｔｒｉｅｎｔｒｅｃｙｃｌｉｎｇｗｅｒｅｃｏｎｓｉｄ－ｅｒｅｄ．Ｉｎｔｈｉ…     

On a theorem of c—wrpp semigroups

In this paper, we define a class of semigroups called C—wrpp semigroups and prove that they are a generalization of C—rpp semigroups. Moreover, Theorem 3.1 generalizes immediately the structure theorem for Clifford semigroups [5] and the main result of J. B. Fountain [3].     

Hypoellipticity in spaces of ultradistributions—Study of a model case

In this work we study C ^??-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying H?rmander??s condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.     

On the Characterization of a Class of π—inverse Semigroups

51.PreliminatriesS.Bogdanovicgaveacharacterizationforstronglyrr-inversesemigroupsinLlJ.TheclassofrightinversesemigroupsisinvestigatedbyP.S-Venkatesanin[2j.Inthispaper,thenotionofstronglyrightff-inversesemigroupsisintroduced.Characteri-zationofthisclassofsemigroupsobtained-Definition1.1[1jAff-regularsemigroupsissaidtobeK-orthodoxifE,,thesetofidem-potentsinS,isasubsemigroupofS.Definition1.2[1jAK-regularsemigroupsiscalledstronglyK-inverseifef=feforev-erye,feE,.Definition1.3Aff-regularsemi…     

Graphlike Manifolds with the Type of the 1—skeleton of a 4—simplex

ＧｒａｐｈｌｉｋｅＭａｎｉｆｏｌｄｓｗｉｔｈｔｈｅＴｙｐｅｏｆｔｈｅ１－ｓｋｅｌｅｔｏｎｏｆａ４－ｓｉｍｐｌｅｘＧｕｏＴｕｏｙｉｎｇ（ＨｕａｚｈｏｎｇＮａｒｍａｌＵｎｉｖｅｒｓｄｙ，Ｗｕｈａｎ，Ｃｈｉｎａ）（郭驼英）ｈＴｈｅｃｏｎｃｅｐｔｏｆｇｒａｐ...     

Boundary values of generalized solutions of a homogeneous sturm — Liouville equation in a space of vector functions

We consider a differential equation of the form ?y” + A²y=0, where A is a self-adjoint operator in a Hilbert space H. We show that each generalized solution of this equation inw _?m (0, b) (0 < b < ∞, m ≥ 0) has boundary values in the spaceH _?m?1/2, where H_J (?∞ ?m(0, b) is the space of continuous linear functionals on ?W_m(0, b), the completion of the space of infinitely differentiable vector functions with compact support with respect to the norm \(\left\| u \right\|_{W_m (0, b) = (\left\| u \right\|_{L_2 (H_{m,} (0, b))} + \left\| u \right\|_{L_2 (H, (0, b))}^{(m)} )} \) . It follows that each function u(t, x) which is harmonic in the strip G = [0, b] x (?∞, ∞) and which is in the space that is dual to ? ₂ ^m (G) has limiting values as t→0 and t→b in the space \(W_2^{ - m - {\raise0.7ex\hbox{\(1\)} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0em}\!\lower0.7ex\hbox{\(2\)}}} ( - \infty ,\infty )\) .     

Generators of Rings of Differential Operators on a Class of k—algebras

51.IlltroductionSmithL4JproPoseaso1vingprocedure,whichcIcternllnesl1olyno[nialresolutlonofI3:lrt1itldifferentialequationcorrespondingbysomecoordinateringsofirreduciblel7l;lnecurves.-I'l1lsprogrameventual1ydependsontl1edeterminationofgeneratorsofringsofdif…     

RESEARCH ANNOUNCEMENTS—On a Generalization of the c—numerical Range

Let Cn×n be the set of n × n complex matrices and An the set of orthonormal n-tuples of vectors in Cn. For a vector c in Cn and a matrix A in Cn×n, the c-numerical range of A is the set Wc(A)={n∑i=1 Ci(Axi,xi):(x1,…xn)∈∧n} When c = (1,0,…,0), Wc(A) is reduced to the classical numerical range W(A) (see [1]). For the classical numerical range and its generalizations, one may see the survey article[2].     

Solutions to a Class of Differential—fuctional Equations

ＳｏｌｕｔｉｏｎｓｔｏａＣｌａｓｓｏｆＤｉｆｆｅｒｅｎｔｉａｌ－ｆｕｃｔｉｏｎａｌ　ＥｑｕａｔｉｏｎｓＬｉＺｅｍｉｎ（李泽民）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＺｈｅｎｇｚｈｏｕＵｎｉｖｅｒｓｉｔｙ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａ...