On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

The Discrepancy of the Sequence (nα + (log n)β)

In this paper we give an upper bound for the discrepancy of the sequence (nα + (log n)β) where α = (α₁, ..., α _s ) R ^s , which satisfies that 1, α₁, ..., α _s are linearly independent over Z, is of finite type η or is of constant type.     

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.     

Two stochastic models of a random walk in the U(n)-spherical duals of U(n + 1)

The random walk to be considered takes place in the δ-spherical dual of the group U(n + 1), for a fixed finite dimensional irreducible representation δ of U(n). The transition matrix comes from the three-term recursion relation satisfied by a sequence of matrix valued orthogonal polynomials built up from the irreducible spherical functions of type δ of SU(n + 1). One of the stochastic models is an urn model and the other is a Young diagram model.     

On the Congruence σ(n)≡1(mod n)

Let k≥2 be an integer,and let σ(n) denote the sum of the positive divisors of an integer n.We call n a quasi-multiperfect number if σ(n)=kn+1.In this paper,we give some necessary properties of them.     

On the invariants of Möbius groupsM(R n )

Suppose that $$\operatorname{Re} (a + d^ * ) \in \left\{ {\begin{array}{*{20}c} {( - 2,2),if g(x) is f.p.f. or elliptic,} \\ {\left[ { - 2,2} \right], if g(x) is parabolic,} \\ {( - \infty ,\infty ), if g(x) is loxodromic.} \\ \end{array} } \right.$$ is a Clifford matrix of dimensionn, g(x)=(ax+b)(cx+d) ^?1. We study the invariant balls and the more careful classifications of the loxodromic and parabolic elements inM(R ⁿ ), prove that the loxodromic elements inM(R ^2k+1 ) certainly have an invariant ball, expound the geometric meaning of Ahlfors' hyperbolic elements, and introduce the uniformly hyperbolic and parabolic elements and give their identifications. We prove that $$\operatorname{Re} (a + d^ * ) \in \left\{ {\begin{array}{*{20}c} {( - 2,2),if g(x) is f.p.f. or elliptic,} \\ {\left[ { - 2,2} \right], if g(x) is parabolic,} \\ {( - \infty ,\infty ), if g(x) is loxodromic.} \\ \end{array} } \right.$$ These results are fundamental in the higher dimensional Möbius groups, especially in Fuchs groups.     

On the mean values of the function τ k (n) in sequences of natural numbers

We obtain an asymptotic formula for the mean value of the function τ _k (n), which is the number of solutions of the equation x ₁ … x _k = n in natural numbers x ₁, …, x _k , in some special sequences of natural numbers.     

Constants of Strong Uniqueness of Minimal Projections onto Hyperplanes in the Space ℓ∞ n (n≥3)

In this paper, we obtain the maximum values of the constants of strong uniqueness of minimal projections with nonunit norm on hyperplanes in the space #x2113; ⁿ (n3)     

Total Chromatic Number of the Join of K_(m,n) and C_n

The total chromatic number XT(G) of a graph G is the minimum number of colors needed to color the elements (vertices and edges) of G such that no adjacent or incident pair of elements receive the same color. G is called Type 1 if XT(G)=Δ(G) 1. In this paper we prove that the join of a complete bipartite graph Km,n and a cycle Cn is of Type 1.     

On the SO(n + 3) to SO(n) branching multiplicity space

We study the decomposition as an $\text{SO}(3)$-module of the multiplicity space corresponding to the branching from $\text{SO}(n+3)$ to $\text{SO}(n)$. Here, $\text{SO}(n)$ (resp. $\text{SO}(3)$) is considered embedded in $\text{SO}(n+3)$ in the upper left-hand block (resp. lower right-hand block). We show that when the highest weight of the irreducible representation of $\text{SO}(n)$ interlaces the highest weight of the irreducible representation of $\text{SO}(n+3)$, then the multiplicity space decomposes as a tensor product of $?(n+2)/2?$ reducible representations of $\text{SO}(3)$.     

On the Congruence σ(n) ≡ 1(mod n), II

Let k ≥ 2 be an integer, and let σ(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if σ(n) = kn + 1. In this paper, we give some necessary properties of quasi-multiperfect numbers with four different prime divisors.     

On the Moments of S_ny/(nφ(n))~(1/p) in p-type Space

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ１９６７，Ｔｅｉｃｈｅｒｐｒｏｖｅｄｔｈａｔ［１］Ｅ（Ｓｕｐｎ｜Ｓｎ｜ｎＬ２ｎ）ｐ＜＋∞，　ｉｆｆＥＸ２ｌｏｇ＋｜Ｘ｜Ｌ２｜Ｘ｜＜＋∞，ｐ＝２Ｅ｜Ｘ｜ｐ＜＋∞，ｐ＞２　　　　．　Ｗｈｅｒｅ｛Ｘ，Ｘｎ，ｎ≥１｝ｉｓａｓｅｑｕｅｎｃｅｏｆｉ．ｉ．ｄｒｅａｌｒａｎｄｏｍｖａｒｉａｂｌｅｓｗｉｔｈｍｅａｎｓｚｅｒｏ．Ｉｎ１９９５，Ｔｈｅｓｉｍｉｌａｒｒｅｓｕｌｔｓｈａｖｅｂｅｅｎｓｅｔｕｐｆｏｒｉ．ｉ．ｄｒａｎｄｏｍｖａｒｉａｂｌｅｓ｛Ｘ，Ｘｎｎ≥１｝ｗｉｔｈｍｅａｎ…     

Invariants of an augmentation of Γ0(n)

Let G _n ^× be the 2-group of primary factors of a positive integer n and fix a direct product decomposition of this group. We define an augmentation of Γ₀(n) based on G _n ^×, paralleling augmentations used by Fricke, Cohn and Knopp, and others. Using the decomposition of G _n ^×, we then define a family of functions based on η-functions and use these functions to construct invariants of the augmented group. Along with proving results analogous to those of Cohn and Knopp, we make a complete determination of the multiplier systems for these new functions.     

Gelfand–Kirillov Dimensions of the Z~2-graded Oscillator Representations of sl(n)

We find an exact formula of Gelfand–Kirillov dimensions for the infinite-dimensional explicit irreducible sl(n, F)-modules that appeared in the Z2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand–Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.     

An inequality for the function π(n)

We prove that the inequality $\pi ^2 \left( m \right) + \pi ^2 \left( n \right) \leqslant \tfrac{5} {4}\pi ^2 \left( {m + n} \right)$ holds for all integers m, n ≥ 2. The constant factor 5/4 is sharp. This complements a result of Panaitopol, who showed in 2001 that ½ π ²(m+ n) ≤ π ²(m) + π ²(n) is valid for all m, n ≥ 2. Here, as usual, π(n) denotes the number of primes not exceeding n.