Removable singularity of the polyharmonic equation

Let x₀∈Ω⊂Rⁿ, n≥2, be a domain and let m≥2. We will prove that a solution u of the polyharmonic equation Δ^mu=0 in Ω?{x₀} has a removable singularity at x₀ if and only if as |x−x₀|→0 for n≥3 and as |x−x₀|→0 for n=2. For m≥2 we will also prove that u has a removable singularity at x₀ if |u(x)|=o(|x−x₀|^2m−n) as |x−x₀|→0 for n≥3 and |u(x)|=o(|x−x₀|^2m−2log(|x−x₀|⁻¹)) as |x−x₀|→0 for n=2.     

All spectral dominant norms are stable

A vector norm |·|on the space of n×n complex valued matrices is called stable if for some constant K&>0, not depending upon A or m, we have |A^m|?K|A|^m We show that such a norm is stable if and only if it dominates the spectralradius.     

Properties of connected graphs having minimum degree distance

The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P₄, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K₁+(K_1,k+1∪(n−k−3)K₁).     

Rational approximations of the number \sqrt[3]{3}

We obtain estimates of the form |ξα ⁿ | > C ^k for all k ≥ K for the values ξ = 3^1/3 and α = 2, where K is an effective constant and | · | denotes the distance to the nearest integer.     

On the stability of the first-order linear recurrence in topological vector spaces

Suppose that X is a sequentially complete Hausdorff locally convex space over a scalar field K, V is a bounded subset of X, (a_n)_n≥0 is a sequence in K?{0} with the property lim inf_n→∞|a_n|>1, and (b_n)_n≥0 is a sequence in X. We show that for every sequence (x_n)_n≥0 in X satisfying

     

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

We show that all eigenfunctions of linear partial differential operators in Rn with polynomial coefficients of Shubin type are extended to entire functions in Cn of finite exponential type 2 and decay like exp(−²|z|) for |z|→∞ in conic neighbourhoods of the form |Imz|?γ|Rez|. We also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip {z∈Cn||Imz|?T} for some T>0. The proofs are based on geometrical and perturbative methods in Gelfand-Shilov spaces. The results apply in particular to semilinear Schrödinger equations of the form

(∗)     

Sperner families over a subset

Let |X| = n > 0, |Y| = k > 0, and Y ? X. A family A of subsets of X is a Sperner family of X over Y if A₁A₂ for every pair of distinct members of A and every member of A has a nonempty intersection with Y. The maximum cardinality f(n, k) of such a family is determined in this paper. $\text{f(n,k)=(}\text{n}\text{[n}\text{2]}\text{)?(}\text{?k}\text{[n}\text{2]}\text{)}$.     

On a problem of Chvátal and Erdös on hypergraphs containing no generalized simplex

Let X be an n-element set and $\text{T}$ a family of k-subsets of X. Let r be an integer, k > r ? 2. Suppose that $\text{T}$ does not contain r + 1 members having empty intersection such that any r of them intersect non-trivially. Chvátal and Erdös conjectured that for (r + 1) k ? rn we have |$\text{F}$|$\text{?}\text{n?1}\text{k?1}$. In this paper we first prove that This conjecture holds asymptotically (Theory 1). In Theorems 4 and 5 we prove it for r = 2, K ? 5, n > n_o(k); k ? 3r, n > n_o(k, r), respectively.     

Characterization of summability methods with high indices

In this paper we establish a set of necessary and sufficient conditions in order that |C, 0| _k ? |R, p _n | _s and |R, p _n | _k ? |C, 0| _s for the case 1 < k ≤ s < ∞. As a corollary, we obtain that a crucial assumption of [BOR, H.: A new result on the high indices theorem, Analysis 29 (2009), 403–405] is omitted and that the other one is not only sufficient but also necessary for his consequence to hold.     

Monotonically Computable Real Numbers

Area number x is called k‐monotonically computable (k‐mc), for constant k > 0, if there is a computable sequence (x_n)_{n ∈ ℕ} of rational numbers which converges to x such that the convergence is k‐monotonic in the sense that k · |x — x_n| ≥ |x — x_m| for any m > n and x is monotonically computable (mc) if it is k‐mc for some k > 0. x is weakly computable if there is a computable sequence (x_s)_{s ∈ ℕ} of rational numbers converging to x such that the sum $\sum _{s \in \mathbb{N}}$|x_s — x_{s + 1}| is finite. In this paper we show that a mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers.     

Rigidity theorems for hypersurfaces with constant mean curvature

Let M ⁿ be a compact oriented hypersurface of a unit sphere \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H. Given an integer k between 2 and n ? 1, we introduce a tensor ? related to H and to the second fundamental form A of M, and show that if |?|² ≤ B _H,k and tr(? ³) ≤ C _n,k |?|³, where B _H,k and C _n,k are numbers depending only on H, n and k, then either |?|² ≡ 0 or |?|² ≡ B _H,k . We characterize all M ⁿ with |?|² ≡ B _H,k . We also prove that if \(\left| A \right|^2 \leqslant 2\sqrt {k(n - k)}\) and tr(? ³) ≤ C _n,k |?|³ then |A|² is constant and characterize all M ⁿ with |A|² in the interval \(\left[ {0,2\sqrt {k\left( {n - k} \right)} } \right] \) . We also study the behavior of |?|², with the condition additional tr(? ³) ≤ C _n,k |?|³, for complete hypersurfaces with constant mean curvature immersed in space forms and show that if sup _M |?|² = B _H,k and this supremum is attained in M ⁿ then M ⁿ is an isoparametric hypersurface with two distinct principal curvatures of multiplicities k y n ? k. Finally, we use rotation hypersurfaces to show that the condition on the trace of ? ³ is necessary in our results; more precisely, for each integer k with 2 ≤ k ≤ n ? 1 and \(H \geqslant 1/\sqrt {2n - 1} \) there is a complete hypersurface M ⁿ in \(\mathbb{S}^{n + 1} \) (1) with constant mean curvature H such that sup _M |?|² = B _H,k , and this supremum is attained in M ⁿ , and which is not a product of spheres.     

Chromatic equivalence classes of complete tripartite graphs

Some necessary conditions on a graph which has the same chromatic polynomial as the complete tripartite graph K_m,n,r are developed. Using these, we obtain the chromatic equivalence classes for K_m,n,n (where 1≤m≤n) and K_m1,m2,m3 (where |m_i−m_j|≤3). In particular, it is shown that (i) K_m,n,n (where 2≤m≤n) and (ii) K_m1,m2,m3 (where |m_i−m_j|≤3, 2≤m_i,i=1,2,3) are uniquely determined by their chromatic polynomials. The result (i), proved earlier by Liu et al. [R.Y. Liu, H.X. Zhao, C.Y. Ye, A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs, Discrete Math. 289 (2004) 175-179], answers a conjecture (raised in [G.L. Chia, B.H. Goh, K.M. Koh, The chromaticity of some families of complete tripartite graphs (In Honour of Prof. Roberto W. Frucht), Sci. Ser. A (1988) 27-37 (special issue)]) in the affirmative, while result (ii) extends a result of Zou [H.W. Zou, On the chromatic uniqueness of complete tripartite graphs K_n1,n2,n3 J. Systems Sci. Math. Sci. 20 (2000) 181-186].     

A subdeterminant inequality for normal matrices

Let A be an n × n normal matrix over $\text{C}$, and Q_m, _n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Q_m, _n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i₁,…, i_k, i_k+1,…, i_m, such that α(i_j) = β(i_j), i = 1,…, k, and {α(i_k+1),…, α(i_m) } ∩ {β(i_k+1),…, β(i_m) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let $\text{U}$_n be the group of n × n unitary matrices. Define the nonnegative number

where | α ∩ β| = k. It is proved that

Let A be semidefinite hermitian. We conjecture that ρ₀(A) ? ρ₁(A) ? ··· ? ρ_m(A). These inequalities have been tested by machine calculations.     

On the absolute Cesàro summability factors of infinite series

The author has established that if {λ _n | is a convex sequence such that the series\(\sum {\frac{{\lambda _n }}{n}} \) is convergent and if Σa _n is bounded [R, logn, 1] with indexk, then\(\sum {a_n \lambda _n } \) is summable |C, 1|k fork>1. The casek=1 of the theorem is due to Pati [3].     

Shadows of ordered graphs

Isoperimetric inequalities have been studied since antiquity, and in recent decades they have been studied extensively on discrete objects, such as the hypercube. An important special case of this problem involves bounding the size of the shadow of a set system, and the basic question was solved by Kruskal (in 1963) and Katona (in 1968). In this paper we introduce the concept of the shadow ∂G of a collection G of ordered graphs, and prove the following, simple-sounding statement: if n∈N is sufficiently large, |V(G)|=n for each G∈G, and |G|<n, then |∂G|?|G|. As a consequence, we substantially strengthen a result of Balogh, Bollobás and Morris on hereditary properties of ordered graphs: we show that if P is such a property, and |Pk|<k for some sufficiently large k∈N, then |Pn| is decreasing for k?n<∞.     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.     

Intersections of q-ary perfect codes

The intersections of q-ary perfect codes are under study. We prove that there exist two q-ary perfect codes C ₁ and C ₂ of length N = qn + 1 such that |C ₁ ? C ₂| = k · |P _i |/p for each k ∈ {0,..., p · K ? 2, p · K}, where q = p ^r , p is prime, r ≥ 1, $n = \tfrac{{q^{m - 1} - 1}}{{q - 1}}$ , m ≥ 2, |P _i | = p ^nr(q?2)+n , and K = p ^{n(2r?1)?r(m?1)}. We show also that there exist two q-ary perfect codes of length N which are intersected by p ^nr(q?3)+n codewords.     

On Hölder maps of cubes

A map of metric spaces f: X → Y satisfying the inequality $$ \left| {f(x) - f(y)} \right| \leqslant C\left| {x - y} \right|^\alpha $$ for some C and α and all x, y ∈ X is called a Hölder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain theorem of this paper gives a general method for constructing Höldermaps of compact metric spaces. This construction yields, in particular, a dimension-raising map f: I ⁿ → I ^m with Hölder exponent arbitrarily close to m/n for m > n > 1 and a map I ¹ → I ^m with Hölder exponent 1/m. The second main theorem states the nonexistence of a regular fractal map f: I ⁿ → I ^m with Hölder exponent n/m from the n-cube onto the m-cube for m < 2n.     

A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence

We consider a stationary time series {X_t} given byX_t=∑^∞_k=−∞ ψ_kZ_t−k, where {Z_t} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {X_t} is squared integrable andm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm ∑_|k|?m ψ²_k→0.