Uniform approximation by some Hermite interpolating splines

In this paper some upper bound for the error ∥ s-f ∥_∞ is given, where f ε C¹[a,b], but s is a so-called Hermite spline interpolant (HSI) of degree 2q ?1 such that f(x_i) = s(x_i), f′(rmx_i) = s′(x_i), s^(j) (x_i) = 0 (i = 0, 1, …, n; j = 2, 3, …, q ?1; n > 0, q > 0) and the knots x_i are such that a = x₀ < x₁ < … < x_n = b. Necessary and sufficient conditions for the existence of convex HSI are given and upper error bound for approximation of the function fε C¹[a, b] by convex HSI is also given.     

On a problem of Katona on minimal completely separating systems with restrictions

Let S be a set of n elements, and k a fixed positive integer $\text{<}\text{1}\text{2}\text{n}$. Katona's problem is to determine the smallest integer m for which there exists a family $\text{A}$ = {A₁, …, A_m} of subsets of S with the following property: |_i| ? k (i = 1, …, m), and for any ordered pair x_i, x_i ∈ S (i ≠ j) there is A₁ ∈ $\text{A}$ such that x_i ∈ A₁, x_j ? A₁. It is given in this note that $\text{m = ?}\text{2n}\text{k}\text{?}\text{if}\text{1}\text{2}\text{k}{}^2\text{? 2}$.     

Weighted inequalities for fractional type operators with some homogeneous kernels

In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.     

Phragmén-Lindelöf conditions for graph varieties

For P_m ∈ ?[z₁, …, z_n], homogeneous of degree m we investigate when the graph of P_m in ?ⁿ⁺¹ satisfies the Phragmén-Lindelöf condition PL(?ⁿ⁺¹, log), or equivalently, when the operator $i{\partial \over \partial_{x_{n+1}}}+P_{m}(D)$ admits a continuous solution operator on C^∞(?ⁿ⁺¹). This is shown to happen if the varieties V_+- ? {z ∈ ?ⁿ: P_m(z) = ±1} satisfy the following Phragmén-Lindelöf condition (SPL): There exists A ≥ 1 such that each plurisubharmonic function u on V_+- satisfying u(z) ≤ ¦z¦+ o(¦z¦) on V_+- and u(x) ≤ 0 on V_+- ∩ ?ⁿ also satisfies u(z) A¦ Im z¦ on V_+-. Necessary as well as sufficient conditions for V_+- to satisfy (SPL) are derived and several examples are given.     

Galois Groups of Generalized Iterates of Generic Vectorial Polynomials

Let q=p^u>1 be a power of a prime p, and let k_q be an overfield of GF(q). Let m>0 be an integer, let J^* be a subset of {1,…,m}, and let E^*_{m, q}(Y)=Y^qm+∑_j∈J^*X_jY^qm−j where the X_j are indeterminates. Let J³ be the set of all m−ν where ν is either 0 or a divisor of m different from m. Let s(T)=∑_0≤i≤ns_iTⁱ be an irreducible polynomial of degree n>0 in T with coefficients s_i in GF(q). Let E^*[s]_{m, q}(Y) be the generalized sth iterate of E^*_{m, q}(Y); i.e., E^*[s]_{m, q}(Y)=∑_0≤i≤ns_iE^*[i]_{m, q}(Y), where E^*[i]_{m, q}(Y), is the ordinary ith iterate. We prove that if J³⊂J^*, m is square-free, and GCD(m,n)=1=GCD(mnu,2p), then Gal(E^*[s]_{m, q},k_q({X_j:j∈j^*})=GL(m, qⁿ). The proof is based on CT (=the Classification Theorem of Finite Simple Groups) in its incarnation as CPT (=the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors).     

Δ-Monotone arrangements of real numbers

Given a set of M × N real numbers, can these always be labeled as x_i,j; i = 1,…, M; j = 1,…, N; such that x_i+1,j+1 ? x_i+1,j ? x_i,j+1 + x_ij ≥ 0, for every (i, j) where 1 ≤ i ≤ M ? 1, 1 ≤ j ≤ N ? 1? For M = N = 3, or smaller values of M, N it is shown that there is a “uniform” rule. However, for max(M, N) > 3 and min(M, N) ≥ 3, it is proved that no uniform rule can be given. For M = 3, N = 4 a way of labeling is demonstrated. For general M, N the problem is still open although, for a special case where all the numbers are 0's and 1's, a solution is given.     

A limit theorem for matrix-solutions of Hamiltonian systems

The following limit theorem on Hamiltonian systems (resp. corresponding Riccati matrix equations) is shown: Given(N, N)-matrices,A, B, C andn ∈ {1,…, N} with the following properties:A and kemelB(x) are constant, rank(I, A, …, A ^n?1) B(x)≠N,B(x)∈C _n(R), andB(x)(A ^T)^j-1 C(x)∈C _n-j(R) forj=1, …, n. Then \(\mathop {\lim }\limits_{x \to x_0 } \eta _1^T \left( x \right)V\left( x \right)U^{ - 1} \left( x \right)\eta _2 \left( x \right) = d_1^T \left( {x_0 } \right)U\left( {x_0 } \right)d_2 \) forx ₀∈R, whenever the matricesU(x), V(x) are a conjoined basis of the differential systemU′=AU + BV, V′=CU?A ^TV, and whenever η_i(x)∈R ^N satisfy η_i(x ₀)=U(x ₀)d _i ∈ imageU(x ₀) η′_i-Aη_ni(x) ∈ imageB(x),B(x)(η′_i(x)-Aη_i(x)) ∈C _n-1 R fori=1,2.     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.     

Existence of Infinitely Many Non-Trivial Bifurcation Points

We consider the non-linear two point boundary value problem where λ > 0,f ∈ C², f′ ≥ 0, f(0) < 0 and lim_{u → ∞} f(u) > 0. By considering the non-negative as well as all sign changing solutions, we establish the existence of infinitely many non-trivial bifurcation points. Further, when f is superlinear, we prove that there exists a constant λ* > 0, such that for each λ ∈ (0, λ*) there are exactly two solutions with m interior zeros for every m = 1,2, …We apply our results to the case when f(u) = u ³ - k; k > 0, and also discuss the evolution of the bifurcation diagram as k → 0.     

Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem $$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$ in a conical domain Ω ? ? ⁿ , n ≥ 3, where 0 ≤ V (x) ∈ L₁(Ω), 0 ≤ ?(x) ∈ L₁(?Ω) and ?(x) is continuous on the boundary ?Ω. It is proved that there exists a constant C _*(n) = (n ? 2)²/4 such that if V ₀(x) = (c + λ ₁)|x|^?2, then, for 0 ≤ c ≤ C _*(n) and V(x) ≤ V ₀(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ?(x) ∈ L ₁(?Ω); for c > C _*(n) and V(x) ≥ V ₀(x) in Ω, this problem has no nonnegative solutions if ?(x) > 0.     

Periods of some nonlinear shift registers

We determine the set of all possible least periods of shift register sequences for non-linear feedback functions of the form f(x₀,…,x_m?1) = x₀ + Π_i=1^k (x_i + b_i) where m ? k + 1 ? 3 and the least period of the k-block b₁…b_k itself.     

A note on an open question on ω- and τ- Matrices

In a recent paper by Engel and Schneider, it was asked if, for every n ? 1, A ∈ τ_<n> implies (A+D) ∈ τ_<n> for every D = diag[d₁, d₂,… d_n] with d_i ? 0, 1 ? i ? n. We answer this question in the negative. More precisely, we show that for, any n ? 3, the set

(τ _{< n>}): = {D ∈C^n,n:(A+D)∈τ < n> for all A∈τ_<n>} is exactly given by

(Gt_<n>) = {γI_n:γ ? 0}.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

Permutation-operator inequalities on certain symmetry classes of tensors

If A∈T(m, N), the real-valued N-linear functions on E_m, and σ∈S_N, the symmetric group on {…,N}, then we define the permutation operator P_σ: T(m, N) → T(m, N) such that P_σ(A)(x₁,x₂,…,x_N = A(x_σ(1),x_σ(2),…, x_σ(N)). Suppose Σ^q_i=1n_i = N, where the n_i are positive integers. In this paper we present a condition on σ that is sufficient to guarantee that 〈P_σ(A₁?A₂???A_q),A₁?A₂?? ? A_q〉 ? 0 for A_i∈S(m, n_i), where S(m, n_i) denotes the subspace of T(m, n_i) consisting of all the fully symmetric members of T(m, n_i). Also we present a broad generalization of the Neuberger identity which is sometimes useful in answering questions of the type described below. Suppose G and H are subgroups of S_N. We let T_G(m, N) denote all A∈T(m, N) such that P_σ(A) = A for all σ∈G. We define the symmetrizer $\text{S}$_G: T(m, N)→T_G(m,N) such that $\text{S}{}_{\mathrm{G}}\text{(A) = 1/|G|Σ}{}_{\mathrm{\sigma \in G}}\text{P}{}_{\mathrm{\sigma }}\text{(A)}$. Suppose H is a subgroup of G and A∈T_H(m, N). Clearly $\text{6}\text{S}{}_{\mathrm{<mtext>G</mtext>}}\text{6(A) 6? 6A6.}$ We are interested in the reverse type of comparison. In particular, if D is a suitably chosen subset of T_H(m,N), then can we explicitly present a constant C>0 such that $\text{6}\text{S}{}_{\mathrm{G}}\text{(A)6?C6A6}$ for all A∈D?     

Partition theorems for Abelian groups

Let A be a finite matrix with integral entries and G be an Abelian group. Define A to be partition regular in G if for every partition of G/(0) into finitely many classes there exist elemens x₁,…,x_m contained in one class such that A(x₁,…,x_m)^T = 0. Theorem. A is partition regular in G iff at least one of the following statements holds. (i) There is x ∈ G/(0) such that A(x,…,x)^T = 0. (ii) A is partition regular in Z_p^?₀ (p prime) and Z_p^?₀ ? G. (iii) A is partition regular in Z and the set of orders of elements in G is unbounded.     

Approximation from Above of Systems of Differential Inclusions with Non-Lipschitzian Right-Hand Side

Suppose that ? ⁿ is the p-dimensional space with Euclidean norm ∥ ? ∥, K (? ^p ) is the set of nonempty compact sets in ? ^p , ?₊ = [0, +∞), D = ?₊ × ? ^m × ? ⁿ × [0, a], D ₀ = ?₊ × ? ^m , F ₀: D ₀ → K (? ^m ), and co F ₀ is the convex cover of the mapping F ₀. We consider the Cauchy problem for the system of differential inclusions $$\dot x \in \mu F(t,x,y,\mu ),\quad \dot y \in G(t,x,y,\mu ),\quad x(0) = x_0 ,\quad y(0) = y_0$$ with slow x and fast y variables; here F: D → K (? ^m ), G: D → K (? ⁿ ), and μ ∈ [0, a] is a small parameter. It is assumed that this problem has at least one solution on [0, 1/μ] for all sufficiently small μ ∈ [0, a]. Under certain conditions on F, G, and F ₀, comprising both the usual conditions for approximation problems and some new ones (which are weaker than the Lipschitz property), it is proved that, for any ε > 0, there is a μ₀ > 0 such that for any μ ∈ (0, μ₀] and any solution (x _μ(t), y _μ(t)) of the problem under consideration, there exists a solution u _μ(t) of the problem ${\dot u}$ ∈ μ co F ₀ (t, u), u(0) = x ₀ for which the inequality ∥x _μ(t) ? u _μ(t)∥ < ε holds for each t ∈ [0, 1/μ].     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

A note on the Fermat equation

Fix one of the base variables x₁, x₂, x₃ in the equation x₁^x4 + x₂^x4 = x₃^x4 subject to x₁, x₂, x₃, x₄ ∈ $\text{Z}$, x₁x₂x₃ ≠ 0, x₄ ≥ 3, gcd(x₁, x₂, x₃) = 1. Then the solutions are bounded. There are no solutions (x₁, x₂, x₃, x₄) with $\text{P(x}{}_{\mathrm{i}}\text{) < ε}{}_0\text{(}\text{log log}\text{|x}{}_{\mathrm{i}}\text{|)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$ for some i ∈ {1, 2, 3}.     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

Construction of a Jacobi matrix from spectral data

A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying b_i>0 for i=1,…,n?1 and ω₁<μ₁<?<μ_n?1<ω_n, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements a_i (i=1,…,m) and off diagonal elements b_i (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new.