Normal families of meromorphic functions and shared functions

The paper is devoted to the normal families of meromorphic functions and shared functions. Generalizing a result of Chang (2013), we prove the following theorem. Let h (≠≡ 0,∞) be a meromorphic function on a domain D and let k be a positive integer. Let F be a family of meromorphic functions on D, all of whose zeros have multiplicity at least k + 2, such that for each pair of functions f and g from F, f and g share the value 0, and f^(k) and g^(k) share the function h. If for every f ∈ F, at each common zero of f and h the multiplicities m_f for f and m_h for h satisfy m_f ≥ m_h + k + 1 for k > 1 and m_f ≥ 2m_h + 3 for k = 1, and at each common pole of f and h, the multiplicities nf for f and nh for h satisfy n_f ≥ n_h + 1, then the family F is normal on D.     

On existence of a universal function for <Emphasis Type="Italic">L</Emphasis> <Superscript> <Emphasis Type="Italic">p</Emphasis> </Superscript>[0, 1] with <Emphasis Type="Italic">p</Emphasis>∈(0, 1)

We show that, for every number p ∈ (0, 1), there is g ∈ L¹[0, 1] (a universal function) that has monotone coefficients c_k(g) and the Fourier–Walsh series convergent to g (in the norm of L¹[0, 1]) such that, for every f ∈ L^p[0, 1], there are numbers δ_k = ±1, 0 and an increasing sequence of positive integers N_q such that the series ∑ _k=0^+∞δ_kc_k(g)W_k (with {W_k} theWalsh system) and the subsequence \(\sigma _{{N_q}}^{\left( \alpha \right)}\), α ∈ (?1, 0), of its Cesáro means converge to f in the metric of L^p[0, 1].     

A cycle cover of a 2-edge-connected graph embedded with large face-width on an orientable surface

In 1985, Alon and Tarsi conjectured that the length of a shortest cycle cover of a bridgeless graph H is at most 7/5 |E(H|). The conjecture is still open. Let G be a 2-edge-connected graph embedded with face-width k on the non-spherical orientable surface S_g. We give an upper bound on the length of a cycle cover of G. In particular, if g = 1 and k ≥ 48, or g = 2 and k ≥ 427, or g ≥ 3 and k ≥ 288(4^g - 1), then the upper bound is 7/5 |E(G|), which means that Alon and Tarsi’s conjecture holds for such a graph.     

Generalized separants of differential polynomials

Let f ∈ K{y} be an element of the ring of differential polynomials in one differential variable y with one differential operator δ. For any variable y _k , the polynomial g = δ ⁿ (f) can be represented in the form g = A _k y _k + go, where go does not depend on y _k . If y _k is the leader of g, then A _k is a separant of the polynomial f. A formula for A _k is obtained for sufficiently large numbers n and k and some applications of this formula are presented.     

Dynamical magnetic susceptibility in the spin-fermion model for cuprate superconductors

Using the method of diagram techniques for the spin and Fermi operators in the framework of the SU(2)-invariant spin-fermion model of the electron structure of the CuO₂plane of copper oxides, we obtain an exact representation of the Matsubara Green’s function D_⊥(k, iω _m ) of the subsystem of localized spins. This representation includes the Larkin mass operator Σ_L(k, iω _m ) and the strength and polarization operators P(k, iω _m ) and Π(k, iω _m ). The calculation in the one-loop approximation of the mass and strength operators for the Heisenberg spin system in the quantum spin-liquid state allows writing the Green’s function D_⊥(k, iω _m ) explicitly and establishing a relation to the result of Shimahara and Takada. An essential point in the developed approach is taking the spin-polaron nature of the Fermi quasiparticles in the spin-fermion model into account in finding the contribution of oxygen holes to the spin response in terms of the polarization operator Π(k, iω _m ).     

The program iteration method in a game problem of guidance

Let a sequence of d-dimensional vectors n _k = (n _k ¹ , n _k ² ,..., n _k ^d ) with positive integer coordinates satisfy the condition n _k ^j = α _j m _k +O(1), k ∈ ?, 1 ≤ j ≤ d, where α ₁ > 0,..., α _d > 0 and {m _k } _k=1 ^∞ is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums \({S_{{m_k}}}\) (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ? and f ∈ φ(L)(ln⁺ L) ^d?1([0, 2π) ^d ), the sequence \({S_{{n_k}}}\) (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.     

Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ ₀ with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem

$$ (P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}}\ \text { in }\ (M,g) . $$

We prove that for any k ∈ ?, there exists ε _k > 0 such that for all ε ∈ (0, ε _k ) the problem (P _?? ) has a symmetric solution u _ε , which looks like the superposition of k positive bubbles centered at the point ξ ₀ as ε → 0. In particular, ξ ₀ is a towering blow-up point.

     

Inequalities between best polynomial approximations and some smoothness characteristics in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ_k(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L₂. On the basis of this, for differentiable functions in the classes L₂^r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω_k. For classes of functions defined by their smoothness characteristics Λ_k(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.     

Engel-type subgroups and length parameters of finite groups

Let g be an element of a finite group G. For a positive integer n, let E _n (g) be the subgroup generated by all commutators [...[[x, g], g],..., g] over x ∈ G, where g is repeated n times. By Baer’s theorem, if E _n (g) = 1, then g belongs to the Fitting subgroup F(G). We generalize this theorem in terms of certain length parameters of E _n (g). For soluble G we prove that if, for some n, the Fitting height of E _n (g) is equal to k, then g belongs to the (k+1)th Fitting subgroup F_k+1(G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1(H)* is the inverse image of the generalized Fitting subgroup F*(H/F*_i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of E _n (g) is equal to k, then g belongs to F*_f(k,m)(G), where f(k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(E _n (g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

Renormalized coupling constants for the three-dimensional scalar <Emphasis Type="Italic">λ</Emphasis><Emphasis Type="Italic">ϕ</Emphasis><Superscript>4</Superscript> field theory and pseudo-<Emphasis Type="Italic">ε</Emphasis>-expansion

The renormalized coupling constants g _2k that enter the equation of state and determine nonlinear susceptibilities of the system have universal values g _2k ^* at the Curie point. We use the pseudo-ε-expansion approach to calculate them together with the ratios R _2k = g _2k /g ₄ ^k-1 for the three-dimensional scalar λ ? ⁴ field theory. We derive pseudo-ε-expansions for g ₆ ^* , g ₈ ^* , R ₆ ^* , and R ₈ ^* in the five-loop approximation and present numerical estimates for R ₆ ^* and R ₈ ^* . The higher-order coefficients of the pseudo-ε-expansions for g ₆ ^* and R ₆ ^* are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives R ₆ ^* = 1.650, while the recent lattice calculation gave R ₆ ^* = 1.649(2). The pseudo-ε-expansions of g ₈ ^* and R ₈ ^* are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for R ₈ ^* gives the estimate R ₈ ^* = 0.890, differing only slightly from the values R ₈ ^* = 0.871 and R ₈ ^* = 0.857 extracted from the results of lattice and field theory calculations.     

Some class 1 graphs on <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g _c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g _c -coloring with k colors is called the g _c -chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δ _g (G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

Metric projection onto subsets of compact connected two-dimensional Riemannian manifolds

The paper is focused on combinatorial properties of the metric projection P _E of a compact connected Riemannian two-dimensional manifold M ² onto its subset E consisting of k closed connected sets E _j . A point x ∈ M ² is called singular if P _E (x) contains points from at least three distinct E _j . An exact estimate of the number of singular points is obtained in terms of k and the type of the manifold M ². A similar estimate is proved for subsets E of a normed plane consisting of a finite number of connected components.     

A gradual rank increasing process for matrix completion

The matrix completion problem is easy to state: let A be a given data matrix in which some entries are unknown. Then, it is needed to assign “appropriate values” to these entries. A common way to solve this problem is to compute a rank-k matrix, B _k , that approximates A in a least squares sense. Then, the unknown entries in A attain the values of the corresponding entries in B _k . This raises the question of how to determine a suitable matrix rank. The method proposed in this paper attempts to answer this question. It builds a finite sequence of matrices \(B_{k}, k = 1, 2, \dots \), where B _k is a rank-k matrix that approximates A in a least squares sense. The computational effort is reduced by using B _k-1 as starting point in the computation of B _k . The ability of B _k to serve as substitute for A is measured with two objective functions: a “training” function that measures the distance between the known part of A and the corresponding part of B _k , and a “probe” function that assesses the quality of the imputed entries. Watching the changes in these functions as k increases enables us to find an optimal matrix rank. Numerical experiments illustrate the usefulness of the proposed approach.     

On the length of a finite group and of its 2-generator subgroups

The nonsoluble length λ(G) of a finite group G is defined as the minimum number of nonsoluble factors in a normal series of G each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group G is the least number h = h* (G) such that F* _h (G) = G, where F* ₁ (G) = F* (G) is the generalized Fitting subgroup, and F* _i+1(G) is the inverse image of F* (G/F*_i (G)). In the present paper we prove that if λ(J) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.     

Conductors in <Emphasis Type="Italic">p</Emphasis>-adic families

Let the function \(s_g\) map a positive integer to the sum of its digits in the base g. A number k is called n-flimsy in the base g if \(s_g(nk)<s_g(k)\). Clearly, given a base g, \(g\geqslant 2\), if n is a power of g, then there does not exist an n-flimsy number in the base g. We give a constructive proof of the existence of an n-flimsy number in the base g for all the other values of n (such an existence follows from the results of Schmidt and Steiner, but the explicit construction is a novelty). Our algorithm for construction of such a number, say k, is very flexible in the sense that, by easy modifications, we can impose further requirements on k—k ends with a given sequence of digits, k begins with a given sequence of digits, k is divisible by a given number (or belongs to a certain congruence class modulo a given number), etc.     

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

Some irreducibility and indecomposability results for truncated binomial polynomials of small degree

In this paper, we show that the truncated binomial polynomials defined by \(P_{n,k}(x)={\sum }_{j=0}^{k} {n \choose j} x^{j}\) are irreducible for each k≤6 and every n≥k+2. Under the same assumption n≥k+2, we also show that the polynomial P _n,k cannot be expressed as a composition P _n,k (x) = g(h(x)) with \(g \in \mathbb {Q}[x]\) of degree at least 2 and a quadratic polynomial \(h \in \mathbb {Q}[x]\). Finally, we show that for k≥2 and m,n≥k+1 the roots of the polynomial P _m,k cannot be obtained from the roots of P _n,k , where m≠n, by a linear map.     

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping \({T:A\longrightarrow B}\) does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

Non-singular Solutions of Normalized Ricci Flow on Noncompact Manifolds of Finite Volume

The main result of this paper shows that if g(t) is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold M of finite volume, then the Euler characteristic number χ(M)≥0. Moreover, if χ(M)≠0, there exists a sequence of times t _k →∞, a double sequence of points \(\{p_{k,l}\}_{l=1}^{N}\) and domains \(\{U_{k,l}\}_{l=1}^{N}\) with p _k,l∈U _k,l satisfying the following:

1. (i)
   \(\mathrm{dist}_{g(t_{k})}(p_{k,l_{1}},p_{k,l_{2}})\rightarrow\infty\) as k→∞, for any fixed l₁≠l₂;
    
2. (ii)
   for each l, (U _k,l,g(t _k ),p _k,l) converges in the \(C_{\mathrm{loc}}^{\infty}\) sense to a complete negative Einstein manifold (M _∞,l ,g _∞,l ,p _∞,l ) when k→∞;
    
3. (iii)
   \(\operatorname {Vol}_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\rightarrow0\) as k→∞.