Abelian integrals and period functions for quasihomogeneous Hamiltonian vector fields

In this paper, we study the number of zeros of Abelian integrals and the monotonicity of period functions for planar quasihomogeneous Hamiltonian vector fields. The result for Abelian integrals extends the recent work of Li et al. [C. Li, W. Li, J. Llibre, Z. Zhang, Polynomial systems: A lower bound for the weakened 16th Hilbert problem, Extracta Math. 16 (3) (2001) 441–447] and Llibre and Zhang [J. Llibre, X. Zhang, On the number of limit cycles for some perturbed Hamiltonian polynomial systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2) (2001) 161–181].     

Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, αi∉(−1/2,1/2), the appropriately defined Riesz transforms , j=1,2,…,d, are Calderón-Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α∈(−1/2,1/2), by means of a local Calderón-Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy-Riemann type equations and to recover the Riesz-Laguerre transforms on the boundary. The two specific values of α, (−1/2,…,−1/2) and (1/2,…,1/2), are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.     

Riesz transforms for multi-dimensional Laguerre function expansions

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of type α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, the appropriately defined Riesz-Laguerre transforms , j=1,2,…,d, are Calderón-Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. Similar results are obtained for all higher order Riesz-Laguerre transforms. The conjugate Poisson integrals are shown to satisfy a system of equations of Cauchy-Riemann type and to recover the Riesz-Laguerre transforms on the boundary.     

First integrals and normal forms for germs of analytic vector fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235-255] on periodic systems. Then in the space of analytic autonomous systems in C2n with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.     

Spectra of principal submatrices of nonnegative matrices

Let A be an n×n nonnegative matrix with the spectrum (λ₁,λ₂,…,λ_n) and let A₁ be an m×m principal submatrix of A with the spectrum (μ₁,μ₂,…,μ_m). In this paper we present some cases where the realizability of (μ₁,μ₂,…,μ_m,ν₁,ν₂,…,ν_s) implies the realizability of (λ₁,λ₂,…,λ_n,ν₁,ν₂,…,ν_s) and consider the question whether this holds in general. In particular, we show that the list

(λ₁,λ₂,…,λ_n,-μ₁,-μ₂,…,-μ_m)     

Two-step-by-two-step PIRK-type PC methods based on Gauss-Legendre collocation points

This paper concerns with parallel predictor-corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of s-stage collocation Gauss-Legendre Runge-Kutta (RK) methods based on c₁,…,c_s and the 2s-stage collocation RK methods based on c₁,…,c_s,1+c₁,…,1+c_s. At nth integration step, the stage values of the 2s-stage collocation RK methods evaluated at t_n+(1+c₁)h,…,t_n+(1+c_s)h can be used as the stage values of the collocation Gauss-Legendre RK method for (n+2)th integration step. By this way, we obtain the corrector methods in which the integration processes can be proceeded two-step-by-two-step. The resulting parallel PC iteration methods which are called two-step-by-two-step (TBT) parallel-iterated RK-type (PIRK-type) PC methods based on Gauss-Legendre collocation points (two-step-by-two-step PIRKG methods or TBTPIRKG methods) give us a faster integration process. Fixed step size applications of these TBTPIRKG methods to the three widely used test problems reveal that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RK methods (PIRK methods) and sequential codes ODEX, DOPRI5 and DOP853 available from the literature.     

Line graph eigenvalues and line energy of caterpillars

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d?3 and n?2(d-1). Let p=[p₁,p₂,…,p_d-1] with p₁?1,p₂?1,…,p_d-1?1 such that

p₁+p₂+?+p_d-1=n-d+1.     

A survey on labeling graphs with a condition at distance two

For positive integers k,d₁,d₂, a k-L(d₁,d₂)-labeling of a graph G is a function f:V(G)→{0,1,2,…,k} such that |f(u)-f(v)|?d_i whenever the distance between u and v is i in G, for i=1,2. The L(d₁,d₂)-number of G, λ_d1,d2(G), is the smallest k such that there exists a k-L(d₁,d₂)-labeling of G. This class of labelings is motivated by the code (or frequency) assignment problem in computer network. This article surveys the results on this labeling problem.     

Bounds on some van der Waerden numbers

For positive integers s and k₁,k₂,…,ks, the van der Waerden number w(k₁,k₂,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided.     

Boundary representations and rigidity of submodules

This paper considers two problems on the Fock type spaces Fs (0<s?1). Firstly, it is shown that on the space Fs (0<s<1), the identity representation of C^∗(I,T₁,…,Tn) is a boundary representation for the Banach subalgebra B(I,T₁,…,Tn), while on the space F₁, it is not. Secondly, it is shown that all the submodules of F₁ are rigid.     

Generalized radix representations and dynamical systems. IV

For r = (r₁,…, r_d) ∈ ?^d the mapping τ_r:?^d →?^d given byτ_r(a₁,…,a_d) = (a₂, …, a_d,−⌊r₁a₁+…+ r_da_d⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?^d there exists an integer k > 0 with τ_r^k(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).     

Pseudo-abelian integrals: Unfolding generic exponential case

We consider functions of the form , with Pi, R, and Q∈R[x,y], which are (generalized Darboux) first integrals of the polynomial system MdlogH₀=0. We assume that H₀ defines a family of real cycles in a region bounded by a polycycle.To each polynomial form η one can associate the pseudo-abelian integrals I(h) of M−1η along γ(h), which is the first order term of the displacement function of the orbits of MdH₀+δη=0.We consider Darboux first integrals unfolding H₀ (and its saddle-nodes) and pseudo-abelian integrals associated to these unfoldings. Under genericity assumptions we show the existence of a uniform local bound for the number of zeros of these pseudo-abelian integrals.The result is a part of a program to extend Varchenko-Khovanskii's theorem from abelian integrals to pseudo-abelian integrals and prove the existence of a bound for the number of their zeros in function of the degree of the polynomial system only.     

Von Neumann’s inequality for noncommuting contractions

Let T₁,…,T_d be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T₁,…,T_d) is bounded by

     

Second order regular variation and conditional tail expectation of multiple risks

For the purpose of risk management, the study of tail behavior of multiple risks is more relevant than the study of their overall distributions. Asymptotic study assuming that each marginal risk goes to infinity is more mathematically tractable and has also uncovered some interesting performance of risk measures and relationships between risk measures by their first order approximations. However, the first order approximation is only a crude way to understand tail behavior of multiple risks, and especially for sub-extremal risks. In this paper, we conduct asymptotic analysis on conditional tail expectation (CTE) under the condition of second order regular variation (2RV). First, the closed-form second order approximation of CTE is obtained for the univariate case. Then CTE of the form E[X₁∣g(X₁,…,X_d)>t], as t→∞, is studied, where g is a loss aggregating function and (X₁,…,X_d)?(RT₁,…,RT_d) with R independent of (T₁,…,T_d) and the survivor function of R satisfying the condition of 2RV. Closed-form second order approximations of CTE for this multivariate form have been derived in terms of corresponding value at risk. For both the univariate and multivariate cases, we find that the first order approximation is affected by only the regular variation index −α of marginal survivor functions, while the second order approximation is influenced by both the parameters for first and second order regular variation, and the rate of convergence to the first order approximation is dominated by the second order parameter only. We have also shown that the 2RV condition and the assumptions for the multivariate form are satisfied by many parametric distribution families, and thus the closed-form approximations would be useful for applications. Those closed-form results extend the study of Zhu and Li (submitted for publication).     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

Equitable specialized block-colourings for 4-cycle systems—I

A block-colouring of a 4-cycle system (V,B) of order v=1+8k is a mapping ?:B→C, where C is a set of colours. Every vertex of a 4-cycle system of order v=8k+1 is contained in blocks and r is called, using the graph theoretic terminology, the degree or the repetition number. A partition of degree r into s parts defines a colouring of type s in which the blocks containing a vertex x are coloured exactly with s colours. For a vertex x and for i=1,2,…,s, let B_x,i be the set of all the blocks incident with x and coloured with the ith colour. A colouring of type s is equitable if, for every vertex x, we have |B_x,i−B_x,j|≤1, for all i,j=1,…,s. In this paper we study bicolourings, tricolourings and quadricolourings, i.e. the equitable colourings of type s with s=2, s=3 and s=4, for 4-cycle systems.     

The 16th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the 16th Hilbert problem on algebraic limit cycles: Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m? In [J. Llibre, R. Ramírez, N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010) 1401-1409] Llibre, Ramírez and Sadovskaia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: Is1+(m−1)(m−2)/2the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have?In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre et al.?s as a special one. For the polynomial vector fields having only non-dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.     

Exponential families of mixed Poisson distributions

If I=(I₁,…,I_d) is a random variable on [0,∞)^d with distribution μ(dλ₁,…,dλ_d), the mixed Poisson distribution MP(μ) on N^d is the distribution of (N₁(I₁),…,N_d(I_d)) where N₁,…,N_d are ordinary independent Poisson processes which are also independent of I. The paper proves that if F is a natural exponential family on [0,∞)^d then MP(F) is also a natural exponential family if and only if a generating probability of F is the distribution of v₀+v₁Y₁+?+v_qY_q for some q?d, for some vectors v₀,…,v_q of [0,∞)^d with disjoint supports and for independent standard real gamma random variables Y₁,…,Y_q.     

Degree complete graphs

Ryser [Combinatorial Mathematics, Carus Mathematical Monograph, vol. 14, Wiley, New York, 1963] introduced a partially ordered relation ‘?’ on the nonnegative integral vectors. It is clear that if S=(s₁,s₂,…,s_n) is an out-degree vector of an orientation of a graph G with vertices 1,2,…,n, then

(Π)