On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

Diagonals and numerical ranges of direct sums of matrices

For any n-by-n matrix A  , we consider the maximum number k=k(A)$k=k(A)$ for which there is a k-by-k compression of A   with all its diagonal entries in the boundary ∂W(A)$\partial W(A)$ of the numerical range W(A)$W(A)$ of A. If A   is a normal or a quadratic matrix, then the exact value of k(A)$k(A)$ can be computed. For a matrix A   of the form B⊕C$B\oplus C$, we show that k(A)=2$k(A)=2$ if and only if the numerical range of one summand, say, B is contained in the interior of the numerical range of the other summand C   and k(C)=2$k(C)=2$. For an irreducible matrix A  , we can determine exactly when the value of k(A)$k(A)$ equals the size of A  . These are then applied to determine k(A)$k(A)$ for a reducible matrix A   of size 4 in terms of the shape of W(A)$W(A)$.     

The elementary symmetric functions of a reciprocal polynomial sequence

Erdös and Niven proved in 1946 that for any positive integers m and d, there are at most finitely many integers n   for which at least one of the elementary symmetric functions of 1/m,1/(m+d),…,1/(m+(n−1)d)$1/m,1/(m+d),\dots ,1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if n?4$n?4$, then none of the elementary symmetric functions of 1/m,1/(m+d),…,1/(m+(n−1)d)$1/m,1/(m+d),\dots ,1/(m+(n-1)d)$ is an integer for any positive integers m and d. Let f   be a polynomial of degree at least 2 and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of 1/f(1),1/f(2),…,1/f(n)$1/f(1),1/f(2),\dots ,1/f(n)$ is an integer except for f(x)=x^m$f(x)=x^m$ with m?2$m?2$ being an integer and n=1$n=1$.     

Simultaneous bifurcation of limit cycles from a linear center with extra singular points

The period annuli of the planar vector field x^′=−yF(x,y)$x^{\prime }=-yF(x,y)$, y^′=xF(x,y)$y^{\prime }=xF(x,y)$, where the set {F(x,y)=0}$\{F(x,y)=0\}$ consists of k   different isolated points, is defined by k+1$k+1$ concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n  . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1$k=1$, the provided upper bound is reached. Finally, the case k=2$k=2$ is also treated.     

Exact finite-difference schemes for two-dimensional linear systems with constant coefficients

An exact finite-difference scheme for a system of two linear differential equations with constant coefficients, (d/dt)x(t)=Ax(t)$(\mathrm{d}/\mathrm{d}t)\mathbf{x}(t)=A\mathbf{x}(t)$, is proposed. The scheme is different from what was proposed by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, New Jersey, 1994, p. 147], in which the derivatives of the two equations are formed differently. Our exact scheme is in the form of (1/φ(h))(_xk+1-_xk)=A[θ_xk+1+(1-θ)_xk]$(1/\phi (h))({\mathbf{x}}_{k+1}-{\mathbf{x}}_k)=A[\theta {\mathbf{x}}_{k+1}+(1-\theta ){\mathbf{x}}_k]$; both derivatives are in the same form of (_xk+1-_xk)/φ(h)$(x_{k+1}-x_k)/\phi (h)$.     

Identities for the Ramanujan zeta function

We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that L(Δ,k)$L(\mathrm{\Delta },k)$ is a period in the sense of Kontsevich and Zagier when k?12$k?12$. As an illustration, we reduce L(Δ,k)$L(\mathrm{\Delta },k)$ to explicit integrals of hypergeometric and algebraic functions when k∈{12,13,14,15}$k\in \{12,13,14,15\}$.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H²${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

Decomposing a graph into bistars

Bárat and the present author conjectured that, for each tree T  , there exists a natural number _kT$k_T$ such that the following holds: If G   is a _kT$k_T$-edge-connected graph such that |E(T)|$|E(T)|$ divides |E(G)|$|E(G)|$, then G has a T-decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T  . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S(k,k+1)$S(k,k+1)$.     

New approach for the existence of pseudo almost periodic solutions for some second order differential equation with piecewise constant argument

The following equation d²/dt²(x(t)+px(t-1))=qx(2[(t+1)/2])+f(t)${\mathrm{d}}^2/\mathrm{d}{t}^2(x(t)+\mathit{px}(t-1))=\mathit{qx}(2[(t+1)/2])+f(t)$ is considered and necessary and sufficient conditions are given in order to ensure the existence and uniqueness of pseudo almost periodic solutions.     

Two-colorings with many monochromatic cliques in both colors

Color the edges of the n-vertex complete graph in red and blue, and suppose that red k-cliques are fewer than blue k-cliques. We show that the number of red k  -cliques is always less than _ckn^k$c_k{n}^k$, where _ck∈(0,1)$c_k\in (0,1)$ is the unique root of the equation z^k=(1−z)^k+kz(1−z)^k−1$z^k={(1-z)}^k+kz{(1-z)}^{k-1}$. On the other hand, we construct a coloring in which there are at least _ckn^k−O(n^k−1)$c_k{n}^k-O(n^{k-1})$ red k-cliques and at least the same number of blue k-cliques.     

A new characterization of comonotonicity and its application in behavioral finance

It is well-known that an Rⁿ${\mathbb{R}}^n$-valued random vector (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ is comonotonic if and only if (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ and (_Q1(U),_Q2(U),?,_Qn(U))$(Q_1(U),Q_2(U),?,Q_n(U))$ coincide in distribution, for any random variable U   uniformly distributed on the unit interval (0,1)$(0,1)$, where _Qk(⋅)$Q_k(\cdot )$ are the quantile functions of _Xk$X_k$, k=1,2,?,n$k=1,2,?,n$. It is natural to ask whether (_X1,_X2,?,_Xn)$(X_1,X_2,?,X_n)$ and (_Q1(U),_Q2(U),?,_Qn(U))$(Q_1(U),Q_2(U),?,Q_n(U))$ can coincide almost surely for some special U. In this paper, we give a positive answer to this question by construction. We then apply this result to a general behavioral investment model with a law-invariant preference measure and develop a universal framework to link the problem to its quantile formulation. We show that any optimal investment output should be anti-comonotonic with the market pricing kernel. Unlike previous studies, our approach avoids making the assumption that the pricing kernel is atomless, and consequently, we overcome one of the major difficulties encountered when one considers behavioral economic equilibrium models in which the pricing kernel is a yet-to-be-determined unknown random variable. The method is applicable to general models such as risk sharing model.