Maximizing Sequence-Submodular Functions and Its Application to Online Advertising

Motivated by applications in online advertising, we consider a class of maximization problems where the objective is a function of the sequence of actions and the running duration of each action. For these problems, we introduce the concepts of sequence-submodularity and sequence-monotonicity, which extend the notions of submodularity and monotonicity from functions defined over sets to functions defined over sequences. We establish that if the objective function is sequence-submodular and sequence-nondecreasing, then there exists a greedy algorithm that achieves [Formula: see text] of the optimal solution. We apply our algorithm and analysis to two applications in online advertising: online ad allocation and query rewriting. We first show that both problems can be formulated as maximizing nondecreasing sequence-submodular functions. We then apply our framework to these two problems, leading to simple greedy approaches with guaranteed performances. In particular, for the online ad allocation problem, the performance of our algorithm is [Formula: see text], which matches the best known existing performance, and for the query rewriting problem, the performance of our algorithm is [Formula: see text], which improves on the best known existing performance in the literature. This paper was accepted by Chung Piaw Teo, optimization.

Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

THE D_p^q (∆^+r )-STATISTICAL CONVERGENCE

Let p(n) and q(n) be nondecreasing sequence of positive integers such that p(n) < q(n) and limn→∞ q(n) = ∞ holds. For any r ∈ Z^+, we define D_p,q^+r- statistical convergence of ∆^+r x where ∆^+r is r- th difference of the sequence (x_n). The main results in this paper consist in determining sets of sequences χ and χ' of the form [D_ p^q]_0 α satisfying χ ⊂ [D_p^q]_0(∆^+r ) ⊂ χ ' and sets φ and φ' of the form [D_p^q]_α satisfying φ ≤ [D_p^q]_∞(∆^+r ) ≤ φ' .

On the spectrum of Robin boundary p-Laplacian problem

We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition\left\{ {\matrix{ { - {\Delta _p}u = \lambda {{\left| u \right|}^{p - 2}}u\,\,\,in\,\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u.v + {{\left| u \right|}^{p - 2}}u = 0\,\,\,on\,\,\Gamma .} \hfill \cr } } \right.We successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λn↗∞. To this end we endow W1,p(Ω) with a norm invoking the trace and use the duality mapping on W1,p (Ω) to apply mini-max arguments on C1-manifold.

Existence of moments of a counting process and convergence in multidimensional time

Starting with independent, identically distributed random variables X1,X2... and their partial sums (Sn), together with a nondecreasing sequence (b(n)), we consider the counting variable N=∑n1(Sn>b(n)) and aim for necessary and sufficient conditions on X1 in order to obtain the existence of certain moments for N, as well as for generalized counting variables with weights, and multi-index random variables. The existence of the first moment of N when b(n)=εn, i.e. ∑n=1∞ℙ(|Sn|>εn)<∞, corresponds to the notion of complete convergence as introduced by Hsu and Robbins in 1947 as a complement to Kolmogorov's strong law.

On the p-Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of the p-biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding.

Approximation and Interpolation by Entire Functions of Several Variables

Let f : ℝn → ℝ be C∞ and let h: ℝn → ℝ be positive and continuous. For any unbounded nondecreasing sequence ﹛ck﹜ of nonnegative real numbers and for any sequence without accumulation points ﹛xm﹜ in ℝn, there exists an entire function g : ℂn → ℂ taking real values on ℝn such thatThis is a version for functions of several variables of the case n = 1 due to L. Hoischen.

On the oscillation of the solutions to delay and difference equations

Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t₀, (1) where p, τ ∈ C ([t₀,∞, ℝ⁺, τ(t) is nondecreasing, τ(t) < t for t ≥ t⁰ and lim_t→∞ τ(t) = ∞, and the (discrete analogue) difference equation Δx(n) + p(n)x(τ(n)) = 0, n= 0, 1, 2,…, (1)ʹ where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and lim_n→∞ τ(n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.

A nonlinear boundary problem involving thep-bilaplacian operator

We show some new Sobolev's trace embedding that we apply to prove that the fourth-order nonlinear boundary conditionsΔp2u+|u|p−2u=0inΩand−(∂/∂n)(|Δu|p−2Δu)=λρ|u|p−2uon∂Ωpossess at least one nondecreasing sequence of positive eigenvalues.

A note on the strong law of large numbers for associated sequences

We prove that the sequence{bn−1∑i=1n(Xi−EXi)}n≥1converges a.e. to zero if{Xn,n≥1}is anassociatedsequence of random variables with∑n=1∞bkn−2Var(∑i=kn−1+1knXi)<∞where{bn,n≥1}is a positive nondecreasing sequence and{kn,n≥1}is a strictly increasing sequence, both tending to infinity asntends to infinity and0<a=infn≥1bknbkn+1−1≤supn≥1bknbkn+1−1=c<1.