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"Wang, Larry X. W."
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Cost-effectiveness analysis of nivolumab plus standard chemotherapy versus chemotherapy alone for the first-line treatment of unresectable advanced or metastatic gastric cancer, gastroesophageal junction cancer, and esophageal adenocarcinoma
Background Nivolumab plus standard chemotherapy has significant clinical benefits for unresectable advanced or metastatic gastric cancer, gastroesophageal junction cancer, and esophageal adenocarcinoma (GC/GEJC/EAC). However, nivolumab is expensive, necessitating a cost-effectiveness evaluation. Aim This study aimed to evaluate the cost-effectiveness of nivolumab plus standard chemotherapy vs. chemotherapy alone for unresectable advanced or metastatic GC/GEJC/EAC from the Chinese healthcare system perspective. This study was based on randomized clinical trial data from the CheckMate-649 clinical trial (NCT02872116) published in Lancet (June 2021). Method A Markov model was used to assess the cost-effectiveness of nivolumab plus standard chemotherapy versus chemotherapy alone for unresectable advanced or metastatic GC/GEJC/EAC. Drug costs were collected from Tianjin Medical Purchasing Center in 2021, and utility values of health states were obtained from the literature. The reliability of model was assessed with one-way and probabilistic sensitivity analyses. Main outcome measure The main outcomes were costs, quality-adjusted life-years (QALYs) and the incremental cost-effectiveness ratio (ICER). Results Over a 10-year horizon, the outputs were 1.19 QALYs at a cost of$78,814.9 and 0.88 QALYs at a cost of $ 19,522.3 with nivolumab plus chemotherapy and chemotherapy alone, respectively. The ICER for nivolumab plus chemotherapy versus chemotherapy alone was$191,266/QALY, exceeding the willingness-to-pay (WTP) threshold ($ 33,436/QALY). One-way sensitivity analysis revealed nivolumab cost was the most influential parameter. Conclusion Adding nivolumab is not cost-effective for unresectable advanced or metastatic GC/GEJC/EAC in the current Chinese healthcare environment.
Journal Article
Context-free Grammars for Triangular Arrays
We consider context-free grammars of the form G = {f → f^b1＋b2＋1g^a1＋a2, g → f^b1 g^a1＋1},where ai and bi are integers sub ject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T（n, k）}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn （x） =∑k=0^n T（n, k）x^k. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn＋1（x） = Pn（Tn（x））. Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Br？ndén, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh.Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers.
Journal Article
Finite differences of the logarithm of the partition function
Let p(n)p(n) denote the partition function. DeSalvo and Pak proved that p(n−1)p(n)(1+1n)>p(n)p(n+1)\\frac {p(n-1)}{p(n)}\\left (1+\\frac {1}{n}\\right )> \\frac {p(n)}{p(n+1)} for n≥2n\\geq 2. Moreover, they conjectured that a sharper inequality p(n−1)p(n)(1+π24n3/2)>p(n)p(n+1)\\frac {p(n-1)}{p(n)}\\left ( 1+\\frac {\\pi }{\\sqrt {24}n^{3/2}}\\right ) > \\frac {p(n)}{p(n+1)} holds for n≥45n\\geq 45. In this paper, we prove the conjecture of Desalvo and Pak by giving an upper bound for −Δ2logp(n−1)-\\Delta ^{2} \\log p(n-1), where Δ\\Delta is the difference operator with respect to nn. We also show that for given r≥1r\\geq 1 and sufficiently large nn, (−1)r−1Δrlogp(n)>0(-1)^{r-1}\\Delta ^{r} \\log p(n)>0. This is analogous to the positivity of finite differences of the partition function. It was conjectured by Good and proved by Gupta that for given r≥1r\\geq 1, Δrp(n)>0\\Delta ^{r} p(n)>0 for sufficiently large nn.
Journal Article
Determinantal inequalities for the partition function
Let p ( n ) denote the partition function. In this paper, we will prove that for$n\\ges 222$,$$\\left| {\\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \\cr {p(n-1)} & {p(n)} & {p(n + 1)} \\cr {p(n-2)} & {p(n-1)} & {p(n)} \\cr } } \\right| > 0.{\\rm }$$As a corollary, we deduce that p ( n ) satisfies the double Turán inequalities, that is, for$n\\ges 222$,$$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$
Journal Article
Families of Sets with Intersecting Clusters
A family of $k$-subsets $A_1,A_2,\\ldots,A_d$ on $[n]=\\{1,2,\\ldots,n\\}$ is called a $(d,c)$-cluster if the union $A_1\\cup A_2 \\cup\\cdots\\cup A_d$ contains at most $ck$ elements with $c
Journal Article
Higher order Turán inequalities for the partition function
The Turán inequalities and the higher order Turán inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre–Pólya class. A sequence {an}n≥0\\{a_{n}\\}_{n\\geq 0} of real numbers is said to satisfy the Turán inequalities or to be log-concave if for n≥1n\\geq 1, an2−an−1an+1≥0a_n^2-a_{n-1}a_{n+1}\\geq 0. It is said to satisfy the higher order Turán inequalities if for n≥1n\\geq 1, 4(an2−an−1an+1)(an+12−anan+2)−(anan+1−an−1an+2)2≥04(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\\geq 0. For the partition function p(n)p(n), DeSalvo and Pak showed that for n>25n>25, the sequence {p(n)}n>25\\{ p(n)\\}_{n> 25} is log-concave, that is, p(n)2−p(n−1)p(n+1)>0p(n)^2-p(n-1)p(n+1)>0 for n>25n> 25. It was conjectured by the first author that p(n)p(n) satisfies the higher order Turán inequalities for n≥95n\\geq 95. In this paper, we prove this conjecture by using the Hardy–Ramanujan–Rademacher formula to derive an upper bound and a lower bound for p(n+1)p(n−1)/p(n)2p(n+1)p(n-1)/p(n)^2. Consequently, for n≥95n\\geq 95, the Jensen polynomials p(n−1)+3p(n)x+3p(n+1)x2+p(n+2)x3p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3 have only distinct real zeros. We conjecture that for any positive integer m≥4m\\geq 4 there exists an integer N(m)N(m) such that for n≥N(m)n\\geq N(m), the Jensen polynomial associated with the sequence (p(n),p(n+1),…,p(n+m))(p(n),p(n+1),\\ldots ,p(n+m)) has only real zeros. This conjecture was posed independently by Ono.
Journal Article
Zeta Functions and the Log Behaviour of Combinatorial Sequences
In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζμ(x) to study the log behaviour of combinatorial sequences. We prove that ζ(x) is log-convex for x > 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n ≥ 1 is log-convex, where Bn is the nth Bernoulli number. We introduce the function θ(x) = (2ζ(x)Γ(x + 1)) 1/x, where Γ(x) is the gamma function, and we show that logθ(x) is strictly increasing for x ≥ 6. This confirms a conjecture of Sun stating that the sequence is strictly increasing. Amdeberhan et al. defined the numbers an(μ) = 2 2n+1 (n + 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence {an(μ)}n≥1 is log-convex for μ = 0 and μ = 1. By proving that ζμ(x) is log-convex for x > 1 and μ > -1, we show that the sequence {an(≥)}n>1 is log-convex for any μ > - 1. We introduce another function θμ,(x) involving ζμ(x) and the gamma function Γ(x) and we show that logθμ(x) is strictly increasing for x > 8e(μ + 2)2. This implies that
Based on Dobinski’s formula, we prove that
where Bn is the nth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of and Holder’s inequality in probability theory.
Journal Article
Schur positivity and the q-log-convexity of the Narayana polynomials
We prove two recent conjectures of Liu and Wang by establishing the strong q-log-convexity of the Narayana polynomials, and showing that the Narayana transformation preserves log-convexity. We begin with a formula of Brändén expressing the q-Narayana numbers as a specialization of Schur functions and, by deriving several symmetric function identities, we obtain the necessary Schur-positivity results. In addition, we prove the strong q-log-concavity of the q-Narayana numbers. The q-log-concavity of the q-Narayana numbers Nq(n,k) for fixed k is a special case of a conjecture of McNamara and Sagan on the infinite q-log-concavity of the Gaussian coefficients.
Journal Article
The Limiting Distribution of the Coefficients of the q-Catalan Numbers
We show that the limiting distributions of the coefficients of the q-Catalan numbers and the generalized q-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small n, we conjecture that for sufficiently large n, the coefficients are unimodal and even log-concave except for a few terms of the head and tail.
Journal Article
Average size of a self-conjugate ( s , t ) (s,t) -core partition
Armstrong, Hanusa and Jones conjectured that if s,ts,t are coprime integers, then the average size of an (s,t)(s,t)-core partition and the average size of a self-conjugate (s,t)(s,t)-core partition are both equal to (s+t+1)(s−1)(t−1)24\\frac {(s+t+1)(s-1)(t-1)}{24}. Stanley and Zanello showed that the average size of an (s,s+1)(s,s+1)-core partition equals (s+13)/2\\binom {s+1}{3}/2. Based on a bijection of Ford, Mai and Sze between self-conjugate (s,t)(s,t)-core partitions and lattice paths in an ⌊s2⌋×⌊t2⌋\\lfloor \\frac {s}{2} \\rfloor \\times \\lfloor \\frac {t}{2}\\rfloor rectangle, we obtain the average size of a self-conjugate (s,t)(s,t)-core partition as conjectured by Armstrong, Hanusa and Jones.
Journal Article