Parametrized variational inequality approaches to generalized Nash equilibrium problems with shared constraints

We consider the generalized Nash equilibrium problem (GNEP), in which each player’s strategy set may depend on the rivals’ strategies through shared constraints. A practical approach to solving this problem that has received increasing attention lately entails solving a related variational inequality (VI). From the viewpoint of game theory, it is important to find as many GNEs as possible, if not all of them. We propose two types of parametrized VIs related to the GNEP, one price-directed and the other resource-directed. We show that these parametrized VIs inherit the monotonicity properties of the original VI and, under mild constraint qualifications, their solutions yield all GNEs. We propose strategies to sample in the parameter spaces and show, through numerical experiments on benchmark examples, that the GNEs found by the parametrized VI approaches are widely distributed over the GNE set.     

Convexity,moduli of smoothness and a Jackson-type inequality

For a Banach space B of functions which satisfies for some m>0

$ \max ({\|F+G\|}_B,{\|F-G\|}_B)\geqq ({\|F\|}^s_B+m{\|G\|}^s_B)^{1/s},\quad \forall \,F,G\in B $

(?)

a significant improvement for lower estimates of the moduli of smoothness ω ^r (f,t) _B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ? ^d or \(\mathbb{T}^{d}\) for which translations are isometries or on S ^d?1 for which rotations are isometries. Results for C ₀ semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An L _p space with 1<p<∞ satisfies (?) where s=max??(p,2), and many Orlicz spaces are shown to satisfy (?) with appropriate s.

     

Moment inequality and Hölder inequality for BSDEs

Under the Lipschitz and square integrable assumptions on the generator g of BSDEs, this paper proves that if g is positively homogeneous in （y, z） and is decreasing in y, then the Moment inequality for BSDEs with generator g holds in general, and if g is positively homogeneous and sub-additive in （y, z）, then the HSlder inequality and Minkowski inequality for BSDEs with generator g hold in general.     

Generalizations of a parameterized Jordan-type inequality,Janous’s inequality and Tsintsifas’s inequality

In this work, we first prove a generalized version of a parameterized Jordan-type inequality. We then use it to prove the generalized versions of Janous’s inequality and Tsintsifas’s inequality which reduce to two inequalities conjectured by Janous and Tsintsifas as special cases.     

Sharpening Jordan’s inequality and the Yang Le inequality

In this work, the following inequality: $\frac{sinx}{x}\le \frac{2}{\pi }+\frac{\pi -2}{{\pi }^3}({\pi }^2-4x^2)\text{,}x\in (0,\pi /2]$ is established. An application of this inequality gives an improvement of the Yang Le inequality [C.J. Zhao, Generalization and strengthening of the Yang Le inequality, Math. Practice Theory 30 (4) (2000) 493–497 (in Chinese)]: where $A_i>0(i=1,2,\dots ,n),{\sum }_{i=1}^n{A}_i\le \pi ,0\le \lambda \le 1$, and $n\ge 2$ is a natural number.     

p-Poincaré inequality versus ∞-Poincaré inequality: some counterexamples

We point out some of the differences between the consequences of p-Poincaré inequality and that of ∞-Poincaré inequality in the setting of doubling metric measure spaces. Based on the geometric characterization of ∞-Poincaré inequality given in Durand-Cartagena et al. (Mich Math J 60, 2011), we obtain a geometric property implied by the support of a p-Poincaré inequality, and demonstrate by examples that an analogous geometric characterization for finite p is not possible. The examples we give are metric measure spaces which are doubling and support an ∞-Poincaré inequality, but support no finite p-Poincaré inequality. In particular, these examples show that one cannot expect a self-improving property for ∞-Poincaré inequality in the spirit of Keith–Zhong (Ann Math 167(2):575–599, 2008). We also show that the persistence of Poincaré inequality under measured Gromov–Hausdorff limits fails for ∞-Poincaré inequality.     

Hölder's inequality

In this paper, refinements of Holder's inequality are obtained and some concave functions are defined.     

Relative isoperimetric inequality and¶linear isoperimetric inequality for minimal submanifolds

We prove an optimal relative isoperimetric inequality

Stopped Doob inequality for p-th moment, 0 <p<∞, stochastic convolution integrals

We give stopped Doob inequalities for p-th moment, 0<p<∞, of stochastic convolution integrals ∫_(0,t] U(t,s)φ _{s ^?} dM _s in a Hilbert space, where M is a Hilbert space-valued cadlag square integrable martingale, φ is an operator-valued predictable process and U(t,s) is a contraction-type evolution operator. We also generalize the previous results and try to get smaller constants.

     

Robustness of the Gaussian concentration inequality and the Brunn–Minkowski inequality

We provide a sharp quantitative version of the Gaussian concentration inequality: for every \(r>0\), the difference between the measure of the r-enlargement of a given set and the r-enlargement of a half-space controls the square of the measure of the symmetric difference between the set and a suitable half-space. We also prove a similar estimate in the Euclidean setting for the enlargement with a general convex set. This is equivalent to the stability of the Brunn–Minkowski inequality for the Minkowski sum between a convex set and a generic one.     

Converse Jensen–Steffensen inequality

In this paper we prove a converse to the Jensen–Steffensen inequality. We also present two inequalities complementary to the Jensen–Steffensen inequality. The equality case conditions are thoroughly investigated.     

Strong Normalizability of Typed Lambda-Calculi for Substructural Logics

The strong normalization theorem is uniformly proved for typed λ-calculi for a wide range of substructural logics with or without strong negation. We would like to thank the referees for their valuable comments and suggestions. This research was supported by the Alexander von Humboldt Foundation. The second author is grateful to the Foundation for providing excellent working conditions and generous support of this research. This work was also supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B) 20700015, 2008.     

On Jordan’s inequality

We present sharp upper and lower bounds for the function \(\sin (x)/x\). Our bounds are polynomials of degree 2n, where n is any nonnegative integer.     

On Bohr’s inequality

It is established that H. Bohr’s inequality \(\sum\nolimits_{k = 0}^\infty {\left| {{{f^{\left( k \right)} \left( 0 \right)} \mathord{\left/ {\vphantom {{f^{\left( k \right)} \left( 0 \right)} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right| \leqslant \sqrt 2 \left\| f \right\|_\infty }\) is sharp on the class H _∞.     

A sub-Riemannian curvature-dimension inequality,volume doubling property and the Poincaré inequality

Let $\mathbb M $ be a smooth connected manifold endowed with a smooth measure $\mu $ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$ , and which is symmetric with respect to $\mu $ . We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in http://arxiv.org/abs/1101.3590, then the following properties hold:

• The volume doubling property;
• The Poincaré inequality;
• The parabolic Harnack inequality.

The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.     

On Hasse’s inequality

We give an elementary exposition of the little known work of Harold Davenport related to Hasse’s inequality. We formulate a new conjecture suggested by this proof that has implications for the classical Riemann hypothesis.     

A Bohr–Nikol'skii inequality

In this paper, we give a new inequality called Bohr–Nikol'skii inequality which combines the inequality of Bohr–Favard and the Nikol'skii idea of inequality for functions in different metrics.     

On Jensen-McShane’s inequality

A sequence of inequalities which include McShane’s generalization of Jensen’s inequality for isotonic positive linear functionals and convex functions are proved and compared with results in [3]. As applications some results for the means are pointed out. Moreover, further inequalities of Hölder type are presented.     

Hlawka’s functional inequality

The paper is devoted to the functional inequality (called by us Hlawka’s functional inequality) $$f(x+y)+f(y+z)+f(x+z)\leq f(x+y+z)+f(x)+f(y)+f(z)$$ for the unknown mapping f defined on an Abelian group, on a linear space or on the real line. The study of the foregoing inequality is motivated by Hlawka’s inequality: $$\|x+y\|+\|y+z\|+\|x+z\|\leq\|x+y+z\|+\|x\|+\|y\|+\|z\|,$$ which in particular holds true for all x, y, z from a real or complex inner product space.