Differentiability of the minimal average action as a function of the rotation number

Let be a finite composition of exact twist diffeomorphisms. For any real number , letA() denote the minimal average action of -invariant measures with angular rotation number . We prove thatA() is differentiable at every irrational number and that for generic it is not differentiable at rational , thus verifying conjectures of S. Aubry. Moreover, we show that these results are valid for a variational principleh which satisfies the condition which we have called elsewhere (H). As a consequence, we generalize a result due to Bangert concerning geodesics on a two dimensional torus with an arbitrary, but sufficiently smooth metric.supported by NSF grant no. DMS-8806067.01 and a Guggenheim Fellowship.     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

Differentiability of a convex function

Translated from Issledovaniya po Prikladnoi Matematike, No. 15, pp. 3–5, 1988.     

Differentiability of probability function

Basic results on differentiability of probability functions are reviewed and systematically presented. Strict mathematical statements are formulated, some basic theorems are informally proved and illustrated with examples. Probability function, from formal point of view, is an expectation of an indicator function or an integral over the domain depending upon the parameter. Gradient of probability function is represented in different forms (integral over the surface, volume, or sum of surface and volume integrals). These results can be used for sensitivity and uncertainty analysis of stochastic models, reliability analysis, Probabilistic Risk Analysis, optimization of the stochastic systems, and other applications involving uncertainties in parameters.     

Differentiability of the value function in continuous-time economic models

In this paper we provide some sufficient conditions for the differentiability of the value function in a class of infinite-horizon continuous-time models of convex optimization arising in economics. We dispense with the assumption of interior optimal paths. This assumption is quite unnatural in constrained optimization, and is usually hard to check in applications. The differentiability of the value function is used to prove Bellman’s equation as well as the existence and continuity of the optimal feedback policy. We also establish the uniqueness of the vector of dual variables. These results become useful for the characterization and computation of optimal solutions.     

Uniqueness of a cycle with discounting that is optimal with respect to the average time profit

For cyclic processes modeled by periodic motions of a continuous control system on a circle, we prove the uniqueness of a cycle maximizing the average one-period time profit in the case of discounting provided that the minimum and maximum velocities of the system coincide at some points only and the profit density is a differentiable function with a finite number of critical points. The uniqueness theorem is an analog of Arnold’s theorem on the uniqueness of such a cycle in the case when the profit gathered along the cycle is not discounted.     

Differentiability of dividends function on jump-diffusion risk process with a barrier dividend strategy

We consider a dividends model with a stochastic jump perturbed by diffusion. First, we prove that the expected discounted dividends function is twice continuously differentiable under the condition that the claim distribution function has continuous density. Then we show that the expected discounted dividends function under a barrier strategy satisfies some integro-differential equation of defective renewal type, and the solution of which can be explicitly expressed as a convolution formula. Finally, we study the Laplace transform of ruin time on the modified surplus process.     

The average of the values of a function at random points

The behavior of (1/N) asN→∞ is considered, wheref is a bounded measurable function on (−∞, ∞) and (S _n) _n ^=1/∞ are the partial sums of a sequence of independent and identically distributed rondom variables.     

Bin covering with a general profit function: approximability results

In the recent paper (Benk? et al. 2010) we introduced a new problem that we call Bin Packing/Covering with Delivery, or BP/CD for short. Mainly we mean under this expression that we look for not only a good, but a “good and fast” packing or covering. In the present paper we investigate the offline case. For the analysis, a novel view on “offline optimum” is introduced, which appears to be relevant concerning all problems where a final solution is ordering-dependent. We prove that if the item sizes are not allowed to be arbitrarily close to zero, then an optimal offline solution can be found in polynomial time. On the other hand, for unrestricted problem instances, no polynomial-time algorithm can achieve an asymptotic approximation ratio better than 6/7 if $P\ne NP$ .     

The average number of photons present in a single-mode squeezed state as function of the mean free path

In this paper, we define a new weight function to represent a single-mode squeezed state. By means of this function, we shall be enabled to construct an interesting relationship between the total mean number of photons occupied a single-mode squeezed state and the mean free path of the atoms of the medium through which the natural light is diffused. This relationship assures the possibility to measure experimentally the average number of photons present in a single-mode squeezed state.     

Limit theorem for the moment of ruin for integrated Gaussian stationary process with power function as profit

The asymptotic distribution of the moment of ruin is obtained for the ruin problem with the loss rate in the form of a Gaussian stationary process and a power function as profit. Mean losses are found in the case of ruin.     

Differentiability of the Arrival Time

For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional 2 Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method, a~priori not even differentiable but only satisfying the equation in the viscosity sense . Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black‐Scholes option pricing. From variations of the Sard and ?ojasiewicz theorems, we relate differentiability to whether singularities all occur at only finitely many times for flows.© 2016 Wiley Periodicals, Inc.     

Differentiability of a convex semigroup on Rn

It is proved that each continuous semigroup {P(t)}_t≥0 of convex operators P(t):Rⁿ→Rⁿ is continuously differentiable with respect to t. This note represents a first step towards a better understanding of semigroups formed by convex operators. We establish the differentiability of a convex semigroup in the finite dimensional case, generalizing a basic result from linear semigroup theory. Our motivation for the study of semigroups of convex operators comes from the theory of Markov decision processes. In [1] and in [2] it was shown that the maximum reward of these processes can be described by a certain nonlinear semigroup. The nonlinear operators are defined as suprema of linear operators (plus a constant), hence they are convex operators. It seems that the convexity assumption keeps its smoothing influence even in the infinite dimensional situation. We hope to discuss this in a future paper.     

The average number of divisors of the Euler function

The upper bound and the lower bound of the average number of divisors of Euler Phi function and Carmichael Lambda function were obtained by Luca and Pomerance (see Publ Math Debr 70(1–2):125–148, 2007). We improve the lower bound and provide a heuristic argument which suggests that the upper bound given by Luca and Pomerance (Publ Math Debr 70(1–2):125–148, 2007) is indeed close to the truth.     

一类分形插值曲面的可微性

在矩形域上,构造一类迭代函数系,并由此生成一类分形曲面,分别给出该分形曲面几乎处处可微和不可微点的条件,得到相应的结论.