Vaidya’s Method for Convex Stochastic Optimization Problems in Small Dimension

Mathematical Notes - The paper deals with a general problem of convex stochastic optimization in a space of small dimension (for example, 100 variables). It is known that for deterministic problems...     

On Balder’s Existence Theorem for Infinite-Horizon Optimal Control Problems

Balder’s well-known existence theorem (1983) for infinite-horizon optimal control problems is extended to the case in which the integral functional is understood as an improper integral. Simultaneously, the condition of strong uniform integrability (over all admissible controls and trajectories) of the positive part max{f₀, 0} of the utility function (integrand) f₀ is relaxed to the requirement that the integrals of f₀ over intervals [T, T′] be uniformly bounded above by a function ω(T, T′) such that ω(T, T′) → 0 as T, T′→∞. This requirement was proposed by A.V. Dmitruk and N.V. Kuz’kina (2005); however, the proof in the present paper does not follow their scheme, but is instead derived in a rather simple way from the auxiliary results of Balder himself. An illustrative example is also given.     

On Ulm’s method for Fréchet differentiable operators

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Izv. Akad. Nauk Est. SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) do not. We also show that under the same hypotheses and computational cost as (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) finer error sequences can be obtained. Numerical examples are also provided further validating the results.     

A Gauss–Newton Approach for Solving Constrained Optimization Problems Using Differentiable Exact Penalties

We propose a Gauss–Newton-type method for nonlinear constrained optimization using the exact penalty introduced recently by André and Silva for variational inequalities. We extend their penalty function to both equality and inequality constraints using a weak regularity assumption, and as a result, we obtain a continuously differentiable exact penalty function and a new reformulation of the KKT conditions as a system of equations. Such reformulation allows the use of a semismooth Newton method, so that local superlinear convergence rate can be proved under an assumption weaker than the usual strong second-order sufficient condition and without requiring strict complementarity. Besides, we note that the exact penalty function can be used to globalize the method. We conclude with some numerical experiments using the collection of test problems CUTE.     

On waring’s problem for cubes

Proceedings - Mathematical Sciences -     

On Berndtsson’s generalization of Prékopa’s theorem

We present a new approach to Berndtssons complex extension of Prékopas theorem. This approach is inspired by the recent local proof of Prékopas theorem obtained by Ball, Barthe and Naor. In the complex setting, this approach leads to a formula expressing in terms of (derivatives of) where e^–(z)= As a consequence, we obtain new conditions ensuring that is subharmonic when is plurisubharmonic.     

On Fréchet’s functional equation

Fréchet’s functional equation \(\Delta _{y_1,y_2,\dots ,y_{n+1}}f=0\) plays a key role in the theory of polynomial functions. A basic theorem of Djokovi? shows that under general conditions the functional equation \(\Delta _y^{n+1}f=0\) is equivalent to Fréchet’s equation. Here we give a short alternative proof for this result using spectral synthesis.     

Malliavin Calculus Approach to Statistical Inference for Lévy Driven SDE’s

By means of the Malliavin calculus, integral representations for the likelihood function and for the derivative of the log-likelihood function are given for a model based on discrete time observations of the solution to equation dX _t = a _θ (X _t )dt + dZ _t with a Lévy process Z. Using these representations, regularity of the statistical experiment and the Cramer-Rao inequality are proved.     

Existence of ∈-Minima for Vector Optimization Problems

The -generalized minima for vector optimization problems are defined and a sufficient condition for the existence of -generalized minima for vector optimization problems is established.     

On Borsuk’s Conjecture for Two-Distance Sets

In this paper, we answer Larman’s question on Borsuk’s conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere $S^{64}\subset\mathbb{R}^{65}$ which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk’s conjecture is known to be false. Other examples of two-distance sets with large Borsuk numbers are given.     

On Yau’s theorem for effective orbifolds

In 1978 Yau (Yau, 1978) confirmed a conjecture due to Calabi (1954) stating the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. A version of this statement for effective orbifolds can be found in the literature (Joyce, 2000; Boyer and Galicki, 2008; Demailly and Kollár, 2001). In this expository article, we provide details for a proof of this orbifold version of the statement by adapting Yau’s original continuity method to the setting of effective orbifolds in order to solve a Monge–Ampère equation. We then outline how to obtain Kähler–Einstein metrics on orbifolds with negative first Chern class by solving a slightly different Monge–Ampère equation. We conclude by listing some explicit examples of Calabi–Yau orbifolds, which consequently admit Ricci flat metrics by Yau’s theorem for effective orbifolds.     

On Stanley’s inequalities for character multiplicities

Let G be a group of automorphisms of a ranked poset \({{\mathcal Q}}\) and let N _k denote the number of orbits on the elements of rank k in \({{\mathcal Q}}\). What can be said about the N _k for standard posets, such as finite projective spaces or the Boolean lattice? We discuss the connection of this question to the representation theory of the group, and in particular to the inequalities of Livingstone-Wagner and Stanley. We show that these are special cases of more general inequalities which depend on the prime divisors of the group order. The new inequalities often yield stronger bounds depending on the order of the group.     

On algebraic relations for Ramanujan’s functions

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(q ⁱ ) (i=1,2,3,4), Q(q ⁱ ) (i=1,2,3), and R(q ⁱ ) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ? of any three-element subset of {P(q),P(q ²),P(q ³),P(q ⁴)} and of any two-element subset of {Q(q),Q(q ²),Q(q ³)} and {R(q),R(q ²),R(q ³)}, respectively. For all the results we use some expressions of $P(q^{i_{1}}), Q(q^{i_{2}}) $ , and $R(q^{i_{3}}) $ in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.     

New Approach to Solving a System of Variational Inequalities and Hierarchical Problems

This paper deals with a viscosity iterative method, in real Hilbert spaces, for solving a system of variational inequalities over the fixed-point sets of possibly discontinuous mappings. Under classical conditions, we prove a strong convergence theorem for our method. The proposed algorithm can be applied for instance to solving variational inequalities in some situations when the projection methods fail. Moreover, the techniques of analysis are novel and provide new tools in designing approximation schemes for combined and bilevel optimization problems.     

Problems for Broome’s Cognitivist Account of Instrumental Reasoning

In this paper, I examine an account of instrumental reasoning recently put forth by John Broome. His key suggestion is that anyone who engages in reasoning about his intentions also believes that he will do what he intends to do and that combined with a belief about necessary means this creates rational pressure towards believing that one will take the necessary means. I argue that Broome’s model has three significant problems; his key premise is false—the sincere expression of an intention does not entail the belief that one will successfully execute that intention; his account yields a model of instrumental reasoning that is uncomfortably reflective; he seems unable to explain the rational pressure towards taking necessary means that arises directly from having an end and an instrumental belief. All three problems, I argue, are a consequence of Broome’s inadequate position on what it is to intend to do something.     

On Sarnak’s conjecture and Veech’s question for interval exchanges

Using a criterion due to Bourgain [10] and the generalization of the self-dual induction defined in [19], for each primitive permutation we build a large family of k-interval exchanges satisfying Sarnak’s conjecture, and, for at least one permutation in each Rauzy class, smaller families for which we have weak mixing, which implies a prime number theorem, and simplicity in the sense of Veech.     

Duality for Nonsmooth Continuous-Time Problems of Vector Optimization

We consider a nonsmooth vector optimization continuous-time problem. We establish weak and strong duality theorems under generalized convexity assumptions. This research was supported by Center of Excellence for Mathematics, University of Isfahan, Isfahan, Iran.     

On Dedekind’s problem for complete simple games

We state an integer linear programming formulation for the unique characterization of complete simple games, i.e. a special subclass of monotone Boolean functions. In order to apply the parametric Barvinok algorithm to obtain enumeration formulas for these discrete objects we provide a tailored decomposition of the integer programming formulation into a finite list of suitably chosen sub-cases. As for the original enumeration problem of Dedekind on Boolean functions we have to introduce some parameters to be able to derive exact formulas for small parameters. Recently, Freixas et al. have proven an enumeration formula for complete simple games with two types of voters. We will provide a shorter proof and a new enumeration formula for complete simple games with two minimal winning vectors.