A game with three players

In the classical gambling problem (two players with initial capital a₁ and a₂, respectively) the expected time to ruin is a₁a₂. Here we extend this problem to three players and prove that the expected time to ruin equals (a₁a₂a₃)/(a₁ + a₂ + a₃ − 2).     

Optimal dividends in the dual model

The optimal dividend problem proposed by de Finetti [de Finetti, B., 1957. Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries, vol. 2. pp. 433-443] is to find the dividend-payment strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the company is ruined or bankrupt. In this paper, it is assumed that the surplus or shareholders’ equity is a Lévy process which is skip-free downwards; such a model might be appropriate for a company that specializes in inventions and discoveries. In this model, the optimal strategy is a barrier strategy. Hence the problem is to determine b^∗, the optimal level of the dividend barrier. A key tool is the method of Laplace transforms. A variety of numerical examples are provided. It is also shown that if the initial surplus is b^∗, the expectation of the discounted dividends until ruin is the present value of a perpetuity with the payment rate being the drift of the surplus process.     

Ruin probabilities of a bidimensional risk model with investment

We consider a classical risk model with the possibility of investment. We study two types of ruin in the bidimensional framework. Using the martingale technique, we obtain an upper bound for the infinite-time ruin probability with respect to the ruin time T_max(u₁,u₂). For each type of ruin, we derive an integral-differential equation for the survival probability, and an explicit asymptotic expression for the finite-time ruin probability.     

Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints

We consider that the surplus of an insurance company follows a Cramér-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks.[Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215-228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890-907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.     

Characterization of unigraphic and unidigraphic integer-pair sequences

Given a graph (digraph) G with edge (arc) set E(G) = {(u₁}, υ₁), (u₂, υ₂),?,(u_q, υ_q, where q = |E(G)|, we can associate with it an integer-pair sequence S_G = ((a₁, b₁), (a₂, b₂),?, (a_q, b_q)) where a_i, b_i are the degrees (indegrees) of u_i, υ_i respectively. An integer- pair sequence S is said to be graphic (digraphic) if there exists a graph (digraph) G such that _SG = S. In this paper we characterize unigraphic and unidigraphic integer-pair sequences.     

The generating function of a family of the sequences in terms of the continuant

Let a₀, a₁, … , a_r−1 be positive numbers and define a sequence {q_m}, with initial conditions q₀ = 0 and q₁ = 1, and for all m ? 2, q_m = a_tq_m−1 + q_m−2 where m ≡ t(mod r). For r = 2, the author called the sequence {q_m} as the generalized Fibonacci sequences and studied it in [1]. But, it remains open to find a closed form of the generating function for general {q_m}. In this paper, we solve this open problem, that is, we find a closed form of the generating function for {q_m}in terms of the continuant.     

Conditions for the algebraic independence of certain series involving continued fractions and generated by linear recurrences

The main theorem of this paper, proved using Mahler's method, gives a necessary and sufficient condition for the values Θ(x,a,q) at any distinct algebraic points to be algebraically independent, where Θ(x,a,q) is an analogue of a certain q-hypergeometric series and generated by a linear recurrence whose typical example is the sequence of Fibonacci numbers. Corollary 1 gives Θ(x,a,q) taking algebraically independent values for any distinct triplets (x,a,q) of nonzero algebraic numbers. Moreover, Θ(a,a,q) is expressed as an irregular continued fraction and Θ(x,1,q) is an analogue of q-exponential function as stated in Corollaries 3 and 4, respectively.     

Partitions with rounded occurrences and attached parts

We introduce the number of (k,i)-rounded occurrences of a part in a partition and use q-difference equations to interpret a certain q-series S _k,i (a;x;q) as the generating function for partitions with bounded (k,i)-rounded occurrences and attached parts. When a=0 these partitions are the same as those studied by Bressoud in his extension of the Rogers-Ramanujan-Gordon identities to even moduli. When a=1/q we obtain a new family of partition identities.     

Linear independence of values of Lerch functions

The number of linearly independent numbers among 1, Φ₁ (z, p/q), ...,Φ _a (z, p/q) is estimated depending on a natural number a, where Φ _s (z, p/q), s = 1, 2, ..., are Lerch functions.     

Surplus analysis for a class of Coxian interclaim time distributions with applications to mixed Erlang claim amounts

Gerber-Shiu analysis with the generalized penalty function proposed by Cheung et al. (in press-a) is considered in the Sparre Andersen risk model with a K_n family distribution for the interclaim time. A defective renewal equation and its solution for the present Gerber-Shiu function are derived, and their forms are natural for analysis which jointly involves the time of ruin and the surplus immediately prior to ruin. The results are then used to find explicit expressions for various defective joint and marginal densities, including those involving the claim causing ruin and the last interclaim time before ruin. The case with mixed Erlang claim amounts is considered in some detail.     

Optimal Lehmer Mean Bounds for the Toader Mean

We find the greatest value p and least value q such that the double inequality L _p (a, b)?<?T(a, b)?<?L _q (a, b) holds for all a, b?>?0 with a?≠ b, and give a new upper bound for the complete elliptic integral of the second kind. Here ${T(a,b)=\frac{2}{\pi}\int\nolimits_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}d\theta}$ and L _p (a, b)?=?(a ^p+1?+?b ^p+1)/(a ^p ?+?b ^p ) denote the Toader and p-th Lehmer means of two positive numbers a and b, respectively.     

The surpluses immediately before and at ruin,and the amount of the claim causing ruin

In the classical compound Poisson model of the collective risk theory we consider X, the surplus before the claim that causes ruin, and Y, the deficit at the time of ruin. We denote by f(u; x, y) their joint density (u initial surplus) which is a defective probability density (since X and Y are only defined, if ruin takes place). For an arbitrary claim amount distribution we find that f(0; x, y) = ap(x + y), where p(z) is the probability density function of a claim amount and a is the ratio of the Poisson parameter and the rate of premium income. In the more realistic case, where u is positive, f(u; x, y) can be calculated explicitly, if the claim amount distribution is exponential or, more generally, a combination of exponential distributions. We are also interested in X + Y, the amount of the claim that causes ruin. Its density h(u; z) can be obtained from f(u; x, y). One finds, for example, that h(0; z) = azp(z).     

An optimal double inequality between geometric and identric means

We find the greatest value p and least value q in (0,1/2) such that the double inequality G(pa+(1−p)b,pb+(1−p)a)<I(a,b)<G(qa+(1−q)b,qb+(1−q)a) holds for all a,b>0 with a≠b. Here, G(a,b), and I(a,b) denote the geometric, and identric means of two positive numbers a and b, respectively.     

Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, φ, s) = p ²/2+ cos q ? 1 + I ²/2 + h(q, φ, s; ε) — proving that for any small periodic perturbation of the form h(q, φ, s; ε) = ε cos q (a ₀₀ + a ₁₀ cosφ + a ₀₁ cos s) (a ₁₀ a ₀₁ ≠ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ π/2μ, μ = a ₁₀/a ₀₁), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any μ). The bifurcations of the scattering map are also studied as a function of μ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.     

Exceptional del Pezzo Hypersurfaces

We compute global log canonical thresholds of a large class of quasismooth well-formed del Pezzo weighted hypersurfaces in ?(a ₀,a ₁,a ₂,a ₃). As a corollary we obtain the existence of orbifold Kähler-Einstein metrics on many of them, and classify exceptional and weakly exceptional quasismooth well-formed del Pezzo weighted hypersurfaces in ?(a ₀,a ₁,a ₂,a ₃).     

Irreducible polynomials over finite fields. I

A function H_n(a₁, a₂, a₃) is found, computing the number of normalized irreducible polynomials of degree n over a finite field F_q, with the first three coefficients a₁, a₂, and a₃ fixed for n = 4. The function is expressed via some sums of characters admitting “good” estimates. In particular, the following theorem is proved: If q = 3m + 1, a ∈ k^*, and N (a) = H₄(0, 0, a), then $$N(a) = \frac{1}{4}(q - 2\operatorname{Re} [(\lambda (a) - \eta ( - 1)\bar \lambda (a/2))J(\lambda ,\lambda )] - \eta ( - 1)),$$ where η is a quadratic character of the field k = F_q, λ is a nontrivial cubic character, and J(λ, λ) is the known Jacobi sum.     

Itération de pliages de quadrilatères

Starting with a quadrilateral q₀=(A₁,A₂,A₃,A₄) of $\text{R}{}^2$, one constructs a sequence of quadrilaterals q_n=(A_4n+1,…,A_4n+4) by iteration of foldings: q_n=?₄°?₃°?₂°?₁(q_n?1) where the folding ?_j replaces the vertex number j by its symmetric with respect to the opposite diagonal.We study the dynamical behavior of this sequence. In particular, we prove that:– The drift $\text{v:=}\text{lim}{}_{\mathrm{n\to \infty }}\text{1}\text{n}\text{q}{}_{\mathrm{n}}$ exists.– When none of the q_n is isometric to q₀, then the drift v is zero if and only if one has $\text{max}\text{a}{}_{\mathrm{j}}\text{+}\text{min}\text{a}{}_{\mathrm{j}}\text{?}\text{1}\text{2}\text{∑a}{}_{\mathrm{j}}$, where a₁,…,a₄ are the sidelengths of q₀.– For Lebesgue almost all q₀ the sequence (q_n?nv)_n?1 is dense on a bounded analytic curve with a center, or an axis of symmetry. However, for Baire generic q₀, the sequence (q_n?nv)_n?1 is unbounded. To cite this article: Y. Benoist, D. Hulin, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

A proof of the q,t-Catalan positivity conjecture

We present here a proof that a certain rational function C_n(q,t) which has come to be known as the “q,t-Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that C_n(q,t) is the Hilbert series of the diagonal harmonic alternants in the variables (x₁,x₂,…,x_n;y₁,y₂,…,y_n). Since C_n(q,t) evaluates to the Catalan number at t=q=1, it has also been an open problem to find a pair of statistics a(π),b(π) on Dyck paths π in the n×n square yielding C_n(q,t)=∑_πt^a(π)q^b(π). Our proof is based on a recursion for C_n(q,t) suggested by a pair of statistics a(π),b(π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363).