Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.
We study the continuous time portfolio optimization model on the market where the mean returns of individual securities or asset categories are linearly dependent on underlying economic factors. We introduce the functional $Q_γ$ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient $γ$ to the variance when we keep the value of the factor levels fixed. The coefficient $γ$ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for $Q_γ$ at any moment of time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring an interest rate which affects a stock index and also serves as a second investment opportunity. We consider two possibilities: the interest rate for the bank account is governed by Vasicek-type and Cox-Ingersoll-Ross dynamics, respectively. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to $γ$.
We consider the problem of initial conditions that lead to the intersection of a satellite orbit with planetocentric sphere of a radius R. The problem is considered in frame of the satellite version of the double-averaged restricted three body problem with taking into account gravitational perturbations caused by the polar oblateness of the planet. For some integrable cases we provide the boundaries of the manifolds of the initial orbital elements leading (or not leading) to the intersection of the satellite orbit with the planet surface.
The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z , relatively prime to z . This group is calculated for z =(3+0 i ) n : it is isomorphic to the product of three cyclic group or orders (8, 3 n −1 and 3 n −1 ).
We give a simple explanation of numerical experiments of V. Arnold with two sequences of symmetric numerical semigroups, S(4,6+4k,87-4k) and S(9,3+9k,85-9k) generated by three elements. We present a generalization of these sequences by numerical semigroups S(r_1^2,r_1r_2+r_1^2k,r_3-r_1^2k), k\in{\mathbb Z}, r_1,r_2,r_3\in{\mathbb Z}^+, r_1\geq 2 and \gcd(r_1,r_2)=\gcd(r_1,r_3)=1, and calculate their universal Frobenius number Phi(r_1,r_2,r_3) for the wide range of k providing semigroups be symmetric. We show that this kind of semigroups admit also nonsymmetric representatives. We describe the reduction of the minimal generating sets of these semigroups up to {r_1^2,r_3-r_1^2k} for sporadic values of k and find these values by solving the quadratic Diophantine equation.
The paper is about the problem of carefully estimating the bounds, which is sometimes missing in the theoretical physics. Possible consequences of the missing of the bounds is discussed on example of the Riemann zeta function. The text of the paper is based on the drafts of A.A. Karatsuba’s lecture “Physical mathematics in number theory”, devoted to the 85th birthday of academician Vasilii Sergeevich Vladimirov.
A natural number is said red if the period of the continued fraction of its square root has odd length. For any quadratic field \(\mathbb{Q}(\sqrt{D})\), we show how the parity of the periods length of the continued fractions of its irrationalities depends on the redness of their discriminant.
The exponential of the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the sum of the inverses of the divisors of n is the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the number of partitions of n. A similar result holds for all pairs of sequences satisfying a special mutual recurrence.
Each degree n polynomial in one variable of the form (x+1)(x n−1+c 1 x n−2+⋅⋅⋅+c n−1) is representable in a unique way as a Schur-Szegő composition of n−1 polynomials of the form (x+1)n−1(x+a i ), see Kostov (2003), Alkhatib and Kostov (2008) and Kostov (Mathematica Balkanica 22, 2008). Set \(\sigma _{j}:=\sum _{1\leq i_{1}<\cdots <i_{j}\leq n-1}a_{i_{1}}\cdots a_{i_{j}}\). The eigenvalues of the affine mapping (c 1,…,c n−1)↦(σ 1,…,σ n−1) are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the roots of these polynomials.
The Fermat–Euler progression of residues modulo n is a geometrical progression, formed by the powers of one residue, a. Such a sequence is always periodic, starting from some place. The object of study of the paper is the minimal period’s length T(a,n) of this progression.
In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form $x^2+p x=q$ with integer $p$ and $q$, $p^2+q^2\le R^2$. Our results concern the average sum of period elements and Gauss--Kuzmin statistics as $R\to\infty$.
For a positive integer Q, we give some conditions to determine whether the continued fraction of \(\sqrt{Q}\) has odd or even period.
With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology.
The Gibbs phenomenon is described for the Fourier series of a function at its jump, the function being defined along the finite circle ℤ/pℤ.
We prove the equivalence of two hierarchies of soliton equations associated to a simply-laced finite Dynkin diagram. The first was defined by Kac and Wakimoto (Proc. Symp. Pure Math. 48:138–177, 1989 ) using the principal realization of the basic representations of the corresponding affine Kac–Moody algebra. The second was defined in Givental and Milanov (The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics, vol. 232, pp. 173–201, Birkhäuser, Basel, 2005 ) using the Frobenius structure on the local ring of the corresponding simple singularity. We also obtain a deformation of the principal realization of the basic representation over the space of miniversal deformations of the corresponding singularity. As a by-product, we compute the operator product expansions of pairs of vertex operators defined in terms of Picard–Lefschetz periods for more general singularities. Thus, we establish a surprising link between twisted vertex operators and deformation theory of singularities.
By Euler-like function we mean a function defined on the positive integers and associating to $n$ the product, over all primes $p$ dividing $n$, of 1 plus (or minus) the inverse of $p$ to the power $s$. We calculate the limit of the Cesaro mean of these functions.
We generalize and prove a conjecture by V.I. Arnold on the parity of Frobenius numbers. For the case of symmetric semigroups with three generators we give an exact formula for Frobenius numbers, which is, in a sense, a sum of two Sylvester’s formulae. We prove that a fraction of symmetric semigroups vanishes in the weak limit.
The matricial Euler congruence \(\mathop{\mathrm{Tr}}\bigl(A^{p^{n}}\bigr)\equiv\mathop{\mathrm{Tr}}\bigl(A^{p^{n-1}}\bigr)\) modulo p n , previously announced in Arnold, Japanese J. Math. 1(1), 1–24, 2006 for A∈M N (ℤ), is given a proof based on extending it to the ring of Witt vectors of length n.
An algebraic permutation \(\hat{A}\in S(N=n^{m})\) is the permutation of the N points of the finite torus ℤ n m , realized by a linear operator A∈SL(m,ℤ n ). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements, whereas \(T(\hat{A})\) is bounded by a power of N for algebraic permutations \(\hat{A}\). The paper also contains a strange mean asymptotics formula for the number of points of the finite projective line P1(ℤ n ) in terms of the zeta function.
Mark all vertices on a curve evolving under a family of curves obtained by intersecting a smooth surface M with the 1-parameter family of planes parallel to the tangent plane of M at a point p . Those vertices trace out a set, called the vertex set through p . We take p to be a generic umbilic point on M and describe the perestroikas of the vertex set under generic n -parameter small deformations of the surface. One of the consequences of our results is that, in some sense, generically the study of the discriminants of small n -deformations, with n greater than or equal to 2, simplifies to that of 2-deformations. Beyond the Mathematical interest in vertices of families of curves, this work was primarily motivated by the medial representation of shapes in Computer Vision and Image Analysis, where the behavior of vertices plays a crucial role in the qualitative changes of the skeleton or Blum medial axis of curves.
