On the postulation of a general projection of a curve inP N,N⩾4

Summary Fix a curve X of genus g and L Pic ^d (X). Let _L(X) be the image of X through the complete linear system H⁰(X, L). Here we prove that a general projection of _L(X) intoP ^N has maximal rank if either (a) N4, 0gN–1, dg+N, or (b) dd (g, N) for suitable d(g, N).     

Distribution of zeros of the Hermite-Padé polynomials for a system of three functions,and the Nuttall condenser

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\Re _3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak{f}_1 ,\mathfrak{f}_2 ,\mathfrak{f}_3$ that are rational on the constructed Riemann surface and satisfy the independence condition det . In the case of m = 3, we refine the main theorem from Nuttall’s paper of 1981. In particular, we show that in this case the complement ?? \ B of the open (possibly, disconnected) set B ? ?? introduced in Nuttall’s paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.     

On the evaluation at (−ι,ι) of the Tutte polynomial of a binary matroid

Vertigan has shown that if M is a binary matroid, then |T _M (?ι,ι)|, the modulus of the Tutte polynomial of M as evaluated in (?ι,ι), can be expressed in terms of the bicycle dimension of M. In this paper, we describe how the argument of the complex number T _M (?ι,ι) depends on a certain $\mathbb{Z}/4\mathbb {Z}$ -valued quadratic form that is canonically associated with M. We show how to evaluate T _M (?ι,ι) in polynomial time, as well as the canonical tripartition of M and further related invariants.     

Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ? ∈ W _M ¹ , we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space W _F ¹ . We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.     

On the Covering of the Vertices of a Graph by Cliques

In conversation I was told by Professor R.Brigham the following conjecture [1].Let G(n) be a graph of n vertices.Denote by f(G(n))=t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote further by h(G(n)) =l the largest integer for which there are l edges of our G(n) no two of Which are in the same clique.Clearly h(G(n)) can be much larger than f(G(n))e.g.if n=2m and G(n) is the complete bipartite graph of m white and m black vertices.Then l(G(n))=m and l(G(n))=m~2. It was conjectured that if G(n).has no isolated vertices then     

Inequality for the Moment of a Function of a Random Variable

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On Sums of a Prime, and a Square of Prime, and a κ-power of Prime

In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation     

What Is the Object of the Encapsulation of a Process?

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?     

Turán Number of the Family Consisting of a Blow-up of a Cycle and a Blow-up of a Star

Let F={H₁,...,H_k}(k> 1) be a family of graphs.The Turán number of the family F is the maximum number of edges in an n-vertex {H₁,...,H_k)-free graph,denoted by ex(n,F) or ex(n,{H₁,H₂,...,H_k}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Turán number of the family cons...     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

The eigenvalue problem for solitary waves of a boussinesq equation,via a generalization of the rouché theorem

We consider the study of an eigenvalue problem obtained by linearizing about solitary wave solutions of a Boussinesq equation. Instead of using the technique of Evans functions as done by Pego and Weinstein in [R. Pego and M. Weinstein, Convective Linear Stability of Solitary Waves for Boussinesq equation. AMS, 99, 311–375] for this particular problem, we perform Fourier analysis to characterize solutions of the eigenvalue problem in terms of a multiplier operator and use the strong relationship between the eigenvalue problem for the linearized Boussinesq equation and the eigenvalue problem associated with the linearization about solitary wave solutions of a special form of the KdV equation. By using a generalization of the Rouché Theorem and the asymptotic behavior of the Fourier symbol corresponding to the eigenvalues problem for the Boussinesq equation and the Fourier symbol corresponding to the eigenvalues problem for the KdV equation, we show nonexistence of eigenvalues with respect to weighted space in a planar region containing the right-half plane.     

The σ-algebra of pasts of a random walk over the orbits of a Bernoulli action of the group Z d

In the present paper, we study the σ-algebra of pasts Ξ = {ξ _n } _n of a random walk T over the orbits of a Bernoulli action of the group Z^d. We calculate the proper scaling and the scaling entropy of this sequence of partitions. We show that the proper scaling entropy of the σ-algebra of pasts is equal to . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 103–112.     

On the number of numerical semigroups 〈a,b〉 of prime power genus

Given g≥1, the number n(g) of numerical semigroups S?? of genus |??S| equal to g is the subject of challenging conjectures of Bras-Amorós. In this paper, we focus on the counting function n(g,2) of two-generator numerical semigroups of genus g, which is known to also count certain special factorizations of 2g. Further focusing on the case g=p ^k for any odd prime p and k≥1, we show that n(p ^k ,2) only depends on the class of p modulo a certain explicit modulus M(k). The main ingredient is a reduction of $\operatorname{gcd}(p^{\alpha}+1, 2p^{\beta}+1)$ to a simpler form, using the continued fraction of α/β. We treat the case k=9 in detail and show explicitly how n(p ⁹,2) depends on the class of $p \operatorname{mod} M(9)=3 \cdot5 \cdot11 \cdot17 \cdot43 \cdot257$ .     

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.     

On the Properties of the Solution of a Strongly Degenerate Parabolic Equation

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The Stablility of a Loop Formed the Separatrix of a Saddle Point and the Condition to Produce a Limit Cycle

In this paper,we discuss a loop by the separatrix of a saddle point of two dimensional autonmoussystem.It contains two sections.In section 1,a short proof of the criterion theorem of stability ofthe loop of saparatrix is given wi~h ~he help o~th~ ~h~orjr 0~ i'otatin~ vector field, and some in~ormatioafor the study of stability of that loop in critical case is supplied.In section 2,we give a sufficientcondition under which a limit cycle is produced from the loop of separatrix.     

Estimates in the Carleson corona theorem,ideals of the algebra H∞, a problem of S.-Nagy

Let E₁, E₂, be Hilbert spaces, H^∞(E₁,E₂) be the space of functions, bounded and analytic in the disk D, with values in the space of bounded linear operators from E₁ to E₂. Estimates are investigated for a solution of the problem of S.-Nagy of finding a left inverse element for a function F, FεH^∞(E₁,E₂). For dim E₁=1 this problem is a generalization of the corona problem. Let C_n(δ)= sup_¦G¦_∞∶FεH^∞(E₁,E₂),dim E₁=n, ¦F¦_∞?1, ¦F(z)a¦₂?δ¦a¦₂(zεD,aεE₁);Gε H^∞(E₂,E₁) is a function of minimal norm for which . Then where a_n, C_n are constants depending only on n. The behavior of the function C₁ as δ→1 is described. Other results are obtained also.     

On solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),