A remark on periodic Tchebyshev systems

The linear hull of a Tchebyshev system is called a Haarspace. Every Haar-space of periodic functions has odd dimension. It is shown that under certain conditions an n-dimensional Haar-space of periodic functions contains i-dimensional Haar-spaces U_i, i=1,3,...,n, with U₁U₃...U_n=U.     

On the zeta-function of systems of nonlinear equations

We introduce the concept of zeta-function for a system of meromorphic functions f = (f ₁,..., f _n) in ?ⁿ. Using residue theory, we give an integral representation for the zeta-function which enables us to construct an analytic continuation of the zeta-function.     

On real solutions of systems of equations

Systems of equations f ₁ = ··· = f _n?1 = 0 in ? ⁿ = {x} having the solution x = 0 are considered under the assumption that the quasi-homogeneous truncations of the smooth functions f ₁,..., f _n?1 are independent at x ≠ 0. It is shown that, for n ≠ 2 and n ≠ 4, such a system has a smooth solution which passes through x = 0 and has nonzero Maclaurin series.     

The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

On form ideals and Artin-Rees condition

We look for conditions which make two ideals and in a noetherian ring A have the same form ideal in an associated graded ring G_A(α). More precisely, when and and f_i?f'_iεα ^m m?0, ? i , we give a necessary and sufficient condition to have , involving the first syzygies modules both of (f₁,...,f_n) and (f'₁,...,f'_n); our proof is based on the Artin-Rees lemma. Finally we show that, when the sequence f₁,...,f_n is regular and for an integer q, then f₁?f'_i ? α^q+1 ? i implies .     

On likelihood ratio ordering of parallel systems with exponential components

Let T(λ₁,...,λ _n ) be the lifetime of a parallel system consisting of exponential components with hazard rates λ₁,...,λ _n , respectively. For systems with only two components, Dykstra et. al. (1997) showed that if (λ₁, λ₂) majorizes (γ₁, γ₂), then, T(λ₁, λ₂) is larger than T(γ₁, γ₂) in likelihood ratio order. In this paper, we extend this theorem to general parallel systems. We introduce a new partial order, the so-called d-larger order, and show that if (λ₁,...,λ _n ) is d-larger than (γ₁,...,γ _n ), then T(λ₁,...,λ _n ) is larger than T(γ₁,...,γ _n ) in likelihood ratio order.     

Multivariate discriminant and iterated resultant

In this paper,we study the relationship between iterated resultant and multivariate discriminant.We show that,for generic form f(x_n) with even degree d,if the polynomial is squarefreed after each iteration,the multivariate discriminant △(f) is a factor of the squarefreed iterated resultant.In fact,we find a factor Hp(f,[x_1,...,x_n]) of the squarefreed iterated resultant,and prove that the multivariate discriminant △(f) is a factor of Hp(f,[x_1,...,x_n]).Moreover,we conjecture that Hp(f,[x_1,...,x_n]) = △(f) holds for generic form/,and show that it is true for generic trivariate form f(x,y,z).     

A generalization of the inversion formulas of systems of power series in systems of implicit functions

Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsx _j =_j(w)=_j(w₁,...,w_m), j=1,...,n, defined by the system of equations f_j(w, x)=F_j (w₁,...,w_m:z₁,...,x _n )=0, j=1,...,n,f _j , (0, 0)=0, F_j(0, 0)/z_k=_jk in a neighborhood of the point (0, 0)C _(w,x) ^m+n , in terms of the coefficients of the power series of the functions F_j(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 47–54, January, 1978.     

Modern thermodynamics as a branch of mathematics (mathematical physics)

In the paper, it is proved that, if f(x₁,..., x_n)g(y₁,..., y_m) is a multilinear central polynomial for a verbally prime T-ideal Γ over a field of arbitrary characteristic, then both polynomials f(x₁,..., x_n) and g(y₁,..., y_m) are central for Γ.     

On the construction of primitive elements in field extensions

The subject of this article are a priori constructions of primitive elements in field extensions. Consider the totality of all separable polynomialsf of degreen over a fieldK with rootsx ₁,...,x _n and prescribed Galois groupG. A vector (b ₁,...,b _n )K ⁿ is called stably primitive (forG), if, for each suchf,b ₁ x ₁+...+b _n x _n generates the splitting field off. We develop representation theoretical devices to investigate the set of stably primitive vectors geometrically. A fundamental observation is that is either very large or very small (or even empty). These two cases are illustrated by various examples. Moreover, criteria are given to decide which case holds. For a number of groups where is recognized to be small we show.     

A generalized mean-value theorem

We give another proof of Seymour and Zaslavsky's theorem: For every familyf ₁,f ₂,...,f _n of continous functions defined on [0, 1], there exists a finite setF[0, 1] such that the average sum off _k overF coincides with the integral off _k for everyk=1, 2,...,n.     

Extremal property of some surfaces in n-dimensional Euclidean space

A surface Γ=(f ₁(X₁,..., x_m),...,f _n(x₁,..., x_m)) is said to be extremal if for almost all points of Γ the inequality $$\parallel a_1 f_1 (x_1 , \ldots ,x_m ) + \ldots + a_n f_n (x_1 , \ldots ,x_m )\parallel< H^{ - n - \varepsilon } ,$$ , where H=max(¦a _i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa ₁, ...,a _n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf ₁, ...,f _n, the extremality of a class of surfaces in n-dimensional Euclidean space.     

Folgenkonvexe funktionen

In this paper functions f:(a,b)R are considered with the property that for all n>2 and all x₁,x₂,...,x_n(a,b) is convex in k. Functions with this property are called sequentially convex. It is proved that if f is convex, twice differentiable, and f is convex then f is sequentially convex. In case f is a continous function defined on the whole ofR these conditions are necessary too.     

Estimate of a complete rational trigonometric sum

Supposef is a polynomial of degree n≥3 with integral coefficientsa ₀,a ₁,...,a _n; q is a natural number; (a ₁,...,a _n, q)=1,f(0) = 0. It is proved that $$\left| {\sum\nolimits_{x = 1}^q {e^{2\pi if(x)/q} } } \right|< e^{5n^2 /\ln n} q^{1 - 1/n} $$ .     

Applications of the Chauvenet Test to Detection of Overliers in Observations Connected into a Homogeneous Markov Chain

Let x₁,...,x_n be random variables connected into a homogeneous Markov chain. The asymptotic behavior of the distribution of the number of overliers is investigated for unknown parameters a, , and . Bibliography: 4 titles.     

Nonhomogeneous recursions and Generalised Continued Fractions

Consider the (n+1)st order nonhomogeneous recursionX _k+n+1=b _k X _k+n +a _k ⁽ⁿ⁾ X _k+n-1+...+a _k ⁽¹⁾ X _k +X _k .Leth be a particular solution, andf ⁽¹⁾,...,f ⁽ⁿ⁾,g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾ andh. If we want to calculate a solutiony which is dominated byg, but dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾, then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations.     

Solving systems of polynomial inequalities over a real closed field in subexponential time

Let _m= (₁,..., _m) denote an ordered field, where _i+1>0 is infinitesimal relative to the elements of _i, 0 < –i < m (by definition, ₀= ). Given a system of inequalities f₁ > 0, ..., f_s > 0, f_s+1 0, ..., f_k 0, where f_{j m} [X₁,..., X_n] are polynomials such that, and the absolute value of any integer occurring in the coefficients of the f_js is at most 2^M. An algorithm is constructed which tests the above system of inequalities for solvability over the real closure of _m in polynomial time with respect to M, ((d)ⁿd₀)^n+m. In the case _m=, the algorithm explicitly constructs a family of real solutions of the system (provided the latter is consistent). Previously known algorithms for this problem had complexity of the order ofM(d d ₀ ^{m 2U(n)} .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 3–36, 1988.     

Two theorems concerning variational systems of smooth dynamical systems

A dynamical system given by a vector field of class C² in an n-dimensional, smooth, closed manifold Vⁿ let us call differentially homogeneous if for every v, w Vⁿ there exists a diffeomorphism of Vⁿ into itself such that it takes v into w and commutes with respect to motion along a trajectory for any time t. It can be shown that all of the variational systems of such a system are almost reducible.Furthermore, the dynamical systems given by the vector fieldsf(v) are considered to be ergodic in that they have the same integral invariant (nearly all of the variational systems of such a system have the same indices ₁(f)₂(f)... _n(f). It is proven that is an upper semicontinuous function off(v) when k=1, 2, ..., n.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 49–54, January, 1969.     

A Question of Nori: Projective Generation of Ideals

Let A be a smooth affine domain of dimension d over an infinite perfect field k and let n be an integer such that 2n d + 3. Let I A[T] be an ideal of height n. Assume that I = (f ₁,...,f _n ) + (I ² T). Under these assumptions, it is proved in this paper that I = (g ₁,...,g _n ) with f _i – g _i (I ² T), thus settling a question of Nori affirmatively.     

Real Solving of Elementary-Algebraic Systems

In this paper we present a new method for enclosing all the real roots in a bounded box of a system of n elementary-algebraic equations depending on n variables. This system is denoted by h(x)=P(x,f ₁(x ₁),...,f _k (x ₁),f ₁(x ₂),...,f _k (x _n )), where x=(x ₁,x ₂,...,x _n ) R ⁿ , P is a system of polynomials depending on n+kn variables and the univariate functions f _i are simple. This method arises from the exclusion method of Dedieu and Yakoubsohn. We provide both a theoretical complexity bound and some numerical experiments.