Article
We consider the resonance set of a real polynomial, i.e. the set of all the points of the coefficient space at which the polynomial has commensurable roots. The resonance set of a polynomial can be considered as a certain generalization of its discriminant set. The structure of the resonance set is useful for investigation of resonances near stationary point of a dynamical system.
The constructive algorithm of computation of polynomial parametrization of the resonance set is provided. The structure of the resonance set of a polynomial of degree \(n\) is described in terms of partitions of the number \(n\).
The main algorithms, described in the paper, are organized as a library of the computer algebra system \(Maple\). The description of the resonance set of a cubic polynomial is given.
Geometric methods are widely used in combinatorial group theory. The theory of small cancellation groups use the diagram method. In particular, it allows to approach various algorithmic problems. One of them is the power conjugacy problem. It is already solved for groups with a presentation satisfying the small cancellation conditions C(3) and T(6). However, it remains open for a similar class of groups, having a presentation satisfying the small cancellation conditions C(3) and T(3).
In this paper we investigate the structure of connected diagrams over presentations satisfying the small cancellation conditions C(3) and T(3) and we indicate how our results may be possible used in the power conjugacy problem.
The main result of this article is the proof of the theorem about lower bound on square of the reduced diagram on the group with small cancellation conditions C(3)-T(6). It is known that for groups with conditions C(p)-T(q) with \((p,q)\in \{(3,6), (4,4), (6,3)\}\), being automatic, isoperimetric inequality is quadratic. The same stated in well-known in small cancellation theory theorem of the square. Both statements restrict the area of the simply connected diagrams in the considered class of groups by the quadratic function of the length of the boundary.
In this article it is proved that the lower bound for the area of the diagram of the specified type also is a quadratic function of the length of the border. The importance of this result is visible from the point of view of evaluation of complexity of the algorithm solves the word problem. It is not less than quadratic complexity of the length of the compared words.
The paper investigates a class of nonlinear dynamic shell models, which non-linearity reflects Gaussian curvature of a surface; in the case when loads are smaller than critical ones in every point in time. Moreover, every unknown function from the system of equations, can be uniquely identified through the deflection function. Domain that is defined by the middle shell surface is bounded with piecewise smooth boundary. Such models as Kirchhoff-Love model (that specify Tymoshenko model, defined both in transferences and mixed forma), a model that reflects the bond between deformation fields and temperature and others can represent that equation class.
The method of subsequent parameters perturbation developed by professor V. Petrov in 1970s is used as a numerical method for such models. This method brings the solution of nonlinear equations to the solution of a sequence of linear equations. The paper discusses problems connected with the realization of this method. It is known, that method of V. Petrov converges slowly. That is why questions of convergence improvement are examined. The usage of variation methods for solving systems of linear equations requires defined convergence speed and orthogonal system of functions that satisfies the boundary conditions. These questions are investigated in the paper as well.
Let in the real \(n\)-dimensional space \(\mathbb{R}^n=\{X\}\) be given \(m\) real homogeneous forms \(f_i(X)\), \(i=1,\dotsc,m\), \(2\leqslant m\leqslant n\). The convex hull of the set of points \(G(X)=(|f_1(X)|,\dotsc,|f_m(X)|)\) for integer \(X\in\mathbb Z^n\) in many cases is a convex polyhedral set. Its boundary for \(||X||<\mathrm{const}\) can be computed by means of the standard program. The points \(X\in\mathbb Z^n\) are called boundary points if \(G(X)\) lay on the boundary. They correspond to the best Diophantine approximations \(X\) for the given forms. That gives the global generalization of the continued fraction. For \(n=3\) Euler, Jacobi, Dirichlet, Hermite, Poincar\'e, Hurwitz, Klein, Minkowski, Brun, Arnold and a lot of others tried to generalize the continued fraction, but without a succes.
Let \(p(\xi)\) be an integer real irreducible in \(\mathbb Q\) polynomial of the order \(n\) and \(\lambda\) be its root. The set of fundamental units of the ring \(\mathbb Z[\lambda]\) can be computed using boundary points of some set of linear and quadratic forms, constructed by means of the roots of the polynomial \(p(\xi)\). Similary one can compute a set of fundamental units of other rings of the field \(\mathbb Q(\lambda)\). Up today such sets of fundamental units were computed only for \(n=2\) (using usual continued fractions) and \(n=3\) (using the Voronoi algorithms).
Our approach generalizes the continued fraction, gives the best rational simultaneous approximations, fundamental units of algebraic rings of the field \(\mathbb Q(\lambda)\) and all solutions of a certain class of Diophantine equations for any \(n\).
Let \(\mathcal{K}\) be an abstract class of groups. Suppose \(\mathcal{K}\) contains at least a non trivial group.
Then \(\mathcal{K}\) is called a root-class if the following conditions are satisfied:
1. If \(A \in \mathcal{K}\) and \(B \leq A\), then \(B \in \mathcal{K}\).
2. If \(A \in \mathcal{K}\) and \(B \in \mathcal{K}\), then \(A\times B \in \mathcal{K}\).
3. If \(1\leq C \leq B \leq A\) is a subnormal sequence and \(A/B, B/C \in \mathcal{K}\), then there exists a normal subgroup \(D\) in group \(A\)
such that \(D \leq C\) and \(A/D \in \mathcal{K}\).
Group \(G\) is root-class residual (or \(\mathcal{K}\)-residual), for a root-class \(\mathcal{K}\) if,
for every \(1 \not = g \in G\),
exists a homomorphism \(\varphi \) of group \(G\) onto a group of root-class \(\mathcal{K}\) such that \(g\varphi \not = 1\).
Equivalently, group \(G\) is \(\mathcal{K}\)-residual if, for every \(1 \not = g \in G\),
there exists a normal subgroup \(N\) of \(G\) such that \(G/N \in \mathcal{K}\) and \(g \not \in N\).
The most investigated residual properties of groups are finite groups residuality (residual finiteness),
\(p\)-finite groups residuality and soluble groups residuality.
All there three classes of groups are root-classes.
Therefore results about root-class residuality have safficiently enough general character.
Let \(\mathcal{K}\) be a root-class of finite groups.
And let \(G\) be a fundamental group of a finite graph of groups with finite edges groups.
The necessary and sufficient condition of virtual \(\mathcal{K}\)-residuality
for the group \(G\) is obtained.
The paper deals with a new object of study --- hyperbolic Hurwitz zeta function, which is given in the right \(\alpha\)-semiplane \( \alpha = \sigma + it \), \( \sigma> 1 \) by the equality
$$
\zeta_H(\alpha; d, b) = \sum_{m \in \mathbb Z} \left(\, \overline{dm + b} \, \right)^{-\alpha},
$$
where \( d \neq0 \) and \( b \) --- any real number.
Hyperbolic Hurwitz zeta function \( \zeta_H (\alpha; d, b) \), when \( \left\| \frac {b} {d} \right\|> 0 \) coincides with the hyperbolic zeta function of shifted one-dimensional lattice \( \zeta_H (\Lambda (d, b) | \alpha) \). The importance of this class of one-dimensional lattices is due to the fact that each Cartesian lattice is represented as a union of a finite number of Cartesian products of one-dimensional shifted lattices of the form \( \Lambda (d, b) = d \mathbb{Z} + b \).
Cartesian products of one-dimensional shifted lattices are in substance shifted diagonal lattices, for which in this paper the simplest form of a functional equation for the hyperbolic zeta function of such lattices is given.
The connection of the hyperbolic Hurwitz zeta function with the Hurwitz zeta function \( \zeta^* (\alpha; b)\) periodized by parameter \(b\) and with the ordinary Hurwitz zeta function \( \zeta (\alpha; b) \) is studied.
New integral representations for these zeta functions and an analytic continuation to the left of the line \( \alpha = 1 + it \) are obtained.
All considered hyperbolic zeta functions of lattices form an important class of Dirichlet series directly related to the development of the number-theoretical method in approximate analysis. For the study of such series the use of Abel's theorem is efficient, which gives an integral representation through improper integrals. Integration by parts of these improper integrals leads to improper integrals with Bernoulli polynomials, which are also studied in this paper.
Proof (or disproof) of the Riemann hypothesis is the central problem of analytic number theory. By now it has not been solved.
In 1985 Karatsuba proved that for any \( 0 <\varepsilon <0,001 \), \( 0,5 <\sigma \leq 1 \), \( T> T_0 (\varepsilon)> 0 \) and \( H = T ^ { 27/82 + \varepsilon} \) in the rectangle with vertices \( \sigma + iT \), \( \sigma + i (T + H) \), \( 1 + i (T + H) \), \( 1 + iT \) contains no more than \( cH / (\sigma-0,5) \) zeros of \( \zeta (s) \). Thereby A.A. Karatsuba significantly strengthened the classical theorem J. Littlewood's.
Decrease in magnitude of \(H\) for individual rectangle has not been obtained. However, by solving this problem <<on average>>, in 1989 L.V. Kiseleva proved that for <<almost all>> \( T \) in the interval \( [X, X + X ^ {11/12 + \varepsilon}] \), \( X> X_0 (\varepsilon) \) in rectangle with vertices \( \sigma + iT \), \( \sigma + i (T + X ^ \varepsilon) \), \( 1 + i (T + X ^ \varepsilon) \), \( 1 + iT \) contains no more than \( O (X ^ \varepsilon / (\sigma-0,5)) \) zeros of \( \zeta (s) \).
In this article, we obtain a result of this kind, but for <<almost all >> \( T \) in the interval \( [X, X + X ^ {7/8 + \varepsilon}] \).
In this paper we consider the behavior of funcions defined by Dirichlet series with multiplicative coefficients and with bounded summatory function when approaching the imaginary axis. We show that the points of the imaginary axis are also the points of continuity in a broad sense of functions defined by Dirichlet series with multiplicative coefficients which are determined by nonprincipal generalized characters. This result is particularly interesting in its connection with a solution of Chudakov hyphotesis, which states that any finite-valued numerical character, which does not vanish on all prime numbers and has bounded summatory function, is a Dirichlet character.
The proof of the main result in this paper is based on the method of reduction to power series, basic principles of which were developed by prof. Kuznetsov in the early 1980s. Ths method establishes a connection between analytical properties of Dirichlet series and boundary properties of the corresponding power series (i.e. a power series with the same coefficients as the Dirichlet series). This allows to obtain new results both for the Dirichlet series and for the power series. In our case this method allowed us to prove the main result using the properties of the power series with multiplicative coefficients determined by the nonprincipal generalized characters, which also were obtained in this work.
Normally in mathematics and physics only point particle systems, which are either finite or countable, are studied. We introduce new formal mathematical object called regular continuum system of point particles (with continuum number of particles). Initially each particle is characterized by the pair: (initial coordinate, initial velocity) in \(R^{2d}\). Moreover, all initial coordinates are different and fill up some domain in \(R^{d}\). Each particle moves via normal newtonian dynamics under influence of sone external force, but there is no interaction between particles. If the external force is bounded then trajectories of any two particles in the phase space do not intersect. More exactly, at any time moment any two particles have either different coordinates or different velocities. The system is called regular if there are no particle collisions in the coordinate space.
The regularity condition is necessary for the velocity of the particle, situated at a given time at a given space point, were uniquely defined. Then the classical Euler equation for the field of velocities has rigorous meaning. Though the continuum of particles is in fact a continuum medium, the crucial notion of regularity was not studied in mathematical literature.
It appeared that the seeming simplicity of the object (absence of interaction) is delusive. Even for simple external forces we could not find simple necessary and sufficient regularity conditions. However, we found a rich list of examples, one dimensional and many dimensional, where we get regularity conditions on different time intervals. In conclusion we formulate many perspective problems for regular systems with interaction.
The paper describes the arithmetic nature of the values at integer points of series from the so-called class of \(F\)--series which constitute a solution of a system of linear differential equations with coefficients --- rational functions in z.
We consider a subclass of the series consisting of the series of the form
\sum_{n=0}^\infty a_n\cdot n!\; z^n
where \(a_n\in\mathbb Q\), \(|a_n|\leq e^{c_1 n}\), \(n=0,1,\ldots\) with some constant \(c_1\). Besides there exists a sequence of positive integers \(d_n\) such that \(d_n\; a_k\in\mathbb Z\), \(k=0,\ldots,n\) and \(d_n=d_{0,n} d_n\), \(d_{0,n}\in\mathbb N\), \mbox{\(n=0,1,\ldots,d\in\mathbb N\)} and for any \(n\) the number \(d_{0,n}\) is divisible only by primes \(p\) such that \(p\leqslant c_2 n\). Moreover
$$ord_p n \leq c_3\left(\log_p n+\frac{n}{p^2}\right).$$
We say then that the considered series belongs to the class \(F(\mathbb{Q},c_1,c_2,c_3,d)\).
Such series converge at a point \(z\in\mathbb Z\), \(z\ne 0\) in the field \(\mathbb Q_p\) for almost all primes \(p\).
The direct product of the rings \(\mathbb Z_p\) of \(p\)--adic integers over all primes \(p\) is called the ring of polyadic integers. It's elements have the form
$$\mathfrak{a} = \sum_{n=0}^\infty a_n\cdot n!,\quad a_n\in\mathbb Z$$
and they can be considered as vectors with coordinates \(\mathfrak{a}^{(p)}\) which are equal to the sum of the series \(\mathfrak{a}\) in the field \(\mathbb Q_p\) (This direct product is infinite).
For any polynomial \(P(x)\) with integer coefficients we define \(P(\mathfrak{a})\) as the vector with coordinates \(P(\mathfrak{a}^{(p)})\) in \(\mathbb Q_p\). According to the classification, described in V. G. Chirskii's works we call polyadic numbers \(\mathfrak{a}_1,\ldots,\mathfrak{a}_m\) infinitely algebraically independent, if for any nonzero polynomial \(P(x_1,\ldots,x_m)\) with integer coefficients there exist infinitely many primes \(p\) such that
$$P\left(\mathfrak{a}_1^{(p)},\ldots,\mathfrak{a}_m^{(p)}\right)\ne 0 $$
in \(\mathbb Q_p\).
The present paper states that if the considered \(F\)--series \(f_1,\ldots,f_m\) satisfy a system of differential equations of the form
$$P_{1,i}y_i^\prime + P_{0,i}y_i = Q_i, i=1,\ldots,m$$
where the coefficients \(P_{0,i}, P_{1,i}, Q_i\) are rational functions in \(z\) and if \(\xi\in\mathbb Z\), \(\xi\ne 0\), \(\xi\) is not a pole of any of these functions and if
$$\exp\left(\int\left(\frac{P_{0,i}(z)}{P_{1,i}(z)}-\frac{P_{0,j}(z)}{P_{1,j}(z)}\right)dz\right)\not\in\mathbb C(z)$$
then
\(f_1(\xi),\ldots,f_m(\xi)\) are infinitely algebraically independent almost polyadic numbers.
For the proof we use a modification of the Siegel-Shidlovsky's method and V. G. Chirskii's. Salikhov's approach to prove the algebraic independence of functions, constituting a solution of the above system of differential equations.
Let \(\Gamma\) be a strongly regular graph with parameters \((v,k,0,2)\). Then \(k=u^2+1\), \(v=(u^4+3u^2+4)/2\)
and \(u \equiv 1, 2, 3(mod 4)\). If \(u=1\), then \(\Gamma\) has parametrs \((4,2,0,2)\) --- tetragonal graph. If \(u=2\), then
\(\Gamma\) has parametrs \((15,5,0,2)\) --- Clebsch graph. If \(u=3\), then \(\Gamma\) has parametrs \((56,10,0,2)\) --- Gewirtz graph.
If \(u=5\) then hypothetical strongly regular graph\(\Gamma\) has parametrs \((352,26,0,2)\) [4]. If \(u=5\) then hypothetical
strongly regular graph\(\Gamma\) has parametrs \((704,37,0,2)\) [5].
Let \(u=7\), then \(\Gamma\) has parametrs \((1276,50,0,2)\). Let \(G\) be the automorphism group of a hypothetical strongly regular
graph with parameters \((1276, 50, 0, 2)\). Possible orders are found and the structure of fixed-point subgraphs
is determined for elements of prime order in \(G\). With the use of theory of characters of finite groups we find the
possible orders and the structures of subgraphs of the fixed points of automorphisms of the graph with parameters
\((1276,50,0,2)\). It proved that if the graph with parametrs (1276,50,0,2) exist, its automorphism group divides
\(2^l\cdot 3\cdot 5^m\cdot 7\cdot 11\cdot 29\). In particulary, \(G\) --- solvable group.
The Arkhipov--Karatsuba's system of congruencies by arbitrary modulo, greater than a degree of forms in it, has a solution for any right-hand parts, and for the number on unknowns exceeding the value
\(8(n+1)^2\log_2n+12(n+1)^2+4(n+1),\) where \(n\) is the degree of forms of this system.
One of essential problems in generating pseudo-random numbers is the problem of periodicity of the resulting numbers. Some generators output periodic sequences. To avoid it several ways are used.
Here we present the following approach: supposed we have some order in the considered set. Let's invent some algorithm which produces disorder in the set. E.g. if we have a periodic sequence of integers, let's construct an irrational number implying the given set. Then the figures of the resulting number form a non-periodic sequence.
Here we can use continued fractions and Lagrange's theorem asserts that the resulting number is irrational.
Another approach is to use series of the form \(\sum_{n=0}^\infty \frac{a_n}{n!}\) with a periodic sequence of integers \(\{a_n\}, a_{n+T}=a_n\) which is irrational.
Here we consider polyadic series \(\sum_{n=0}^\infty a_n n!\) with a periodic sequence of positive integers \(\{a_n\},a_{n+T} = a_n\) and describe some of their properties.
Suppose there is subset \(A\) of positive integers from the interval \([1,q]\) with the following condition. If the elements \(a,b\) of \(A\) and \(ab\) is at most \(q\), then \(ab\) belongs to \(A\). In additition let also know that \(|A|<q^{\nu}\), \(\nu\) - is some fixed number, not exceeding 1. In this paper we consider the question of the number of elements belonging to \(A\) on the interval with length substantially less than \(q\), - on the interval \([1, x]\), where \(x\) is much smaller than an arbitrary power of \(q\).
In this task, in the case when \(A\) - is a special set and with certain restrictions on \(|A|\) and \(x\), there exists some results. So, from the work of J. Bourgain, S. Konyagin and I. Shparlinskii there are nontrivial estimates in the case when \(A\) - a multiplicative subgroup of invertible elements of the residue ring modulo prime.
The initial problem generalize it to the case of semigroups instead of multiplicative subgroups. It should be noted that there are quite definite results on this task. The main result of this work is to derived a new estimate on the number of elements of the semigroup of natural numbers given short interval from 1 to \(x\). These estimates are meaningful when \(x\) is much smaller than any power of \(q\). More precisely, let \(A\) - our semigroup, \(g: =\frac{\log{\log x}}{\log{\log q}}, x = q^{o (1)}\) for \(q\) tends to infinity. Then the number of elements of \(A\) in the interval \((1,x)\) does not exceed \(x^{1-C (g,\nu)+o(1)}\), where \(C(g,\nu )\) - some clearly written positive function. Previous result relates to the estimation of function \(C(g,\nu)\), a new estimate for the \(C(g,\nu)\) improves the previous result for a certain range of parameters \((g,\nu)\).
We essentially use in the proof the distribution of smooth numbers, the numbers with a large part of the smooth part, estimates on the number of divisors of a fixed number in a given interval. We use some results of J. Bourgain, S. Konyagin and I. Shparlinski.
