On the number of nonstationary bounded trajectories of a class of autonomous systems on the plane

We study the number of nonstationary bounded trajectories of autonomous systems of the form z′ = \(\overline {P_n (z)} \), z = x + iy ∈ C, where P _n (z) is a polynomial of degree n with complex coefficients that has k distinct roots, n, k > 1. We prove that the number N of nonstationary bounded trajectories of this system satisfies the following assertions (Theorem 1): (a) N = n + k ? N ₊, N ₊ = N _?, n + 1 ≤ N ₊ ≤ n + k, where N ₊ and N _? are the numbers of system trajectories unbounded as t → +∞ and t → ?∞, respectively; (b) if some r distinct roots \(c_{j_1 } \), ..., \(c_{j_r } \) of the polynomial P _n satisfy the relations V _n+1 (\(c_{j_1 } \)) = ··· = V _n+1 (\(c_{j_r } \)), where V _n+1 is the imaginary part of the indeterminate integral of P _n , then N ≥ \(m_{j_1 } \) + ··· + \(m_{j_r } \) + r ? n ? 1; (c) if k = 2, then the conditions N = 1 and V _n+1 (c ₁) = V _n+1 (c ₂) are equivalent. For n = k = 3, we derive a formula for the number of nonstationary bounded trajectories (Theorem 2).     

Systems that generate solutions with a small period

Let (j₁,..., j_n) be a permutation of the n-tuple (1, ..., n). A system of differential equations \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) in which each function f_i is continuous on ? is considered. This system is said to have the property of generation of solutions with a small period if, for any number M > 0, there exists a number ω₀ = ω₀(M) > 0 such that if 0 < ω ≤ ω₀ and h_i(t, x₁, ..., x_n) are continuous functions on ? × ?ⁿ ω-periodic in t that satisfy the inequalities |h_i| ≤ M the system \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) has an ω-periodic solution. It is shown that a system has the property of generation of solutions with a small period if and only if f_i(?) = ? for i = 1,..., n. It is also shown that the smallness condition on the period is essential.     

An extended class of stabilizable uncertain systems

The system of equations \(\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u\), where A(·) ∈ ?^{n × n}, B(·) ∈ ?^{n × m}, S(·) ∈ R^{n × m}, is considered. The elements of the matrices A(·), B(·), S(·) are uniformly bounded and are functionals of an arbitrary nature. It is assumed that there exist k elements \({\alpha _{{i_i}{j_l}}}\left( \cdot \right)\left( {l \in \overline {1,k} } \right)\) of fixed sign above the main diagonal of the matrix A(·), and each of them is the only significant element in its row and column. The other elements above the main diagonal are sufficiently small. It is assumed that m = n ?k, and the elements β_ij(·) of the matrix B(·) possess the property \(\left| {{\beta _{{i_s}s}}\left( \cdot \right)} \right| = {\beta _0} > 0\;at\;{i_s}\; \in \;\overline {1,n} \backslash \left\{ {{i_1}, \ldots ,{i_k}} \right\}\). The other elements of the matrix B(·) are zero. The positive definite matrix H = {h_ij} of the following form is constructed. The main diagonal is occupied by the positive numbers h_ii = h_i, \({h_{{i_l}}}_{{j_l}}\, = \,{h_{{j_l}{i_l}}}\, = \, - 0.5\sqrt {{h_{{i_l}}}_{{j_l}}} \,\operatorname{sgn} \,{\alpha _{{i_l}}}_{{j_l}}\left( \cdot \right)\). The other elements of the matrix H are zero. The analysis of the derivative of the Lyapunov function V(x) = x*H^–1x yields h_i\(\left( {i \in \overline {1,n} } \right)\) and λ_i ≤ 0 \(\left( {i \in \overline {1,n} } \right)\) such that for S(·) = H^?1ΛB(·), Λ = diag(λ₁, ..., λ_n), the system of the considered equations becomes globally exponentially stable. The control is robust with respect to the elements of the matrix A(·).     

An isometrical ℂP<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-theorem

Let M ⁿ (n ? 3) be a complete Riemannian manifold with sec _M ? 1, and let \(M_i^{n_i }\) (i = 1, 2) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n ? 2 and if the distance |M₁M₂| ? π/2, then M _i is isometric to \(\mathbb{S}^{n_i } /\mathbb{Z}_h\), \(\mathbb{C}P^{n_i /2}\), or \(\mathbb{C}P^{n_i /2} /\mathbb{Z}_2 \) with the canonical metric when n _i > 0; and thus, M is isometric to S ⁿ /? _h , ?P^n/2, or ?P^n/2/?₂ except possibly when n = 3 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{S}^1 /\mathbb{Z}_h \) with h ? 2 or n = 4 and \(M_1 (or M_2 )\mathop \cong \limits^{iso} \mathbb{R}P^2 \).     

Coherence relative to a weak torsion class

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules

$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$

such that F₀,..., F_n?1 are finitely generated free and K_n?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext ⁿ _R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor ^R _n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

     

Complete hypersurfaces with constant laguerre scalar curvature in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let x: M ^n?1 → R ⁿ be an umbilical free hypersurface with non-zero principal curvatures. Two basic invariants of M under the Laguerre transformation group of R ⁿ are Laguerre form C and Laguerre tensor L. In this paper, n > 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R ⁿ , denote the trace-free Laguerre tensor by ?\(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\) · Id. If \(\widetilde L = L - \frac{1}{{n - 1}}tr\left( L \right)\), then M is Laguerre equivalent to a Laguerre isotropic hypersurface; and if \({\sup _M}\left\| {\widetilde L} \right\| = \frac{{\sqrt {\left( {n - 1} \right)\left( {n - 2} \right)} R}}{{\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right)}},\), M is Laguerre equivalent to the hypersurface ?x: H ¹ × S ^n?2 → R ⁿ .     

The Number of <Emphasis Type="Italic">k</Emphasis>-Sumsets in an Abelian Group

Let G be an abelian group of order n. The sum of subsets A₁,...,A_k of G is defined as the collection of all sums of k elements from A₁,...,A_k; i.e., A₁ + A₂ + · · · + A_k = {a₁ + · · · + a_k | a₁ ∈ A₁,..., a_k ∈ A_k}. A subset representable as the sum of k subsets of G is a k-sumset. We consider the problem of the number of k-sumsets in an abelian group G. It is obvious that each subset A in G is a k-sumset since A is representable as A = A₁ + · · · + A_k, where A₁ = A and A₂ = · · · = A_k = {0}. Thus, the number of k-sumsets is equal to the number of all subsets of G. But, if we introduce a constraint on the size of the summands A₁,...,A_k then the number of k-sumsets becomes substantially smaller. A lower and upper asymptotic bounds of the number of k-sumsets in abelian groups are obtained provided that there exists a summand A_i such that |A_i| = n log^qn and |A₁ +· · ·+ A_i-1 + A_i+1 + · · ·+A_k| = n log^qn, where q = -1/8 and i ∈ {1,..., k}.     

Stabilization of a Class of Uncertain Systems

We consider the problem to synthesize a stabilizing control u synthesis for systems \(\frac{{dx}}{{dt}} = Ax + Bu\) where A ∈ ?^n×n and B ∈ ?^n×m, while the elements α_i,j(·) of the matrix A are uniformly bounded nonanticipatory functionals of arbitrary nature. If the system is continuous, then the elements of the matrix B are continuous and uniformly bounded functionals as well. If the system is pulse-modulated, then the elements of the matrix B are differentiable uniformly bounded functions of time. It is assumed that k isolated uniformly bounded elements \({\alpha _{{i_l},{j_l}}}\left( \cdot \right)\) satisfying the condition \(\mathop {\inf }\limits_{\left( \cdot \right)} \left| {{\alpha _{{i_l},{j_l}}}\left( \cdot \right)} \right|{\alpha _ - } > 0,\quad l \in \overline {1,k}\) are located above the main diagonal of the matrix A(·), where G _k is the set of all isolated elements of the system, J₁ is the set of indices of rows of matrix A(·) containing isolated elements, and J₂ is the set of indices of its rows free of isolated elements. It is assumed that other elements located above the main diagonal are sufficiently small provided that their row indices belong to J₁, i.e., \(\mathop {\sup }\limits_{\left( \cdot \right)} \left| {{\alpha _{i,j}}\left( \cdot \right)} \right| < \delta ,\quad {\alpha _{i,j}} \notin {G_k},\quad i \in {J_1},\quad j > i\). All other elements located above the main diagonal are uniformly bounded. The relation u = S(·)x is satisfied in the continuous case, while the relation u = ξ(t) is satisfied in the pulse-modulated case; here the components of the vector ξ are outputs of synchronous pulse elements. Constructing a special quadratic Lyapunov function, one can determine a matrix S(·) such that the closed system becomes globally exponentially stable in the continuous case. In the pulse-modulated case, input pulses are synthesized such that the system becomes globally asymptotically stable.     

Linear and nonlinear methods of relief approximation

In this paper, we compare the effectiveness of free (nonlinear) relief approximation, equidistant relief approximation, and polynomial approximation {ie129-01}, and {ie129-02} of an individual function ?(x) in the metric {ie129-03}, where {ie129-04} is the unit ball |x| ≤ 1 in the plane ?². The notation we use is the following: {fx129-01}. Here {ie129-05} is the set of all N-term linear combinations of functions of the plane-wave type {fx129-02} with arbitrary profiles W _j (x), x ∈ ?¹ and transmission directions {θ _j } ₁ ^N ; {ie129-06} is the subset of {ie129-07} associated with N equidistant directions; {fx129-03} denotes the subspace of algebraic polynomials of degree less than or equal to N ? 1 in two real variables. Obviously, the inequalities {ie129-08} hold.We state the following model problem. What are the functions which satisfy the relation {ie129-09}, i.e., where the nonlinear approximation {ie129-10} is more effective than a linear one? This effect has been proved for harmonic functions, namely, for any ε > 0 there exists c _ε > 0 such that if Δ?(x) = 0, |x| < 1, and ? ∈ {ie129-11}, then {fx129-04}. On the other hand, {ie129-12}. Thus, {ie129-13} has an “almost squared effectiveness” of {ie129-14} for ? = ?_harm. However, this ultra-high order of approximation is obtained via a collapse of wave vectors.On the other hand, the nonlinearity of {ie129-15} which corresponds to the freedom of choice of wave vectors does not much improve the order of approximation, for instance, for all the radial functions. If {ie129-16}, then {ie129-17} and {ie129-18}.The technique we use is the Fourier-Chebyshev analysis (which is related to the inverse Radon transform on {ie129-19}) and a duality between the relief approximation problem and the optimization of quadrature formulas in the sense of Kolmogorov-Nikolskii [14] for trigonometric polynomial classes.     

Bounded indecomposable semigroups of non-negative matrices

A semigroup \({\mathfrak{S}}\) of non-negative n × n matrices is indecomposable if for every pair i, j ≤ n there exists \({S\in\mathfrak{S}}\) such that (S) _ij ≠ 0. We show that if there is a pair k, l such that \({\{(S)_{kl} : S\in\mathfrak{S}\}}\) is bounded then, after a simultaneous diagonal similarity, all the entries are in [0, 1]. We also provide quantitative versions of this result, as well as extensions to infinite-dimensional cases.     

The maximum and minimum degree of the random <Emphasis Type="Italic">r</Emphasis>-uniform <Emphasis Type="Italic">r</Emphasis>-partite hypergraphs

In this paper we consider the random r-uniform r-partite hypergraph model H(n ₁, n ₂, ···, n _r; n, p) which consists of all the r-uniform r-partite hypergraphs with vertex partition {V ₁, V ₂, ···, V _r} where |V _i| = n _i = n _i(n) (1 ≤ i ≤ r) are positive integer-valued functions on n with n ₁ +n ₂ +···+n _r = n, and each r-subset containing exactly one element in V _i (1 ≤ i ≤ r) is chosen to be a hyperedge of H _p ∈ H (n ₁, n ₂, ···, n _r; n, p) with probability p = p(n), all choices being independent. Let

$${\Delta _{{V_1}}} = {\Delta _{{V_1}}}\left( H \right)$$

and

$${\delta _{{V_1}}} = {\delta _{{V_1}}}\left( H \right)$$

be the maximum and minimum degree of vertices in V ₁ of H, respectively;

$${X_{d,{V_1}}} = {X_{d,{V_1}}}\left( H \right),{Y_{d,{V_1}}} = {Y_{d,{V_1}}}\left( H \right)$$

,

$${Z_{d,{V_1}}} = {Z_{d,{V_1}}}\left( H \right)and{Z_{c,d,{V_1}}} = {Z_{c,d,{V_1}}}\left( H \right)$$

be the number of vertices in V ₁ of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H(n ₁, n ₂, ···, n _r; n, p),

$${X_{d,{V_1}}},{Y_{d,{V_1}}},{Z_{d,{V_1}}}and{Z_{c,d,{V_1}}}$$

all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D(n) such that in the space H(n ₁, n ₂, ···, n _r; n, p),

$$\mathop {\lim }\limits_{n \to \infty } P\left( {{\Delta _{{V_1}}} = D\left( n \right)} \right) = 1$$

? What is the range of p such that a.e., H _p ∈ H (n ₁, n ₂, ···, n _r; n, p) has a unique vertex in V ₁ with degree

$${\Delta _{{V_1}}}\left( {{H_p}} \right)$$

? Both answers are p = o (log n ₁/N), where

$$N = \mathop \prod \limits_{i = 2}^r {n_i}$$

. The corresponding problems on

$${\delta _{{V_i}}}\left( {{H_p}} \right)$$

also are considered, and we obtained the answers are p ≤ (1 + o(1))(log n ₁/N) and p = o (log n ₁/N), respectively.

     

Volume Growth and the Topology of Manifolds with Nonnegative Ricci Curvature

Let M ⁿ be a complete, open Riemannian manifold with Ric≥0. In 1994, Grigori Perelman showed that there exists a constant δ _n >0, depending only on the dimension of the manifold, such that if the volume growth satisfies \(\alpha_{M}:=\lim_{r\rightarrow \infty}\frac{\operatorname{Vol}(B_{p}(r))}{\omega_{n}r^{n}}\geq 1-\delta_{n}\), then M ⁿ is contractible. Here we employ the techniques of Perelman to find specific lower bounds for the volume growth, α(k,n), depending only on k and n, which guarantee the individual k-homotopy group of M ⁿ is trivial.     

Ramanujan-type congruences modulo powers of 5 and 7

Let b _? (n) denote the number of ?-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of ?. In particular, they showed that for α, n ≥ 0, b ₂₅ (3^2α+3 n+2 · 3^2α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c _N (n) counts the number of bipartitions (λ₁,λ₂) of n such that each part of λ₂ is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, \({c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0\) (mod 5 ^j+1), \({c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0\) (mod 7 ^j+1) and b ₂₅ (3^2α+3 · n+2 · 3^2α+2-1) ≡ 0 (mod 3 · 5^2j-1).     

Extensions of representations of integral quadratic forms

Let N and M be quadratic ?-lattices, and K be a sublattice of N. A representation σ:K→M is said to be extensible to N if there exists a representation ρ:N→M such that ρ | _K =σ. We prove in this paper a local–global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite ?-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132–141, 1978) and Jöchner and Kitaoka (J. Number Theory 48:88–101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed.     

On EA-equivalence of certain permutations to power mappings

In this paper we investigate the existence of permutation polynomials of the form x ^d  + L(x) on \({{\mathbb{F}_{2^n}}}\) , where \({{L(x)\in\mathbb{F}_{2^n}[x]}}\) is a linearized polynomial. It is shown that for some special d with gcd(d, 2 ⁿ ?1) > 1, x ^d  + L(x) is nerve a permutation on \({{\mathbb{F}_{2^n}}}\) for any linearized polynomial \({{L(x)\in\mathbb{F}_{2^n}[x]}}\) . For the Gold functions \({{x^{2^i+1}}}\) , it is shown that \({{x^{2^i+1}+L(x)}}\) is a permutation on \({{\mathbb{F}_{2^n}}}\) if and only if n is odd and \({{L(x)=\alpha^{2^i}x+\alpha x^{2^i}}}\) for some \({{\alpha\in\mathbb{F}_{2^n}^{*}}}\) . We also disprove a conjecture in (Macchetti Addendum to on the generalized linear equivalence of functions over finite fields. Cryptology ePrint Archive, Report2004/347, 2004) in a very simple way. At last some interesting results concerning permutation polynomials of the form x ^?1 + L(x) are given.     

Conjugacy classes in Möbius groups

Let \({\mathbb H^{n+1}}\) denote the n + 1-dimensional (real) hyperbolic space. Let \({\mathbb {S}^{n}}\) denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of \({\mathbb {S}^{n}}\) is denoted by M(n). Let M _o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M _o (n) depends on the conjugacy of T and T ^?1 in M _o (n). For an element T in M(n), T and T ^?1 are conjugate in M(n), but they may not be conjugate in M _o (n). In the literature, T is called real if T is conjugate in M _o (n) to T ^?1. In this paper we classify real elements in M _o (n). Let T be an element in M _o (n). Corresponding to T there is an associated element T _o in SO(n + 1). If the complex conjugate eigenvalues of T _o are given by \({\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}\) , then {θ₁, . . . , θ _k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M _o (n) we have parametrized the conjugacy classes of regular elements in M _o (n). In the parametrization, when T is not conjugate to T ^?1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.     

Dynamic control for non-linear systems with delay

We consider some class of non-linear systems of the form

$\dot x = A( \cdot )x + \sum\limits_{i = 1}^l {A_i ( \cdot )x(t - \tau _i (t)) + b( \cdot )u} ,$

where A(·) ∈ ? ^{n × n} , A _i (·) ∈ ? ^{n × n} , b(·) ∈ ? ⁿ , whose coefficients are arbitrary uniformly bounded functionals.

A special type of the Lyapunov-Krasovskii functional is used to synthesize dynamic control described by the equation

$\dot u = \rho ( \cdot )u + (m( \cdot ),x),$

where ρ(·) ∈ ?¹, m(·) ∈ ? ⁿ , which makes the system globally asymptotically stable. Also, the situation is considered where the control u enters into the system not directly but through a pulse element performing an amplitude-frequency modulation.

     

On the Baire Classification of Positive Characteristic Exponents in the Perron Effect of Change of Their Values

In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y₀) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y₀)] of these solutions is a function of the second Baire class of their initial vectors y₀ ∈ ?ⁿ {0}.     

Classification of smooth structures on a homotopy complex projective space

We classify, up to diffeomorphism, all closed smooth manifolds homeomorphic to the complex projective n-space \(\mathbb {C}\textbf {P}^{n}\), where n=3 and 4. Let M²ⁿ be a closed smooth 2n-manifold homotopy equivalent to \(\mathbb {C}\textbf {P}^{n}\). We show that, up to diffeomorphism, M⁶ has a unique differentiable structure and M⁸ has at most two distinct differentiable structures. We also show that, up to concordance, there exist at least two distinct differentiable structures on a finite sheeted cover N²ⁿ of \(\mathbb {C}\textbf {P}^{n}\) for n=4,7 or 8 and six distinct differentiable structures on N¹⁰.     

Linear preservers of row-dense matrices

Let M_m,n be the set of all m × n real matrices. A matrix A ∈ M_m,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_m,n → M_m,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ M_n,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.