Structure and Properties of Electrodeposited Composite Metal–Nanodisperse Inorganic Compound Coatings

By using the electrodeposition method, composite Sn-inorganic coatings have been obtained. As the inorganic compounds, ultrafine PbMo₆S₈, NbC_xN_y, Ti_xN_y, and Ti_0.63Nb_0.34C_0.38N_0.58 powders were utilized. The Sn–PbMo₆S₈ coatings containing 28-88 wt.% PbMo₆S₈ were superconductive. The highly electroconductive Sn–NbC_xN_y coatings containing 1-88 wt.% of the disperse phase and the Sn–Ti_xN_y coatings with 42-70 wt.% Ti_xN_y were structurally stable at a low temperature (–50°C), solderable, and corrosion resistant on copper. The formation of the structure of the coatings is determined by the electrolyte composition, the cathode current density during the electrolysis, and the codeposition of the disperse phase. By electrodeposition of the chromium carbonitride Cr₃C_1.6N_0.4, in the presence of H₄P₂O₇ in an CrO₃-based electrolyte, mat composite Cr–Cr₃C_1.6N_0.4 coatings with 6.8 wt.% Cr₃C_1.6N_0.4 were obtained, which had a finely uneaven surface structure. Such surfaces are needed in the cases where lubricants are used.     

Remarks on a stability problem

The paper considers a system of three differential equations of the following form:x ₁=a₁₁x₁+₂(x₂)+₃(x₃),x ₂=a₂₁x₁+a₂₂x₂+a₂₃x₃,x ₃=a₃₁x₁+a₃₂x₂+a₃₃x₃.It is established that the given system can have periodic solutions if the generalized Routh-Hurwitz conditions hold.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 533–536, November, 1971.     

Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study

We continue the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation u_ty = u_xu_xy ? u_yu_xx, (2) the Pavlov equation u_yy = u_tx + u_yu_xx ? u_xu_xy, (3) the universal hierarchy equation u_yy = u_tu_xy ? u_yu_tx, and (4) the modified Veronese web equation u_ty = u_tu_xy ? u_yu_tx. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.     

Product spaces and semi-separation axioms

In this paper new characterizations of semi-R₀ and semi-R₁ spaces are obtained and used to prove that the product space of semi-R₀, semi-T₁, and semi-T₀ spaces is, respectively, semi-R₀, semi-T₁, and semi-T₀; and that the product space of semi-R₁ space need not be semi-R₁. An example is given where the product space is semi-T₂ and one of the factor spaces is not semi-T₀ or semi-R₀.     

The minimum increment of f-divergences given total variation distances

Let (P _i , Q _i ), i = 0, 1, be two pairs of probability measures defined on measurable spaces (Ω _i ,F _i ) respectively. Assume that the pair (P₁, Q₁) is more informative than (P₀,Q₀) for testing problems. This amounts to say that If (P₁,Q₁) ≥ If (P₀,Q₀), where If (·, ·) is an arbitrary f-divergence. We find a precise lower bound for the increment of f-divergences I_f(P₁,Q₁) ? I_f(P₀,Q₀) provided that the total variation distances ||Q₁ ? P₁|| and ||Q₀ ? P₀|| are given. This optimization problem can be reduced to the case where P₁ and Q₁ are defined on the space consisting of four points, and P₀ and Q₀ are obtained from P₁ and Q₁ respectively by merging two of these four points. The result includes the well-known lower and upper bounds for I_f(P,Q) given ||Q ? P||.     

Distribution of lattice points on surfaces of second order

Let f₁(x₁,..., xl₁) and f₂(y₁,...,yl₂) be positive definite primitive quadratic forms in l₁ and l₂ variables, respectively. We obtain new results in the well-known problem on the number of lattice points on the cone f₁(x₁,...,xl₁)=f₂(y₁,...,yl₂), in the domain f₁(x₁,...,xl₁)≦N for N»∞. Our technical tool is the Rankin-Selberg convolution. In several special cases the results can be sharpened by other methods. In addition, new facts concerning the uniform distribution of lattice points on ellipsoids in l variables, l odd, l≧5 are obtained.     

Regular sequences in Rees and symmetric algebras,II

In continuing [7] we study necessary and sufficient conditions for a system of elements b₁,...,b_s,a₁,...,a_t of a local Noetherian ring A such that the sequence b₁,...,b_s,a₁T,a₁-a₂T,...,a_t–1-a_tT,a_t in the Rees algebra A[a₁ T,...,a_t T], T is an indeterminate, constitutes a regular sequence.     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁ +A₂X₂B₂ +A₃X₃B₃ =C,(ii) A₁XB₁ =C₁ A₂XB₂ =C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A ₁ X ₁ B ₁+A ₂ X ₂ B ₂+A ₃ X ₃ B ₃+A ₄ X ₄ B ₄=C, (iv) A ₁ XB ₁=C ₁ A ₂ XB ₂=C ₂ A ₃ XB ₃=C ₃ A ₄ XB ₄=C ₄, (v) AXB+CXD=E are also considered.     

The Poincaré series of the algebras of simultaneous invariants and covariants of two binary forms

Let ? _{d 1, d 2} and 𝒞 _{d 1, d 2} be the algebras of simultaneous invariants and simultaneous covariants of the two binary forms of degrees d ₁ and d ₂. Formulas for computation of the Poincaré series 𝒫? _{d 1, d 2} (z), 𝒫𝒞 _{d 1, d 2} (z) of the algebras are found. By using these formulas, we have computed the series for d ₁, d ₂ ≤ 20.     

由分数次导数所确定的L2(T)中的一元周期函数类在Lq(T)中的相对宽度

作者研究了相对宽度K_n(W₂^α(T), MW₂^β(T), L₂(T)), T=[0,2π], 确定了使等式K_n(W₂^α(T), MW₂^β(T), L₂(T))=d_n(W₂^α(T), L₂(T))成立的最小M值, 得到了相对宽度K_n(W₂^α(T), W₂^α(T), L_q(T))的渐近阶, 其中α≥β>0, 1≤q≤∞, K_n(., ., L_q(T)) 和 d_n(., L_q(T))分别表示Kolmogorov意义下L_q(T)尺度下的相对宽度和宽度, MW_p^α(T), 1≤ p≤∞, 表示有如下卷积表达式的2π 周期函数类, f(t)=c+(B_α* g)(t),c∈ R, B_α*g 表示 B_α 和g 的卷积, g∈L_p(T) 满足∫₀^2πg(τ)dτ=0 和||g||_p≤M, B_α∈ L₁(T) 有如下Fourier展开: B_α(t)=1/2π∑' _{k∈ Z}(ik)^-αe^ikt,∑^'表示去掉 k=0的项.     

Geometry of the motion of robot manipulators

A p-parametric robot is a mapping g of R^p into the homogeneous space P=C₆×C₆/Diag(C₆×C₆) given by the formula g(u₁,...,u_p=exp u₁X₁..... exp u_pX_p, where C₆, is the Lie group of all congruences of E₃ and X₁,..., X_p are fixed vectors from the Lie algebra of C₆. We characterize the set g(R^p) locally by a system of PDE and give some geometrical properties of g as a p-dimensional motion for p<6. We also characterize the Frenet frame of g and show how to construct it for the robot manipulator given by its axes X₁,...,X_p.     

Some New Optimal Ternary Linear Codes

Let d₃(n,k) be the maximum possible minimum Hamming distance of a ternary [ n,k,d;3]-code for given values of n and k. It is proved that d₃(44,6)=27, d₃(76,6)=48,d₃(94,6)=60 , d₃(124,6)=81,d₃(130,6)=84 , d₃(134,6)=87,d₃(138,6)=90 , d₃(148,6)=96,d₃(152,6)=99 , d₃(156,6)=102,d₃(164,6)=108 , d₃(170,6)=111,d₃(179,6)=117 , d₃(188,6)=123,d₃(206,6)=135 , d₃(211,6)=138,d₃(224,6)=147 , d₃(228,6)=150,d₃(236,6)=156 , d₃(31,7)=17 and d₃(33,7)=18 . These results are obtained by a descent method for designing good linear codes.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

一四元数矩阵方程组的广义(反)反射解

该文给出了四元数矩阵方程组X_1B_1=C_1,X_2B_2=C2,A_1X_1B_3+A_2X_2B_4=C_b可解的充要条件及其通解的表达式,利用此结果建立了四元数矩阵方程组XB_a=C_a,A_bXB_b=C_b有广义(反)反射解的充要条件及其有此种解时通解的表达式.     

On characterization of probability distributions by means of independent statistics

Summary The probability density functions f_k(x_k)=A_k|x_k|^p _k−1 e^−aφ _k(x_k) of independent random variables x₀, x₁, ..., x_n, are characterized by independence of two functions of them. Entrata in Redazione il 12 aprile 1969.     

On minimal embedding of two graphs as center and periphery

For any graphs G_i and G₂, and an integer d (2 ≤ d ≤ r(G₂)), define β(G₁,G₂) (or β(G₁, G₂; d)) to be the minimum number of vertices of the graph H which contains G₁ as its center and G₂ as its periphery (and dia(H) = d, respectively). In this paper, the values of β(G₁,G₂) and the upper bounds for β(G₁, G₂; d) are obtained when G₂ is not 3-self-centered.     

Criteria for embedding of spaces constructed by the method of means with arbitrary quasi-concave functional parameters

We consider a generalization ?(X₀,X₁)_p0,p1 of the method of means to arbitrary non-degenerate functional parameter. In this case non-trivial embedding ?(X₀,X₁)_p0,p1⊂ψ(X₀,X₁)_q0,q1 take place. We find necessary and sufficient condition for such embedding if 1?q₀?p₀?∞ and 1?q₁?p₁?∞ or 1?p₀?q₀?∞ and 1?p₁?q₁?∞.     

从属解析函数族的有限阶哈达玛乘积

设P[A,B]={f(z):f(z)(?)(1+Az)/(1-Bz),A+B≠0,|B|≤1,f(z)在单位开圆盘内解析}.定义函数族H_1,H_2,…,H_n的有限阶哈达玛乘积为H_1*H_2*H_3…*H_n(z)={f_1*f_2*f_3*…f_n(z):f_i∈H_i,i=1,2,…,n,n∈Z~+}.讨论并得到了P(A_1,B_1)*P(A_2,B_2)*P(A_3,B_3)*…*P(A_n,B_n)=P(X,Y)的充分必要条件,主要结果是对先前相应研究内容的直接推广.     

Regular sequences in Rees and symmetric algebras I

In this paper we describe necessary and sufficient conditions for a system of elements a₁,...,a_t of a local Noetherian ring A such that the sequence a₁T,a₁–a₂T,...,a_t–1– a_tT, a_tin the Rees algebra A[a₁T,...,a_tT], T is an indeterminate, constitutes a regular sequence.     

Excellent extensions and homologieal dimensions

In this paper ve shov that if A=(a₁,....,a₁) is R-regular and A.I=I then (a₁,...a₂):is not in general R/I-ragiuar. On ths otter hand VB shov if F₁∑_j>0in R[[X₁...,X_m]] with (f₁₀...,f₁₀) is an R-regular sequence then (F₁...,F_n) is regular in R [[ X₁..,X_n]]. These results are ansvers to questions asked in [P]