Electron paramagnetic resonance of Mn2+ in (NH4)2SO4 single crystal

Electron paramagnetic resonance investigation of Mn²⁺ in (NH₄)₂SO₄ single crystal is discussed both in paraelectric and ferroelectric phases of the crystal. Mn²⁺ is found to substitute one of the two possible types (α andβ) of NH ₄ ⁺ ions and get associated with the second type of NH ₄ ⁺ vacancy, the vacancy being the second distant neighbour in thebc-plane. As The line joining Mn²⁺ substituted NH ₄ ⁺ site and NH ₄ ⁺ vacancy lies at an angle of 18° from the crystallographicb-axis in thebc-plane. As the temperature is lowered to ? 56° C the crystal becomes ferroelectric and the spectrum in the paraelectric phase splits into two from which it appears that two sets of Mn²⁺ sites which are magnetically equivalent in the paraelectric phase become inequivalent in the ferroelectric phase. The spin Hamiltonian analysis is presented for the spectrum in the paraelectric phase.     

Paramagnetic resonance of Mn2+ in KNO3 single crystal: Motional effects

The electron paramagnetic resonance of Mn²⁺ in KNO₃ single crystal is investigated over a temperature cycle through transition temperatures. The hyperfine coupling constant, A, half width,Δ H, and the line intensity, I, are found to show sudden changes at the transition temperature, at whichα-KNO₃ changes intoβ-KNO₃. The lines are much sharper in the high temperatureβ-phase than inα-phase of the crystal. They are explained, qualitatively, in terms of structure change and rotation of NO ₃ ^? ions. The spectra in the two phases,α andβ, are analysed in terms of usual spin-Hamiltonian. A search of metastableγ-phase is also made and probable indications for the same are found.     

Solution of the Ulam Stability Problem for an Euler Type Quadratic Functional Equation

In 1968 S.M. Ulam proposed the problem: “When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?’’. In 1978 according to P.M. Gruber this kind of problems is of particular interest in probability theory and in the case of functional equations of different types. In 1997 W. Schuster established a new vector quadratic identity on the basis of the well-known Euler type theorem on quadrilaterals: If ABCD is a quadrilateral and M, N are the mid-points of the diagonals AC, BD as well as A′, B′, C′, D′ are the mid-points of the sides AB, BC, CD, DA, then |AB|² + |BC|² + |CD|² + |DA|² = 2|A′C′|² + 2|B′D′|² + 4|MN|². Employing in this paper the above geometric identity we introduce the new Euler type quadratic functional equation

$\begin{array}{l}2{[}Q(x_{0} - x_{1}+Q(x_{1}-x_{2})+Q(x_{2}- x_{3})+Q(x_{3}-x_{0}){]}\\\qquad = Q(x_{0}-x_{1}-x_{2}+x_{3})+Q(x_{0} + x_{1}-x_{2}-x_{3})+2Q(x_{0}-x_{1}+ x_{2}-x_{3})\end{array}$

for all vectors (x₀, x₁, x₂, x₃) X⁴, with X and Y linear spaces. For every x ∈ R set Q(x) = x². Then the mapping Q : X → Y is quadratic. Note also that if Q : R → R is quadratic, then we have Q(x) = Q(1)x². Besides note that the geometric interpretation of the special example

$\begin{array}{l}2{[}(x_{0} - x_{1})^{2}+ (x_{1}-x_{2})^{2}+ (x_{2}-x_{3})^{2}+(x_{3}-x_{0})^{2}{]}\\\qquad = (x_{0}-x_{1}-x_{2} + x_{3})^{2}+(x_{0} + x_{1}-x_{2}-x_{3})^{2} + 2(x_{0}-x_{1}+ x_{2}-x_{3})^{2}\end{array}$

leads to the above-mentioned Euler type theorem on quadrilaterals ABCD with position vectors x₀, x₁, x₂, x₃ of vertices A, B, C, D, respectively. Then we solve the Ulam stability problem for the afore-mentioned equation, with non-linear Euler type quadratic mappings Q : X → Y.

     

Fast Algorithm of Square Rooting in Some Finite Fields of Odd Characteristic

It was proved that the complexity of square root computation in the Galois field GF(3^s), s = 2^kr, is equal to O(M(2^k)M(r)k + M(r) log₂r) + 2^kkr^1+o(1), where M (n) is the complexity of multiplication of polynomials of degree n over fields of characteristics 3. The complexity of multiplication and division in the field GF(3^s) is equal to O(M(2^k)M(r)) and O(M(2^k)M(r)) + r^1+o(1), respectively. If the basis in the field GF(3^r) is determined by an irreducible binomial over GF(3) or is an optimal normal basis, then the summands 2^kkr^1+o(1) and r^1+o(1) can be omitted. For M(n) one may take n log₂nψ(n) where ψ(n) grows slower than any iteration of the logarithm. If k grow and r is fixed, than all the estimates presented here have the form O_r (M (s) log ₂s) = s (log ₂s)²ψ(s).     

A Method for the Construction of Wavelet Analogs by Means of Trigonometric <Emphasis Type="Italic">B</Emphasis>-Splines

We construct an analog of two-scale relations for basis trigonometric splines with uniform knots corresponding to a linear differential operator of order 2r + 1 with constant coefficients L_2r+1(D) = D(D₂ + α₁² )(D² + α₂² )... (D² + α _r ² ), where α₁, α₂,..., α _r are arbitrary positive numbers. The properties of nested subspaces of trigonometric splines are analyzed.     

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).     

Lattices generated by joins of the flats in orbits under finite affine-singular symplectic group and its characteristic polynomials

Let ASG(2ν + l, ν;F _q ) be the (2ν + l)-dimensional affine-singular symplectic space over the finite field F _q and ASp_2ν+l,ν (F _q ) be the affine-singular symplectic group of degree 2ν + l over F _q . Let O be any orbit of flats under ASp_2ν+l,ν (F _q ). Denote by L ^J the set of all flats which are joins of flats in O such that O ? L ^J and assume the join of the empty set of flats in ASG(2ν + l, ν;F _q ) is ?. Ordering L ^J by ordinary or reverse inclusion, then two lattices are obtained. This paper firstly studies the inclusion relations between different lattices, then determines a characterization of flats contained in a given lattice L ^J , when the lattices form geometric lattice, lastly gives the characteristic polynomial of L ^J .     

Sharpened forms of the generalized Schwarz inequality on the boundary

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function f(z) = 1 + c_pz^p + c_{p + 1}z^{p + 1}…, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account c_{p + 1} and zeros of the function f(z) ? 1. The sharpness of these inequalities is also proved.     

Exact Laplace-type asymptotic formulas for the Bogoliubov Gaussian measure: The set of minimum points of the action functional

We prove a theorem on the exact asymptotic relations of large deviations of the Bogoliubov measure in the L ^p norm for p = 4, 6, 8, 10 with p > p ₀, where p ₀ = 2+4π ²/β ² ω ² is a threshold value, β > 0 is the inverse temperature, and ω > 0 is the natural frequency of the harmonic oscillator. For the study, we use the Laplace method in function spaces for Gaussian measures.     

On a Diophantine Inequality with Prime Numbers of a Special Type

We consider the Diophantine inequality |p ₁ ^c + p ₂ ^c + p ₃ ^c ? N| < (logN)^?E, where 1 < c < 15/14, N is a sufficiently large real number and E > 0 is an arbitrarily large constant. We prove that the above inequality has a solution in primes p₁, p₂, p₃ such that each of the numbers p₁ + 2, p₂ + 2 and p₃ + 2 has at most [369/(180 ? 168c)] prime factors, counted with multiplicity.     

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Let {p _n (t)} _n=0 ^t8 be a system of algebraic polynomials orthonormal on the segment [?1, 1] with a weight p(t); let {x _n,ν ^(p) } _ν=1 ⁿ be zeros of a polynomial p _n (t) (x _x,ν ^(p) = cosθ _n,ν ^(p) ; 0 < θ _n,1 ^(p) < θ _n,2 ^(p) < ... < θ _n,n ^(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θ _n,1 ^(p) and 1 ? x _n,1 ^(p) coincide in order with n ^?1 and n ^?2, respectively. In the present paper, we prove that, if the weight p(t) has the form p(t) = 4(1 ? t ²)^?1{ln²[(1 + t)/(1 ? t)] + π ²}^?1, then the following asymptotic formulas are valid as n → ∞:

$$\theta _{n,1}^{(p)} = \frac{{\sqrt 2 }}{{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1}{{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1}{{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1}{{n^2 \ln ^2 (n + 1)}}} \right).$$

     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

A tracing of the fractional temperature field

This paper is devoted to a study of L~q-tracing of the fractional temperature field u(t, x)—the weak solution of the fractional heat equation(?_t +(-?_x)~α)u(t, x) = g(t, x) in L~p(R_+~(1+n)) subject to the initial temperature u(0, x) = f(x) in L~p(R~n).     

Small prime solutions to cubic equations

Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a_1,..., a_9, n) such that the cubic equation a_1p_1~3+ + a9p_9~3= b is solvable with p_j 《 P for all 1 ≤ j ≤ 9. It is proved that one can take P = max{|a_1|,..., |a_9|}~c+ |b|~(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.     

Non-triviality of a product in the Adams <Emphasis Type="Italic">E</Emphasis> <Subscript>2</Subscript>-term

Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E₂-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E₂-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p² + (s + 4)p³ + pⁿ + p^m].     

Results of Diophantine approximation by unlike powers of primes

Let k be an integer with k ≥ 6: Suppose that λ₁, λ₂,..., λ₅ be nonzero real numbers not all of the same sign, satisfying that λ₁/λ₂ is irrational, and suppose that η is a real number. In this paper, for any ε > 0; we consider the inequality |λ₁p₁ + λ₂p ₂ ² + λ₃p ₃ ³ + λ₄p ₄ ⁴ + λ₅p ₅ ^k + η | < (max p_j)^-σ(k)+ε has infinitely many solutions in prime variables p₁, p₂,...,p₅, where σ(k) depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

<Emphasis Type="Italic">L</Emphasis>-Approximation of a linear combination of the poisson kernel and its conjugate kernel by trigonometric polynomials

A linear combination Π _q,α = cos(απ/2)P + sin(απ/2)Q of the Poisson kernel P(t) = 1/2 + q cos t + q ² cos 2t + ... and its conjugate kernel Q(t) = q sin t + q ² sin 2t + ... is considered for α ∈ ? and |q| < 1. A new explicit formula is found for the value E _n?1(Π _q,α ) of the best approximation in the space L = L _2π of the function Π _q,α by the subspace of trigonometric polynomials of order at most n ? 1. More exactly, it is proved that \(E_{n - 1} \left( {\prod _{q,\alpha } } \right) = \left. {\frac{{\left| q \right|^n \left( {1 - q^2 } \right)}}{{1 - q^{4n} }}} \right\|\left. {\frac{{\cos \left( {nt - {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) - q^{2n} \cos \left( {nt + {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{1 - q^2 - 2q \cos t}}} \right\|_L\). In addition, the value E _n?1(Π _q,α ) is represented as a rapidly convergent series.     

Division Algebras with Left Algebraic Commutators

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a ₀ + a ₁ x + ? + a _n x ⁿ (resp. right polynomial a ₀ + x a ₁ + ? + x ⁿ a _n ) over K such that a ₀ + a ₁ a + ? + a _n a ⁿ = 0 (resp. a ₀ + a a ₁ + ? + a ⁿ a _n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.