On the dispersion law of the form ɛ(p) = ħ 2 p 2/2m + \tilde V(p) − \tilde V(0) for elementary excitations of a nonideal fermi gas in the pair interaction approximation (i ↔ j), V(|x x − x j |)

We give a derivation of the dispersion law ?(p) = ? ² p ²/2m + $\tilde V$ (p) ? $\tilde V$ (0), where $\tilde V$ (p) is the Fourier transform of the pair interaction potential V(r). (The interaction between particles x _x and x _j is V(|x _i ? x _j |).)     

Approximation of the functional classes\tilde W_p^\alpha (L) in the spaces ℒp [−π, π] by the Fejér method

Asymptotic estimates for the approximation of the functional classes by means of the Fejér means in the metric of the spaces _s[–,] are obtained.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 343–349, March, 1978.In conclusion, the author thanks V. M. Tikhomirov for assistance with this note and discussion of the results.     

On the\bar \partial equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

For a large class of subharmonicφ, the equation is studied in . Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inL ^p norms are derived as a corollary. A class of counterexamples, related to the failure of to be analytic-hypoelliptic on certain CR manifolds, is discussed. Communicated by Steven Krantz     

General fractional differential equations of order $$\alpha \in (1,2)$$ and Type $$\beta \in [0,1]$$β∈[0,1] in Banach spaces

In this paper, we are concerned with general fractional differential equations of order \(\alpha \in (1,2)\) and type \(\beta \in [0,1]\) in Banach spaces. We define and develop a theory of general fractional sine functions and show that they are essentiality equivalent to a general fractional resolvent. We use such theory to study the well-posedness of the above general fractional differential equations. An illustration example is presented.     

The complex 2-sphere in ℂ3 and Schrödinger flows

Science China Mathematics - By using holomorphic Riemannian geometry in ?3, the coupled Landau-Lifshitz equation (CLL) is proved to be exactly the equation of Schrödinger flows from...     

The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)

Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n ² ? 1 and 12n ² + 1 are odd primes, then the general elliptic curve y ² = x ³+(36n ²?9)x?2(36n ²?5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y ² = x ³ + 27x ? 62 really is an unusual elliptic curve which has large integral points.     

( B0−$\bar B^0 $) Mixing in the next-to-leading order of the 1/ mb-expansion

We present results of calculating the matrix elements of the (B⁰−[`(B)]⁰\bar B^0 )-mixing operators in the next-to-leading order of the expansion in 1/m _b .     

On the Diophantine Equations (2 n − 1)(6 n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

In this paper we prove that the equation (2 ⁿ – 1)(6 ⁿ – 1) = x ² has no solutions in positive integers n and x. Furthermore, the equation (a ⁿ – 1) (a ^kn – 1) = x ² in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).     

On the diophantine equation X 2 − (1 + a 2)Y 4 = −2a

Let a≥1 be an integer.In this paper,we will prove the equation in the title has at most three positive integer solutions.     

格蕴涵代数的区间值$(\in, \in\vee\,q)$-模糊$LI$-理想格

In the present paper, the interval-valued (∈,∈∨q)-fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties of interval-valued (∈,∈∨q)-fuzzy LI-ideals are given. Representation theorem of interval-valued (∈,∈∨q)-fuzzy LI-ideal which is generated by an interval-valued fuzzy set is established. It is proved that the set consisting of all interval-valued (∈,∈∨q)-fuzzy LI-ideals in a lattice implication algebra, under the partial order ?, forms a complete distributive lattice.     

Über die Differentialungleichung 0<α≦rt−S 2≦\<∞

Ohne Zusammenfassung Leon Lichtenstein zum GedächtnisDiese Arbeit wurde teilweise von einem Kontrakt des Office of Naval Research in Stanford University unterstützt.     

Sharp estimates for derivatives of functions in the Sobolev classes $$ \mathop {W_2^r }\limits^ \circ $$(−1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f ^(k)(x), x ∈ (−1, 1), k ∈ {0, 1, ..., r − 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r − 1 at the points ±1 and are such that $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 . As a corollary, it is shown that the first eigenvalue λ _1,r of the operator (−D ²) ^r with these boundary conditions is $ \sqrt 2 $ \sqrt 2 (2r)! (1 + O(1/r)), r → ∞.