Measuring the Size(1)

How do you measure the size of a one-dimensional figure?The size of a line segment is given by its length.How do you measure the size of two-dimensional figures?Rectangles and other polygons are two-dimensional.There are two common ways of measuring these figures.Perimeter measures the boundary.Area measures the space inside the figure.     

Computing the function GM(υ)

Tables are presented on the function G_M() which describes head waves or waves propagating near a concave surface.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 85–92, 1975.     

On the Chebyshev Function ψ(x)

Suppose that the Riemann hypothesis is valid, and let be the Chebyshev function. In this paper, we obtain the following bound for all >1 and positive integers .     

On the Evaluation of the Tutte Polynomial at the Points (1, –1) and (2, –1)

Motivated by the identity t (K _n+2; 1, –1) = t (K _n ; 2, –1), where t(G; x, y) is the Tutte polynomial of a graph G, we search for graphs G having the property that there is a pair of vertices u, v such that t(G; 1, –1) = t(G – {u, v}; 2, –1). Our main result gives a sufficient condition for a graph to have this property; moreover, it describes the graphs for which the property still holds when each vertex is replaced by a clique or a coclique of arbitrary order. In particular, we show that the property holds for the class of threshold graphs, a well-studied class of perfect graphs.     

Operators on the orthogonal complement of the Dirichlet space(Ⅱ)

In this paper we first prove that a dual Hankel operator R φ on the orthogonal complement of the Dirichlet space is compact for φ∈ W 1,∞(D),and then that a semicommutator of two Toeplitz operators on the Dirichlet space or two dual Toeplitz operators on the orthogonal complement of the Dirichlet space in Sobolev space is compact.We also prove that a dual Hankel operator R φ with φ∈ W 1,∞(D) is of finite rank if and only if B φ is orthogonal to the Dirichlet space for some finite Blaschke product B,and give ...     

On the size of the set A(A + 1)

Let F _p be the field of a prime order p. For a subset A ì F_p{A \subset F_p} we consider the product set A(A + 1). This set is an image of A ×  A under the polynomial mapping f(x, y) = xy +  x : F _p ×  F _p → F _p . In the present note we show that if |A| <  p ^1/2, then

The grothendieck property in the spacel ∞ (E) and the weak Banach-Saks property inc 0 (E)

We prove the following: 1) if E is a B-convex Banach lattice, the space l(E) is a Grothendieck space; 2) if the space E has the p-Banach-Saks property for some p>1, the space c₀(E) also has this property. It is shown by examples that these conditions are close to being necessary. These results are applied to study the geometric structure of limits of symmetric spaces.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 39–44.     

On the inequality μ(mM) ≥ μ(M) over polynomial rings

In [4], Macaulay proved that μ(mI) ≥ Q(μ(I)) ≥ μ(I) for any homogeneous ideal I in k[X ₁, …,X_n ] with minimal generators of a single degree where m is the homogeneous maximal ideal (X ₁, …,X_n ) and Q is the function called the Binomial Expansion base n - 1. The inequelity can be generalised to the case of torsion-fre modules with generators of a single degree. The proof follows the lines of Robbiano [5]. A family of counterexamples to μ(mI) ≥ μ(I) in which I has minimal generators of just two different degrees is given.     

An inequality for the function π(n)

We prove that the inequality $\pi ^2 \left( m \right) + \pi ^2 \left( n \right) \leqslant \tfrac{5} {4}\pi ^2 \left( {m + n} \right)$ holds for all integers m, n ≥ 2. The constant factor 5/4 is sharp. This complements a result of Panaitopol, who showed in 2001 that ½ π ²(m+ n) ≤ π ²(m) + π ²(n) is valid for all m, n ≥ 2. Here, as usual, π(n) denotes the number of primes not exceeding n.     

Series representations for the Apery constant $$\zeta (3)$$ ζ ( 3 ) involving the values $$\zeta (2n)$$ ζ ( 2 n )

The Ramanujan Journal - In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant $$\zeta (3)$$ . In...     

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

Hermitian geometry on the resolvent set(Ⅱ)

For an element A in a unital C*-algebra B, the operator-valued 1-form ω_A(z) =(z-A)~(-1) dz is analytic on the resolvent set ρ(A), which plays an important role in the functional calculus of A. This paper defines a class of Hermitian metrics on ρ(A) through the coupling of the operator-valued(1, 1)-form ?_A=-ω_A~*∧ω_A with tracial and vector states. Its main goal is to study the connection between A and the properties of the metric concerning curvature, arc length, completeness and singularity. A particular example is when A is quasi-nilpotent, in which case the metric lives on the punctured complex plane C \ {0}. The notion of the power set is defined to gauge the "blow-up" rate of the metric at 0, and examples are given to indicate a likely link with A's hyper-invariant subspaces.     

The Integral Mean of the Discrepancy of the Sequence (nα)

 For a real number x let be the fractional part of x and for any set M let c _M be the characteristic function of M. For and a positive integer N let

The Discrepancy of the Sequence (nα + (log n)β)

In this paper we give an upper bound for the discrepancy of the sequence (nα + (log n)β) where α = (α₁, ..., α _s ) R ^s , which satisfies that 1, α₁, ..., α _s are linearly independent over Z, is of finite type η or is of constant type.     

On the Integral Curvature of Curve-(III)

In this paper we prove the following theorem.It is a generalization of Tenchel's theorem on theintegral curvature of curve.Theorem.If 1 is the length of a curve C=AB and φ is the angle between the tangent vectors ofC at A,B,then the integral curvature of C     

Property(ω) and the Single-valued Extension Property

By the new spectrum originated from the single-valued extension property, we give the necessary and sufficient conditions for a bounded linear operator defined on a Banach space for which property(ω) holds. Meanwhile, the relationship between hypercyclic property(or supercyclic property)and property(ω) is discussed.     

Extremal problems in the class σ(Τ)

Let L_f(r)={w=f(x), ¦z¦=r}, 1 < r < , be a level line of the function f(z) . Sharp upper bounds are obtained for the diameter of the curve L_f(r) in the class () for functions f(z)=z+₀+₁z^–1+... for which there exists a domain _f that complements the exterior of the unit disk and has conformal radius at the point w=0 satisfying the condition R(_f,0) , 0 < < 1. Also, a set of values is found for the coefficient ₁ in the class ().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 196, pp. 138–153, 1991.     

On the Congruence σ(n)≡1(mod n)

Let k≥2 be an integer,and let σ(n) denote the sum of the positive divisors of an integer n.We call n a quasi-multiperfect number if σ(n)=kn+1.In this paper,we give some necessary properties of them.