Trace formulae for irreducible polynomials over with minimal order roots in

The extension field where q is a prime divisor of (P−1), has a unique structure. This paper describes this unique structure and uses it to derive formulas relating the trace values for elements in . These formulas can be refined for certain elements to produce a formula for the trace.     

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of “mixed” products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type . A realization of the basic representation that is of “”-type plays the central role.     

On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

The spectrum of the Leray transform for convex Reinhardt domains in

The Leray transform and related boundary operators are studied for a class of convex Reinhardt domains in . Our class is self-dual; it contains some domains with less than C²-smooth boundary and also some domains with smooth boundary and degenerate Levi form. L²-regularity is proved, and essential spectra are computed with respect to a family of boundary measures which includes surface measure. A duality principle is established providing explicit unitary equivalence between operators on domains in our class and operators on the corresponding polar domains. Many of these results are new even for the classical case of smoothly bounded strongly convex Reinhardt domains.     

Plancherel–Polya-type inequalities for entire functions of exponential type in

The goal of the paper is to prove generalizations of the classical Plancherel–Polya inequalities in which point-wise sampling of functions (δ-distributions) is replaced by more general compactly supported distributions on . As an application it is shown that a function , 1p∞, which is an entire function of exponential type is uniquely determined by a set of numbers {Ψ_j(f)}, , where {Ψ_j}, , is a countable sequence of compactly supported distributions. In the case p=2 a reconstruction method of a Paley–Wiener function f from a sequence of samples {Ψ_j(f)}, , is given. This method is a generalization of the classical result of Duffin–Schaeffer about exponential frames on intervals.     

Pointwise -convergence and -convergence in measure of sequences of functions

Let be an ideal. We say that a sequence of real numbers is -convergent to if for every neighborhood U of y the set of n's satisfying y_nU is in . Basing upon this notion we define pointwise -convergence and -convergence in measure of sequences of measurable functions defined on a measure space with finite measure. We discuss the relationship between these two convergences. In particular we show that for a wide class of ideals including Erdős–Ulam ideals and summable ideals the pointwise -convergence implies the -convergence in measure. We also present examples of very regular ideals such that this implication does not hold.     

The operator equation

The solution of the linear operator equation:A^n-1X+A^n-2XB++AXB^n-2+XB^n-1=Y is given by if the spectra of A and B are in the sector {z:z≠0,-π/n<argz<π/n}.     

Compactness in

This paper is concerned with compactness for some topologies on the collection of bounded linear operators on Banach spaces. New versions of the Eberlein–Šmulian theorem and Day's lemma in the collection are established. Also we obtain a partial solution of the dual problem for the quasi approximation property, that is, it is shown that for a Banach space X if X^** is separable and X^* has the quasi approximation property, then X has the quasi approximation property.     

A Hardy type inequality in the half-space on and Heisenberg group

We prove a Hardy type inequality in the half-space on the Heisenberg group and show that a Hardy inequality given by J. Tidblm in [J. Tidblm, A Hardy inequality in the half-space, J. Funct. Anal. 221 (2005) 482–492] is sharp.     

Global well-posedness of Korteweg–de Vries equation in

We prove that the Korteweg–de Vries initial-value problem is globally well-posed in and the modified Korteweg–de Vries initial-value problem is globally well-posed in . The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation in H^−3/4 by constructing some special resolution spaces in order to avoid some ‘logarithmic divergence’ from the high–high interactions. Our local solution has almost the same properties as those for H^s (s>−3/4) solution which enable us to apply the I-method to extend it to a global solution.     

On fractional maximal function and fractional integrals associated with the Dunkl operator on the real line

In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the Dunkl-type fractional maximal operator Mβ, and the Dunkl-type fractional integral operator Iβ from the spaces Lp,α(R) to the spaces Lq,α(R), 1<p<q<∞, and from the spaces L1,α(R) to the weak spaces WLq,α(R), 1<q<∞. In the case , we prove that the operator Mβ is bounded from the space Lp,α(R) to the space L∞,α(R), and the Dunkl-type modified fractional integral operator is bounded from the space Lp,α(R) to the Dunkl-type BMO space BMOα(R). By this results we get boundedness of the operators Mβ and Iβ from the Dunkl-type Besov spaces to the spaces , 1<p<q<∞, 1/p−1/q=β/(2α+2), 1?θ?∞ and 0<s<1.     

The difference in a problem of rations from the Rhind mathematical papyrus

In an ancient Egyptian problem of bread distribution from the Rhind mathematical papyrus (dated between 1794 and 1550 B.C.), a procedure of “false position” is used in the calculation of a series of five rations. The algorithm is only partially illustrated in the problem text, and last century's prevailing interpretations suggested a determination of the series by trial and error. The missing part of the computational procedure is reconstructed in this article as an application of the algorithm, exemplified in the preceding section of the papyrus, to calculate an unknown quantity by means of the method of “false position.”     

On the existence of a proper conformal maximal disk in

In this paper we construct an example of a properly immersed maximal surface in the Lorentz–Minkowski space with the conformal type of a disk.     

On the -module structure of the noncommutative harmonics

Using a noncommutative analog of Chevalley's decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius characteristic for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the derived free Lie algebra in n variables.     

Double point self-transverse immersions of

A self-transverse immersion of a smooth manifold M^8k in has a double point self-intersection set which is the image of an immersion of a smooth four-dimensional manifold, cobordent to P⁴, P²×P², P⁴+P²×P² or a boundary. We will prove that for any value of k>1 the double point self-intersection set is a boundary. If k=1, then there exists an immersion of P²×P²×P²×P² in with double point manifold boundary and odd number of triple points. In particular any immersion of oriented manifold in this dimension has double point manifold cobordant to a boundary.     

APN permutations on and Costas arrays

We study PN and APN functions over the integers modulo n. We give some construction techniques based on Costas arrays, which allow us to construct APN permutations on where p is a prime. Although PN permutations do not exist, one set of our functions is very close to being a set of PN permutations.     

Properties of multilevel block -circulants

In a previous paper we characterized unilevel block α-circulants , , 0mn-1, in terms of the discrete Fourier transform of , defined by . We showed that most theoretical and computational problems concerning A can be conveniently studied in terms of corresponding problems concerning the Fourier coefficients F₀,F₁,…,F_n-1 individually. In this paper we show that analogous results hold for (k+1)-level matrices, where the first k levels have block circulant structure and the entries at the (k+1)-st level are unstructured rectangular matrices.     

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.