Selberg’s integral for Ramanujan automorphic representation

Let π _Δ be the automorphic representation of GL(2,ℚ_A) associated with Ramanujan modular form Δ and L(s, π _Δ) the global L-function attached to π _Δ. We study Selberg’s integral for the automorphic L-function L(s, π _Δ) under GRH. Our results give the information for the number of primes in short intervals attached to Ramanujan automorphic representation.     

A reliable treatment of Abel’s second kind singular integral equations

The central idea of this paper is to construct a new mechanism for the solution of Abel’s type singular integral equations that is to say the two-step Laplace decomposition algorithm. The two-step Laplace decomposition algorithm (TSLDA) is an innovative adjustment in the Laplace decomposition algorithm (LDA) and makes the calculation much simpler. In this piece of writing, we merge the Laplace transform and decomposition method and present a novel move toward solving Abel’s singular integral equations.     

A note on Abel’s partial summation formula

Several applications of Abel’s partial summation formula to the convergence of series of positive vectors are presented. For example, when the norm of the ambient ordered Banach space is associated with a strong order unit, it is shown that the convergence of the series \(\sum x_{n}\) implies the convergence in density of the sequence \((nx_{n})_{n}\) to 0. This is done by extending the Koopman–von Neumann characterization of convergence in density. Also included is a new proof of the Jensen–Steffensen inequality based on Abel’s partial summation formula and a trace analogue of the Tomi?–Weyl inequality of submajorization.     

Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral

In this paper we introduce a nonlinear integral equation such that the system of global solutions to this equation represents the class of a very narrow beam as T → ∞ (an analog of the laser beam) and this sheaf of solutions leads to an almost-exact representation of the Hardy-Littlewood integral (1918). The accuracy of our result is essentially better than the accuracy of related results of Balasubramanian, Heath-Brown, and Ivic.     

A remark on the Abel’s and Dirichlet’s criterions concerning generalizations to monotonicity

The Abel’s and Dirichlet’s criterions for convergence of series in analysis are very basic classical results and both require the monotonicity condition. In this note we show that the monotonicity condition in these criterions can be generalized to RBV condition, while cannot be generalized to quasimonotonicity.     

A note on Burer’s copositive representation of mixed-binary QPs

In an important paper, Burer (Math. Program Ser. A 120:479–495, 2009) recently showed how to reformulate general mixed-binary quadratic optimization problems (QPs) into copositive programs where a linear functional is minimized over a linearly constrained subset of the cone of completely positive matrices. In this note we interpret the implication from a topological point of view, showing that the Minkowski sum of the lifted feasible set and the lifted recession cone gives exactly the closure of the former. We also discuss why feasibility of the copositive program implies feasibility of the original mixed-binary QP, which can be derived from the arguments in Burer (Math. Program Ser. A 120:479–495, 2009) without any further condition.     

Multidimensional Jensen’s operator on a Hilbert space and applications

Motivated by a joint concavity of connections, solidarities and multidimensional weighted geometric mean, in this paper we extend an idea of convexity (concavity) to operator functions of several variables. With the help of established definitions, we introduce the so called multidimensional Jensen’s operator and study its properties. In such a way we get the lower and upper bounds for the above mentioned operator, expressed in terms of non-weighted operator of the same type. As an application, we obtain both refinements and converses for operator variants of some well-known classical inequalities. In order to obtain the refinement of Jensen’s integral inequality, we also consider an integral analogue of Jensen’s operator for functions of one variable.     

Maslov’s canonical operator in arbitrary coordinates on the Lagrangian manifold

We present the construction of the Maslov canonical operator adapted to an arbitrary coordinate system on the corresponding Lagrangian manifold. The construction does not require any additional choice of the phase function.     

Blowup solutions of Grushin’s operator

In this note, we consider the blowup phenomenon of Grushin’s operator. By using the knowledge of probability, we first get an expression of heat kernel of Grushin’s operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solution will blow up in finite time.     

Waring’s problem for integral quaternions

A (Lipschitz) integral quaternion is a Hamiltonian quaternion of the form $a+bi+cj+dk$ with all of $a,b,c,d\in \mathbb{Z}$. In 1946, Niven showed that the integral quaternions expressible as a sum of squares of integral quaternions are precisely those for which $2\mid b,c,d$; moreover, all of these are expressible as sums of three squares. Now let $m$ be an integer with $m>2$, and suppose $2^r\parallel m$. We show that if $r=0$ (i.e., $m$ is odd), then all integral quaternions are sums of $m$th powers, while if $r>0$, then the integral quaternions that are sums of $m$th powers are precisely those for which $2^r\mid b,c,d$ and $2^{r+1}\mid b+c+d$. Moreover, all of these are expressible as a sum of $g\left(m\right)$$m$th powers, where $g\left(m\right)$ is a positive integer depending only on $m$.     

Cayley’s hyperdeterminant: A combinatorial approach via representation theory

Cayley’s hyperdeterminant is a homogeneous polynomial of degree 4 in the 8 entries of a $2\times 2\times 2$ array. It is the simplest (nonconstant) polynomial which is invariant under changes of basis in three directions. We use elementary facts about representations of the 3-dimensional simple Lie algebra $\mathfrak{s}{l}_2(\mathbb{C})$ to reduce the problem of finding the invariant polynomials for a $2\times 2\times 2$ array to a combinatorial problem on the enumeration of $2\times 2\times 2$ arrays with non-negative integer entries. We then apply results from linear algebra to obtain a new proof that Cayley’s hyperdeterminant generates all the invariants. In the last section we discuss the application of our methods to general multidimensional arrays.     

Über Abel’s Summation endlicher Differenzenreihen

Ohne Zusammenfassung     

Convergence field of Abel’s summation method

Let F(A) denote the set of all bounded sequences summable by Abel’s method. It is known, that F(A) is a linear subspace of the linear metric space (S, ρ) of all bounded sequences endowed with the sup metric. It is shown in [KOSTYRKO, P.: Convergence fields of regular matrix transformations 2, Tatra Mt. Math. Publ. 40 (2008), 143–147] that the convergence field of a regular matrix transformation is a σ-porous set. We show that F(A) is very porous in S.     

Numerical evaluation of Goursat’s infinite integral

The infinite integral ò₀^￥x dx/(1+x⁶sin²x)\int_0^{\infty}x\,dx/(1+x^6\sin^2x) converges but is hard to evaluate because the integrand f(x) = x/(1 + x ⁶sin² x) is a non-convergent and unbounded function, indeed f(kπ) = kπ→ ∞ (k→ ∞). We present an efficient method to evaluate the above integral in high accuracy and actually obtain an approximate value in up to 73 significant digits on an octuple precision system in C++.