相关论文: $q$-norms are really norms
We study decompositions of natural numbers into triangular summands. For instance, we prove that any natural number can be represented as a sum of four triangular numbers, two of them having even indices and the other two having odd…
We introduce a bulging triangle like the generalization of the Reuleaux triangle. We may be able to propose various ways to bulge a triangle, but this paper presents the way so that its vertices are the same as them of the original…
In the present paper we construct normal numbers in base $q$ by concatenating $q$-ary expansions of pseudo polynomials evaluated at the primes. This extends a recent result by Tichy and the author.
We consider a class of two-dimensional functions f(x,y) with the property that the smallness of its rectangular norm implies the smallness of rectangular norm for f(x,x+y). Also we study a family of functions f(x,y) having a similar…
A well-known conjecture asserts that there are infinitely many primes $p$ for which $p - 1$ is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes $p,q \le x$ for which $(p - 1)(q -…
In this note one tries to venture into a study of some notions, in the context of a (unital) normed algebra, in particular the algebra of operators on a Hilbert space. Namely, one considers ``moving norms'', i.e.\ norming an element minus a…
Euclid uses an undefined notion of "equal figures", to which he applies the common notions about equals added to equals or subtracted from equals. When (in previous work) we formalized Euclid Book~I for computer proof-checking, we had to…
The so called $q$-triplets were conjectured in 2004 and then found in nature in 2005. A relevant further step was achieved in 2005 when the possibility was advanced that they could reflect an entire infinite algebra based on combinations of…
In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…
For a positive semidefinite matrix $H= \begin{bmatrix} A&X\\ X^{*}&B \end{bmatrix} $, we consider the norm inequality $ ||H||\leq ||A+B|| $. We show that this inequality holds under certain conditions. Some related topics are also…
We consider two classes of $q$-continued fraction whose odd and even parts are limit 1-periodic for $|q|>1$, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the…
This paper establishes some equivalent conditions of a uninorm, extending an arbitrary triangular norm on [0, e] or an arbitrary triangular conorm on [e, 1] to the whole lattice.
We present a generalization of the notion of an algebra norm relevant to real finite-dimensional unital associative algebras. Among other things, this leads to a novel set of algebra isomorphism invariants, some of which are computationally…
We present a natural extension of the process of taking a group quotient to arbitrary subgroups. We first review basic concepts from group theory. This will allow us to see the relationship between our new, more general quotient operation…
The rearrangement inequality states that the sum of products of permutations of 2 sequences of real numbers are maximized when the terms are similarly ordered and minimized when the terms are ordered in opposite order. We show that similar…
In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalises strong $q$-additivity and -multiplicativity, respectively. We show that…
In this paper, we provide criteria for the log-concavity of rows and the strong $q$-log-convexity of the generating functions of rows in more generalized triangles. Additionally, we prove that the bi$^s$nomial transformation not only…
To determine whether a number is congruent or not is an old and difficult topic and progress is slow. The paper presents a new theorem when a prime number is a congruent number or not. The proof is not necessarily any simpler or shorter…
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We…
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…