It is known that the hybrid numbers are generalizations of complex, hyperbolic and dual numbers. Recently, they have attracted the attention of many scientists. At this paper, we provide the Euler’s and De Moivre’s formulas for the 4×4 matrices associated with hybrid numbers by using trigonometric identities. Also, we give the roots of the matrices of hybrid numbers. Moreover, we give some illustrative examples to support the main formulas.

Large language models trained on a mixture of NLP tasks that are converted into a textto-text format using prompts, can generalize into novel forms of language and handle novel tasks. A large body of work within prompt engineering attempts to understand the effects of input forms and prompts in achieving superior performance. We consider an alternative measure and inquire whether the way in which an input is encoded affects social biases promoted in outputs. In this paper, we study T0, a large-scale multi-task text-to-text language model trained using prompt-based learning. We consider two different forms of semantically equivalent inputs: question-answer format and premise-hypothesis format. We use an existing bias benchmark for the former BBQ (Parrish et al., 2021) and create the first bias benchmark in natural language inference BBNLI with hand-written hypotheses while also converting each benchmark into the other form. The results on two benchmarks suggest that given two different formulations of essentially the same input, T0 conspicuously acts more biased in question answering form, which is seen during training, compared to premisehypothesis form which is unlike its training examples. Code and data are released under https://github.com/feyzaakyurek/bbnli. 1

In this paper, we obtain Euler’s and De Moivre’s formulas for the [Formula: see text] matrix representation of Pauli quaternions. Moreover, we provide De Moivre’s formula for the light-like Pauli quaternions. Additionally, we give the [Formula: see text] roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas.

The Pell numbers, named after the English diplomat and mathematician John Pell, are studied by many authors. At this work, by inspiring the definition harmonic numbers, we define harmonic Pell numbers. Moreover, we construct one type of symmetric matrix family whose elements are harmonic Pell numbers and its Hadamard exponential matrix. We investigate some linear algebraic properties and obtain inequalities by using matrix norms. Furthermore, some summation identities for harmonic Pell numbers are obtained. Finally, we give a MATLAB-R2016a code which writes the matrix with harmonic Pell entries and calculates some norms and bounds for the Hadamard exponential matrix.

No abstract

This paper focuses on a specially constructed matrix whose entries are harmonic Fibonacci numbers and considers its Hadamard exponential matrix. A lot of admiring algebraic properties are presented for both of them. Some of them are determinant, inverse in usual and in the Hadamard sense, permanents, some norms, etc. Additionally, a MATLAB-R2016a code is given to facilitate the calculations and to further enrich the content.

Metallic ratio is a root of the simple quadratic equation x2  =  kx + 1 for k is any positive integer which is the characteristic equation of the recurrence relation of k‐Fibonacci (k‐Lucas) numbers. This paper is about the metallic ratio in Zp. We define k‐Fibonacci and k‐Lucas numbers in Zp, and we show that metallic ratio can be calculated in Zp if and only if p≡ ± 1  mod (k2 + 4), which is the generalization of the Gauss reciprocity theorem for any integer k. Also, we obtain that the golden ratio, the silver ratio, and the bronze ratio, the three together, can be calculated in Z79 for the first time. Moreover, we introduce k‐Fibonacci and k‐Lucas quaternions with some algebraic properties and some identities for them.

At this work, we give a method for constructing the Perrin and Padovan sequences and obtain the De Moivre-type identity for Padovan numbers. Also, we define a Padovan sequence with new initial conditions and find some identities between all of these auxiliary sequences. Furthermore, we give quadratic approximations for these sequences.