Abstract:
Quantum computation has received great attention in recent years for its possible application to difficult problem in classical calculation. Despite the experimental problems of implementing quantum devices, theoretical physicists have tried to conceive some implementations for quantum algorithms. We present here some explicit schemes for executing elementary arithmetic operations.

Abstract:
Quantum computers require quantum arithmetic. We provide an explicit construction of quantum networks effecting basic arithmetic operations: from addition to modular exponentiation. Quantum modular exponentiation seems to be the most difficult (time and space consuming) part of Shor's quantum factorising algorithm. We show that the auxiliary memory required to perform this operation in a reversible way grows linearly with the size of the number to be factorised.

Abstract:
Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and of quadric surfaces. These results are applied to deduce new instances of "elementary equivalence implies isomorphism": for all genus zero curves over a number field, and for certain genus one curves over a number field, including some which are not elliptic curves.

Abstract:
Let $a$ and $b$ be positive integers. In 1946, Erd\H{o}s and Niven proved that there are only finitely many positive integers $n$ for which one or more of the elementary symmetric functions of $1/b, 1/(a+b),..., 1/(an-a+b)$ are integers. In this paper, we show that for any integer $k$ with $1\le k\le n$, the $k$-th elementary symmetric function of $1/b, 1/(a+b),..., 1/(an-a+b)$ is not an integer except that either $b=n=k=1$ and $a\ge 1$, or $a=b=1, n=3$ and $k=2$. This refines the Erd\H{o}s-Niven theorem and answers an open problem raised by Chen and Tang in 2012.

Abstract:
Thinking is one of the most interesting mental processes. Its complexity is sometimes simplified and its different manifestations are classified into normal and abnormal, like the delusional and disorganized thought or the creative one. The boundaries between these facets of thinking are fuzzy causing difficulties in medical, academic, and philosophical discussions. Considering the dopaminergic signal-to-noise neuronal modulation in the central nervous system, and the existence of semantic maps in human brain, a self-organizing neural network model was developed to unify the different thought processes into a single neurocomputational substrate. Simulations were performed varying the dopaminergic modulation and observing the different patterns that emerged at the semantic map. Assuming that the thought process is the total pattern elicited at the output layer of the neural network, the model shows how the normal and abnormal thinking are generated and that there are no borders between their different manifestations. Actually, a continuum of different qualitative reasoning, ranging from delusion to disorganization of thought, and passing through the normal and the creative thinking, seems to be more plausible. The model is far from explaining the complexities of human thinking but, at least, it seems to be a good metaphorical and unifying view of the many facets of this phenomenon usually studied in separated settings.

Abstract:
Intuition as an element of human cognitive activity is always attracted the attention of researchers, regardless of their ideological orientation. The difficulty of studying intuition is primarily concerned with its inaccessibility in terms of rational perception. The article discusses various estimates of the content and the form of intuition in the philosophical context.

Abstract:
Elementary proofs of unique factorization in rings of arithmetic functions using a simple variant of Euclid's proof for the fundamental theorem of arithmetic.

Abstract:
In this essay I will examine the role that intuition plays in Russell's paradox; showing how different approaches to intuition will license different treatments of the paradox. In addition, I will argue for a specific approach to the paradox, one that follows from the most plausible account of intuition. On this account, intuitions, though fallible, have epistemic import. In addition, the intuitions involved in paradoxes point to something wrong with concept that leads to paradox. In the case of Russell's paradox, this is an ambiguity in the notion of a class.

Abstract:
In this article I wish to share how I learned about intuition through personal experiences and why it is important in education. Intuition is linked to epistemology, language, emotions, health, memory and involves the inner life of the person. For most of my life, I had very little understanding about intuition and deemed this phenomenon as useless in education. I started to learn about intuition as a result of my spouse’s catastrophic death. The grief I experienced precipitated a loss in my belief system and I felt a significant decrease in my ability to function in a logical or rational manner. My journey into intuition enabled me to forge a new way to live my life. I believe that each person has the ability to learn about intuition and how it can be useful in guiding one’s life. The mainstream school system however fails to recognize intuition as a valid way of learning despite the research in this field. Students are therefore being given a partial education. Given this, I feel there is a serious problem which emerges if people are led to believe that the mainstream education system is offering students a complete education.

Abstract:
Suppose a projectile collides perpendicularly with a stationary rigid rod on a smooth horizontal table. We show that, contrary to what one naturally expects, it is not always the case that the rod acquires maximum angular velocity when struck at an extremity. The treatment is intended for first year university students of Physics or Engineering, and could form the basis of a tutorial discussion of conservation laws in rigid body dynamics.