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Papers by Aleksander Zujev

Note on Non-Unitary Quantum Gates in Quantum Computing

arXiv: Number Theory, 2015

We give solutions of a Diophantine equation containing factorials, which can be written as a cubi... more We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable Fermat Last Theorem equation.

arXiv: Number Theory, 2016

We give a new appraisal of a famous oscillating power series considered by Hardy and Ramanujan re... more We give a new appraisal of a famous oscillating power series considered by Hardy and Ramanujan related to the erroneous theory of distribution of primes by Ramanujan.

We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^... more We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.

The famous open problem of finding positive integer solutions to a5+b5=c5+d5a^5 + b^5 = c^5 + d^5a5+b5=c5+d5 is consi... more The famous open problem of finding positive integer solutions to a5+b5=c5+d5a^5 + b^5 = c^5 + d^5a5+b5=c5+d5 is considered, and related solutions are found in two distinct settings: firstly, where aaa and bbb are both positive integers with ccc and ddd both Gaussian integers; secondly, where all of aaa, bbb, ccc, and ddd are Gaussian integers.

The study examines gaps in primary health policy designed to enable Ghana to achieve universal ac... more The study examines gaps in primary health policy designed to enable Ghana to achieve universal access to health. The policy has existed for over 15 years with remarkable achievements, but data shows gaps between the procedure and what is going on. The researchers use a qualitative technique to explore the gap by focusing on persons directly involved in policy implementation as participants. Three main japs were identified, insufficient collaboration between community health officers and community leaders, favoritism in promoting CHOs, and inadequate provision of medical consumables and tools to CHPS-compounds as stated by the policy. The study concluded with the call for a review of the procedures and examining of the outlined gaps

The use of heuristical estimate in number theory

arXiv: Number Theory, 2015

We give new nested radical equations of similar kind to Ramanujan's questions to the Indian M... more We give new nested radical equations of similar kind to Ramanujan's questions to the Indian Mathematical Society 100 years ago. While many have since considered these from the perspectives of the Notebooks of Ramanujan and from the theory of Class numbers and Units, there seems no comprehensive theory to cover off the results, and it seems always possible to find new and surprizing elementary equations of this kind. We also consider a few more methods of constructing nested radicals.

arXiv: Number Theory, 2016

Author(s): Campbell, Geoffrey B.; Zujev, Aleksander | Abstract: In this paper we consider integer... more Author(s): Campbell, Geoffrey B.; Zujev, Aleksander | Abstract: In this paper we consider integers in base 10 like abcabcabc, where aaa, bbb, ccc are digits of the integer, such that abc2−(abccdotcba);=;pmn2abc^2 - (abc \cdot cba) \; = \; \pm n^2abc2−(abccdotcba);=;pmn2, where nnn is a positive integer, as well as equations abc2−(abccdotcba);=;pmn3abc^2 - (abc \cdot cba) \; = \; \pm n^3abc2−(abccdotcba);=;pmn3, and abc3−(abccdotcba);=;pmn2abc^3 - (abc \cdot cba) \; = \; \pm n^2abc3−(abccdotcba);=;pmn2 We consider asymptotic density of solutions. We also compare the results with ones with bases different from 10.

Induced magnetism versus Kondo screening in alternating Mott-metal layers

Physical Review B, 2013

Studies of systems with two fermionic bands with repulsive interaction strength U have a long his... more Studies of systems with two fermionic bands with repulsive interaction strength U have a long history, with the Periodic Anderson Model (PAM) being one of the most frequently considered Hamiltonians. In this paper, we use Quantum Monte Carlo to study analogous issues for attractive interactions. As in the Periodic Anderson Model, we focus on a case where one band is uncorrelated (U = 0), and focus on the effect of hybridization V between the bands on the pairing correlations. A key difference with the PAM is that there is no sign problem, so that we are able to explore the physics of doped multi-band attractive systems at low temperatures whereas ground state properties of repulsive models can be determined only at half-filling. For small V , pairing in the U < 0 layer induces pairing in the U = 0 layer. At larger V the ground state of the coupled system loses its superconducting character. The Quantum Monte Carlo data are complemented by results obtained with the Bogoliubov-de Gennes approximation.

Mean Field Theory Calculation of Isentropic Curves of the Fermion Hubbard Model

Recent experiments on optical lattices have focussed attention on understanding how many body cor... more Recent experiments on optical lattices have focussed attention on understanding how many body correlations change when the entropy (rather than the temperature) is varied as a control parameter. Quantum Monte Carlo (QMC) simulations have addressed some of the issues involved, but, for fermions, are limited by the sign problem. In this talk, we present results for the isentropic curves of the square lattice fermion Hubbard model in mean field theory (MFT). The topology of these curves on the phase diagram is explored, and compared to what is found in QMC when the latter is available. We also compare to MFT calculations for other models.

We study the ground state phases of Bose-Fermi mixtures in one-dimensional optical lattices with ... more We study the ground state phases of Bose-Fermi mixtures in one-dimensional optical lattices with quantum Monte Carlo simulations using the Canonical Worm algorithm. Depending on the filling of bosons and fermions, and the on-site intra-and inter-species interaction, different kinds of incompressible and superfluid phases appear. On the compressible side, correlations between bosons and fermions can lead to a distinctive behavior of the bosonic superfluid density and the fermionic stiffness, as well as of the equal-time Green functions, which allow one to identify regions where the two species exhibit anticorrelated flow. We present here complete phase diagrams for these systems at different fillings and as a function of the interaction parameters.

We present Quantum Monte Carlo simulations of a generalization of the Feynman-Kikuchi model which... more We present Quantum Monte Carlo simulations of a generalization of the Feynman-Kikuchi model which includes the possibility of vacancies and interactions between the particles undergoing exchange. By measuring the winding number (superfluid density) and density structure factor, we determine the phase diagram, and show that it exhibits regions which possess both superfluid and charge ordering.

Determinant Quantum Monte Carlo method applied to the t-J model

The usual approach to simulating the t-J model with the Determinant Quantum Monte Carlo (DQMC) me... more The usual approach to simulating the t-J model with the Determinant Quantum Monte Carlo (DQMC) method starts with the Hubbard model with a finite on-site interaction U which is then increased to ``almost'' infinity. This approach, however, has considerable difficulties with large round-off errors (stability) and variances, and also a very bad fermion sign problem. In this talk, I will describe a different approach which starts with (almost) infinite U by means of a projector operator and further prohibiting double occupancy by using a modified creation operator. The new technique will be shown to solve some of these difficulties. Unfortunately, the sign problem remains significant. I will discuss the different attempts we have made to reduce it.

Experiments on cold atom systems in which a lattice potential is ramped up on a confined cloud ha... more Experiments on cold atom systems in which a lattice potential is ramped up on a confined cloud have raised intriguing questions about how the temperature varies along isentropic curves, and how these curves intersect features in the phase diagram. In this paper, we study the isentropic curves of two models of magnetic phase transitions- the classical Blume-Capel Model (BCM) and the Fermi Hubbard Model (FHM). Both Mean Field Theory (MFT) and Monte Carlo (MC) methods are used. The isentropic curves of the BCM generally run parallel to the phase boundary in the Ising regime of low vacancy density, but intersect the phase boundary when the magnetic transition is mainly driven by a proliferation of vacancies. Adiabatic heating occurs in moving away from the phase boundary. The isentropes of the half-filled FHM have a relatively simple structure, running parallel to the temperature axis in the paramagnetic phase, and then curving upwards as the antiferromagnetic transition occurs. However, in the doped case, where two magnetic phase boundaries are crossed, the isentrope topology is considerably more complex.

Note on Non-Unitary Quantum Gates in Quantum Computing

arXiv: Number Theory, 2015

We give solutions of a Diophantine equation containing factorials, which can be written as a cubi... more We give solutions of a Diophantine equation containing factorials, which can be written as a cubic form, or as a sum of binomial coefficients. We also give some solutions to higher degree forms and relate some solutions to an unsolvable Fermat Last Theorem equation.

arXiv: Number Theory, 2016

We give a new appraisal of a famous oscillating power series considered by Hardy and Ramanujan re... more We give a new appraisal of a famous oscillating power series considered by Hardy and Ramanujan related to the erroneous theory of distribution of primes by Ramanujan.

We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^... more We give an infinite number of integer solutions to the Diophantine equation x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, and some solutions to some similar equations.

The famous open problem of finding positive integer solutions to a5+b5=c5+d5a^5 + b^5 = c^5 + d^5a5+b5=c5+d5 is consi... more The famous open problem of finding positive integer solutions to a5+b5=c5+d5a^5 + b^5 = c^5 + d^5a5+b5=c5+d5 is considered, and related solutions are found in two distinct settings: firstly, where aaa and bbb are both positive integers with ccc and ddd both Gaussian integers; secondly, where all of aaa, bbb, ccc, and ddd are Gaussian integers.

The study examines gaps in primary health policy designed to enable Ghana to achieve universal ac... more The study examines gaps in primary health policy designed to enable Ghana to achieve universal access to health. The policy has existed for over 15 years with remarkable achievements, but data shows gaps between the procedure and what is going on. The researchers use a qualitative technique to explore the gap by focusing on persons directly involved in policy implementation as participants. Three main japs were identified, insufficient collaboration between community health officers and community leaders, favoritism in promoting CHOs, and inadequate provision of medical consumables and tools to CHPS-compounds as stated by the policy. The study concluded with the call for a review of the procedures and examining of the outlined gaps

The use of heuristical estimate in number theory

arXiv: Number Theory, 2015

We give new nested radical equations of similar kind to Ramanujan's questions to the Indian M... more We give new nested radical equations of similar kind to Ramanujan's questions to the Indian Mathematical Society 100 years ago. While many have since considered these from the perspectives of the Notebooks of Ramanujan and from the theory of Class numbers and Units, there seems no comprehensive theory to cover off the results, and it seems always possible to find new and surprizing elementary equations of this kind. We also consider a few more methods of constructing nested radicals.

arXiv: Number Theory, 2016

Author(s): Campbell, Geoffrey B.; Zujev, Aleksander | Abstract: In this paper we consider integer... more Author(s): Campbell, Geoffrey B.; Zujev, Aleksander | Abstract: In this paper we consider integers in base 10 like abcabcabc, where aaa, bbb, ccc are digits of the integer, such that abc2−(abccdotcba);=;pmn2abc^2 - (abc \cdot cba) \; = \; \pm n^2abc2−(abccdotcba);=;pmn2, where nnn is a positive integer, as well as equations abc2−(abccdotcba);=;pmn3abc^2 - (abc \cdot cba) \; = \; \pm n^3abc2−(abccdotcba);=;pmn3, and abc3−(abccdotcba);=;pmn2abc^3 - (abc \cdot cba) \; = \; \pm n^2abc3−(abccdotcba);=;pmn2 We consider asymptotic density of solutions. We also compare the results with ones with bases different from 10.

Induced magnetism versus Kondo screening in alternating Mott-metal layers

Physical Review B, 2013

Studies of systems with two fermionic bands with repulsive interaction strength U have a long his... more Studies of systems with two fermionic bands with repulsive interaction strength U have a long history, with the Periodic Anderson Model (PAM) being one of the most frequently considered Hamiltonians. In this paper, we use Quantum Monte Carlo to study analogous issues for attractive interactions. As in the Periodic Anderson Model, we focus on a case where one band is uncorrelated (U = 0), and focus on the effect of hybridization V between the bands on the pairing correlations. A key difference with the PAM is that there is no sign problem, so that we are able to explore the physics of doped multi-band attractive systems at low temperatures whereas ground state properties of repulsive models can be determined only at half-filling. For small V , pairing in the U < 0 layer induces pairing in the U = 0 layer. At larger V the ground state of the coupled system loses its superconducting character. The Quantum Monte Carlo data are complemented by results obtained with the Bogoliubov-de Gennes approximation.

Mean Field Theory Calculation of Isentropic Curves of the Fermion Hubbard Model

Recent experiments on optical lattices have focussed attention on understanding how many body cor... more Recent experiments on optical lattices have focussed attention on understanding how many body correlations change when the entropy (rather than the temperature) is varied as a control parameter. Quantum Monte Carlo (QMC) simulations have addressed some of the issues involved, but, for fermions, are limited by the sign problem. In this talk, we present results for the isentropic curves of the square lattice fermion Hubbard model in mean field theory (MFT). The topology of these curves on the phase diagram is explored, and compared to what is found in QMC when the latter is available. We also compare to MFT calculations for other models.

We study the ground state phases of Bose-Fermi mixtures in one-dimensional optical lattices with ... more We study the ground state phases of Bose-Fermi mixtures in one-dimensional optical lattices with quantum Monte Carlo simulations using the Canonical Worm algorithm. Depending on the filling of bosons and fermions, and the on-site intra-and inter-species interaction, different kinds of incompressible and superfluid phases appear. On the compressible side, correlations between bosons and fermions can lead to a distinctive behavior of the bosonic superfluid density and the fermionic stiffness, as well as of the equal-time Green functions, which allow one to identify regions where the two species exhibit anticorrelated flow. We present here complete phase diagrams for these systems at different fillings and as a function of the interaction parameters.

We present Quantum Monte Carlo simulations of a generalization of the Feynman-Kikuchi model which... more We present Quantum Monte Carlo simulations of a generalization of the Feynman-Kikuchi model which includes the possibility of vacancies and interactions between the particles undergoing exchange. By measuring the winding number (superfluid density) and density structure factor, we determine the phase diagram, and show that it exhibits regions which possess both superfluid and charge ordering.

Determinant Quantum Monte Carlo method applied to the t-J model

The usual approach to simulating the t-J model with the Determinant Quantum Monte Carlo (DQMC) me... more The usual approach to simulating the t-J model with the Determinant Quantum Monte Carlo (DQMC) method starts with the Hubbard model with a finite on-site interaction U which is then increased to ``almost'' infinity. This approach, however, has considerable difficulties with large round-off errors (stability) and variances, and also a very bad fermion sign problem. In this talk, I will describe a different approach which starts with (almost) infinite U by means of a projector operator and further prohibiting double occupancy by using a modified creation operator. The new technique will be shown to solve some of these difficulties. Unfortunately, the sign problem remains significant. I will discuss the different attempts we have made to reduce it.

Experiments on cold atom systems in which a lattice potential is ramped up on a confined cloud ha... more Experiments on cold atom systems in which a lattice potential is ramped up on a confined cloud have raised intriguing questions about how the temperature varies along isentropic curves, and how these curves intersect features in the phase diagram. In this paper, we study the isentropic curves of two models of magnetic phase transitions- the classical Blume-Capel Model (BCM) and the Fermi Hubbard Model (FHM). Both Mean Field Theory (MFT) and Monte Carlo (MC) methods are used. The isentropic curves of the BCM generally run parallel to the phase boundary in the Ising regime of low vacancy density, but intersect the phase boundary when the magnetic transition is mainly driven by a proliferation of vacancies. Adiabatic heating occurs in moving away from the phase boundary. The isentropes of the half-filled FHM have a relatively simple structure, running parallel to the temperature axis in the paramagnetic phase, and then curving upwards as the antiferromagnetic transition occurs. However, in the doped case, where two magnetic phase boundaries are crossed, the isentrope topology is considerably more complex.