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Articles 5641 - 5670 of 10319
Full-Text Articles in Physical Sciences and Mathematics
Kinetics Study And Influence Of Water-Soluble Polymer On The Electrodeposition Of Iron From A Citrate-Chloride Electrolyte On The Basis Of Fe(Iii), Vyacheslav Protsenko, Felix Danilov
Kinetics Study And Influence Of Water-Soluble Polymer On The Electrodeposition Of Iron From A Citrate-Chloride Electrolyte On The Basis Of Fe(Iii), Vyacheslav Protsenko, Felix Danilov
Turkish Journal of Chemistry
The kinetics of the iron electrodeposition reaction from a plating bath containing trivalent iron ions is investigated in the present work. The electrochemical reaction is stated to proceed via the formation of relatively stable intermediates: Fe(II) ions. Only a part of the total amount of Fe(II) ions formed at the first stage of Fe(III) discharge is reduced further, producing metal iron. The effect of polyhexamethyleneguanidine hydrochloride (with an average molecular weight of 1000) on the iron deposition is ascertained. The presence of polymer additive in the iron electrolyte appreciably affects the rate of iron deposition, while the polymer additive does …
Characterization Of A Cu$^{2+}$-Selective Fluorescent Probe Derived From Rhodamine B With 1,2,4-Triazole As Subunit And Its Application In Cell Imaging, Na Li, Chunwei Yu, Yuxiang Ji, Jun Zhang
Characterization Of A Cu$^{2+}$-Selective Fluorescent Probe Derived From Rhodamine B With 1,2,4-Triazole As Subunit And Its Application In Cell Imaging, Na Li, Chunwei Yu, Yuxiang Ji, Jun Zhang
Turkish Journal of Chemistry
A rhodamine B derivative containing 1,2,4-triazole as subunit was characterized as an "off-on" type Cu$^{2+}$-selective fluorescent probe. It exhibited high selectivity and sensitivity for Cu$^{2+}$ in ethanol-water solution (9:1, v:v, pH 7.0, 20 mM HEPES) and underwent ring opening. A prominent fluorescence enhancement at 570 nm was observed in the presence of Cu$^{2+}$ with the change in the absorption spectrum, and a 1:1 metal-ligand complex was formed. With the optimized experimental conditions, the probe exhibited a dynamic response range for Cu$^{2+}$ from 8.0 $\times $ 10$^{-7}$ to 7.5 $\times $ 10$^{-6}$ M with a detection limit of 2.3 $\times $ …
Acid-Free Synthesis Of S-Nitrosothiols At Neutral Ph By Shock-Freezing In Liquid Nitrogen, Dimitrios Tsikas, Anke Böhmer, Arne Trettin, Arslan Arinç Kayaçelebi̇
Acid-Free Synthesis Of S-Nitrosothiols At Neutral Ph By Shock-Freezing In Liquid Nitrogen, Dimitrios Tsikas, Anke Böhmer, Arne Trettin, Arslan Arinç Kayaçelebi̇
Turkish Journal of Chemistry
A laboratory method for the acid-free synthesis of S-nitrosothiols (RSNO) from NaNO$_{2}$ and thiols in aqueous buffer of neutral pH is reported. Equimolar amounts of NaNO$_{2}$ and thiol are added to a 100-$\mu $M Na$_{4}$EDTA-containing Na$_{2}$HPO$_{4}$ buffer (100 mM, pH 7.4). Samples are set into liquid nitrogen for 1 min and then thawed to room temperature or in an ice bath. Dependent upon the RSNO, repetition of the procedure may be required. This method may be of particular importance when the classical acid-based method fails or releases harmful gases such as H$_{2}$S.
Sustainable Activity Of Hydrothermally Synthesized Mesoporous Silicates In Acetic Acid Esterification, Veli̇ Şi̇mşek, Levent Deği̇rmenci̇, Kirali̇ Mürtezaoğlu
Sustainable Activity Of Hydrothermally Synthesized Mesoporous Silicates In Acetic Acid Esterification, Veli̇ Şi̇mşek, Levent Deği̇rmenci̇, Kirali̇ Mürtezaoğlu
Turkish Journal of Chemistry
A hydrothermal method was applied in the synthesis of mesoporous silicates containing silicotungstic acid (STA). The synthesis procedures were developed by modification of procedures previously applied in the synthesis of MCM-41 and SBA-15. The synthesized catalysts were named MCM-41-S and SBA-15-S based on MCM-41 and SBA-15. Their activities were investigated in ethyl acetate production, which was selected as the model reaction. The results indicated that the activity of SBA-15-S catalysts increased with increasing STA in the catalyst. Reaction experiments were also conducted in identical conditions in the presence of catalysts recovered after the reaction. The results indicated that sustainable activity …
Some Concrete Operators And Their Properties, Mehmet Gürdal, Mubariz T. Garayev, Suna Saltan
Some Concrete Operators And Their Properties, Mehmet Gürdal, Mubariz T. Garayev, Suna Saltan
Turkish Journal of Mathematics
We consider integration and double integration operators, the Hardy operator, and multiplication and composition operators on Lebesgue space $L_{p}\left[ 0,1\right] $ and Sobolev spaces $W_{p}^{\left( n\right) }\left[ 0,1\right] $ and $W_{p}^{\left( n\right) }\left( \left[ 0,1\right] \times\left[ 0,1\right] \right) ,$ and we study their properties. In particular, we calculate norm and spectral multiplicity of the Hardy operator and some multiplication operators, investigate its extended eigenvectors, characterize some composition operators in terms of the extended eigenvectors of the Hardy operator, and calculate the numerical radius of the integration operator on the real $L_{2}\left[ 0,1\right] $ space. The main method for our investigation …
On Zeros Of Certain Dirichlet Polynomials, Kajtaz H. Bllaca
On Zeros Of Certain Dirichlet Polynomials, Kajtaz H. Bllaca
Turkish Journal of Mathematics
In this article we establish the zero-free region of certain Dirichlet polynomials $L_{F, X}$ arising in approximate functional equation for functions in the Selberg class and we prove an asymptotic formula for the number of zeros of $L_{F, X}$.
A New Heuristic Method To Solve Unit Commitment By Using A Time-Variant Acceleration Coefficients Particle Swarm Optimization Algorithm, Arsalan Najafi, Mohsen Farshad, Hamid Falaghi
A New Heuristic Method To Solve Unit Commitment By Using A Time-Variant Acceleration Coefficients Particle Swarm Optimization Algorithm, Arsalan Najafi, Mohsen Farshad, Hamid Falaghi
Turkish Journal of Electrical Engineering and Computer Sciences
Unit commitment is one of the most important problems in power system operation. Because of the large amount of parameters and constraints, it contains a high level of complexity. In this paper a new method based on a time-variant acceleration coefficients particle swarm optimization algorithm has been proposed to solve the unit commitment problem. Integer coding (for satisfying minimum up/down constraints) and binary coding (for satisfying spinning reserve constraint) have been utilized in the proposed method. Simulations in the different cases have been done with different sizes. Numerical results have shown the superiority and better convergence of the proposed method …
Electric And Magnetic Field Induced Geometric Phases For The 2d Harmonic Oscillator In Noncommutative Phase Space, Mai-Lin Liang, Li-Fang Xu
Electric And Magnetic Field Induced Geometric Phases For The 2d Harmonic Oscillator In Noncommutative Phase Space, Mai-Lin Liang, Li-Fang Xu
Turkish Journal of Physics
In this work, we point out the phenomenon whereby the noncommutative corrections to the geometric phases induced by an electric field and a magnetic field have different orders in terms of the noncommutative parameters. The first order correction is zero for the electric field induced geometric phase and it is nonzero for the magnetic field induced geometric phase. In our calculation, the system is in coherent states when the electric field is applied, so the corresponding geometric phase calculated is that of the coherent states. Considering that the noncommutative parameters are very small, it is better to use the magnetic …
Dye-Sensitized Solar Cells Based On Dyes Extracted From Dried Plant Leaves, Sofyan A. Taya, Taher M. El-Agez, Kamal S. Elrefi, Monzir S. Abdel-Latif
Dye-Sensitized Solar Cells Based On Dyes Extracted From Dried Plant Leaves, Sofyan A. Taya, Taher M. El-Agez, Kamal S. Elrefi, Monzir S. Abdel-Latif
Turkish Journal of Physics
In this work, natural dyes were extracted from dried plant leaves of plant cream, apricot, figs, apples, sage, thyme, mint, Ziziphus jujuba, orange, shade tree, basil, berry, Mirabelle plum, Victoria plum, peach, mango, pomegranate, banana, guava, and fluoridation-treated plant. The extracts were used as photosensitizers for dye-sensitized solar cells (DSSCs). The cells were assembled using nanostructured TiO_2 films. The best performance was observed for the DSSC sensitized with Ziziphus jujuba with an efficiency of 1.077%. The impedance spectroscopy of 3 cells was carried out and the equivalent circuits were found.
Extension Of The Class Of Exactly Solvable Nonstationary Quantum Mechanics Problems, Rana Gazanfar Aghayeva
Extension Of The Class Of Exactly Solvable Nonstationary Quantum Mechanics Problems, Rana Gazanfar Aghayeva
Turkish Journal of Physics
A modified factorization method for a charged particle in a time-dependent magnetic field has been developed by virtue of motion quantum integrals. Application of this method to the considered system helps to find exact solutions for a range of nonstationary potentials.
A Theoretical Way To Determine Gamma-Ray Mass Attenuation Coefficients Of Materials, Eli̇f Ebru Ermi̇ş, Ercan Pi̇li̇çer, Cüneyt Çeli̇ktaş
A Theoretical Way To Determine Gamma-Ray Mass Attenuation Coefficients Of Materials, Eli̇f Ebru Ermi̇ş, Ercan Pi̇li̇çer, Cüneyt Çeli̇ktaş
Turkish Journal of Physics
The gamma-ray mass attenuation coefficients of various absorber materials such as Ag, Al, Au, Bakelite, Cu, Fe, Pb, Plexiglas, Si, Sn, water, and Zn were determined theoretically at different gamma-ray energies and different absorber thicknesses in order to investigate how the number of gamma photons and their energies affect the calculation of mass attenuation coefficients of the absorbers since no study such a comprehensive work here was encountered. For this purpose, the FLUKA Monte Carlo (MC) and XCOM programs were used. Calculated coefficients were compared to the literature values and found to agree well with them. The FLUKA MC program …
Polaron Effects On The Second-Harmonic Generation In Asymmetrical Semiexponential Quantum Wells, Bo Xiao, Kangxian Guo, Sen Mou, Zhongmin Zhang
Polaron Effects On The Second-Harmonic Generation In Asymmetrical Semiexponential Quantum Wells, Bo Xiao, Kangxian Guo, Sen Mou, Zhongmin Zhang
Turkish Journal of Physics
Polaron effects on the second-harmonic generation (SHG) in asymmetrical semiexponential quantum wells (ASEQWs) are investigated theoretically. By using the framework of the compact-density-matrix approach and iterative method, the analytical expression of the SHG coefficients in ASEQWs is presented. Numerical results are illustrated for a typical GaAs/AlGaAs. By considering the electron--LO--phonon interaction (ELOPI), the energy levels and the wave functions of an electron confined in ASEQWs are obtained. It is found that whether we consider the ELOPI or not, the incident photon frequency $\omega$ and parameters $U_0$ and $\sigma$ also affect the SHG coefficient obviously.
Search For Mixed-Symmetry State In Even-Even $^{130-138}$Ce Isotopes Within The Interacting Boson Model-2, Ali Mahdi Abdul Hussain, Falih Hussain Al-Khudair, Abdul Ridha Hussain Subber
Search For Mixed-Symmetry State In Even-Even $^{130-138}$Ce Isotopes Within The Interacting Boson Model-2, Ali Mahdi Abdul Hussain, Falih Hussain Al-Khudair, Abdul Ridha Hussain Subber
Turkish Journal of Physics
The energy levels and reduced transition probabilities B(M1) and B(E2) of the even-even $^{130-138}$Ce isotopes were calculated by using the interacting boson model-2. We have analyzed the F-spin values and produced the symmetry labeling of the states. The calculated results were compared with the available experimental data. It was proved that the proton--neutron interacting boson model is a reasonable model for calculating low-lying spectra in the set of Ce isotopes and the quality of fit presented in this work is acceptable.
The Third Kind Of Darboux Transformation And Multisoliton Solutions For Generalized Broer-Kaup Equations, Sheng Zhang, Dong-Dong Liu
The Third Kind Of Darboux Transformation And Multisoliton Solutions For Generalized Broer-Kaup Equations, Sheng Zhang, Dong-Dong Liu
Turkish Journal of Physics
In this paper, the third kind of Darboux transformation of generalized Broer-Kaup equations is derived from the corresponding spectral problem. By virtue of this Darboux transformation, new 2$N$-soliton solutions with parameters of the generalized Broer-Kaup equations are obtained. Although 2$N$ is an even number, it is graphically shown that in the cases of $N $= 1 and $N$ = 2 the obtained 2$N$-soliton solutions can degenerate into $M$-soliton solutions for any positive integer $M$ less than 2$N$.
A Full Relativistic Treatment In Analyzing $\Alpha $-$^{12}$C And $\Alpha $-$^{40,42,44,48}$Ca Elastic Scattering Data At 1.37 Gev, Zuhair Fayyad Mahmoud Shehadeh
A Full Relativistic Treatment In Analyzing $\Alpha $-$^{12}$C And $\Alpha $-$^{40,42,44,48}$Ca Elastic Scattering Data At 1.37 Gev, Zuhair Fayyad Mahmoud Shehadeh
Turkish Journal of Physics
A simple optical potential of Woods-Saxon form is used within the framework of the Klein-Gordon equation, for the first time, to analyze the elastic differential cross sections for 1.37 GeV (lab) incident $\alpha $-particles on $^{12}$C and calcium isotopes $^{40,42,44,48}$Ca. The relativistic energy-momentum relation assures the need for relativistic calculations. As such, the previously determined parameters of all optical potential forms, used in the nonrelativistic Schrödinger equation and successfully analyzing the $\alpha $-$^{12}$C and $\alpha $-$^{40,42,44,48}$Ca data at 1.37 GeV, are to be revised. In the presence of Coulomb potential, as in the cases under consideration, the asymptotic solution of …
Solution Processable Graphene Oxide Hole Transport Layers And Their Application In P3ht:Hhper Active Layer Based Bhjscsolution Processable Graphene Oxide Hole Transport Layers And Their Application In P3ht:Hhper Active Layer Based Bhjsc, Görkem Memi̇şoğlu, Hali̇de Di̇ker, Canan Varlikli
Solution Processable Graphene Oxide Hole Transport Layers And Their Application In P3ht:Hhper Active Layer Based Bhjscsolution Processable Graphene Oxide Hole Transport Layers And Their Application In P3ht:Hhper Active Layer Based Bhjsc, Görkem Memi̇şoğlu, Hali̇de Di̇ker, Canan Varlikli
Turkish Journal of Physics
Graphene oxide (GO) material was synthesized by an improved Hummers method and characterized by FT-IR, XPS, XRD, AFM, SEM, and UV-VIS analyses. The thickness of the GO layer was measured as 1.5 nm. Solution processed bulk heterojunction solar cells comprising poly(3-hexylthiophene) (P3HT) as the electron donor and N,N'-bis-2-(1-hydoxyhexyl)-3,4,9,10-perylenebis(dicarboximide) (HHPER) as the electron acceptor component of the active layer were produced with and without the GO doped PEDOT-PSS hole transport layers. The optical investigations of the active layer were performed by ground state absorption and photoluminescence measurements. Optimized blend w/w was determined as P3HT:HHPER, 3:1. It was found that the presence …
Multiwave Solutions Of Fractional 4th And 5th Order Burgers Equationsmultiwave Solutions Of Fractional 4th And 5th Order Burgers Equations, Emad A-B. Abdel-Salam, Gamal Fargahly Hassan
Multiwave Solutions Of Fractional 4th And 5th Order Burgers Equationsmultiwave Solutions Of Fractional 4th And 5th Order Burgers Equations, Emad A-B. Abdel-Salam, Gamal Fargahly Hassan
Turkish Journal of Physics
Fractional 4th and 5th order Burgers equations were studied by improving the generalized Exp-function method. The chief aim here was to discuss single-wave, double-wave, three-wave, and four-wave solutions for the space-time fractional (STF), time-fractional (TF), and space-fractional (SF) 4th and 5th order Burgers equations. The arbitrariness of the fractional orders allows much richer structures. It was observed that the wave amplitude of STF and SF Burgers equations is directly proportional to the noninteger derivative order, but the wave amplitude of TF Burgers equations is inversely proportional to the noninteger derivative order. From the graphical study of the obtained solution, a …
A Comparative Study: Synthetic Dyes As Photosensitizers For Dye-Sensitized \\Solar Cellsa Comparative Study: Synthetic Dyes As Photosensitizers For Dye-Sensitized \\Solar Cells, Amal M. Al-Kahlout, Hatem S. El-Ghamri, Naji Al Dahoudi, Taher M. El-Agez, Sofyan A. Taya, Monzir S. Abdel-Latif
A Comparative Study: Synthetic Dyes As Photosensitizers For Dye-Sensitized \\Solar Cellsa Comparative Study: Synthetic Dyes As Photosensitizers For Dye-Sensitized \\Solar Cells, Amal M. Al-Kahlout, Hatem S. El-Ghamri, Naji Al Dahoudi, Taher M. El-Agez, Sofyan A. Taya, Monzir S. Abdel-Latif
Turkish Journal of Physics
In this paper, dye-sensitized solar cells (DSSCs) were fabricated using a zinc oxide (ZnO) semiconducting layer and different synthetic dyes. Eight different synthetic dyes were used to fabricate the DSSCs. Nanocrystalline ZnO powder was coated on transparent conducting fluorine-doped tin oxide glass using the doctor blade method to form a thin layer. The absorption spectra of the synthetic dyes were investigated. DSSCs were characterized with $J$-$V$ characteristic curves. The parameters reflecting the cell performance were calculated. The results revealed that eosin Y dye corresponds to the highest conversion efficiency. The dependence of the sintering temperature and the thicknesses of the …
Dust-Ion Acoustic Shock Waves For Nonplanar Geometry With Nonthermal Electrons And Adiabatic Dusty Plasmadust-Ion Acoustic Shock Waves For Nonplanar Geometry With Nonthermal Electrons And Adiabatic Dusty Plasma, Louis Ete Akpabio
Turkish Journal of Physics
The nonplanar Burgers' equation for dust-ion-acoustic shock waves was derived for unmagnetized adiabatic dusty plasmas, comprising static negatively charged dust fluid, nonthermal distributed electrons, and Boltzmann distributed positron and adiabatic ion fluid, by employing the standard reductive perturbation method. The solution of the modified Burgers' equation for nonplanar geometry is numerically analyzed. The dependence of dust-ion-acoustic shock waves on some plasma parameters is explored. It is observed that inclusion of the nonthermal electron distribution modifies the shock wave profile significantly. Likewise, it is found that the nonplanar geometry effects have an important role on the establishment of shock waves. It …
Bifurcation And Dynamics Of A Normal Form Map, Reza Khoshsiar Ghaziani
Bifurcation And Dynamics Of A Normal Form Map, Reza Khoshsiar Ghaziani
Turkish Journal of Mathematics
This paper investigates the dynamics and stability properties of a so-called planar truncated normal form map. This kind of map is widely used in the applied context, especially in normal form coefficients of n-dimensional maps. We determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the system. We first analyze stability of the fixed points and the existence of local bifurcations. Our analysis shows the presence of a rich variety of local bifurcations, namely stable fixed points, periodic cycles, quasiperiodic cycles that are constraints to stable attractors called invariant closed curves, and chaos, where dynamics of …
On Ampleness And Pseudo-Anosov Homeomorphisms In The Free Group, Rizos Sklinos
On Ampleness And Pseudo-Anosov Homeomorphisms In The Free Group, Rizos Sklinos
Turkish Journal of Mathematics
We use pseudo-Anosov homeomorphisms of surfaces in order to prove that the first-order theory of non-Abelian free groups, T_{fg}, is n-ample for any n \in \omega. This result adds to the work of Pillay, which proved that T_{fg} is non-CM-trivial. The sequence witnessing ampleness is a sequence of primitive elements in F_{\omega}. Our result provides an alternative proof to the main result of a recent preprint by Ould Houcine and Tent.
Rational Schubert Polynomials, Kürşat Aker, Nesri̇n Tutaş
Rational Schubert Polynomials, Kürşat Aker, Nesri̇n Tutaş
Turkish Journal of Mathematics
We define and study the rational Schubert, rational Grothendieck, rational key polynomials in an effort to understand Molev's dual Schur functions from the viewpoint of Lascoux.
Groups With The Given Set Of The Lengths Of Conjugacy Classes, Neda Ahanjideh
Groups With The Given Set Of The Lengths Of Conjugacy Classes, Neda Ahanjideh
Turkish Journal of Mathematics
We study the structures of some finite groups such that the conjugacy class size of every noncentral element of them is divisible by a prime $p$.
Random Process Generated By The Incomplete Gauss Sums, Emek Demi̇rci̇ Akarsu
Random Process Generated By The Incomplete Gauss Sums, Emek Demi̇rci̇ Akarsu
Turkish Journal of Mathematics
In this paper we explore a random process generated by the incomplete Gauss sums and establish an analogue of weak invariance principle for these sums. We focus our attention exclusively on a generalization of the limit distribution of the long incomplete Gauss sums given by the family of periodic functions analyzed by the author and Marklof.
On The Computation Of Generalized Division Polynomials, Ömer Küçüksakalli
On The Computation Of Generalized Division Polynomials, Ömer Küçüksakalli
Turkish Journal of Mathematics
We give an algorithm to compute the generalized division polynomials for elliptic curves with complex multiplication. These polynomials can be used to generate the ray class fields of imaginary quadratic fields over the Hilbert class field with no restriction on the conductor.
The Strong ``Zero-Two" Law For Positive Contractions Of Banach--Kantorovich $L_P$-Lattices, Inomjon Ganiev, Farrukh Mukhamedov, Dilmurod Bekbaev
The Strong ``Zero-Two" Law For Positive Contractions Of Banach--Kantorovich $L_P$-Lattices, Inomjon Ganiev, Farrukh Mukhamedov, Dilmurod Bekbaev
Turkish Journal of Mathematics
In the present paper we study dominated operators acting on Banach--Kantorovich $L_p$-lattices, constructed by a measure $m$ with values in the ring of all measurable functions. Using methods of measurable bundles of Banach--Kantorovich lattices, we prove the strong ``zero-two" law for positive contractions of Banach--Kantorovich $L_p$-lattices. \vspace{1mm}
Fiber Product Preserving Bundle Functors On Fibered-Fibered Manifolds, Wlodzimierz M. Mikulski
Fiber Product Preserving Bundle Functors On Fibered-Fibered Manifolds, Wlodzimierz M. Mikulski
Turkish Journal of Mathematics
We introduce the concept of modified vertical Weil functors on the category $\F_2\M_{m_1,m_2}$ of fibered-fibered manifolds with $(m_1,m_2)$-dimensional bases and their local fibered-fibered maps with local fibered diffeomorphisms as base maps. We then describe all fiber product preserving bundle functors on $\F_2\M_{m_1,m_2}$ in terms of modified vertical Weil functors.
Planar Embedding Of Trees On Point Sets Without The General Position Assumption, Asghar Asgharian Sardroud, Alireza Bagheri
Planar Embedding Of Trees On Point Sets Without The General Position Assumption, Asghar Asgharian Sardroud, Alireza Bagheri
Turkish Journal of Mathematics
The problem of point-set embedding of a planar graph $G$ on a point set $P$ in the plane is defined as finding a straight-line planar drawing of $G$ such that the nodes of $G$ are mapped one to one on the points of $P$. Previous works in this area mostly assume that the points of $P$ are in general position, i.e. $P$ does not contain any three collinear points. However, in most of the real applications we cannot assume the general position assumption. In this paper, we show that deciding the point-set embeddability of trees without the general position assumption …
T-Spaces, Hassan Maleki, Mohammadreza Molaei
T-Spaces, Hassan Maleki, Mohammadreza Molaei
Turkish Journal of Mathematics
In this paper, using generalized groups and their generalized actions, we define and study the notion of $T$-spaces. We study properties of the quotient space of a $T$-space and we present the conditions that imply the Hausdorff property for it. We also prove some essential results about topological generalized groups. As a main result, we show that for each positive integer $n$ there is a topological generalized group $T$ with $n$ identity elements. Moreover, we study the maps between two $T$-spaces and we consider the notion of $T$-transitivity.
A Note On The Unit Distance Problem For Planar Configurations With $\Mathbb{Q}$-Independent Direction Set, Mark Herman, Jonathan Pakianathan
A Note On The Unit Distance Problem For Planar Configurations With $\Mathbb{Q}$-Independent Direction Set, Mark Herman, Jonathan Pakianathan
Turkish Journal of Mathematics
Let $T(n)$ denote the maximum number of unit distances that a set of $n$ points in the Euclidean plane $\mathbb{R}^2$ can determine with the additional condition that the distinct unit length directions determined by the configuration must be $\mathbb{Q}$-independent. This is related to the Erd\"os unit distance problem but with a simplifying additional assumption on the direction set that holds ``generically''. We show that $T(n+1)-T(n)$ is the Hamming weight of $n$, i.e. the number of nonzero binary coefficients in the binary expansion of $n$, and find a formula for $T(n)$ explicitly. In particular, $T(n)$ is $\Theta(n log(n))$. Furthermore, we describe …