Value N0 Research Articles

Proving the conjecture of O’Donnell in certain cases and disproving its general validity

For a function f:{−1,1}n→{−1,1} the relationship between the sum of its linear Fourier coefficients f̂(i) (defined by f̂(i)≔12n∑x∈{−1,1}nf(x)xi for i=1,2,…,n and x=(x1,…,xn)) and its degree d is a problem in theoretical computer science related to social choice. In that regard, in 2012, O’Donnell conjectured that ∑i=1nf̂(i)≤dd−1⌊d−12⌋21−d. In 2020, Wang (Wang, 2020) proved that the conjecture is equivalent to a conjecture about cryptographic resilient Boolean functions and proved that the conjecture is true for every n, when d=1 and d=n−1. In this paper, we contribute theoretically to this problem by showing that if the conjecture is true for some specific relatively small value n0, which depends on d, then it is valid for any n≥n0. In particular, we show that the conjecture is correct for every n, when d=2 and d=3. However, despite these promising results, we present a counterexample when d=4, and so the conjecture is not true in general.

Non-poissonian Distribution of Point Mutations in DNA.

In general, for chemical reactions occurring in systems, where fluctuations are not negligibly small, it is necessary to introduce a master equation for distribution of probability of fluctuations. It has been established that the monomolecular reactions of a type as A ↔ X are described by the master equation, which leads to a Poisson distribution with the variance equal to the average value N0. However, the consideration of the Löwdin mechanism as autocatalytic non-linear chemical reactions such as A + X ↔ 2X and the corresponding master equation lead to a non-Poissonian probability distribution of fluctuations. In the presented work, first-order autocatalysis has been applied to the Löwdin's mechanism of spontaneous mutations in DNA. Describing double proton transfers between complimentary nucleotide bases along the chain by first-order autocatalytic reactions, the corresponding master equation for protons in tautomeric states becomes non-linear, and at non-equilibrium conditions this leads to the non-Poissonian distribution of spontaneous mutations in DNA. It is also suggested that the accumulation of large fluctuations of successive cooperative concerted protons along the chain may produce higher non-linearities which could have a significant impact on some biochemical processes, occurring in DNA.

Extremal binary matrices without constant 2-squares

In this paper we solve, by computational means, an open problem of Erickson: denoting [n] = {1, . . . , n}, what is the smallest integer n0 such that, for every n ≥ n0 and every 2-coloring of the grid [n]× [n], there is a constant 2-square, i.e. a 2 × 2 subgrid S = {i, i + t} × {j, j + t} whose four points are colored the same? It has been shown recently that 13 ≤ n0 ≤ min(W (2, 8), 5·2 240), whereW (2, 8) is the still unknown 8th classical van der Waerden number. We obtain here the exact value n0 = 15. In the process we display 2-colorings of [13]× Z and [14]× [14] without constant 2-squares, and show that this is best possible.

RETRACTED ARTICLE: A proof of the Collatz conjecture

An elementary proof by contradiction of the Collatz Conjecture (CC) (also known as the '3X + 1' Conjecture), is presented. A modified form of the Collatz transformation is formulated, leading to the concept of a modified Collatz chain. A smallest counterexample N0 is hypothesized; the existence of N0 implies that N0 must generate an infinite sequence {Nk}, each of whose elements is at least as large as N0. A formula for Nk is derived, in terms of an auxiliary sequence {Ek} and the starting value N0. It is shown that each Ek satisfies k = Ek < 1.585k; this, in turn, leads us to conclude that N0 is unbounded, which is a contradiction of its definition, thereby establishing CC.

Optimizing Dual-Time and Serial Positron Emission Tomography and Single Photon Emission Computed Tomography Scans for Diagnoses and Therapy Monitoring

A region's early and late tracer uptake activities, QE and QL, within a dual-time scan (i.e. using two frames) or in serial scans (as for monitoring therapeutic response), are popular quantitative diagnostic aids, especially in oncology. In this paper, maximum performance is sought from their joint use. QL/QnE is introduced as a tumor marker with an empirical n. This generalizes traditional data weighting having n=1 for QL/QE, the retention index (RI), with its associated % difference. Using patient data, iterative guessing finds an optimal n that maximizes a measure of diagnostic performance: D=(difference of normal and abnormal marker means)/(their combined SD), which may be computed from values of QL/QnE, as well as of QL, QE, and RI each used alone. For 2-deoxy-2-[F-18]fluoro-D-glucose(FDG)-positron emission tomography (PET) dual-time protocols, another approach to optimization-selection of scan times-is investigated by simulations using the Sokolov model. A meta-analysis of 12 PET and single photon emission computed tomography (SPECT) studies with various tracers, cancers, and scan classes (dual-time or serial) finds ns from 0.5 to 1.1. The optimal D necessarily exceeds the best (or any) computed using QE, QL, or RI: negligibly to by as much as 0.6 (or 1.5). The increases in optimal receiver operating curve area (Az) over the best (or any) traditional marker range from negligible to 0.07 (or 0.4). QE alone usually has the lowest D and Az. Statistically significant performance improvement of QL/QnE over QE and QL is shown for most studies. Contrasting with an optimal n, another value n0 can also be found where D=0. Occasionally, n0 can be close to 1, and RI then will have a small D and poor performance. Simulation with kinetic modeling of FDG dual-time scans for liver and liver metastases demonstrates worst and best scan times. Indicated for these imaging protocols are QE at very early cellular transport associated times and QL rather late when phosphorylation/dephosphorylation dominate. Benefits from choosing optimal times in dual-time protocols, especially in combination with choosing optimal ns, can be significant. A protocol-dependent optimizing parameter n in an improved classification marker can easily be identified in a learning set of scans having normals and abnormals. Finding this parameter below 1.0 in most all studies suggests that a popularly used QL/QE may often overweight early activities. Additionally, QL/QE may sometimes be a poor marker choice and underestimate a protocol's diagnostic capability. Subsequent use of the proposed QL/QnE in settings similar to that of the learning set gives improved diagnostic performance over traditional approaches, although by widely varying amounts. Additionally, a method of seeking optimal scan times is demonstrated and suggests significant gains in dual-time protocol performances are possible.

Millimeter Observations and Modeling of the AB Aurigae System

We present the results of millimeter observations and a suitable chemical and radiative transfer model of the AB Aurigae (HD 31293) circumstellar disk and surrounding envelope. The integral molecular content of this system is studied by observing CO, C18O, CS, HCO+, DCO+, H2CO, HCN, HNC, and SiO rotational lines with the IRAM 30 m antenna, while the disk is mapped in the HCO+ (1-0) transition with the Plateau de Bure Interferometer. Using a flared disk model with a vertical temperature gradient and an isothermal spherical envelope model with a shadowed midplane and two unshielded cones together with a gas-grain chemical network, time-dependent abundances of observationally important molecules are calculated. Then a two-dimensional non-LTE line radiative transfer code is applied to compute excitation temperatures of several rotational transitions of HCO+, CO, C18O, and CS molecules. We synthesize the HCO+ (1-0) interferometric map along with single-dish CO (2-1), C18O (2-1), HCO+ (1-0), HCO+ (3-2), CS (2-1), and CS (5-4) spectra and compare them with the observations. Our disk model successfully reproduces observed interferometric HCO+ (1-0) data, thereby constraining the following disk properties: (1) the inclination angle i=17+6-3 deg, (2) the position angle φ=80deg+/-30deg, (3) the size Rout=400+/-200 AU, (4) the mass Mdisk=1.3×10-2 Msolar (with a factor of ~7 uncertainty), and (5) that the disk is in Keplerian rotation. Furthermore, indirect evidence for a local inhomogeneity of the envelope at >~600 AU is found. The single-dish spectra are synthesized for three different cases, namely, for the disk model, for the envelope model, and for their combination. An overall reasonable agreement between all modeled and acquired line intensities, widths, and profiles is achieved for the latter model, with the exception of the CS (5-4) data that require the presence of high-density clumpy structures in the model. This allows us to constrain the physical structure of the AB Aur inner envelope: (1) its mass-average temperature is about 35+/-14 K; (2) the density goes inversely down with the radius, ρ~r-1.0+/-0.3, starting from an initial value n0~3.9×105 cm-3 at 400 AU; and (3) the mass of the shielded region within 2200 AU is about 4×10-3 Msolar (the latter two quantities are uncertain by a factor of ~7). In addition, evolutionary nature and lifetime for dispersal of the AB Aur system and Herbig Ae/Be systems in general are discussed.

Anodic spark deposition of P, Me(II) or Me(III) containing coatings on aluminium and titanium alloys in electrolytes with polyphosphate complexes

The aim of this study was to characterise the correlation between the molar ratio n=[polyphosphate]/[metal cation] in aqueous electrolyte and the growth parameters, composition and surface distribution of elements of anodic coatings on aluminium and titanium alloys. Coatings were formed in electrochemical cell using the anodic spark deposition method. The two-component electrolytes contained sodium hexametaphosphate (Na6P6O18) or sodium tripolyphosphate (Na5P3O10) and acetates of Mg(II), Zn(II), Ba(II), Mn(II), Ni(II), Pb(II), Y(III), Eu(III) were used. Coatings were characterised by electron probe microanalysis, laser mass spectrometry and X-ray diffraction. It was shown that the n value essentially affects the composition and thickness of coatings when lower than a certain value n0, which lies in the range from 2.5 to 10 depending on the metal cation in the electrolyte. Coatings formed in electrolytes of n<1 consist mostly of electrolyte species. The influence of n on composition and parameters of the coatings is explained in terms of deposition and thermolysis of polyphosphate complexes on the anode surface owing to electric breakdown.

Charge-carrier exclusion and accumulation intensified by ohmic contacts

The effect of the high generation-recombination power of an ohmic contact (S contact) on exclusion-accumulation processes in structures with an antiblocking contact (asymmetric structures of the type p+-p-S) was studied theoretically and experimentally. It is shown that, in contrast to the ordinarily studied symmetric structure p+-p-p+, the asymmetric structures, which is being studied, forms a region of accumulation or exclusion, depending on the direction of the current. The nonequilibrium carrier density in the accumulation layer is much higher while the exclusion region is longer than in the symmetric structure. A density n 100 times higher than the equilibrium value n0 was obtained experimentally for Ge at 300 K. The length of the exclusion region reached 96% of the sample length. Applications of structures with antiblocking and ohmic contacts based on narrow-gap materials are proposed.

Plasma channel formation by short pulse laser

A short pulse laser propagating through a preformed plasma exerts a radial ponderomotive force on the electrons and creates a low density plasma channel via ambipolar diffusion. If the radial intensity profile of the laser beam is clamped, i.e., assumed Gaussian in r with fixed half-width r0, then closed form analytic solution of the evolution of plasma density in the channel is obtained in the perturbation approximation, Δn≪n0, where Δn is the modification in electron density over its equilibrium value n0, by solving the linearized equation of continuity. The relevance of the channel for low spot size beams self-focusing is examined.

Asymptotic behavior of high‐order differences of the partition function

AbstractThe partition function p(n) is the number of partitions of an integer n into positive integers, and p(n, k) = Δkp(n) is its k‐th difference. We obtain a recursion equation for p(n, k) which we solve asymptotically for n and k both large by using the WKB method and the method of matched asymptotic expansions. The result shows that p(n, k) is positive for n greater than a critical value n0(k). while for n &lt; n0(k), p(n, k) alternates in sign as n increases. This confirms the recent result of A. M. Odlyzko. It also yields a refinement of his approximation for n0(k). The improved approximation agrees well with exact numerical values.

Angle-dependent nonlinear modes of a cylindrical waveguide

Solutions of the wave equation for a cylindrical waveguide with cladding in which the dielectric constant depends nonlinearly on the optical field are obtained by numerical methods. We consider both saturable and nonsaturable nonlinearities. Asymmetrical solutions for which the field distributions have no axial symmetry are found. It is shown that the energy of such modes is less than the energy of the axially symmetric modes. The asymmetric modes exist in an interval of wave vectors n above a minimum value n0 at which the branching of the asymmetric mode away from the symmetric one first occurs. The dependences of the total energy and the value n0 on the core diameter are obtained by expanding the field in powers of (n − n0). Solutions that are periodic in angle with several maxima and minima along the interface of the core and the cladding are also considered.

A test for the median, combining the sign and signed-rank tests

A test is proposed for testing the null hypothesis that the median of a distribution is a specified value n0 without any assumption about the symmetry of the distribution. The signed-rank test is generally more powerful than the sign test for testing the null hypothesis that the distribution is symmetric around n0, but for our problem it has the disadvantage that it may reject a correct null hypothesis due to the skewness of the distribution. To guard against this possibility, the following procedure is proposed: If the signed-rank test has a highly significant value, reject H0 if it has an intermediate value, calculate the value of the sign test statistic and reject H0 if this value is significant. It is shown that for suitable criti-cal values this test is robust and has good power properties compared to the sign test.

The Statistics of Charge Carrier Fluctuations in Semiconductors

The statistical fluctuations in the number of charge carriers (conduction electrons of valence band holes) in a given volume of semiconductor in the steady state is calculated from the transition probabilities of electrons between the conduction and valence bands or between either band and impurity levels The case in which one independent fluctuating variable (either n or p) specifies the electronic state of the system is considered. Simple general formulae are developed for the mean value n0 of n and its variance and it is shown that for large numbers n tends to be normally distributed about n0. The relaxation time for small deviations from equilibrium is evaluated. These results are applied to three cases: (i) the intrinsic semiconductor, (ii) the strongly extrinsic semiconductor and (iii) the slightly extrinsic semiconductor with all impurity atoms ionized. The relation between the statistical approach and the thermodynamical treatment (given in another paper) is indicated.