Integrals evaluated in terms of Catalan's constant

2017 ◽  
Vol 101 (550) ◽  
pp. 38-49
Author(s):  
Graham Jameson ◽  
Nick Lord
Keyword(s):  
Unsolved Problem ◽  
Close Relative ◽  
Catalan’S Constant ◽  
Image Position

Catalan's constant, named after E. C. Catalan (1814-1894) and usually denoted by G, is defined byIt is, of course, a close relative ofThe numerical value is G ≈ 0.9159656. It is not known whether G is irrational: this remains a stubbornly unsolved problem. The best hope for a solution might appear to be the method of Beukers [1] to prove the irrationality of ζ (2) directly from the series, but it is not clear how to adapt this method to G.

Greek Kinship Terminology

1953 ◽  
Vol 73 ◽  
pp. 46-52 ◽  
Author(s):  
M. Miller
Keyword(s):  
Close Association ◽  
Close Relative ◽  
The Common ◽  
Image Position ◽  
Kinship Terminology ◽  
Exact Term

Classical Greek kinship terminology, as it is used for example by Isaios, offers few difficulties of meaning in its terms, and describes a bilateral family rather like our own. The principal usages mav be shown in genealogical form as follows:The noteworthy terms are: (i) kedestes, (2) anepsios, anepsiadous, exanepsios, and (3) adelphos and adelphe. Kedestes is applicable to any male who is a close relative by marriage, but who does not belong to the circle of heirs within the anchisteia: the term thus covers our father-in-law, stepfather, brother-in-law, and son-in-law. The close association of the term with words for ‘mourning’ suggests that this name arose from the duties performed in the funerals of members of their wives' anchisteia, even though they were outside the circle of heirs. The terms pentheros and gambros are, apparently, influenced by the usage of kedestes, and tend to the same classificatory employment. The meaning of nyos similarly tends to wander. Anepsios varies between cousin-german and nephew, and each of these relationships also has its exact term, in both cases a compound of adelphos. Anepsiadous and its synonym anepsiou pais are used not only for the cousin's child (the first cousin once removed), but also for Ego's parent's cousin (also a first cousin once removed): so Theopompos, the mother's cousin and heir of Hagnias, calls himself anepsiou pais to Hagnias. The exanepsios was outside the Attic anchisteia, and the term is rarely found. The terms for blood relatives are of the common IE vocabulary except adelphos and adelphe, which have replaced phrater (surviving to mean ‘member of a phratry’), and the lexicographers' eor.

scholarly journals An unsolved problem on the powers of 3/2

1968 ◽  
Vol 8 (2) ◽  
pp. 313-321 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Unsolved Problem ◽  
Fractional Part ◽  
Arbitrary Positive Number ◽  
Image Position

Let α be an arbitrary positive number. For every integer n ≦ 0 we can write where is the largest integer not greater than, i.e the integral part of, and rn is its fractional part and so satisfies the inequality

K-Theory and Asymptotically Commuting Matrices

1988 ◽  
Vol 40 (1) ◽  
pp. 197-216 ◽  
Author(s):  
Terry A. Loring
Keyword(s):  
Original Problem ◽  
Unsolved Problem ◽  
Point Of View ◽  
Commuting Matrices ◽  
Theoretic Point ◽  
K Theory ◽  
Image Position ◽  
Shed Light

To shed light on the following unsolved problem, several authors have considered related problems. The problem is that of finding commuting approximants to pairs of asymptotically commuting self-adjoint matrices:Suppose that Hn and Kn are self-adjoint matrices of dimension m(n), with ║Hn║, ║Kn║ ≦ 1, which commute asymptotically in the sense thatMust there then exist commuting self-adjoint matrices H′n and K′n for whichOne may alter the conditions imposed on Hn and Kn, for example, by requiring Hn to be normal and Kn to be self-adjoint, and ask whether commuting approximants H′n and K′n can be found satisfying the same conditions. Some of these related problems have been solved. This paper will examine their solutions from a K-theoretic point of view, illustrating the difficulty inherent in modifying them to work for the original problem.

An infinite class of periodic solutions of ẍ + 2x3 = p(t)

1965 ◽  
Vol 61 (1) ◽  
pp. 157-164 ◽  
Author(s):  
G. R Morris
Keyword(s):  
General Solution ◽  
Periodic Solutions ◽  
Unsolved Problem ◽  
Bounded Solutions ◽  
Infinite Class ◽  
Non Linear ◽  
Image Position

An important unsolved problem in the theory of non-linear oscillations is to establish the boundedness or unboundedness of the general solution ofwhere dots denote differentiation with respect to t. When p(t) is periodic, we may seek periodic solutions. This search is interesting for its own sake, and of course leads us to special bounded solutions. In three previous papers (2, 3, 4) I have exhibited the equationas tractable: on the assumption that e(t) is even and periodic, it was shown that the equation has an infinity of periodic solutions.

A Catalan constant inspired integral odyssey

2020 ◽  
Vol 104 (561) ◽  
pp. 449-459
Author(s):  
Seán M. Stewart
Keyword(s):  
Definite Integrals ◽  
Catalan Constant ◽  
Catalan’S Constant ◽  
Image Position

There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by $${\rm{G}} = \sum\limits_{n = 0}^\infty {{{{{\left( { - 1} \right)}^n}} \over {{{\left( {2n + 1} \right)}^2}}} = 1 - {1 \over {{3^2}}} + {1 \over {{5^2}}} \ldots = 0.915\,965\,594 \ldots \,\,,} $$ Scott in [1] quipped that this constant seemed almost as useful as the more widely known Euler–Mascheroni constant γ, particularly in the evaluation of definite integrals. And like γ, Catalan's constant continues to remain one of the most inscrutable constants in mathematics where the question concerning its irrationality is not settled.

scholarly journals NEW REDUCTIONS AND LOGARITHMIC LOWER BOUNDS FOR THE NUMBER OF CONJUGACY CLASSES IN FINITE GROUPS

2012 ◽  
Vol 87 (3) ◽  
pp. 406-424
Author(s):  
EDWARD A. BERTRAM
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Lower Bounds ◽  
Finite Groups ◽  
Unsolved Problem ◽  
Critical Role ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
Other Information ◽  
Image Position

AbstractThe unsolved problem of whether there exists a positive constant $c$ such that the number $k(G)$ of conjugacy classes in any finite group $G$ satisfies $k(G) \geq c \log _{2}|G|$ has attracted attention for many years. Deriving bounds on $k(G)$ from (that is, reducing the problem to) lower bounds on $k(N)$ and $k(G/N)$, $N\trianglelefteq G$, plays a critical role. Recently Keller proved the best lower bound known for solvable groups: \[ k(G)\gt c_{0} \frac {\log _{2}|G|} {\log _{2} \log _{2} |G|}\quad (|G|\geq 4) \] using such a reduction. We show that there are many reductions using $k(G/N) \geq \beta [G : N]^{\alpha }$ or $k(G/N) \geq \beta (\log [G : N])^{t}$ which, together with other information about $G$ and $N$ or $k(N)$, yield a logarithmic lower bound on $k(G)$.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

Atomic orientation effects in proton stopping at low velocity

1983 ◽  
Vol 41 ◽  
pp. 708-709
Author(s):  
Kin Lam
Keyword(s):  
Polycrystalline Material ◽  
Charged Particles ◽  
Target Material ◽  
Stopping Power ◽  
Low Velocity ◽  
Electronic Density ◽  
Interatomic Distances ◽  
Frame Work ◽  
Image Position ◽  
Atomic Orientation

The energy of moving ions in solid is dependent on the electronic density as well as the atomic structural properties of the target material. These factors contribute to the observable effects in polycrystalline material using the scanning ion microscope. Here we outline a method to investigate the dependence of low velocity proton stopping on interatomic distances and orientations.The interaction of charged particles with atoms in the frame work of the Fermi gas model was proposed by Lindhard. For a system of atoms, the electronic Lindhard stopping power can be generalized to the formwhere the stopping power function is defined as

A Magnetic Spectrometer for a Scanning Transmission Electron Microscope

1974 ◽  
Vol 32 ◽  
pp. 436-437
Author(s):  
A. Kosiara ◽  
J. W. Wiggins ◽  
M. Beer
Keyword(s):  
Scanning Transmission Electron Microscope ◽  
Magnetic Spectrometer ◽  
Fringe Field ◽  
Curvature Radius ◽  
Magnetic Pole ◽  
Quadrupole Field ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
Under Construction ◽  
Image Position

A magnetic spectrometer to be attached to the Johns Hopkins S. T. E. M. is under construction. Its main purpose will be to investigate electron interactions with biological molecules in the energy range of 40 KeV to 100 KeV. The spectrometer is of the type described by Kerwin and by Crewe Its magnetic pole boundary is given by the equationwhere R is the electron curvature radius. In our case, R = 15 cm. The electron beam will be deflected by an angle of 90°. The distance between the electron source and the pole boundary will be 30 cm. A linear fringe field will be generated by a quadrupole field arrangement. This is accomplished by a grounded mirror plate and a 45° taper of the magnetic pole.

Optimizing conditions for x-ray microchemical analysis in analytical electron microscopy

1978 ◽  
Vol 36 (1) ◽  
pp. 548-549 ◽  
Author(s):  
N. J. Zaluzec
Keyword(s):  
Solid Angle ◽  
Analytical Electron Microscopy ◽  
Incident Beam ◽  
Thin Foil ◽  
Experimental Conditions ◽  
X Ray ◽  
Microchemical Analysis ◽  
Analytical Electron ◽  
Image Position ◽  
Incident Beam Energy

The ultimate sensitivity of microchemical analysis using x-ray emission rests in selecting those experimental conditions which will maximize the measured peak-to-background (P/B) ratio. This paper presents the results of calculations aimed at determining the influence of incident beam energy, detector/specimen geometry and specimen composition on the P/B ratio for ideally thin samples (i.e., the effects of scattering and absorption are considered negligible). As such it is assumed that the complications resulting from system peaks, bremsstrahlung fluorescence, electron tails and specimen contamination have been eliminated and that one needs only to consider the physics of the generation/emission process.The number of characteristic x-ray photons (Ip) emitted from a thin foil of thickness dt into the solid angle dΩ is given by the well-known equation

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