Determinantal inequalities for the partition function

2019 ◽  
Vol 150 (3) ◽  
pp. 1451-1466 ◽  
Author(s):  
Dennis X.Q. Jia ◽  
Larry X.W. Wang
Keyword(s):  
Partition Function ◽  
Image Position ◽  
Determinantal Inequalities

AbstractLet p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$, $$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$As a corollary, we deduce that p(n) satisfies the double Turán inequalities, that is, for $n\ges 222$, $$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$

Ramanujan Congruences For p-k(n) Modulo Powers Of 17

1991 ◽  
Vol 43 (3) ◽  
pp. 506-525 ◽  
Author(s):  
Kim Hughes
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Ramanujan Congruences ◽  
Image Position

For each integer r we define the sequence pr(n) by We note that p-1(n) = p(n), the ordinary partition function. On account of this some authors set r = — k to make positive values of k correspond to positive powers of the generating function for p(n): We follow this convention here. In [3], Atkin proved the following theorem.

Congruences for the Coefficients of Modular forms and Some New Congruences for the Partition Function

1957 ◽  
Vol 9 ◽  
pp. 549-552 ◽  
Author(s):  
Morris Newman
Keyword(s):  
Partition Function ◽  
Modular Forms ◽  
Negative Integer ◽  
Image Position

If n is a non-negative integer, define pr(n) as the coefficient of xn in;otherwise define pr(n) as 0. In a recent paper (2) the author established the following congruence:Let r = 4, 6, 8, 10, 14, 26. Let p be a prime greater than 3 such that r(p + l) / 24 is an integer, and set Δ = r(p2 − l)/24.

Ramanujan Congruences for p-k(n)

1968 ◽  
Vol 20 ◽  
pp. 67-78 ◽  
Author(s):  
A. O. L. Atkin
Keyword(s):  
Partition Function ◽  
Ramanujan Congruences ◽  
Image Position ◽  
Congruence Properties

Let12Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.

The diamagnetism of quasi-bound conduction electrons

1953 ◽  
Vol 49 (2) ◽  
pp. 292-298 ◽  
Author(s):  
A. H. Wilson
Keyword(s):  
Magnetic Field ◽  
Partition Function ◽  
Power Series ◽  
Density Matrix ◽  
Complete Solution ◽  
Exact Expression ◽  
Free Electrons ◽  
Conduction Electrons ◽  
Image Position ◽  
Bound Electrons

The diamagnetism of the conduction electrons gives rise to some of the most difficult problems in the theory of metals, the complete solution of which has not yet been found. Formally, the problem is equivalent to determining the density matrixand the exact expression for ψ(r′, r, γ) for perfectly free electrons in a constant magnetic field H has recently been found by Sondheimer and Wilson(2). The extension of the theory to deal with quasi-bound electrons for all values of H seems to be out of the question, but an approximate partition function was given by Peierls (1), excluding terms of higher order than H2. In The theory of metals ((3), referred to as T.M.) I gave a more powerful and simpler method of dealing with the problem, based upon the properties of ψ(r′, r, γ), but since the solution was obtained as a power series in μ0Hγ, where it could at best determine only the normal diamagnetism.

On one-dimensional regular assemblies

1948 ◽  
Vol 44 (2) ◽  
pp. 263-271 ◽  
Author(s):  
G. S. Rushbrooke ◽  
H. D. Ursell
Keyword(s):  
Partition Function ◽  
Simple Formula ◽  
Chemical Potential ◽  
Absolute Temperature ◽  
Grand Partition Function ◽  
One Dimensional ◽  
Image Position

The grand partition function of any statistical assembly may be defined by the equationwhere E denotes any value of the energy of the assembly, k is Boltzmann's constant, T the thermodynamic absolute temperature and λi a parameter which is later to be connected with the chemical potential, μi, of the ith species in the assembly by the simple formula

The rotational specific heat of a polyatomic molecule for high temperatures—correction

1933 ◽  
Vol 29 (3) ◽  
pp. 407-407 ◽  
Author(s):  
Irene E. Viney
Keyword(s):  
Asymptotic Expansion ◽  
Partition Function ◽  
Specific Heat ◽  
High Temperatures ◽  
Polyatomic Molecule ◽  
Department Of Commerce ◽  
Image Position

I am indebted to Mr L. Kassel of the Department of Commerce, U.S.A., for pointing out an error of computation in a recent paper in these Proceedings.The asymptotic expansion for the partition function should readwhere

Asymptotic expansions of the expressions for the partition function and the rotational specific heat of a rigid polyatomic molecule for high temperatures

1933 ◽  
Vol 29 (1) ◽  
pp. 142-148 ◽  
Author(s):  
Irene E. Viney
Keyword(s):  
Partition Function ◽  
Specific Heat ◽  
High Temperatures ◽  
Asymptotic Expansions ◽  
Polyatomic Molecule ◽  
Absolute Temperature ◽  
Fundamental Importance ◽  
Molecular Gas ◽  
Perfect Gas ◽  
Image Position

The partition function F(θ) is of fundamental importance in the theory of the specific heat of gases. Once it is known, the rotational specific heat of a perfect gas is given bywhere R is the gram-molecular gas constant, and θ bears the relationto the absolute temperature T, k being Boltzmann's constant.

Congruence properties of the partition function and associated functions

1952 ◽  
Vol 48 (3) ◽  
pp. 402-413 ◽  
Author(s):  
J. M. Rushforth
Keyword(s):  
Partition Function ◽  
Unpublished Manuscript ◽  
Indian Mathematician ◽  
Associated Functions ◽  
The Subject ◽  
Image Position ◽  
Congruence Properties

The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in the possession of Prof. G. N. Watson. It is entitled‘Properties of p(n) and τ(n) defined by the relationsand was sent to the late Prof. G. H. Hardy by Ramanujan a few months before the latter's death in 1920. The work in the manuscript is concerned with the congruence properties of p(n) and τ(n), and Hardy extracted from it (Ramanujan (16)) proofs of the theorems

Adsorption Isotherms. Critical Conditions

1936 ◽  
Vol 32 (1) ◽  
pp. 144-151 ◽  
Author(s):  
R. H. Fowler
Keyword(s):  
Partition Function ◽  
Adsorption Isotherm ◽  
Critical Conditions ◽  
Adsorbed Atom ◽  
Adsorbed State ◽  
Adsorbed Gas ◽  
Internal States ◽  
Adsorbed Atoms ◽  
Image Position ◽  
Necessary And Sufficient

1. In a recent note I showed that Langmuir's adsorption isothermwhere θ is the fraction of the surface covered by adsorbed gas, p the gas pressure in equilibrium with it, and A (T) a specified function of the temperature, can be derived as a theorem in statistical mechanics without any appeal to the mechanism of deposition and re-evaporation. Necessary and sufficient assumptions for the truth of (1) are that the atoms (or molecules) of the gas are adsorbed as wholes on to definite points of attachment on the surface of the adsorber, that each point of attachment can accommodate one and only one adsorbed atom, and that the energies of the states of any adsorbed atom are independent of the presence or absence of other adsorbed atoms on neighbouring points of attachment. Under these assumptions the explicit form of (1) iswhere m is the mass of the adsorbed atom or molecule, bg(T) the partition function for its internal states in the gas phase, and vs(T) the partition function for its set of adsorbed states. These sets of states are to be so specified that the energy zero is assigned tot the lowest state of each set in constructing bg(T) and vs(T), and then X is the energy required to transfer a molecule from the lowest adsorbed state tot the lowest gas state. Quite another adsorption isotherm was shown to hold when adsorption of a molecule takes place as atoms and requires two or more points of attachment.

Some arithmetical properties of m-ary partitions

1970 ◽  
Vol 68 (2) ◽  
pp. 447-453 ◽  
Author(s):  
Öystein Rödseth
Keyword(s):  
Partition Function ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Fixed Integer ◽  
De Bruijn ◽  
Image Position

We denote by tm(n) the number of partitions of the positive integer n into non-decreasing parts which are positive or zero powers of a fixed integer m > 1 and we call tm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula for tm(n), the first term of which isMahler's result was later improved by de Bruijn (2).

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