Power partitions and a generalized eta transformation property

In their famous paper on partitions, Hardy and Ramanujan also raised the question of the behaviour of the number $p_s(n)$ of partitions of a positive integer~$n$ into $s$-th powers and gave some preliminary results. We give first an asymptotic formula to all orders, and then an exact formula, describing the behaviour of the corresponding generating function $P_s(q) = \prod_{n=1}^\infty \bigl(1-q^{n^s}\bigr)^{-1}$ near any root of unity, generalizing the modular transformation behaviour of the Dedekind eta-function in the case $s=1$. This is then combined with the Hardy-Ramanujan circle method to give a rather precise formula for $p_s(n)$ of the same general type of the one that they gave for~$s=1$. There are several new features, the most striking being that the contributions coming from various roots of unity behave very erratically rather than decreasing uniformly as in their situation. Thus in their famous calculation of $p(200)$ the contributions from arcs of the circle near roots of unity of order 1, 2, 3, 4 and 5 have 13, 5, 2, 1 and 1 digits, respectively, but in the corresponding calculation for $p_2(100000)$ these contributions have 60, 27, 4, 33, and 16 digits, respectively, of wildly varying sizes

Development of a Caterpillar-Type Walker for the Elderly People

A walker assists elderly people with age-related reduced walking ability and helps to improve stability and balance ability. However, if the general-type walker (GTW) is used on an uneven, obstacle, or sloped terrain, it may cause excessive muscle use and falls. Therefore, in this study, we developed a caterpillar-type walker (CTW) that elderly people can safely use in various terrains. Twelve elderly who were able to walk normally participated in the study. The activity of upper and lower extremity muscles, the number of obstacles overcome, and walking speed was compared and analyzed when using two types of walkers in uneven terrain, obstacle terrain, and sloped terrain. In addition, satisfaction with the use of these walkers was evaluated. When CTW was used, the activity of the muscles of the upper and lower extremities was significantly reduced compared to the use of GTW on all terrains. The walker developed in this study overcame obstacles of all heights, but the GTW failed to overcome obstacles starting from the 2 cm section. In terms of walking speed, when the CTW was used, the walking speed was higher than that of the GTW in uneven terrain and obstacle terrain. In satisfaction, there were significant differences in safety, durability, simplicity of use, comfort, and effectiveness. Through these results, it was confirmed that the CTW can efficiently and safely assist the elderly in walking on uneven terrain, obstacle terrain, and inclined terrain.

Comparative Analysis of Ports on the Eastern Baltic Sea Coast

Ports are an important part of the global and regional freight supply chain and transport network. As port activities have a significant impact on the economic growth of these countries, it is necessary to constantly analyse and plan port activities, anticipate market changes and improve the ability of ports to withstand the growing general competitive pressure. This article analyses studies conducted by researchers on the topic of seaports, thus, and find that there are no analyses comparing more than two ports. A comparative analysis of the ports on the eastern shore of the Baltic Sea was conducted using the analysis of statistical data. The comparative analysis of ports on the eastern Baltic Sea revealed that Klaipeda port is the most diversified port in the eastern Baltic Sea region, given that it does not have any single most important type of cargo. The largest ports in terms of bulk cargo are in Tallinn, Riga and Ventspils. Primorsk is the largest port for liquid cargo and St. Petersburg handles the greatest volumes of cargo of a general type andwhile the distribution of cargo flows in the port of Visotsk is best correlated with the selected parameters, which allows us to state that the infrastructure of this port is used to the maximum.

A supersingular coincidence

The 15 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 are called the supersingular primes: they occur in several contexts in number theory and also, strikingly, they are the primes that divide the order of the Monster. It is also known that the moduli space of (1, p)-polarised abelian surfaces is of general type for these primes. In this note, we explain that apparently coincidental fact by relating it to other number-theoretic occurences of the supersingular primes.

Explicit volume formula for a hyperbolic tetrahedron in terms of edge lengths

We consider a compact hyperbolic tetrahedron of a general type. It is a convex hull of four points called vertices in the hyperbolic space [Formula: see text]. It can be determined by the set of six edge lengths up to isometry. For further considerations, we use the notion of edge matrix of the tetrahedron formed by hyperbolic cosines of its edge lengths. We establish necessary and sufficient conditions for the existence of a tetrahedron in [Formula: see text]. Then we find relations between their dihedral angles and edge lengths in the form of a cosine rule. Finally, we obtain exact integral formula expressing the volume of a hyperbolic tetrahedron in terms of the edge lengths. The latter volume formula can be regarded as a new version of classical Sforza’s formula for the volume of a tetrahedron but in terms of the edge matrix instead of the Gram matrix.

Canonical Coordinates and Natural Equation for Lorentz Surfaces in R13

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

FUJITA DECOMPOSITION AND MASSEY PRODUCT FOR FIBERED VARIETIES

Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization

An inverse methodology is proposed to estimate a time-varying heat transfer coefficient (HTC) for a hollow cylinder with time-dependent boundary conditions of different kinds on inner and outer surfaces. The temperatures at both the inner surface and the interior domain are measured for the hollow cylinder, while the time history of HTC of the outer surface will be inversely determined. This work first expressed the unknown function of HTC in a general form with unknown coefficients, and then regarded these unknown coefficients as the estimated parameters which can be randomly searched and found by the self-learning particle swarm optimization (SLPSO) method. The objective function which wants to be minimized was found with the absolute errors between the measured and estimated temperatures at several measurement times. If the objective function converges toward the null, the inverse solution of the estimated HTC will be found eventually. From numerical experiments, when the function of HTC with exponential type is performed, the unknown coefficients of the HTC function can be accurately estimated. On the contrary, when the function of HTC with a general type is conducted, the unknown coefficients of HTC are poorly estimated. However, the estimated coefficients of an HTC function with the general type can be regarded as the equivalent coefficients for the real function of HTC.

Lang–Vojta conjecture over function fields for surfaces dominating $${{\mathbb {G}}}_m^2$$

We prove the nonsplit case of the Lang–Vojta conjecture over function fields for surfaces of log general type that are ramified covers of $${{\mathbb {G}}}_m^2$$ G m 2 . This extends the results of Corvaja and Zannier (J Differ Geom 93(3):355–377, 2013), where the conjecture was proved in the split case, and the results of Corvaja and Zannier (J Algebr Geom 17(2):295–333, 2008), Turchet (Trans Amer Math Soc 369(12):8537–8558, 2017) that were obtained in the case of the complement of a degree four and three component divisor in $${{\mathbb {P}}}^2$$ P 2 . We follow the strategy developed by Corvaja and Zannier and make explicit all the constants involved.