Abstract:
There are few approaches to the solution of a system of nonlinear differential equations in partial derivatives, for example $\cite{NK87} - \cite{EK98}$. In our paper we propose an approach that was used to solve the Navier-Stokes equations in three dimensional space. This solution is described in details in article "Existence, uniqueness and smoothness of solution for 3D Navier-Stokes equations with any smooth initial velocity" $\cite{TT12}$. The authors expect that it can be successfully applied to other systems of nonlinear differential equations in partial derivatives.

Abstract:
usually, primary hyperparathyroidism is the first endocrinopathy to be diagnosed in patients with multiple endocrine neoplasia type 1, and is also the most common one. the timing of the surgery and strategy in multiple endocrine neoplasia type 1/hyperparathyroidism are still under debate. the aims of surgery are to: 1) correct hypercalcemia, thus preventing persistent or recurrent hyperparathyroidism; 2) avoid persistent hypoparathyroidism; and 3) facilitate the surgical treatment of possible recurrences. currently, two types of surgical approach are indicated: 1) subtotal parathyroidectomy with removal of at least 3-3 k glands; and 2) total parathyroidectomy with grafting of autologous parathyroid tissue. transcervical thymectomy must be performed with both of these procedures. unsuccessful surgical treatment of hyperparathyroidism is more frequently observed in multiple endocrine neoplasia type 1 than in sporadic hyperparathyroidism. the recurrence rate is strongly influenced by: 1) the lack of a pre-operative multiple endocrine neoplasia type 1 diagnosis; 2) the surgeon's experience; 3) the timing of surgery; 4) the possibility of performing intra-operative confirmation (histologic examination, rapid parathyroid hormone assay) of the curative potential of the surgical procedure; and, 5) the surgical strategy. persistent hyperparathyroidism seems to be more frequent after subtotal parathyroidectomy than after total parathyroidectomy with autologous graft of parathyroid tissue. conversely, recurrent hyperparathyroidism has a similar frequency in the two surgical strategies. to plan further operations, it is very helpful to know all the available data about previous surgery and to undertake accurate identification of the site of recurrence.

Abstract:
The purpose of the critical path method (CPM) is to identify the critical activities in the critical path of an activity network. In the real world for many projects we have to use human judgment for estimating the duration of activities. However, the unknowns or vagueness about the time duration for activities in network planning, has led to the development of fuzzy CPM. A way to deal with this imprecise data is to employ the concept of fuzziness, where the vague activity times can be represented by fuzzy sets. In this paper a new method based on fuzzy theory is developed to solve the project scheduling problem under fuzzy environment. Assuming that the duration of activities are triangular fuzzy numbers, in this method we compute total float time of each activity and fuzzy criticalpath without computing forward and backward pass calculations. Through a numerical example, calculation steps in this method and the results are illustrated. Compare with other fuzzy critical method the proposed method is simple, fast and effective to find total float time of each activity and fuzzy critical path in a fuzzy project network.

Abstract:
Partial
formalization, which involves the development of deductive connections among statements,
can be used to examine assumptions, definitions and related methodologies that
are used in science. This approach has been applied to the study of nucleic
acids recovered from natural microbial assemblages (NMA) by the use of bulk
extraction. Six pools of bulk-extractable nucleic acids (BENA) are suggested to
be present in a NMA: (pool 1) inactive microbes (abiotic-limited); (pool 2) inactive
microbes (abiotic permissive, biotic-limited); (pool 3) dormant microbes
(abiotic permissive, biotic-limited, but can become biotic permissive); (pool
4) in situ active microbes (the
microbial community); (pool 5) viruses (virocells/virions/cryptic viral genomes);
and (pool 6) extracellular nucleic acids including extracellular DNA (eDNA).
Definitions for cells, the microbial community (in situ active cells), the rare biosphere, dormant cells (the
microbial seed bank), viruses (virocells/virions/cryptic viral genomic), and
diversity are presented, together with methodology suggested to allow their
study. The word diversity will require at least 4 definitions, each involving a
different methodology. These suggested definitions and methodologies should
make it possible to make further advances in bulk extraction-based molecular
microbial ecology.

Abstract:
We give an alternative proof of the optimality of the solution for the deterministic EPQ with partial backordering (EPQ-PBO) [Omega, vol. 37, no. 3, pp. 624–636, 2009]. Our proof is based on the mathematical programming theory. We also demonstrate the determination of the optimal decision policy through solving the corresponding mathematical programming problem. We indicate that the same approach can be used within other inventory models with partial backordering, and we consider additional models. 1. Introduction A basic inventory control model is the economic order quantity (EOQ) model known by its simplicity and its restrictive modeling assumptions. Managing the inventory of a single item the purpose of the EOQ model is to determine how much to purchase (order quantity) and when to place the order (the reorder point). By relaxing the assumption that shortages are not allowed and allowing stockouts with partial backordering, the EOQ will extend to EOQ with partial backordering (EOQ-PBO). But when the assumption of instantaneous replenishment is replaced with the assumption that the replenishment order is received at a constant finite rate over time, EOQ is extended to the economic production quantity (EPQ) model, and by allowing stockouts with partial backordering it will extend to EPQ with partial backordering (EPQ-PBO). There are many papers that propose an inventory model with partial backordering; a very few of them are [1–7]. For a survey of the deterministic models for the EOQ and EPQ with partial backordering, see Penticio’s and Drake’s paper [8]. Pentico et al. [5] proposed EPQ-PBO model, whose main characteristics are that the derived equations are more like those for the classical EPQ model and it gives the optimal solution in a closed form that helps in understanding the behavior of the inventory system. Let us briefly give the notations from [5] that are used in the total cost function and in the solution of the Pentico et al.'s EPQ-PBO as well. They are？ : demand per year,？ : production rate per year if constantly producing,？ : the fixed cost of placing and receiving an order,？ : the cost to hold a unit in inventory for a year,？ : the cost to keep a unit backordered for a year,？ : the cost for a lost sale,？ : the backordering rate,？ : the length of an order cycle (a decision variable),？ : the fill rate (a decision variable). The derived total averaged cost per year for the Pentico et al.’s EPQ-PBO model, according to [5], that has to be minimized is where , , and . Setting the first partial derivatives of equal to zero, Pentico et al. [5]

Abstract:
Ambiguity is shown in the context of the differential calculus of several variables and with the help of the language of category theory, a way to solve it in its most general form is offered. It is also shown that this new definition is related to other well-known definitions in the literature.

Abstract:
The total derivatives in the gravitational action are usually disregarded as non-producing any non-trivial dynamics. In the context of the gravitational entropy, within Wald's approach, these terms are considered irrelevant as non-contributing to the entropy. On the other hand, the total derivatives are usually present in the trace anomaly in dimensions higher than 2. As the trace anomaly is related to the logarithmic term in the entanglement entropy it is natural to ask whether the total derivatives make any essential contribution to the entropy or they can be totally ignored. In this note we analyze this question for some particular examples of total derivatives. Rather surprisingly, in all cases that we consider the total derivatives produce non-trivial contributions to the entropy. Some of them are non-vanishing even if the extrinsic curvature of the surface is zero. We suggest that this may explain the earlier observed discrepancy between the holographic entanglement entropy and Wald's entropy.

Abstract:
The natural forms of the Leibniz rule for the $k$th derivative of a product and of Fa\`a di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,...,x_k$ are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular, if all variables are distinct then all coefficients are 1). The computation of the multiplicities in this generalization of Fa\`a di Bruno's formula is a combinatorial enumeration problem that, although completely elementary, seems to have been neglected. We apply the results to cumulants of probability distributions.

Abstract:
Fractional variational approach has gained much attention in recent years. There are famous fractional derivatives such as Caputo derivative, Riesz derivative and Riemann-Liouville derivative. Several versions of fractional variational principles are proposed. However, it becomes difficult to apply the existing fractional variational theories to fractional differential models, due to the definitions of fractional variational derivatives which not only contain the left fractional derivatives but also appear right ones. In this paper, a new definition of fractional variational derivative is introduced by using a modified Riemann-Liouville derivative and the fractional Euler-Lagrange principle is established for fractional partial differential equations.

Abstract:
A new integral identity for functions with continuous second partial derivatives is derived. It is shown that the value of any function f(r,t) at position r and time t is completely determined by its previous values at all other locations r' and retarded times t'