Abstract:
Toll-like receptor (TLR) signaling pathways are strictly coordinated by several mechanisms to regulate adequate innate immune responses. Recent lines of evidence indicate that the suppressor of cytokine signaling (SOCS) family proteins, originally identified as negative-feedback regulators in cytokine signaling, are involved in the regulation of TLR-mediated immune responses. SOCS1, a member of SOCS family, is strongly induced upon TLR stimulation. Cells lacking SOCS1 are hyperresponsive to TLR stimulation. Thus, SOCS1 is an important regulator for both cytokine and TLR-induced responses. As an immune organ, the liver contains various types of immune cells such as T cells, NK cells, NKT cells, and Kupffer cells and is continuously challenged with gut-derived bacterial and dietary antigens. SOCS1 may be implicated in pathophysiology of the liver. The studies using SOCS1-deficient mice revealed that endogenous SOCS1 is critical for the prevention of liver diseases such as hepatitis, cirrhosis, and cancers. Recent studies on humans suggest that SOCS1 is involved in the development of various liver disorders in humans. Thus, SOCS1 and other SOCS proteins are potential targets for the therapy of human liver diseases. 1. Introduction Proper and coordinated activation of immune signal pathways is required for immune responses, including eradication of invading pathogens. Toll-like receptor (TLR)- and cytokine receptor-mediated signaling are involved in innate and subsequent adoptive immunity. Aberrant and/or sustained activation of immune signal pathways may result in serious disorders such as septic shock, autoimmunity, and cancer. Thus, immune signals must be tightly regulated for preventing overactivated immune responses. A number of regulatory mechanisms on immune signaling pathways have been reported. A family named suppressor of cytokine signaling (SOCS) represents a negative regulator for various cytokine signaling (Table 1) [1]. SOCS proteins play important roles in maintaining organ homeostasis by preventing the harmful cytokine responses in various organs [2]. In this paper, we will focus on SOCS1, a member of SOCS family, which plays a key role in the negative regulation of both cytokine receptor- and TLR-mediated signaling. We will further discuss the importance of SOCS1 in the pathogenesis of liver diseases. Table 1: Inducing factors of SOCS family proteins and suppressed signaling by SOCS family proteins (see [ 1, 3, 4]). 2. Regulation of Immune Signal Pathways by Suppressor of Cytokine Signaling (SOCS) 2.1. Inhibition of Cytokine Signaling by

Abstract:
We present another expression to regularize the Euler product representation of the Riemann zeta function. % in this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a new one as the limit of the product of some terms derived from the usual Euler product. We also refer to the relation between the Bernoulli number and $P(z)$, which is an infinite summation of a $z$ power of the inverse primes. When we apply the same technique to the $L$-function associated to an elliptic curve, we can evaluate the power of the Taylor expansion for the function even in the critical strip, which is deeply related to problems known as the Birch and Swinnerton-Dyer conjecture and the Beilinson conjecture.

Abstract:
We consider a variant expression to regularize the Euler product representation of the zeta functions, where we mainly apply to that of the Riemann zeta function in this paper. The regularization itself is identical to that of the zeta function of the summation expression, but the non-use of the M\"oebius function enable us to confirm a finite behavior of residual terms which means an absence of zeros except for the critical line. Same technique can be applied to the $L$-function associated to the elliptic curve, and we can deal with the Taylor expansion at the pole in critical strip which is deeply related to the Birch--Swinnerton-Dyer conjecture.

Abstract:
We propose a regularization technique and apply it to the Euler product of zeta functions, mainly of the Riemann zeta function, to make unknown some clear. In this paper that is the first part of the trilogy, we try to demonstrate the Riemann hypotheses by this regularization technique and show conditions to realize them. In part two, we will focus on zeros of the Riemann zeta function and the nature of prime numbers in order to prepare ourselves for physical applications in the third part.

Abstract:
We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note.

Abstract:
We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random distribution of solutions for the quadratic residue problem.

Abstract:
We have proposed a regularization technique and apply it to the Euler product of zeta functions in the part one. In this paper that is the second part of the trilogy, we give another evidence to demonstrate the Riemann hypotheses by using the approximate functional equation. Some other results on the critical line are also presented using the relations between the Euler product and the deformed summation representions in the critical strip. In part three, we will focus on physical applications using these outcomes.

Abstract:
We have dealt with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and showed some evidence to indicate the hypothesis. We briefly review the essential points and we also define a finite ratio in the functional equation from divergent quantities in this note.

Abstract:
We give an estimation of the existence density for the $2d$ different primes by using a new and simple algorithm for getting the $2d$ different primes. The algorithm is a kind of the sieve method, but the remainders are the central numbers between the $2d$ different primes. We may conclude that there exist infinitely many $2d$ different primes including the twin primes in case of $d=1$ because we can give the lower bounds of the existence density for the $2d$ different primes in this algorithm. We also discuss the Hardy-Littlewood conjecture and the Sophie Germain primes.

Abstract:
We deal with the Hill's spherical vortex, which is an exact solution to the Euler equation, and manage the solution to satisfy the incompressible Navier-Stokes(INS) equations with a viscous term. Once we get a viscous solution to the INS equations, we will be able to analyze the flows with discontinuities in vorticity. In the same procedure, we also present a time developing exact solution to the INS equations, which has a rotation on the axis besides the Hill's vortex.