Symmetrical parallel RC Multiple-T Frequency Combinations

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

Multicast is an important communication method used in many emerging multimedia networking applications. Providing efficient security method to support the distribution of multimedia multicast is a challenging issue, since the group membership in such applications requires dynamic key generation and updation which takes more computation time. Moreover, the key must be sent securely to the group members. In this paper, we propose a new Key Distribution Protocol that provides more security and also reduces computation complexity. To achieve higher level of security, we use Euler's Totient Function φ(n) in the key distribution protocol. Therefore, it increases the key space while breaking the re-keying information. Two major operations in this scheme are join and leave operations for managing group memberships. An N-ary tree is used to reduce number of multiplications needed to perform the member leave operation. Using this tree, we reduce the computation time when compared with the existing key management schemes.

We investigate some equations involving the number of divisors d(n); the sum of divisors σ(n); Euler's totient function ϕ(n); the number of distinct prime factors ω(n); and the number of all prime factors (counted with multiplicity) Ω(n). The first part deals with equation f(xy) + f(xz) = f(yz). In the second part, as an analogy to x2 + y2 = z2, we study equation f(x2) + f(y2) = f(z2) and its generalization to higher degrees and more terms. We use just elementary methods and basic facts about the above functions and indicate why and how to discuss this topic in group study sessions or special maths classes of secondary schools in the framework of inquiry based learning.
 Subject Classification: 97F60, 11A25

This letter introduces a real valued summation known as Complex Conjugate Pair Sum (CCPS). The space spanned by CCPS and its one circular downshift is called {\em Complex Conjugate Subspace (CCS)}. For a given positive integer $N\geq3$, there exists $\frac{\varphi(N)}{2}$ CCPSs forming $\frac{\varphi(N)}{2}$ CCSs, where $\varphi(N)$ is the Euler's totient function. We prove that these CCSs are mutually orthogonal and their direct sum form a $\varphi(N)$ dimensional subspace $s_N$ of $\mathbb{C}^N$. We propose that any signal of finite length $N$ is represented as a linear combination of elements from a special basis of $s_d$, for each divisor $d$ of $N$. This defines a new transform named as Complex Conjugate Periodic Transform (CCPT). Later, we compared CCPT with DFT (Discrete Fourier Transform) and RPT (Ramanujan Periodic Transform). It is shown that, using CCPT we can estimate the period, hidden periods and frequency information of a signal. Whereas, RPT does not provide the frequency information. For a complex valued input signal, CCPT offers computational benefit over DFT. A CCPT dictionary based method is proposed to extract non-divisor period information.

On greatest common divisor matrices and their applications

In this paper, we indicate a method to obtain asymptotic evaluations for double sums from the asymptotic evaluations of a single sum. As applications, we show that all results in the recent paper [New extensions of some classical theorems in number theory, J. Number Theory133(13) (2013) 3771–3795] can be used to obtain some asymptotic evaluations for double sums. Further, we give asymptotic evaluations for double sums which involve the divisor and the Euler totient function.

In the present paper, by using the regular matrix given by Euler Totient function, we give a new paranormed sequence space l(Φ,p) and prove that the spaces l(Φ,p) and l(p) are linearly isomorphic. Also, we compute some dual spaces and the Schauder basis of this space.

The matrix formed using the Euler totient function together with its summatory function is employed to generate new sequence spaces. After establishing several features of these spaces, their duals are explicitly identified. Subsequent sections provide the characterization of a number of matrix transformations acting on the new domains. The paper concludes with an investigation of the compactness of operators associated with the aforementioned matrix on the space of null sequences.

The main purpose of this manuscript is to introduce Banach spaces $$c_0^\Phi $$ and $$c^\Phi $$ as the matrix domain of a regular matrix $$\Phi $$ derived by the Euler totient function. These spaces consist of $$\varphi $$-convergent to zero and $$\varphi $$-convergent sequences, respectively. After determining $$\alpha $$-, $$\beta $$- and $$\gamma $$-duals of these spaces, some matrix classes are characterized. Finally, using the Hausdorff measure of noncompactness, the characterization of some classes of compact operators on the space $$c_0^\Phi $$ is given.

Objective To investigate the effect of combinational hypothermia on growth and proliferation of temperature sensitive stem cells in hypoxia microenvironment after traumatic brain injury (TBI).Methods Temperature sensitive human umbilical cord mesenchymal stem cell line (tsUCMSCs) was generated and labeled by BrdU.In vivo study,severe traumatic brain injury mice model was established by fluid percussion.Then,PBS,UCMSCs or tsUCMSCs was implanted into injured region,and normothermia or mild hypothermia therapy (33℃) was adopt.The expression of BrdU in post -injury area was examined by immunohistochemistry.In vitro study,hypoxia or low serum in low metling agarose gel model was used,by which UCMSCs or tsUCMSCs in 33℃ or 37℃ was cultured.Cell number was counted with hematocytometer and PCNA or caspase 3 was measured with immunofluorecent staining.The level of hypoxia - inducible factor - 1 α ( HIF - 1 α) in vitro and in vivo was examined by western blot.Results The BrdU- positive cells in post - injury area at 7d of transplantation were significantly increased in combinational tsUCMSCs and hypothermia group.Compared with cells in 37℃,cells in 33℃ showed more cell number,more PCNA and less caspase 3 expression in hypoxia or low serum culture condition,among of which tsUCMSCs in 33℃ group showed the most enhancement of cell number and PCNA or the most decrease of caspase 3.The level of HIF - 1α was elevated both in vitro and in vivo model.Conclusion Combination of tsUCMSCs and hypothermia could increase the ability of grafted stem cells adapting to the hypoxia microenvironment and facilitate more survival of transplanted stem cells after TBI.This combinational therapy may provide a new and feasible strategy for TBI. Key words: Temperature sensitive stem cells; Hypothermia; Hypoxia; Microenvironment; HIF - 1α

Blood count is one of the most common medical laboratory tests performed today. It provides information on patient’s oxygen carrying capacity, immune system functionality, and the overall hemapoiesis process for disease diagnosis and drug side-effect monitoring. Traditionally blood count is performed either manually or by conventional automated blood analyzers. With the advance of microfabrication, on-chip blood count has become a target for miniaturization aiming at providing cost-effective, functional, capable point-of-care devices and systems that use less than 100 nL of blood sample and generate measurement results within minutes. The focus of this thesis is on investigation of erythrocyte/leukocyte counting and leukocyte differential, which are the two key components in blood count, in microdevices. Due to the large number ratio of erythrocytes to leukocytes and their property overlap, conventional automated blood analyzers accomplish blood count in several different channels by measuring different parameters. Similarly, in microdevices, it is desirable that erythrocytes and leukocytes can be separated before further analysis. Two types of hydrodynamic separation devices were investigated to continuously separate erythrocytes and leukocytes based on size. The principle of the device exploits the laminar flow in microdevices and design of streamlines which particles follow. Pillar-shaped devices with single geometrical design demonstrate a binary separation profile. With proper design, separation efficiency over 90% can be achieved. Channel-shaped devices, an improved design, can achieve similar separation efficiency with the added benefits of a smaller footprint, fewer requirements on flow control, and easier integration with downstream components. Erythrocyte and leukocyte count is accomplished with electrical impedance sensing, which is one of the most accurate ways to measure particle volume. The well-known problem of small double-layer capacitance inherent to micro impedance sensors is solved by two methods: platinum black electroplating on the electrode surface and inductor-induced resonance sensing. In the first method, platinum black is electroplated in situ on the electrode surfaces, which increases the effective surface area by two orders of magnitude and thus increases the double-layer capacitance significantly. The other innovative way, inductor-induced resonance sensing, nullifies the capacitive components in the system at the resonance frequency by connecting a parallel inductor to the system. In this way the sensitivity can be greatly improved and the optimal sensing frequency can be chosen from the inductance value. For both methods, polystyrene beads of different diameters were used for validation, while diluted blood samples and leukocyte-rich plasma were used to successfully demonstrate the feasibility. Two-part leukocyte differential is demonstrated in microflow cytometers with fluorescence sensing. Unlike methods used in conventional blood analyzers, undiluted blood samples are stained with nucleic acid stain acridine orange. Lymphocytes and granulocytes emit fluorescent light at different peak frequency after interaction with the dye due to the difference in cellular composition. Using the undiluted sample greatly minimizes sample preparation procedure, and reduces the overall measurement time, the reagent, and the waste volume. These benefits make it a practical method for implementation in microdevices. A throughput of one thousand leukocytes per second was demonstrated, which means the leukocyte differential could be accomplished in a couple of seconds.

Evaluating the mean square averages of the Dirichlet $L$-functions over Dirichlet characters $\chi$ of the same parity is an active problem in number theory. Here we explicitly evaluate $\sum_{\chi\text{ odd}}L(3,\chi)$ using certain trigonometric sums and Bernoulli polynomials and express the sum in terms of the Euler totient function $\phi$ and the Jordan totient function $J_s$.

The discrete logarithm problem with auxiliary inputs (DLP-wAI) is a special discrete logarithm problem. Cheon first proposed a novel algorithm to solve the discrete logarithm problem with auxiliary inputs. Given a cyclic group $${\mathbb {G}}=\langle P\rangle $$G=?P? of order p and some elements $$P,\alpha P,\alpha ^2 P,\ldots , \alpha ^d P\in {\mathbb {G}}$$P,?P,?2P,?,?dP?G, an attacker can recover $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? in the case of $$d|(p\pm 1)$$d|(p±1) with running time of $${\mathcal {O}}(\sqrt{(p\pm 1)/d}+d^i)$$O((p±1)/d+di) group operations by using $${\mathcal {O}}(\text {max}\{\sqrt{(p\pm 1)/d}, \sqrt{d}\})$$O(max{(p±1)/d,d}) storage ($$i=\frac{1}{2}$$i=12 or 1 for $$d|(p-1)$$d|(p-1) case or $$d|(p+1)$$d|(p+1) case, respectively). In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs (ECDLP-wAI). We show that if some points $$P,\alpha P,\alpha ^k P,\alpha ^{k^2} P,\alpha ^{k^3} P,\ldots ,\alpha ^{k^{\varphi (d)-1}}P\in {\mathbb {G}}$$P,?P,?kP,?k2P,?k3P,?,?k?(d)-1P?G and multiplicative cyclic group $$K=\langle k \rangle $$K=?k? are given, where d is a prime, $$\varphi (d)$$?(d) is the order of K and $$\varphi $$? is the Euler totient function, the secret key $$\alpha \in {\mathbb {Z}}_p^*$$??Zp? can be solved in $${\mathcal {O}}(\sqrt{(p-1)/d}+d)$$O((p-1)/d+d) group operations by using $${\mathcal {O}}(\sqrt{(p-1)/d})$$O((p-1)/d) storage.

A Von Neumann Algebra over the Adele Ring and the Euler Totient Function

This thesis investigates three distinct questions concerning the asymptotic average order of certain group theoretic objects. The three main chapters can be read independently. In chapter zero, we give a high level overview of our results. For a polynomial f [is an element of] [finite field]q[x] with nonzero constant term, define the order of f to be the order of x in the quotient ring [finite field]q[x]/(f). In Chapter 1, we consider the average order of several classes of polynomials as the degree n [right arrow] [infinity]. We show that the average order of an irreducible polynomial is asymptotically [theta]([phi](q^n - 1)) with [phi] the Euler Totient function. Building on work of Stong [32], we show that the asymptotic average order of both squarefree and nonsquarefree polynomials is q^n / n^(1+o(1)). As an application we relate this to the expected orbit size of a vector under iteration by a random invertible matrix. In chapter two, we count the number of compositions of n with k parts whose pairwise gcds are coprime to a given finite set P of primes. We obtain asymptotic results when n [right arrow] [infinity], showing that the number of such compositions is ([product](p[is an element of]P|(n)) h[k](p) + (-1)^(k-1)(k-1)/p^(k-1) (1-1/p)) x ([product](p[is an element of P[crossed pipe](n)) h[k](p) + (-1)^k(k-1)/p^k) n^(k-1)/(k-1)! + O(n^(k-2)) where h[k](p) := (1-1/p)^(k-1)(1+(k-1)/p). Building on work of Bubbolini, Luca and Spiga [6], we also consider the set of compositions whose parts are pairwise coprime. We use a geometric argument to strengthen their result when k = 3. We apply their results to obtain bounds on the average order of a permutation with a given number of cycles. Chapter three is joint work with Huseyin Acan, Charles Burnette, Sean Eberhard, and Eric Schmutz. Motivated by a question of Thibault Godin [17], we show that the probability P₁ that two random permutations in the symmetric group S[n] have the same order is asymptotically n^(-2+o(1)). We compare this to the probability P₂ that two permutations are conjugate, showing by construction that lim sup(n) P₁/P₂ = [infinity]. Additionally, we show by more direct methods that with probability O(log log n / log n) there exists some prime dividing the order of one of the permutations but not the other.