The Minimum Numbers for Certain Positive Operators

Sol gel transition in aqueous DNA solutions was probed to determine the gelation time tgel and temperature Tgel It was remarkable to observe that Tgel linearly increased from to deg C and tgellinearly decreased from to s with increase in DNA concentration The hydrogels were studied by using small angle neutron scattering SANS DNA le w v to determine the underlying hierarchy of length scales From structure factor profile analysis three distinct signatures were obtained a nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Radius of gyration Rg asymp plusmn nm which assigned a length of nm to DNA strand b nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Geometry of scattering moiety defined by the value of alpha asymp plusmn indicating Gaussian chain behavior c nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Correlation length xi increased from to nm and the inter chain spacing d decreased from to nm with increase in DNA concentration Physical networks were formed for concentration c ge w v and the system involved at least five identifiable length scales A revisit has shown that hierarchical structure of DNA hydrogel owes its origin to considerable self organization at the molecular level dependent on biopolymer concentration

We generalize several inequalities involving powers of the numerical radius for the product of two operators acting on a Hilbert space. Moreover, we give a Jensen operator inequality for strongly convex functions. As a corollary, we improve the operator Holder-McCarthy inequality under suitable conditions. In particular, we prove that if $f:Jrightarrow mathbb{R}$ is strongly convex with modulus $c$ and differentiable on ${rm int}(J)$ whose derivative is continuous on ${rm int}(J)$ and if $T$ is a self-adjoint operator on the Hilbert space $cal{H}$ with $sigma(T)subset {rm int}(J)$, then $$langle T^2x,xrangle-langle Tx,xrangle^2leq dfrac{1}{2c}(langle f'(T)Tx,xrangle -langle Tx,xrangle langle f'(T)x,xrangle)$$ for each $xincal{H}$, with $|x|=1$.

Depression being a state of sadness may be defined as a psychoneurotic disorder characterised by mental and functional activity sadness reduction in activity difficulty in thinking loss of concentration perturbations in appetite sleeping and feelings of dejection hopelessness and generation of suicidal tendencies It is a common and recurrent disorder causing significant morbidity and mortality worldwide The antidepressant compounds used against depression are reported to be used also for treating pain anxiety syndromes etc They have been grouped in five different categories such as nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp I nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Tricyclic antidepressants TCAs nbsp nbsp nbsp nbsp nbsp nbsp II nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Selective serotonin reuptake inhibitors SSRIs nbsp nbsp nbsp nbsp III nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Monoamine oxidase inhibitors MAOIs nbsp nbsp nbsp nbsp IV nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Serotonin norepinephrine reuptake inhibitor SNRI and nbsp nbsp nbsp nbsp nbsp nbsp V nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Non TCA antidepressants based on their mode of action Most of the antidepressants have been reported to possess adverse effects on the health of users The present review article focuses on an updated current of antidepressants their mechanism of actions pathophysiology of these compounds their side effects and the strategies to combat the drug induced toxicity An account of phytochemicals found to be acting as antidepressant is also included

We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternionic compact normal operators. Finally, we show that every quaternionic compact operator is norm attaining and prove the Lindenstrauss theorem on norm attaining operators, namely, the set of all norm attaining quaternionic operators is norm dense in the space of all bounded quaternionic operators defined between two quaternionic Hilbert spaces.

Let B(H) denote the algebra of bounded linear operators on a complex Hilbert space H. The (classical) numerical range of T ∈ B(H) is the set
\n W(T) = {〈T x; x〉: x ∈ H; ‖x‖ = 1}
\n
\nWriting T= T_1 + iT_2 for self-adjoint T_1, T_2 ∈ B(H), W(T) can be identified with the set 
\n{(〈T_1 x, x〉,〈T_2 x, x〉) : x ∈ H, ‖x‖ = 1}.
\n
\nThis leads to the notion of the joint numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈ 〖B(H)〗^n. It is defined by 
\nW(T) = {(〈T_1 x, x〉,〈T_2 x, x〉, …, 〈T_n x, x〉) : x ∈ H, ‖x‖ = 1}.
\nThe joint numerical range has been studied extensively in order to understand the
\njoint behaviour of operators.
\n
\nLet K(H) be the set of all compact operators on a Hilbert space H. The essential numerical range of T ∈ B(H) is defined by 
\nW_(e ) (T)=∩{W(T+K) :K∈K(H) }.
\n
\nThe joint essential numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈〖 B(H)〗^n is defined analogously by 
\nW_(e ) (T)=∩{ /W(T+K) :K∈〖K(H)〗^n }.
\nThese notions have been generalized to operators on a Banach space. In Chapter 1 of this thesis, the joint spatial essential numerical range were introduced. Also the notions of the joint algebraic numerical range V(T) and the joint algebraic essential numerical range Ve(T) were reviewed. Basic properties of these sets were given.
\n
\nIn 2010, Müller proved that each n-tuple of operators T on a separable Hilbert space has a compact perturbation T + K so that We(T) = W(T + K). In Chapter 2, it was shown that any n-tuple T of operators on lp has a compact perturbation T +K so that Ve(T) = V (T +K), provided that Ve(T) has an interior point. A key step was to find for each n-tuple of operators on lp a compact perturbation and a sequence of finite-dimensional subspaces with respect to which it is block 3 diagonal. This idea was inspired by a similar construction of Chui, Legg, Smith and Ward in 1979.
\n
\nLet H and L be separable Hilbert spaces and consider the operator D_AB=A⨂I_L⨂B on the tensor product space H ⨂▒L. In 1987 Magajna proved that W_(e ) (D_AB )=co[W_(e ) (A)- /(W(B)))∪/W(A) - W_(e ) (B))] by considering quasidiagonal operators. An alternative proof of the equality was given in Chapter 3 using block 3 diagonal operators. 
\nThe maximal numerical range and the essential maximal numerical range of T ∈ B(H) were introduced by Stampi in 1970 and Fong in 1979 respectively. In 1993, Khan extended the notions to the joint essential maximal numerical range.
\nHowever the set may be empty for some T ∈ B(H). In Chapter 4, the kth joint essential maximal numerical range, spatial maximal numerical range and algebraic numerical range were introduced. It was shown that kth joint essential maximal numerical range is non-empty and convex. Also, it was shown that the kth joint algebraic maximal numerical range is the convex hull of the kth joint spatial maximal numerical range. This extends the corresponding result of Fong.

Aim: To evaluate neurodevelopmental outcome of premature infants born with a gestational age less than 32 weeks and or very low birth weight lt;1500gr at the adjusted age of three years Material and Method: nbsp; In this study neurodevelopmental characteristics Denver II Developmental screening Test DDST II nbsp; and neurological sequelaes Amiel Tison were examined retrospectively Results: Seventeen hundred infants were evaluated at a mean age of 35 8 plusmn;2 3 months 27 2 nbsp; There were 57 female and 60 male patients and mean birth weight and gestational age were 1271 22 09 gr and 31 2 1 weeks respectively nbsp; Most important acute morbidities included respiratory distress syndrome nbsp; nbsp; RDS nbsp; 21 nbsp; bronchopulmonary dysplasia BPD 8 nbsp; severe retinopathy of prematurity nbsp; ROP nbsp; 5 nbsp; sepsis nbsp; 21 severe intraventricular hemorrhage IVH nbsp; 7 and an Apgar score of lt;6 24 nbsp; 9 4 nbsp; of the patients had major neurological sequelae; ten cases had cerebral palsy five had hydrocephalus and two had blindness Denver II Developmental Screening Test was abnormal in 23 nbsp; cases Conclusions: Our neurodevelopmental assessment results were similar to the literature One of the most important risk factors in neurodevelopmental retardation was transport Turk Arch Ped 2011; 46: 199 206

The purpose of this study was to determine the psychometric qualities of a Turkish adaptation of The Five Factor Wellness Inventory Teen Version 5F Wel T; Myers amp; Sweeney 2004 This study was conducted at the suggestion and with the encouragement of Drs Myers and Sweeney The participants in this investigation were 328 6th to 9th grade students nbsp; Initial factor analysis failed to demonstrate a good correspondence with the originally identified factors reported by Myers and Sweeney nbsp; Ultimately and with the permission of the original authors a highly modified version containing 41 items was developed and it was this instrument that was administered to the study sample nbsp; Exploratory factor analysis was used to establish a validity structure that resulted in the identification of 5 subscales nbsp; Confirmatory factor analysis demonstrated a good fit with the revised model of 41 items and 5 subscales The original 5F Wel T was composed of 5 subscales; Creative Coping Social Essential and Physical The resulting Turkish adaptation generated a quite different set of subscales that were identified as Positive Self Esteem Social Life Control and Meaning of Life Coping and Problem Solving and Physical and Emotional Health nbsp; Stability and internal consistency reliabilities were determined during a subsequent study n=131 with 6th 9th grade students nbsp; A nbsp; Chronbach rsquo;s alpha coefficient for Total Wellness was 86 subscale alphas ranging from 52 62 nbsp; Test Retest reliability with a 21 day interval resulted in a reliability coefficient of 84 for the total scale and subscale ranging from 62 79 nbsp; Although this study sought to adapt the 5F Wel T scale for use with a Turkish population factor analysis revealed a substantially different factor structure nbsp; These findings resulted in a five factor model with different scale labels and the rational for this was discussed nbsp; This initial investigation suggests that this revised scale may have useful application for use with Turkish youth nbsp; Additional investigation with larger and more diverse samples was suggested nbsp;

Frıedman Okul Müdürleri Tükenmişlik Ölçeği: Türkçeye Uyarlaması Gecerlik ve Güvenirlik Çalışmaları

Let denote the C* –algebra of all bounded linear operators on a Hilbert space . Among other results, we show that if are nonzero, then where denote the operator norm and the numerical radius, respectively. This provides a significant improvement and reverse of the known bound To achieve this, we treat the triangle inequality and a well-known identity for the numerical radius of an operator matrix. Using the same technique, we then present an inequality for the spectral radius of the sum of two operators. The results obtained are compared with those of the existing literature. Many other results are shown as refinements or reverses of other known facts about the numerical radius and the usual operator norm.

Aim: Epidemic conjunctivitis caused by adenovirus frequently spreads by direct contact during ophtalmologic examination This report is written nbsp; to share our nbsp; experience of an adenovirus epidemic in our neonatal intensive care unit Material and Method: The detailed medical and infection control committee records of the newborns cared in Ege University neonatal intensive care unit and the personnel together in the period between 09 14 2009 10 17 2009 were used in this report Conjunctiva nasopharynx and stool samples were collected from 15 patients hospitalized in the service 9 girls 6 boys and from 25 personnel Weekly samples were collected from infants during the epidemic Direct fluorescence antibody test and nbsp; ldquo;shell vial rdquo; cell culture method were conducted synchronously in all samples Adenovirus conjunctivitis was diagnosed according to clinical evidence and or detection of adenovirus in direct fluorescence antibody test and cell cultures SPSS 13 0 statistics programme was used for analysis Results: Severe conjunctivitis occurred within 3 days of fundus examination for screening of retinopathy of prematurity in 5 newborns who were examined in the same day in the ophtalmologist who examined the infants and in 2 newborns who were not examined nbsp; Ten of 15 infants who were cared in the neonatal intensive care unit during this epidemic had conjunctivitis Five of these patients had also gastroenteritis Five nurses had conjunctivitis and had a leave for therapy and isolation Adenovrus was detected in direct fluorescence antibody test and cell culture in one nurse Adenovirus antigen was detected in 3 conjunctival swabs and 1 tracheal aspirate in symptomatic nbsp; infants nbsp; The diagnosis was confirmed with tissue cultures In addition to standard precautions droplet precautions were also taken with the recommendation of Infection Control Committee Conclusions: The meticulous infection control measures including nbsp; closing the unit to new admissions detailed cleaning and decontamination of neonatal intensive care unit prevented the epidemic to spread further and limited the duration of the epidemic A second epidemic did not occur and all patients recovered Turk Arch Ped 2011; 46: 194 8

Sustained/controlled release is designed to release a drug at a predetermined rate in which a constant drug concentration maintained for a specific period with minimal side effects and drug consumption.This can be achieved via various formulations, including liposomes, solid lipid microspheres and drug-polymer conjugates.The role of polymer in drug delivery is to provide weight, consistency and volume for the right and proper administration of the drug.It provides synergistic functions, stability, drug release, targeting, enhancing its bioavailability and patient acceptability.Beeswax, a natural polymer is considered as a base biomaterial for preparation of microspheres and loading with Capecitabine, an anti-cancer drug for drug delivery.The Beeswax microspheres via coacervation and phase separation method were in the mean diameter of 3441nm and SEM micrograph reveals the distribution of beeswax microsphere in Poly Lactic Glycolic Acid (PLGA) medium uniformly.The interaction between beeswax, PLGA and drug were interpreted by FTIR and confirming the drugpolymer interactions.In-vitro drug release of Beeswax microspheres confirms a linear behaviour with 26% of the drug released in the dissolution medium at pH 6.8.The cell culture confirms the good biocompatibility of beeswax microspheres concluding the beeswax could be used to fabricate the drug delivery and targeting vehicles in cancer therapy.

We develop several $\ell ^p$ -operator norm inequalities for $k\times k$ block matrices defined on the $\ell ^p$ -sum of Banach spaces. Using these inequalities, we obtain p -numerical radius and spectral radius bounds for $k\times k$ block matrices. We deduce a p -numerical radius bound for the Kronecker product $A\otimes B$ , where $A\in {M}_k(\mathbb {C})$ is a $k\times k$ complex matrix and $B\in \mathcal {L}(\mathbb {H})$ is a bounded linear operator on a complex Hilbert space $\mathbb {H}$ . This improves and extends Holbrook’s bound $w(A\otimes B)\leq w(A)\|B\|.$ If $\|A\|_{\ell ^p}$ and $w_p(A)$ denote the $\ell ^p$ -operator norm and p -numerical radius of $A\in {M}_k(\mathbb {C})$ , respectively, then it is shown that $$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$ where $\mu _p(A)$ is a positive real number that involves the $\ell ^p$ -operator norms of the Cartesian decomposition of A . In addition, a complete characterization of the equality case $\frac {1}{2}\|A\|_p= w_p(A)$ is given.

The numerical range W(T) of a linear bounded operator T on a Hilbert space is the collection of complex numbers of the form (Tx,x) with x ranging over the unit vectors in the Hilbert space. In this study, firstly, the connection between numerical range of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the first investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical range of the operator A is in the form W(A)= co (⋃_{n=1}^{∞}W(A_{n}) ), where co (Ω), Ω⊂C denotes the convex hull of Ω. The numerical radius w(T) of a linear bounded operator T on a Hilbert space is a number which is given by w(T)=sup {|λ| : λ∈W(T) }. In this study, secondly, the connection between numerical radius of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the second investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then numerical radius of the operator A is in the following form w(A)= sup w(A_{n}). The Crawford number c(T) of a linear bounded operator T on a Hilbert space is given by c(T)=inf {|λ| : λ∈W(T) }. In this study, thirdly, the connection between Crawford number of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the third investigation the result obtained is as follows: If for any n≥1, H_{n} is a Hilbert space, A_{n}∈L(H_{n}), Re(A_{n})≥0 (≤0), H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n}, A ∈L(H), then Crawford number of the operator A is in the following form c(A)=inf c(A_{n}). Moreover, It is known that for a linear bounded accretive operator T if W(T)⊂{z ∈C:|argz| < ϕ} for any ϕ∈[0, π/ 2), then it is called a sectorial operator with vertex γ=0 and semi-angle ϕ. In this case, T ∈S_{ϕ}(H). In this study, finally, the connection between sectoriality of direct sum of operators in the direct sum of Hilbert spaces and their coordinate operators has been investigated. At the end of the last investigation the result obtained is as follows: If for any n≥1,H_{n} is a Hilbert space, A_{n}∈L(H_{n}),A_{n}∈S_{ϕ_{n}}(H_{n}),H=⊕_{n=1}^{∞}H_{n} and A=⊕_{n=1}^{∞}A_{n},A∈L(H), then for some ϕ∈[0,π/2), A∈S_{ϕ}(H) the necessary and sufficient condition is sup ϕ_{n}<ϕ. At the end of this study, an example including all these connections has been given.

Let $mathcal H$ be a complex Hilbert space and let $B(mathcal H)$ be the algebra of all bounded linear operators on $mathcal H$. Let $Tin B(mathcal H)$.In this paper, we determine the norm of the inner Jordan $*$-derivation $Delta_T:Xmapsto TX-X^*T$ acting on the Banach algebra $B(mathcal{H})$. More precisely, we show that $$big{|}Delta_Tbig{|}geq 2sup_{lambdain W_0(T)}|{rm Im}(lambda)|$$in which $W_0(T)$ is the maximal numerical range of operator $T$.

This work is devoted to rigorous results about the adiabatic theorem of quantum mechanics. This theorem deals with the time dependent Schrodinger equation when the hamiltonian is a slowly varying function of time, characterizing the so-called adiabatic regime. Mathematically, the adiabatic theorem describes the solutions ψe(t) in an Hilbert space H of the rescaled Schrodinger equation in the limit e→ 0. Suppose the hamiltonian possesses for any time t two spectral projectors, P1(t) and P2(t), which are spectrally isolated. Let us consider a normalized solution which belongs at time t = - ∞ to the spectral subspace P1(-∞)H, i.e. which satisfies the boundary condition Then the transition probability P21(e) from P1(-∞)H to P2(+∞)H between the times -∞ and +∞ is defined by The adiabatic theorem states that P21(e) tends to zero in the limit e→ 0. Our main concern is the study of the decay of P21(e) as e→ 0. We fist show that if H(t) is an analytic unbounded operator then P21(e) decays exponentially fast to zero in the adiabaticity parameter e: for some positive constant τ. Then we turn to two-level systems for which we have a finer control on the behaviour of P21(e) as e→ 0. Indeed, in the generic case we give an explicit asymptotic formula for the transition probability P21(e) which reads The prefactor exp {2Imθ1} is of geometrical nature and the exponential decay rate 21m ∫γ e1(z)dz is computed by means of the integral of the analytic continuation of the eigenvalue e1 (t) along a suitable path γ in the complex plane. This expression constitutes a generalization of the so-called Dykhne formula which does not contain the geometric prefactor. Moreover, we improve this result and compute the leading term of P21(e) up to a correction of order Ο(eq) for any q, instead of Ο(e). This result shows as well that the logarithm of P21(e) admits an asymptotic power series in e up to any order. Finally we push the estimates to get the leading term of P21(e) up to a correction of order Ο(e-τ/c). We consider also cases where the 2 X 2 hamiltonian possesses some symmetry, as the time reversal symmetry for example. In these situations, the leading term of P21(e) changes qualitatively since it is given by a decreasing exponential multiplying an oscillatory function of 1 /e. Then we come back to general systems driven by unbounded hamiltonians and study the case where P1(t) and P2(t) are both one-dimensional. These projectors are thus associated with non-degenerate instantaneous eigenvalues el(t) and e2(t) of the hamiltonian H(t). We prove that, in this case too, an asymptotic formula for P21(e) exists, provided the two levels el(t) and e2(t) are sufficiently isolated in the spectrum of H(t). This formula turns out to be the same as the formula valid for two-level systems. Finally, we consider the situation frequently encountered in applications where the two levels el(t) and e2(t) display an avoided crossing during the evolution. For an avoided crossing located at time t = 0, this means that the levels behave as where δ ≪ 1. As a consequence, the gap between el(t) and e2(t) is minimum for where its value is In this case, we show that for e and δ small enough, the above formula for P21(e) reduces to the well-known Landau-Zener formula When c = 0 we recover the familiar Landau-Zener formula. This gives a rigorous mathematical status to a formula which has been widely applied for years in a variety of circumstances.