The 𝐿₂-system of a unimodular group. I

The unbounded Self-adjoint operators that strongly commute on a common dense subset of their domain commute pointwise. When the operators commute pointwise on the same dense subset, there is to guarantee that they will commute strongly. By imposing some conditions, we on the operators as well as the underlying space, we get pointwise commuting unbounded operators that commute strongly. This article shows that by suitably selecting two unbounded positive Self-adjoint operators with compact inverses we get a set of pointwise commuting self-adjoint operators that commute on common core. then prove that it strongly commutes on the same subspace. Keywords: Unbounded operators, Self-adjoint operators, Commutative operators DOI: 10.7176/MTM/10-8-03 Publication date: December 31 st 2020

In a recent article [K. H. Hofmann and F. G. Russo, ‘The probability that$x$and$y$commute in a compact group’,Math. Proc. Cambridge Phil Soc., to appear] we calculated for a compact group$G$the probability$d(G)$that two randomly selected elements$x, y\in G$satisfy$xy=yx$, and we discussed the remarkable consequences on the structure of$G$which follow from the assumption that$d(G)$is positive. In this note we consider two natural numbers$m$and$n$and the probability$d_{m,n}(G)$that for two randomly selected elements$x, y\in G$the relation$x^my^n=y^nx^m$holds. The situation is more complicated whenever$n,m\gt 1$. If$G$is a compact Lie group and if its identity component$G_0$is abelian, then it follows readily that$d_{m,n}(G)$is positive. We show here that the following condition suffices for the converse to hold in an arbitrary compact group$G$: for any nonopen closed subgroup$H$of$G$, the sets$\{g\in G: g^k\in H\}$for both$k=m$and$k=n$have Haar measure$0$. Indeed, we show that if a compact group$G$satisfies this condition and if$d_{m,n}(G)\gt 0$, then the identity component of$G$is abelian.

Because of their flexibility, trees and other plants deform with great amplitude (reconfigure) when subjected to fluid flow. Hence the drag they encounter does not grow with the square of the flow velocity as it would on a classical bluff body, but rather in a less pronounced way. The reconfiguration of actual plants has been studied abundantly in wind tunnels and hydraulic canals, and recently a theoretical understanding of reconfiguration has been brought by combining modelling and experimentation on simple systems such as filaments and flat plates. These simple systems have a significant difference with actual plants in the fact that they are not porous: fluid only flows around them, not through them. We present experimentation and modelling of the reconfiguration of a simple poroelastic system. Proper scaling of the drag and the fluid loading allows comparing the reconfiguration regimes of porous systems to those of simple systems. It is found that in the large reconfiguration regime, the scaling of the drag with flow velocity is independent of the porosity for a range of parameter values.

Bases play a prominent role in the analysis of vector spaces, as well in the finite-dimensional as in the infinite-dimensional case. The idea is the same in both cases, namely to consider a family of elements such that all vectors in the considered space can be expressed in a unique way as a linear combination of these elements. In the infinite-dimensional case the situation is complicated: we are forced to work with infinite series, and different concepts of a basis are possible, depending on how we want the series to converge. For example, are we asking for the series to converge with respect to a fixed order of the elements (conditional convergence) or do we want it to converge regardless of how the elements are ordered (unconditional convergence)? We define the relevant types of bases in general Banach spaces in Section 3.1, but besides this we mainly consider Hilbert spaces. In Section 3.4 we discuss the most important properties of orthonormal bases in Hilbert spaces; we expect the reader to have some basic knowledge about this subject. A slight (but useful) modification leads to the definition of Riesz bases, which are treated in detail in Section 3.6. Orthonormal bases and Riesz bases satisfy the so-called Bessel inequality, which is the key to the observation that they deliver unconditionally convergent expansions and can be ordered in an arbitrary way. Sequences satisfying the Bessel inequality are therefore discussed already in Section 3.2.

Novel waste heat recovery-assisted heat pump dryer is designed, developed, and experimented in different operating modes. Energy, exergy, economic, and exergoeconomic parameters of the dryer with and without waste heat recovery are experimentally compared. Radish chips are dried to remove moisture from 93.5% to 10.5% at fixed dryer inlet air velocity. Both energy and exergy efficiencies are found highest for the waste heat recovery-assisted (hybrid) system (56.26% and 35.9%, respectively). Specific moisture extraction rate (2.4 kg/kWh) and coefficient of performance (6.72) are highest for the hybrid system. The payback period for the hybrid system over the simple system is 33 months. The lowest exergoeconomic factor is for expansion devices in both simple and hybrid systems (0.0918 and 0.1348, respectively). Total exergy destruction costs for simple and hybrid systems are 0.10148 and 0.1266 $/h, respectively. The most important component that needs to improve based on the exergoeconomic factor is the drying chamber. Practical applications In the present study, a novel heat pump dryer assisted with waste heat recovery from the diesel engine is designed and developed. Various energy, exergy, economic, and exergoeconomic performance parameters with and without waste heat recovery are experimentally compared. To check the applicability, the radish chips have been dried to remove moisture from 93.5% to 10.5% at fixed dryer inlet air velocity. The testing shows that the proposed system is superior as compared to the simple heat pump dryer in terms of both energy and exergy efficiencies, specific moisture extraction rate, and overall heating coefficient of performance. The payback period for the proposed system over a simple one is found 3 years approximately. The proposed system can be a futuristic and promising solution for the drying applications, which satisfies two purposes: An effective way to utilize the engine waste heat and superior system performance.

Extract Although examples drive this book, we first provide a whirlwind survey of the general concepts of operator theory. We do not expect the student to master these topics now since they are covered in future chapters. A Hilbert space H is a complex vector space endowed with an inner product ⟨x,y⟩ that defines a norm ‖x‖=⟨x,x⟩ with respect to which H is (Cauchy) complete. The inner product on a Hilbert space satisfies the Cauchy–Schwarz inequality |⟨x,y⟩|⩽‖x‖‖y‖ for all x,y∈H⁠. Examples of Hilbert spaces include Cn (complex Euclidean space), ℓ2 (the space of square-summable complex sequences), and L2[0,1] (the Lebesgue space of square-integrable, complex-valued functions on [0,1]⁠). Vectors x,y in a Hilbert space H are orthogonal if ⟨x,y⟩=0⁠. The dimension of a Hilbert space H is the cardinality of a maximal set of nonzero orthogonal vectors. This book is almost exclusively concerned with Hilbert spaces of countable dimension. Every such Hilbert space has an orthonormal basis (un)n=1∞⁠, a (possibly finite) maximal orthogonal set such that ⟨um,un⟩=δmn for all m,n⩾1⁠. With respect to an orthonormal basis (un)n=1∞⁠, each x∈H enjoys a generalized Fourier expansion x=∑n=1∞⟨x,un⟩un that satisfies Parseval’s identity ‖x‖2=∑n=1∞|⟨x,un⟩|2. A subspace (a norm-closed linear submanifold) of H is itself a Hilbert space with the operations inherited from H⁠. If (wn)n=1∞ is a (possibly finite) orthonormal basis for a subspace M of H and x∈H⁠, then PMx=∑n=1∞⟨x,wn⟩wn belongs to M and satisfies ‖x−PMx‖⩽‖x−y‖ for every y∈M⁠. In short, PMx is the unique closest vector to x in M⁠. Furthermore, PM defines a linear transformation on H whose range is M⁠. It is called the orthogonal projection of H onto M and it satisfies PM2=PM and ⟨PMx,y⟩=⟨x,PMy⟩ for all x,y∈H⁠. Let H and K be Hilbert spaces. A linear transformation A:H→K is bounded if ‖A‖=sup{‖Ax‖K:‖x‖H=1} is finite. Let B(H,K) denote the set of bounded linear operators from H to K⁠. We write B(H) for B(H,H)⁠. The quantity ‖A‖ is the norm of A. Since B(H) is closed under addition and scalar multiplication, it is a vector space. Furthermore, since ‖A+B‖⩽‖A‖+‖B‖ and ‖cA‖=|c|‖A‖ for all A,B∈B(H) and c∈C⁠, it follows that B(H) is a normed vector space. Endowed with this norm, B(H) is complete and thus forms a Banach space. Moreover, the composition AB belongs to B(H) and ‖AB‖⩽‖A‖‖B‖ for all A,B∈B(H)⁠. Therefore, B(H) is a Banach algebra. For each A∈B(H,K)⁠, there is a unique A*∈B(K,H) such that ⟨Ax,y⟩K=⟨x,A*y⟩H for all x∈H and y∈K⁠. The operator A* is the adjoint of A; it is the analogue of the conjugate transpose of a matrix. One can show that A↦A* is conjugate linear, that A**=A⁠, ‖A‖=‖A*‖⁠, and ‖A*A‖=‖A‖2⁠. This additional structure upgrades B(H) from a Banach algebra to a C*-algebra. One can exploit adjoints to obtain information about the kernel and range of an operator. For most of the operators A∈B(H) covered in this book, we give the matrix representation [A]=[⟨Auj,ui⟩]i,j=1∞ with respect to an orthonormal basis (un)n=1∞ for H⁠. This matrix representation [A] defines a bounded operator x↦[A]x on the Hilbert space ℓ2 of square summable sequences that is structurally identical to A. Schur’s theorem helps us determine which infinite matrices define bounded operators on ℓ2⁠. Many of the operators covered in this book, such as the Cesàro operator, the Volterra operator, weighted shifts, Toeplitz operators, and Hankel operators, have fascinating structured-matrix representations. An important class of operators is the compact operators. These are the A∈B(H) such that (Axn)n=1∞ has a convergent subsequence whenever (xn)n=1∞ is a bounded sequence in H⁠. Equivalently, an operator is compact if it takes each bounded set to one whose closure is compact. Each finite-rank operator is compact and every compact operator is the norm limit of finite-rank operators. The compact operators form a norm-closed, *-closed ideal within B(H)⁠. Some operators have a particularly close relationship with their adjoint. For example, the operator M on L2[0,1] defined by (Mf)(x)=xf(x) satisfies M*=M⁠. Such operators are selfadjoint. If µ is a positive finite compactly supported Borel measure on C⁠, then the operator N on L2(μ) defined by (Nf)(z)=zf(z) satisfies (N*f)(z)=z―f(z)⁠, and thus N*N=NN*⁠. Such operators are normal. The operator (Uf)(eiθ)=eiθf(eiθ) on L2(T) satisfies U*U=UU*=I⁠. Such operators are unitary. Unitary operators preserve the ambient structure of Hilbert spaces and can serve as a vehicle to relate A∈B(H) with B∈B(K)⁠. We say that A is unitarily equivalent to B if there is a unitary U∈B(H,K) such that UAU*=B⁠. Unitary equivalence is often used to identify seemingly complicated operators with relatively simple ones.

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

A nuclear operator T on a Banach space is an operator admitting a representation T = (SIGMA) T(,n) where each operator T(,n) has rank one and (SIGMA) (VERT.BAR)(VERT.BAR)(TAU)(,n)(VBAR)(VBAR) < (INFIN). A. Grothendieck proved that the sequence of eigenvalues, repeated according to their multiplicities, of such a nuclear operator must be square summable and that, in fact, two is the best possible exponent of summability. This work uses operators of diagonal type, i.e., operators which admit a diagonal representation with respect to a biorthogonal system, to construct examples of eigenvalue behavior. In particular we show that all of the known results concerning eigenvalue summability of nuclear operators can be obtained using operators of diagonal type. In Chapter 1, we present the terminology and notation to be used in this work. Chapter 2 gives a summary of results concerning nuclear (trace class) operators on Hilbert space. In Chapter 3, some of the important known results concerning eigenvalue behavior of nuclear operators on general Banach spaces are presented. The convolution operators on L(,p)(0,1), 1 < p < (INFIN), p (NOT=) 2, are diagonal operators since the trigonometric system (e('2(pi)inx)) is a conditional basis of these spaces. In Chapter 4 we use these important operators to exhibit extremal behavior of eigenvalue summability of nuclear operators on the L(,p)-spaces. In Chapter 5, we study nuclear cyclic diagonal operators. If X(,1),... ,X(,n) are sequence spaces, an operator D on (CRPLUS)X(,i) is a cyclic diagonal operator if its restriction to each X(,i) is a diagonal operator into X(,i+1) (if i = n into X(,1)). Certain eigenspaces of these operators have important applications to Banach space theory. Finally, in Chapter 6, we use a recently discovered space of G. Pisier to answer affirmatively a long outstanding question of. Pe l cynski and Saphar which is converse to Grothendieck's result: Given a (nonzero) sequence ((lamda)(,n)) in l(,2), is there a Banach space X and a nuclear operator on X whose eigenvalue sequence is ((lamda)(,n))?

In this paper, we study growth rates for strongly continuous semigroups. We fixate that a growth rate for the resolvent estimate on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroupis positive and the underlying space is an -space or a space of continuous functions. Also proved variations of the main results on fractional domains; these are valid on more general Banach spaces by Jan Rozendaal and Mark Veraar. In the second part apply the main theorem to prove optimality in a classical example of a perturbed wave equation which shows unusual sequence of spectral behavior.

In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.

Soils represent the largest terrestrial carbon (C) sink and understanding its dynamics is crucial. The metabolic degradation and stabilization of soil organic matter (SOM) follow the rules of thermodynamics. In the catabolic reaction, SOM is oxidized to CO2 and the part of the energy delivered by this reaction is used in anabolism, during which biomass formation and, thereby, energy and C conservation take place. C and energy fluxes are thus linked and contribute to the C transformation and stabilization in natural soil systems. These processes are among others largely influenced by environmental conditions (e.g. temperature, soil moisture, C/N ratio).Due to the complexity and heterogeneity of soil, thermodynamic models and experimental approaches to study the linkage of C and energy fluxes in soil systems are rare and still in their infancy. To establish it, we use calorimetric and carbon mass balancing methods to study both C and energy fluxes in artificial soil systems in incubation experiments over 64 days with cellulose and over 16 days with glucose as substrates. This simplified system allows reliable measurement and interpretation of energy input, accumulation and output and their interaction with SOM turnover processes. Carbon and Energy Use Efficiency (CUE and EUE) are studied under varying environmental conditions. The heat production rate and the reaction enthalpy of metabolism in artificial soil systems are monitored with isothermal microcalorimeters. C mass balances consist of mineralization (measured using gas chromatography coupled with thermal conductivity detector), changes in total carbon (quantified by elemental analysis - isotope ratio mass spectrometry), and carbohydrates (recorded via a phenol sulphuric acid assay). In addition, biomass and necromass contents are quantified by phospholipid fatty acid and amino sugar analysis.EUE will be calculated from calorimetric data and further we will build an energy balance model. Furthermore, evolution of carbon input and output measurements will be further utilized for carbon balance model. Calorimetric and respirometric data provide the calorespirometric (CR) ratio of the soil system, which is closely related CUE (Chakrawal et al., 2020; Hansen et al., 2004). Experimentally determined CUE will be compared to that derived theoretically from CR ratio through calorimetric data and biomass yield modelling (Brock et al., 2017). Preliminary results on the linkage between carbon and energy balance in soil systems will be presented.Brock, A. L., K&amp;#228;stner, M., &amp; Trapp, S. (2017). Microbial growth yield estimates from thermodynamics and its importance for degradation of pesticides and formation of biogenic non-extractable residues. SAR and QSAR in Environmental Research, 28 (8), 629&amp;#8211;650. https://doi.org/10.1080/1062936X.2017.1365762Chakrawal, A., Herrmann, A. M., &amp;#352;antr&amp;#367;&amp;#269;kov&amp;#225;, H., &amp; Manzoni, S. (2020). Quantifying microbial metabolism in soils using calorespirometry &amp;#8212; A bioenergetics perspective. Soil Biology and Biochemistry, 148 (May), 107945. https://doi.org/10.1016/j.soilbio.2020.107945Hansen, L. D., MacFarlane, C., McKinnon, N., Smith, B. N., &amp; Criddle, R. S. (2004). Use of calorespirometric ratios, heat per CO2 and heat per O2, to quantify metabolic paths and energetics of growing cells. Thermochimica Acta, 422 (1&amp;#8211;2), 55&amp;#8211;61. https://doi.org/10.1016/J.TCA.2004.05.033

The relationship between streamwater mean residence time (MRT) and landscape characteristics is poorly understood. We used tritium (3H) to define our MRT. We tested the hypothesis that baseflow water MRT increases with increasing absolute catchment size at the Maimai catchments. These catchments are simple hydrologic systems relative to many catchments around the world, with uniformly wet climatic conditions, little seasonality, uniform and nearly impermeable bedrock, steep short hillslopes, shallow soils, and well‐characterized hillslope and catchment hydrology. As a result, this is a relatively simple system and an ideal location for new MRT‐related hypothesis testing. Whilst hydrologists have used 3H to estimate water age since the 1960s nuclear testing spike, atmospheric 3H levels have now approached near background levels and are often complicated by contamination from the nuclear industry. We present results for 3H sampled from our set of nested catchments in nuclear‐industry‐free New Zealand. Because of high precision analysis, near‐natural atmospheric 3H levels, and well‐characterized rainfall 3H inputs, we were able to estimate the age of young (i.e. less than 3 years old) waters. Our results showed no correlation between MRT and catchment size. However, MRT was correlated to the median sub‐catchment size of the sampled watersheds, as shown by landscape analysis of catchment area accumulation patterns. These preliminary findings suggest that landscape organization, rather than total area, is a first‐order control on MRT and points the way forward for more detailed analysis of how landscape organization affects catchment runoff characteristics. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Bounded and periodic solutions of infinite delay evolution equations

Characterizations of the metric and generalized metric projections on subspaces of Banach spaces

Reductionism assumes that causation in the physical world occurs at the micro level, excluding the emergence of macro-level causation. We challenge this reductionist assumption by employing a principled, well-defined measure of intrinsic cause-effect power–integrated information (Φ), and showing that, according to this measure, it is possible for a macro level to “beat” the micro level. Simple systems were evaluated for Φ across different spatial and temporal scales by systematically considering all possible black boxes. These are macro elements that consist of one or more micro elements over one or more micro updates. Cause-effect power was evaluated based on the inputs and outputs of the black boxes, ignoring the internal micro elements that support their input-output function. We show how black-box elements can have more common inputs and outputs than the corresponding micro elements, revealing the emergence of high-order mechanisms and joint constraints that are not apparent at the micro level. As a consequence, a macro, black-box system can have higher Φ than its micro constituents by having more mechanisms (higher composition) that are more interconnected (higher integration). We also show that, for a given micro system, one can identify local maxima of Φ across several spatiotemporal scales. The framework is demonstrated on a simple biological system, the Boolean network model of the fission-yeast cell-cycle, for which we identify stable local maxima during the course of its simulated biological function. These local maxima correspond to macro levels of organization at which emergent cause-effect properties of physical systems come into focus, and provide a natural vantage point for scientific inquiries.