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Journal articles on the topic 'Bs to Phi Phi'

Author: Grafiati

Published: 11 March 2023

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1

Giunti, C., J. Levraut, P. P.A.yan, and D. Grimaud. "R478 Bicarbonate de sodium (BS), pHi et pouvoir tampon extracellulaire." Annales Françaises d'Anesthésie et de Réanimation 17, no. 8 (January 1998): 1050. http://dx.doi.org/10.1016/s0750-7658(98)80595-9.

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2

Heming, Thomas A., Gregory Boyarsky, Divina M. Tuazon, and Akhil Bidani. "pHi responses to osmotic cell shrinkage in the presence of open-system buffers." Journal of Applied Physiology 89, no. 4 (October 1, 2000): 1543–52. http://dx.doi.org/10.1152/jappl.2000.89.4.1543.

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Changes in plasma volume in vivo cause rapid changes in extracellular pH by altering the plasma bicarbonate concentration at a constant Pco2 (Garella S, Chang BS, and Kahn SI. Kidney Int 8: 279, 1975). Few studies have examined the possibility that changes in cell volume produce comparable changes in intracellular pH (pHi). In the present study, alveolar macrophages were exposed to hyperosmotic medium in the absence or presence of the open-system buffers CO2-HCO3 −, propionic acid-propionate, or NH3-NH4 +. In the absence of open-system buffers, exposure to twice-normal osmolarity (2T) produced a slow cellular alkalinization [change in pHi(ΔpHi) ≈ 0.38; exponential time constant (τ) ≈ 120 s]. In the presence of 5% CO2, 2T caused a biphasic pHi response: a rapid increase (ΔpHi ≈ 0.10, τ ≈ 15 s) followed by a slower pHi increase. Identical rapid pHiincreases were produced by 2T in the presence of propionic acid (20 mM). Conversely, 2T caused a rapid pHi decrease (ΔpHi ≈ −0.21, τ ≈ 10 s) in the presence of NH3 (20 mM). Thus osmotic cell shrinkage caused rapid pHi changes of opposite direction in the presence of a weak acid buffer (contraction alkalosis with CO2 or propionic acid) vs. a weak base buffer (contraction acidosis with NH3). Graded ΔpHi were produced by varying extracellular osmolarity in the presence of open-system buffers; osmolarity increases of as little as 5–10% produced significant ΔpHi. The rapid pHi responses to 2T were insensitive to inhibitors of membrane H+ transport (ethylisopropylamiloride and bafilomycin A1). The results are consistent with shrinkage-induced disequilibria in the total cellular buffer system (i.e., intrinsic buffers plus added weak acid-base buffer).

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3

Hong, Qing, Guozhen Lu, and Lu Zhang. "L p boundedness of rough bi-parameter Fourier integral operators." Forum Mathematicum 30, no. 1 (January 1, 2018): 87–107. http://dx.doi.org/10.1515/forum-2016-0221.

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Abstract In this paper, we will investigate the boundedness of the bi-parameter Fourier integral operators (or FIOs for short) of the following form: T(f\/)(x)=\frac{1}{(2\pi)^{2n}}\int_{\mathbb{R}^{2n}}e^{i\varphi(x,\xi,\eta)}% \cdot a(x,\xi,\eta)\cdot\widehat{f}(\xi,\eta)\,d\xi\,d\eta, where {x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{n}} and {\xi,\eta\in\mathbb{R}^{n}\setminus\{0\}} , {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} is the amplitude, and the phase function is of the form \varphi(x,\xi,\eta)=\varphi_{1}(x_{1},\xi\/)+\varphi_{2}(x_{2},\eta) , with \varphi_{1},\varphi_{2}\in L^{\infty}\Phi^{2}(\mathbb{R}^{n}\times\mathbb{R}^{% n}\setminus\{0\}) , and satisfies a certain rough non-degeneracy condition (see (2.2)). The study of these operators are motivated by the {L^{p}} estimates for one-parameter FIOs and bi-parameter Fourier multipliers and pseudo-differential operators. We will first define the bi-parameter FIOs and then study the {L^{p}} boundedness of such operators when their phase functions have compact support in frequency variables with certain necessary non-degeneracy conditions. We will then establish the {L^{p}} boundedness of the more general FIOs with amplitude {a(x,\xi,\eta)\in L^{\infty}BS^{m}_{\rho}} and non-smooth phase function {\varphi(x,\xi,\eta)} on x satisfying a rough non-degeneracy condition.

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4

McLean, Allan J., Lisa L. Demos, and Constantine G. Berbatis. "George Michael Eckert PhC, MB BS(Hons), MSc, PhD, MD." Medical Journal of Australia 181, no. 9 (November 2004): 503. http://dx.doi.org/10.5694/j.1326-5377.2004.tb06411.x.

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5

Coyle-Rogers, Patricia, and Christie Putman. "Using Experiential Learning: Facilitating Hands-On Basic Patient Skills." Journal of Nursing Education 45, no. 4 (April 1, 2006): 142–43. http://dx.doi.org/10.3928/01484834-20060401-11.

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6

Jelti, L., C. Joachim, C. Boivin, F. Tetart, A. Lesourd, and E. Fiaux. "Souvenir de Ko Phi Phi." Annales de Dermatologie et de Vénéréologie 144, no. 12 (December 2017): S288—S289. http://dx.doi.org/10.1016/j.annder.2017.09.478.

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7

Fenton, P. C. "The growth of $\Phi \Phi''/\Phi'^{2}$ for convex functions." Illinois Journal of Mathematics 37, no. 3 (September 1993): 502–7. http://dx.doi.org/10.1215/ijm/1255987064.

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8

Grampp, K. H., and W. Huckenbeck. "Das Schicksal von Ko Phi Phi Don." Rechtsmedizin 15, no. 6 (December 2005): 447–55. http://dx.doi.org/10.1007/s00194-005-0353-z.

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9

Bertolotto, L., A. Buzzo, P. T. Debevec, D. Drijard, S. Easo, R. A. Eisenstein, W. Eyrich, et al. "Results from $$\bar pp \to \phi \phi $$." Il Nuovo Cimento A 107, no. 11 (November 1994): 2329–37. http://dx.doi.org/10.1007/bf02733999.

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10

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 32 (2005): 27–28. http://dx.doi.org/10.5840/tpm2005328.

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11

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 52 (2011): 31–33. http://dx.doi.org/10.5840/tpm2011528.

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12

Pizzani, Lori. "Why Phi?" CFA Institute Magazine 28, no. 1 (March 2017): 44–45. http://dx.doi.org/10.2469/cfm.v28.n1.19.

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13

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 12 (2000): 22. http://dx.doi.org/10.5840/tpm200012119.

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14

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 13 (2001): 27. http://dx.doi.org/10.5840/tpm20011313.

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15

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 14 (2001): 29. http://dx.doi.org/10.5840/tpm200114131.

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16

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 15 (2001): 29. http://dx.doi.org/10.5840/tpm20011593.

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17

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 16 (2001): 51. http://dx.doi.org/10.5840/tpm20011662.

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18

Iredale, Matthew. "Sci-Phi." Philosophers' Magazine, no. 17 (2002): 25. http://dx.doi.org/10.5840/tpm200217117.

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19

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 18 (2002): 22. http://dx.doi.org/10.5840/tpm20021878.

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20

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 19 (2002): 22. http://dx.doi.org/10.5840/tpm20021942.

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21

Iredale, Matthew. "Sci-Phi." Philosophers' Magazine, no. 20 (2002): 18. http://dx.doi.org/10.5840/tpm2002206.

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22

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 21 (2003): 21. http://dx.doi.org/10.5840/tpm200321111.

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23

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 22 (2003): 21. http://dx.doi.org/10.5840/tpm20032277.

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24

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 23 (2003): 21. http://dx.doi.org/10.5840/tpm20032344.

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25

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 24 (2003): 22. http://dx.doi.org/10.5840/tpm2003248.

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26

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 25 (2004): 20. http://dx.doi.org/10.5840/tpm200425105.

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27

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 26 (2004): 19. http://dx.doi.org/10.5840/tpm20042672.

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28

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 27 (2004): 22. http://dx.doi.org/10.5840/tpm20042740.

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29

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 28 (2004): 29–30. http://dx.doi.org/10.5840/tpm2004288.

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30

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 29 (2005): 24–25. http://dx.doi.org/10.5840/tpm200529107.

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31

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 30 (2005): 22–23. http://dx.doi.org/10.5840/tpm20053074.

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32

Iredale, Mathew. "Sci-Phi." Philosophers' Magazine, no. 31 (2005): 29–31. http://dx.doi.org/10.5840/tpm20053139.

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33

Turner, Mark. "PHI Reserving." Journal of the Staple Inn Actuarial Society 33, no. 1 (1993): 1–36. http://dx.doi.org/10.1017/s2049929900010527.

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Papers devoted to the subject of reserving for Permanent Health Insurance (PHI) are not exactly numerous. The author could not find one published in the U.K., although there were papers in the U.S.A., Australia and South Africa.Most U.K. PHI papers have sought to cover the topic in its entirety and therefore do give views on reserving issues. Ironically, this paper in seeking to focus on reserving did, incidentally, find itself wandering over a large part of the PHI actuarial countryside. Perhaps this is unavoidable.PHI reserving is not just a question of ensuring adequate solvency. The reserving basis we choose can aid or hinder our understanding of the business. After all, the recognition of profits is defined by the reserving basis. If management is going to track profits in order to assist decisions on marketing position, financial strength of the office, etc. then it needs to be concerned that the reserving basis is not distorting the emerging results. An equivalent argument can be put forward in respect of the capital requirements of the business.

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34

Hebert, Kirsten. "Phi Omicron." Hindsight: Journal of Optometry History 50, no. 4 (October 16, 2019): 84. http://dx.doi.org/10.14434/hindsight.v50i4.28032.

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35

Sahami, A., and A. Pourabbas. "On $\phi$-biflat and $\phi$-biprojective Banach algebras." Bulletin of the Belgian Mathematical Society - Simon Stevin 20, no. 5 (November 2013): 789–801. http://dx.doi.org/10.36045/bbms/1385390764.

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36

Chen, Quanyuan, Xiaochun Fang, and Changjing Li. "The characterization of generalized Jordan centralizers on algebras." Boletim da Sociedade Paranaense de Matemática 35, no. 3 (October 25, 2017): 225. http://dx.doi.org/10.5269/bspm.v35i3.30008.

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In this paper, it is shown that if $\mathcal {A}$ is a CSL subalgebra of a von Neumann algebr and $\phi$ is a continuous mapping on $\mathcal {A}$ such that $(m+n+k+l)\phi(A^{2})-(m\phi(A)A+nA\phi(A)+k\phi(I)A^2+l A^2 \phi(I))\in \mathbb{F}I $ for any $A\in \mathcal {A}$, where $\mathbb{F}$ is the real field or the complex field, then $\phi$ is a centralizer. It is also shown that if $\phi$ is an additive mapping on $\mathcal {A}$ such that $(m+n+k+l)\phi(A^{2})=m\phi(A)A+nA\phi(A)+k\phi(I)A^2+l A^2 \phi(I) $for any $A\in\mathcal{A}$, then $\phi$ is a centralizer.

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37

Świtalska, Alicja. "IMPACT OF ANTROPOPRESSION ON LANDSCAPE OF PHI PHI ISLANDS." space&FORM 2017, no. 31 (October 10, 2017): 255–72. http://dx.doi.org/10.21005/pif.2017.31.d-02.

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38

Marechal, Y., G. Meunier, and H. Ben Harara. "A new 3D AV- Phi - Phi /sub r/ formulation." IEEE Transactions on Magnetics 28, no. 2 (March 1992): 1204–7. http://dx.doi.org/10.1109/20.123902.

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39

Dodds, Rachel. "Koh Phi Phi: Moving Towards or Away from Sustainability?" Asia Pacific Journal of Tourism Research 15, no. 3 (September 2010): 251–65. http://dx.doi.org/10.1080/10941665.2010.503615.

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40

Granucci, Tiziano. "$L^{\Phi }-L^{\infty }$\ Inequalities and Applications." Journal of Mathematics Research 7, no. 2 (May 23, 2015): 201. http://dx.doi.org/10.5539/jmr.v7n2p201.

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In this paper we prove some $L^{\Phi }-L^{\Phi }$ and $L^{\Phi }-L^{\infty }$inequalities for quasi-minima of scalar integral functionals defined inOrlicz-Sobolev space $W^{1}L^{\Phi }\left( \Omega \right) $, where $\Phi $\is a N-function and $\Phi \in \triangle _{2}$. Moreover, if $\Phi \in\triangle ^{^{\prime }}$ or if $\Phi \in \triangle _{2}\cap \nabla _{2}$, weprove that quasi-minima are H\"{o}lder continuous functions.

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41

Barnett-Vanes, Ashton, and Joseph Shalhoub. "MB/PhD: don't forget the ‘BS’." British Journal of Hospital Medicine 74, no. 9 (September 2013): 535. http://dx.doi.org/10.12968/hmed.2013.74.9.535.

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42

Zin, Nur Shaylinda Mohd, Hamidi Abdul Aziz, Mohd Nordin Adlan, Azlan Ariffin, Mohd Suffian Yusoff, and Irvan Dahlan. "A Comparative Study of Floc and Sludge of Leachate under Different Types of Coagulants." Applied Mechanics and Materials 802 (October 2015): 406–11. http://dx.doi.org/10.4028/www.scientific.net/amm.802.406.

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This study compared the floc and sludge formed during the coagulation of leachate by different types of coagulants. The coagulants tested in this study were Ferric chloride (FC), pre-hydrolyzed iron (PHI), dual coagulant (PHI+TF(tapioca flour)), and composite coagulant (pre mix of PHI and TF (PHITF)). The floc and sludge were characterized by measuring the sludge volume index (SVI), sludge velocity (SV), and floc size. Results showed that the hierarchy for SVI, SV, and floc size were FC>PHI>PHI+TF>PHITF, PHITF>PHI+TF>PHI>FC, and PHI+TF>PHITF>PHI>FC, respectively. PHITF demonstrated the lowest SVI and the shortest sludge settling time among the tested coagulants. PHI+TF produced the largest floc size. The addition of TF in PHI as a composite and dual coagulant significantly improved the floc and sludge characteristics. PHITF improved the floc and sludge formation of partially stabilized leachate.This study compared the floc and sludge formed during the coagulation of leachate by different types of coagulants. The coagulants tested in this study were Ferric chloride (FC), pre-hydrolyzed iron (PHI), dual coagulant (PHI+TF(tapioca flour)), and composite coagulant (pre mix of PHI and TF (PHITF)). The floc and sludge were characterized by measuring the sludge volume index (SVI), sludge velocity (SV), and floc size. Results showed that the hierarchy for SVI, SV, and floc size were FC>PHI>PHI+TF>PHITF, PHITF>PHI+TF>PHI>FC, and PHI+TF>PHITF>PHI>FC, respectively. PHITF demonstrated the lowest SVI and the shortest sludge settling time among the tested coagulants. PHI+TF produced the largest floc size. The addition of TF in PHI as a composite and dual coagulant significantly improved the floc and sludge characteristics. PHITF improved the floc and sludge formation of partially stabilized leachate.

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43

Stout, R. D., and K. Bottomly. "Antigen-specific activation of effector macrophages by IFN-gamma producing (TH1) T cell clones. Failure of IL-4-producing (TH2) T cell clones to activate effector function in macrophages." Journal of Immunology 142, no. 3 (February 1, 1989): 760–65. http://dx.doi.org/10.4049/jimmunol.142.3.760.

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Abstract IFN-gamma-producing (TH1) and IL-4-producing (TH2) clones were assayed for their ability to directly induce cytostatic activity in macrophages generated from splenic myeloid precursors (M phi-c). In the presence, but not in the absence, of antigen, TH1 clones activated the M phi-c to inhibit the growth of P815 tumor cells in vitro. TH2 clones were not able to activate such effector activity in the M phi-c. The M phi-c did effectively present Ag to the TH2 clones as evidenced by the proliferation of TH2 cells cultured with Ag in the presence, but not in the absence, of M phi-c. Therefore, although both TH1 and TH2 were activated by cognate interaction with antigen presenting M phi-c, only TH1:M phi-c interactions displayed reciprocity resulting in activation of the M phi-c. TH1-derived lymphokines or rIFN-gamma, in the presence of LPS, could activate proteose-peptone elicited M phi, resident peritoneal M phi, and M phi-c whereas neither TH2-derived lymphokines nor rIL4 could induce detectable activity in any of the 3 M phi populations. IFN-gamma, in the absence of LPS, could activate the elicited M phi and to a lesser and more variable degree, the resident M phi Only the M phi-c consistently required both IFN-gamma and LPS for induction of cytostatic activity. Since M phi-c consistently required at least two signals for activation, the ability of TH1-derived lymphokines to synergize with TH2 cells in M phi activation was examined. TH2 could activate the Ag-presenting M phi-c in the presence of IFN-gamma. The ability of added IFN-gamma to synergize with TH2 indicates that the cognate interaction between TH2 and antigen presenting M phi-c does result in delivery of at least one of the signal required for M phi activation.

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44

Сатоми, Такаси, and Takashi Satomi. "Неравенство для композиций выпуклых функций со свертками и альтернативное доказательство неравенства Брунна-Минковского-Кемпермана." Trudy Matematicheskogo Instituta imeni V.A. Steklova 319 (December 2022): 280–97. http://dx.doi.org/10.4213/tm4268.

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Пусть $m(G)$ - точная нижняя грань объемов всех открытых подгрупп унимодулярной локально компактной группы $G$. Предположим, что интегрируемые функции $\phi _1,\phi _2: G\to [0,1]$ удовлетворяют условиям $\|\phi _1\|\leq \|\phi _2\|$ и $\|\phi _1\| + \|\phi _2\| \leq m(G)$, где $\|\kern 1pt{\cdot }\kern 1pt\|$ - норма пространства $L^1$ относительно меры Хаара $dg$ на $G$. В работе для любой выпуклой функции $f: [0,\|\phi _1\|]\to \mathbb R $ такой, что $f(0) = 0$, доказано неравенство $\int _{G} f \circ (\phi _1 * \phi _2)(g) dg \leq 2 \int _{0}^{\|\phi _1\|} f(y) dy + (\|\phi _2\| - \|\phi _1\|) f(\|\phi _1\|)$. Как следствие выводится несколько усиленная версия неравенства Брунна-Минковского-Кемпермана, а именно $\mathrm {vol}_*(B_1 B_2) \geq \mathrm {vol}(\{g\in G \mid 1_{B_1} * 1_{B_2}(g) > 0\}) \geq \mathrm {vol}(B_1) + \mathrm {vol}(B_2)$ для любых измеримых множеств $B_1,B_2 \subset G$ положительного объема таких, что $\mathrm {vol}(B_1) + \mathrm {vol}(B_2) \leq m(G)$, где $\mathrm {vol}_*$ - внутренняя мера, а $1_B$ - характеристическая функция множества $B$.

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45

Størmer, Erling. "Multiplicative properties of positive maps." MATHEMATICA SCANDINAVICA 100, no. 1 (March 1, 2007): 184. http://dx.doi.org/10.7146/math.scand.a-15020.

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Let $\phi$ be a positive unital normal map of a von Neumann algebra $M$ into itself. It is shown that with some faithfulness assumptions on $\phi$ there exists a largest Jordan subalgebra $C_{\phi}$ of $M$ such that the restriction of $\phi$ to $C_{\phi}$ is a Jordan automorphism and each weak limit point of $(\phi^n (a))$ for $a\in M$ belongs to $C_{\phi}$.

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46

Park, Choonkil, Kandhasamy Tamilvanan, Ganapathy Balasubramanian, Batool Noori, and Abbas Najati. "On a functional equation that has the quadratic-multiplicative property." Open Mathematics 18, no. 1 (August 4, 2020): 837–45. http://dx.doi.org/10.1515/math-2020-0032.

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Abstract In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation: \phi (st-uv)+\phi (sv+tu)={[}\phi (s)+\phi (u)]{[}\phi (t)+\phi (v)] by using the direct method and the fixed point method.

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47

Kwant, G., B. Oeseburg, A. Zwart, and W. G. Zijlstra. "Human whole-blood O2 affinity: effect of CO2." Journal of Applied Physiology 64, no. 6 (June 1, 1988): 2400–2409. http://dx.doi.org/10.1152/jappl.1988.64.6.2400.

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The proton Bohr factor (phi H = alpha log PO2/alpha pH), the carbamate Bohr factor (phi C = alpha log PO2/alpha log PCO2), the total Bohr factor (phi HC = d log PO2/dpH[base excess) and the CO2 buffer factor (d log PCO2/dpH) were determined in the blood of 12 healthy donors over the whole O2 saturation (SO2) range. All three Bohr factors proved to be dependent on SO2, although to a lesser extent than reported in some of the recent literature. At SO2 = 50% and 37 degrees C, we found phi H = -0.428 +/- 0.010 (SE), phi C = 0.054 +/- 0.006, and phi HC = -0.488 +/- 0.007. The values obtained for phi H, phi C, and d log PCO2/dpH were used to calculate phi HC. Calculated and measured values of phi HC proved to be in good agreement. In an additional series of 12 specimens of human blood we determined the influence of PCO2 on phi H and the influence of pH on phi C. At SO2 = 50%, phi H varied from -0.49 +/- 0.009 at PCO2 = 15 Torr to -0.31 +/- 0.010 at PCO2 = 105 Torr and phi C from 0.157 +/- 0.015 at pH = 7.80 to 0.006 +/- 0.009 at pH = 7.00. When on the basis of these data a second-order term is taken into account, a still slightly better agreement between measured and calculated values of phi HC can be attained.

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48

Kuczma, M., and A. Zajtz. "On the real form of real solutions of the matrix functional equation $\Phi(x)\Phi(y)=\Phi(xy)$ for non-singular matrices $\Phi$." Publicationes Mathematicae Debrecen 13, no. 1-4 (July 1, 2022): 257–62. http://dx.doi.org/10.5486/pmd.1966.13.1-4.31.

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49

Havlin, John L., and Alan J. Schlegel. "Review of Phosphite as a Plant Nutrient and Fungicide." Soil Systems 5, no. 3 (August 30, 2021): 52. http://dx.doi.org/10.3390/soilsystems5030052.

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Phosphite (Phi)-containing products are marketed for their antifungal and nutritional value. Substantial evidence of the anti-fungal properties of Phi on a wide variety of plants has been documented. Although Phi is readily absorbed by plant leaves and/or roots, the plant response to Phi used as a phosphorus (P) source is variable. Negative effects of Phi on plant growth are commonly observed under P deficiency compared to near adequate plant P levels. Positive responses to Phi may be attributed to some level of fungal disease control. While only a few studies have provided evidence of Phi oxidation through cellular enzymes genetically controlled in plant cells, increasing evidence exists for the potential to manipulate plant genes to enhance oxidation of Phi to phosphate (Pi) in plants. Advances in genetic engineering to sustain growth and yield with Phi + Pi potentially provides a dual fertilization and weed control system. Further advances in genetic manipulation of plants to utilize Phi are warranted. Since Phi oxidation occurs slowly in soils, additional information is needed to characterize Phi oxidation kinetics under variable soil and environmental conditions.

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Li, Chi-Kwong, and Nam-Kiu Tsing. "NORMS ON CARTESIAN PRODUCT OF LINEAR SPACES." Tamkang Journal of Mathematics 21, no. 1 (March 1, 1990): 35–39. http://dx.doi.org/10.5556/j.tkjm.21.1990.4692.

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Let $X_i, (i=1, \cdots, n)$ be real or complex linear spaces, each equipped with a norm $||\cdot||_i$. Standard ways of constructing norms $||\cdot||$ on the Cartestian product $X =X_1 \times \cdots \times X_n$ are to define \[ ||(x_1, \cdots, x_n)||=\phi(||x_1||_1, \cdots, ||x_n||_n)\] via some functions $\phi$ on $\mathbb{R}^n$. Common examples of $\phi$ in standard texbooks are norms on $\mathbb{R}^n$. This may mislead peoples to think that any norm $\phi$ on $\mathbb{R}^n$ can induce a norm on the product space $X$ in the above way. In this note we show that this is actually false and characterize the functions $\phi$ that can give rise to norms on $X$ in the above manner. It turns out that a necessary and sufficient condition on $\phi$ is : for any $a_1, \cdots, a_n, b_1, \cdots, b_n\ge 0$, (I) $\phi(a_1, \cdots, a_n)>0$ if $(a_1, \cdots, a_n)\neq (0, \cdots,0)$; (II) $\phi(\alpha(a_1, \cdots, a_n))= \alpha \phi(a_1, \cdots, a_n)$ if $\alpha\ge 0$; (III) $\phi(c_1, \cdots, c_n)\le \phi(a_1, \cdots, a_n)+ \phi(b_1, \cdots, b_n)$ if $(c_1, \cdots, c_n)= (a_1, \cdots, a_n)+ (b_1, \cdots, b_n)$;(IV) $\phi(a_1, \cdots, a_n) \le \phi(b_1, \cdots, b_n)$ if $a_i \le b_i$ for all $i$. Several interesting consequences of the result are discussed.

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