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Q: Why is there no test instrumentation for BroadcastReceiver? Maybe I'm missing something. I want to write test cases for a BroadcastReceiver; specifically, it is for receiving the BOOT_COMPLETED event and setting an alarm for another receiver to handle later; it doesn't seem to be setting it properly, but the point is that I have no obvious way to test it. I can't exactly attach a debugger and wait for BOOT_COMPLETED, and I can't send a fake BOOT_COMPLETED broadcast.
Why are there instrumentation classes for Activity, Service, and Provider, but not BroadcastReceiver? Any advice for testing this?

A: For most cases I agree completely with https://stackoverflow.com/a/5181010/527016
There are however cases when extending AndroidTestCase is not suitable (and can cause surprises). In particular, if you are doing more complex integration testing and want to test your BroadcastReceiver with an actual Intent sent by the system. The main reason is that the onReceive method in the broadcast receiver runs on the main application thread while the tests in AndroidTestCase run in another thread. This can cause test-related threading issues in code that was not intended to run on multiple threads.
The solution to this is to subclass your test from InstrumentationTestCase instead and use the @UiThreadTest annotation to make the tests run on the same thread as the onReceive method.
For more info (and an example) see:  http://olafurhelgason.blogspot.com/2012/12/threading-and-android-integration.html

A: There is nothing magical about the life cycle for the BroadcastReceiver. It's enough to test it with an AndroidTestCase. In a test case, instantiate your BroadcastReceiver, create whatever Intent you want to send and call onReceive using the Context available from AndroidTestCase or some mock Context.
E.g.
public class TestMyBroadcastReceiver extends AndroidTestCase {
  public void testReceive() {
    MyBroadcastReceiver r = new MyBroadcastReceiver();
    Intent i = new Intent("MY_ACTION");
    // TODO put extras
    r.onReceive(getContext(), i);
    // TODO query application state to verify results
  }
}

redpajama_set_name: RedPajamaStackExchange


I'm very selective about the toys we get for Little Dude. Battery-operated and flashy toys simply irritate me, and from the beginning, I got him toys which at least required him to use basic skills. I think that the purpose of a toy is to spark creativity and imagination, and become whatever the child using it wants it to.
That's why, I love the Biddoo store - an online portal based in the UAE which has an amazing collection of toys that are definitely not run of the mill. Biddoo prides itself in being different, and not stocking the "latest craze" or plastic toys which run on batteries.
They have a huge collection of creative and educational toys, apart from arts and crafts kits aimed at children from 1 to 15 years of age.
Founder Isabelle Wessollek wrote to me about the site, "I wanted to create an alternative for mums looking for sophisticated, inspirational and also educational gifts for children which you can't find easily at the mall. And I also wanted to offer a convenient, fast and simple shopping possibility for busy mums. At the same time, I was always looking for creative activities for my three children!"
Biddoo currently has three exclusive brands, and other exclusive products you won't find in shops. "We try to distinguish ourselves with this hand-picked product selection," adds Isabelle. Some of the things they look for in the toys they stock are creativity, inspiration for mums and kids, high quality standards, sophistication, great materials and design, and uniqueness.
We tried a few of the products available on Biddoo, and these will give you a snapshot into the creative range available on their portal.
I have also linked directly to the products, if you wish to order them.
This was the first product Little Dude wanted to open! Clearly, he was attracted by the monsters and the fact that it involved finger painting. This kit comes with good quality finger paints in eight colours, a palette, four scenes for painting (two each) and a guide book to paint from. One of the things I liked was that it directs kids to use different fingers for painting, thus improving fine motor skills. This is definitely a fun product for any child who has an imagination.
This has become another favourite of ours. There are 24 brightly coloured, and beautifully illustrated big puzzle pieces. The puzzle pieces are sturdy and have not frayed or bent despite the number of times we used them. Little Dude loves the box so much, he plays with it too!
I love how clever this product is! It's a fabric casing for a 9-inch balloon, which converts it into a firm ball. That way, you can carry around this ball anywhere and inflate it as needed. I think it would be great to take on a family vacation.
This one is actually my favourite toy! This LED torch gets charged by pushing the lever, which retracts inside. It lasts for quite a while too. I find that it's a really handy tool to have around. Little Dude is also fascinated by it, and has found many uses for the torch, such as "finding treasure".
We like this leopard sticker as Little Dude is a Go Diego Go fan and loves anything about animals! We just haven't found a special place to stick it yet (lots of art projects in our house), but we are saving it for a good spot.
I must say that I was delighted to find a website that stocks toys which really encourage children to get creative and be imaginative. I know that I'm going to be shopping with them often - not just for Little Dude, but also for gifts for his friends. The prices are also reasonable, and in fact, a bang for a buck for such good products!
Visit Biddoo.ae to see their range of products, and follow them on Facebook or Instagram to stay in touch.
We received the above products for the purpose of review, but all views and opinions are my own.
These toys are really creative! I'm not a fan of the battery operated toys either.


This is a pretty old video I'd forgotten all about. It's from the first year we ran the Dribble Drive, when I was in England. We were playing undefeated Leeds Carnegie, in probably the most perfect game any of my teams have ever played.
After the first quarter we had been called for 10 fouls and Leeds had only been called for one and we were down only 21-18. At the break I had a quiet word with the referees, where they didn't get to say much, and things evened out.
At the end of the game we were down one, with Leeds having the ball, but also having to shoot before time expired. We blocked them twice in the key with 10 seconds to go (I don't know why they shot the second shot), got the ball back, and a kick-back from point guard Simon Mitchell to shooting guard James Harris won the game.
It's just a reminder that the fundamentals of the Dribble Drive can win you unexpected games.


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Q: Anylogic: how to Batch agents with similar parameters? I have an agent called products, and in this agent, I assigned a parameter called sp; in the simulation, I have the same agent with a different sp range from 1 to 5. I want to batch the agents with the same sp in the same batch, which is 10. So if I have 200 agents, 49 of them with sp equals 1, I would like to batch them in 5 batches (10,10,10,10,9), and sp equals 2 with another batch and so on.
I really appreciate any help you can provide.

A: There are a number of ways in which you can do this, but given the fact that you have non-consistent batches (i.e. you have batches of 10 and then the last one is 9) I would first put all the agents inside a wait block (or you can use a queue) and then programmatically control them
Take the following small example

I wait until 200 of my "Product" agents arrive at the wait block and then I press the button "Check Batches" which calls the function batchReleaseCheck()
Here is the code
LinkedHashMap<Integer, List<Product>> productsWaiting = new LinkedHashMap<Integer, List<Product>>();
for (int i = 0; i < wait.size(); i ++){
    Product product = wait.get(i);
    int sp = product.sp;
    if (!productsWaiting.containsKey(sp)) productsWaiting.put(sp, new ArrayList<Product>());
    productsWaiting.get(sp).add(product);

    //Check the batch size if sufficient we release it
    if (productsWaiting.get(sp).size() == batchSize) {
        for (Product p:productsWaiting.get(sp)) {
            wait.free(p);
        }
        return; // we exit the loop since we have released a batch
    }
}

// If we get to the end of the loop we were not able to release any batch we now release each SP type regardless of their current batch size

for (int i = 1; i < 6; i ++) {
    if (productsWaiting.get(i) == null) continue;
    for (Product p:productsWaiting.get(i)) {
        wait.free(p);
    }
    batch.set_batchSize(productsWaiting.get(i).size()); //Since the batch is less than the standard we need to change it to what ever we are releasing
    return; // we exit the loop since we have released a batch
}

You create a map to store the products in a list based on their sp number. If at any time you find you have enough units to create a batch we stop and release them from the wait block.
If we get to the end of the for loop and we did not have any sp number that had enough units to make up a whole batch we simple release what we have.
In this example you will need to click the button every time to release a batch, alternatively you can add the batch check function call to the on release code of the batch object. This will all happen sequentially, but at the same time step


The energy from last week's Lunar Eclipse in Libra is still alive and kicking. Libra is about balance, and finding that balance through relating to others. There's a lot to take in…courage will not be an option, but a necessity. Although you might not see it yet, change has already happened, it's just a matter of time (sooner than later), for a new door to present its self. It'll be up to you to open it! Once you take that first step, you may notice a special relationship awakening a sweet spot within you that you've been craving to satisfy.


You are in vacation and there is one thing you want to do: dance, dance, and dance! So where do you go in Venice? Here, there is not exactly a rich crop of dance clubs and discotheques. But don't despair, there is one quiet famous, or maybe even infamous one: Piccolo Mondo. This dance club first opened in the 1960s and it might have never seen a renovation or new interior design since. If you make it past the bouncer, who has his very own ideas about who should be allowed inside, you will enter into a retro world. Essentially there is only one thing to do and everyone is doing it: to dance. The club opens at 10 pm, but you shouldn't expect large crowds before midnight. It all ends at 4 am, when the last night owls are sent home.


a) 3D surgical anatomy of nose and paranasal sinuses, hands on, Tubingen, Germany.
b) Transnasal corridors to skull base and orbit, Vienna, Austria,since 2010.
c) Transnasal corridors to pitutary, Vienna, Austria ,since 2007.
d) Basic dissection of paranasal sinuses and orbit,Paris ,France since 1995.
e) Basic dissection of paranasal sinus ,Marselille,France since 1996.
f) The modern sinonasal surgery:Anatomy,Diagnostics and operative techniques (Basic and advance course)-English and Italian Language,since 1995 in Belgium.
- Guest speaker and Faculty Member in several meetings throughout the world, giving more than 350 official presentations and lectures.
- Guest operating surgeon in more than 50 national and International courses with live-TV surgery.
- Director of International Biennale Milano Masterclass, since 1999.(7 editions).
- Fellowship training programm in Rhinology and endoscopic skull base(International Candidate) since 2006.
- 2014 - He has organized the 6th World Congress for Endoscopic Surgery of the brain, skull base and spine that will be held in Milan in April 14th – 17th,.
- Non profit orginization for scientific research and training courses (www.attingo-edu.it).
- He has performed more than 7000 surgical operations of which more than 500 major operations in collaboration with neurosurgical team.
- September 17, 1979- Degree in Medicine and Surgery at the Universita 'degli Studi di Milano.
- July 8, 1982- Award of the Diploma of Specialist in Otolaryngology at the University of Pavia.
- 29 October 1991- Attainment of the Diploma of Specialist in Audiology at the University of Pavia.
- December 24, 1986 - appointed assistant professor in Otorhinolaryngology, Dept. of Otorhinolaryngology , University of Pavia, Italy.
- April 16, 1990- appointed university researcher in Otorhinolaryngology at the Faculty of Medicine of the University 'of Pavia.
- 1998 was appointed Deputy Director of the ENT Clinic of the University of Pavia for three academic years 1999/2002.
- 5 April 2005 - appointed Director of the School of Specialization in Otorhinolaryngology , University of Insubria-Varese.
-1st December 2006 - appointed Full Professor in Otorhinolaryngology, at the Faculty of Medicine and Surgery of the University of Insubria-Varese.
- July 2013 – appointed the Director of Surgical Specialties Department at Azienda Ospedaliero- Universitaria, Ospedale di Circolo- Varese.
His main interests are focused on endoscopic sinus and skull base surgery since the late 80's. Besides inflammatory lesions of the paranasal sinuses, he had been gaining an extensive surgical experience and skills in the endoscopic management of neoplasms of the sinonasal tract and skull base (CSF leaks, sellar and parasellar) lesions.
Sinonasal and Skull base oncology, Endoscopic sinus and skull base surgery, challenging oncological cases and CSF leak repair.
- Italian Society of Otorhinolaryngology (SIOeChCF).
- European Rhinologic Society (ERS).
- European Skull Base Society (ESBS).
- European Academy of Facial Plastic Surgery (EAFPS).
- American Rhinologic Society (ARS).
- Italian Society of AICEF.
Publication and presentation more than (491) scientific papers (132 original works printed in full, 97 international relations, national reports 132 and more than 130 video communications, all presented at conferences and symposia) .
- Publish book: Surgical Anatomy of the internal carotid artery: An Atlas for Skull Base Surgeons 2013.
- More than 150 chapters in published book.
- Series of anatomical teaching booklets published by Storz company.
- Roma, Italy, 1993, Italian society of Otorhinolaryngology, head and neck surgery: Dedicated to "Caliceti, Citelli e Pietrantoni": III prize for research of: "Cholesteatoma of middle ear"
- Padova, Italy, Video larynx 1997, award Prof. Teatini - 1° prize video: Endoscopic treatment of nasal Hemangiopericitoma.
- Padova, Italy, Video Larynx 1998, award Prof. Teatini - 1° prize video: Isolated pathology of sphenoid sinus.
More than 491 scientific participations, in attachment you find the published PubMed which are more than 120 article.


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Home / Bible Commentaries / Sermon Bible Commentary/ Acts
Sermon Bible Commentary
Acts Acts 4
Haydock's Catholic Commentary
Vincent's Studies
Box's Commentaries on Selected Books
Gary Hampton Commentary
Everett's Study Notes
Gray's Commentary
The People's Bible
Wells of Living Water
Commentary on Acts
Birdgeway Bible Commentary
(with Acts 4:4)
St. Peter's Second Apology
If the latter portion of this speech of St. Peter's be examined, it will be found that its central point, on which is thrown the chief weight of exhortation, is precisely the same as in Luke's abridged version of the former speech. "Repent and be converted, that your sins may be blotted out," he says here. "Repent and be baptized for the remission of sins," he said then. As though God were beseeching his countrymen through his lips, Peter here prayed them in Christ's stead to be reconciled to God; and he used such motives as, in the mouth of a Jew speaking to Jews, were most fit and likely to persuade.
I. In the first place he suggested, as an extenuation of their guilt, that it had been contracted in ignorance. It may not have been quite true of all, but it certainly was true of the vast mass of the people, who wheeled so readily from applauding Jesus to execrating Him, that neither when they did the one nor the other had they real knowledge on solid grounds who He was. Such ignorance as this does not excuse a crime, but it palliates it. It makes it more pardonable.
II. Further to open his hearers' hearts to penitence, he reminded them, as he had done at Pentecost, how their very crime had been the fulfilment, all unknown to themselves, of those predicted sufferings which it had been God's will to inflict upon Messiah. Through their slaying of the Christ, God had ordained that the Christ should become their Saviour.
III. But the most singular motive by which Peter here pressed his countrymen to repent, is that, upon their doing so, had been made to hinge the return of Christ in glory, and that predicted era of blessedness which is to enter when His personal presence is restored to the earth. The object of our Lord's retirement into the heavens he took to be the conversion of Israel to faith in Himself. So long as He was here they had denied Him; now, in His absence, they were to return and call with tears upon Him whom they had pierced. The faster Israel turned to Jesus, the sooner would Jesus return to Israel; for as Peter wrote a great many years later, "The Lord is not slack concerning his promise" to return, He is only "longsuffering, not willing that any should perish." With urgency, therefore, did the preacher that day press upon his brethren, as Israelites, to turn every one from his iniquities, so that there might come the sooner those times of national reviving and restoration, which had so often been predicted to their fathers.
J. Oswald Dykes, From Jerusalem to Antioch, p. 123.
Acts 3:1-11
Look (1) at the social side, and (2) at the Apostolic side of this incident.
I. The social side. (1) We may be able to carry the cripple while we are unable to heal him. Do what you can. (2) The commonest minds, as well as the highest, have always associated the idea of charity with the idea of religion. (3) Look at the compensations of the poorest life. The man was carried daily by friendly hands. The man had the temple as his daily hope.
II. The Apostolic side. (a) The Apostles never attempted to do without public worship. (b) The Apostles never neglected human want in their anxiety to render Divine worship. (c) The Apostles never attended even to physical necessities in their own name.
Parker, City Temple, vol. ii., p. 117.
I. Man is, by nature, morally crippled and helpless; a beggar, a bondman, carried about at another's will. Great bodily infirmities are the shadows of the sins and weaknesses of the soul. What a cripple is among men, a sinner is before the angels and pure spirits on high. All sin works by privation. It shuts up senses and organs which God meant to be inlets of joy and channels of life.
II. There is a Name which can make us whole again, sound, glad, and free. Your soul wants precisely what that poor cripple's body wanted, power to stand, to walk, to leap, and to utter forth the praises of God. And that power is in Christ, and in Christ alone. Light to the blind, strength to the impotent, life to the dead is He. The more you think of it earnestly, the more you will find that life is just what you need. A man whose system is worn out can be patched up for a while by the physicians, but a new gush of life into it is what it needs. Give it that, or you patch and prop in vain. This is what Christ can truly do for your soul. I am not speaking now of the solace of His compassion, of the joy of His communion, of the sweetness of His love, of the glory of the hope which He inspires. I sum it all up when I say, "In Him is life." That life, God's life, He can give to man, He will give to you. It will be a power in the end, all-mastering, all-ruling, a power unto salvation.
III. This is the time to believe on that Name, and to rise up and walk. Does God care for wrecks? Let that poor cripple answer. Let the Lord's works of mercy answer. They were mostly fragments, broken fragments of humanity that He gathered; they were mostly wrecks that He saved. Publicans, harlots, thieves, prodigals, whatever the world flung out as worthless, He gathered. Such life is in Him, such power of quickening and re-creating souls, that wretched ones, whom Scribes and Pharisees cast out from the decent fellowships of earth, shall be reigning among the angels, white-robed, palm-crowned, through eternity. Lay hold on Christ and the Lord will lift you; you will stand up as a man, and look your tempters and tyrants in the face; you will find strength to defy them, and to win at first, at any rate, an easy victory. You will go forth to the old drudgery with a new and wonderful joy.
J. Baldwin Brown, Christian World Pulpit, Oct. 18th, 1876, p. 248.
I. The lame man. It is a fact that almost all the alms of the world are administered at the gate of the Temple. Almost all the charitable institutions of the world are dependent for their moral and pecuniary support; and almost all the benevolent movements of society are dependent for their success, on them that go up to the Temple at the hour of prayer. When money is needed to assuage the world's grief, to relieve the world's distress, men go straight to the gate of the Temple to beg. Christianity is founded not so much in the powers as in the needs of the race.
II. The cure of the lame man. The man sought alms—but the Apostles gave him what was better; they gave him health. Health without money is infinitely better than money without health. Moreover, by endowing him with health, they were conferring on him the ability to earn money; by imparting the greater they were also giving the lesser. In this the miracle was a sign, and typifies to us the Divine method of saving the world. The Gospel does not aim directly at improving men's circumstances, it aims at improving men themselves. But no sooner does it bring about a moral improvement in the men, than the men bring about a noticeable improvement in their surroundings. The Gospel converts the man, the man converts the house. Men need better houses and purer air, and more wholesome water; but the great want of men is life—more life; and Jesus Christ came that they might have life, and have it more abundantly. Utilitarianism does men good, Christianity makes them good.
J. Cynddylan Jones, Studies in the Acts, p. 52.
I. At that which Peter had not, "Silver and gold have I none." The question as to whether the Church's power is increased by worldly possessions is one of the very last importance. It was an essential condition of the prevalence of the Gospel that men should be holy, and if they were determined to cling to their sins it was a necessary consequence that the only way to peace should lie through contention. So, though it be the future of the Church to inherit the glory of the Gentiles, it is an essential condition of her power that she shall abandon all selfishness and covetousness, and if men will cling to selfishness and covetousness, then the only way to power is by stripping herself of her earthly possessions. God only knows whether that be necessary for the Church. Now, if ever, we must gird up our loins and trim our lamps, taking up the pilgrim's staff and scallop-shell, seeking upon the hard rough sands of the world's desert the way to the heavenly Jerusalem.
II. Notice next the positive aspect of the text. (1) The completeness of the miracle. The Apostle did but speak, and straightway by the Almighty power of God—it was as if an electric shock had passed through him—the impotent man could leap and walk. And so it is in the conversion of the soul. "If any man is in Christ, he is a new creation." (2) The name and means by which the miracle was wrought. "In the name of Jesus Christ of Nazareth, rise up and walk." There is the explanation of all; there is the explanation of the miracle; there is the secret spell of Apostolic power. Just at the time when His people are passing through gloom and sorrow, the highest heavenly power is wielded with a tenderest human pity, so that when we come in our weakness, our sin, our loneliness, and look up to heaven, we see not the naked blinding glory of the Deity, but the face of the High Priest who can be touched with the feeling of our infirmities bending over us in pity and love.
Bishop Moorhouse, Penny Pulpit, No. 407.
References: Acts 3:6.—Clergyman's Magazine, vol. i., p. 189; Christian World Pulpit, vol. vi., p. 3. Acts 3:6-8.—G. Brooks, Outlines of Sermons, p. 60. Acts 3:7.—New Outlines on the New Testament, p. 81.
Verse 10
The architecture of the old Jewish Temple may serve us for a parable today. The truth that it suggests will be the harmony between a noble undertaking and a beautiful beginning—that every true temple ought to have a beautiful gate. The importance of beginnings is the veriest commonplace of practical virtue. Think of the wisdom and love of God who has put the beauty of youth at the entrance of every human life, and especially now consider the child's religion.
I. The religion of childhood is not only possible, but it is the normal type of religion; is that which Christianity most contemplates; and that which, when Christianity shall have really entered into her power, all men shall accept as the very image and pattern of religion. The current idea of the Churches, that adult conversion is the type and intended rule of Christianity, comes largely from the fact that the first preachers of Christianity had of necessity to be largely occupied with men who had known nothing of Christianity in their youth. The evident design of God's creation, the comprehensive form of the incarnation, the clear presence in children of the power and the need of religion, these are the forces which, in spite of every tendency of the grown people to make children wait till they grow up, has always kept alive a hope, a trust, however blind, that a child's religion was a possible reality; that a child might serve, and love, and live for God.
II. What is the true character of the religion of a child? Certainly to be sweet and real, it must be the possession by God of the faculties and qualities that belong especially to childhood. (1) The first and most prominent of them all is the faculty of genuine, unhesitating, unqualified admiration. (2) Another thing in a child's religion is the perfect healthiness of his traditionalism, of his belonging to a certain sect, and holding certain opinions. Grown people often cling to the faith of their fathers controversially. Their love for it is mixed up with jealousy and spite and pride. A child knows nothing of all that. (3) The simplest and primary form of the presentation of the Gospel is the one which is preserved most truly and necessarily in the teaching of children. The child is a little Athenian, always listening for some new thing. And so the child is ready, if it can be rightly told him, to hear, above all the other messages that come to him out of this ever-opening and surprising world, the best and highest news of all, the Gospel, simply as glad tidings of the love of God and the salvation of the world by Jesus.
Phillips Brooks, Twenty Sermons, p. 127.
References: Acts 3:11, Acts 3:12.—Homiletic Quarterly, vol. iii., p. 459. Acts 3:12.—R. W. Dale, The Evangelical Revival, p. 171.
Verses 12-26
The speech of Peter may be regarded in four aspects:—
I. As showing the false method of looking at human affairs. "As though through our own power of holiness we had made this man to walk."
II. As showing the true method of regarding the most extraordinary events. "God hath glorified His Son Jesus."
III. As showing the only method of setting man right with God. "Repent ye therefore and be converted."
IV. As showing the sublime object of Jesus Christ's Incarnation. "To bless you, in turning away every one of you from his iniquities."
References: Acts 3:12-26.—R. W. Dale, Evangelical Revival, p. 171; Homilist, 3rd series, vol. iii., p. 314. Acts 3:13, Acts 3:15—Homiletic Quarterly, vol. iii., p. 460. Acts 3:14, Acts 3:15.—J. Baldwin Brown, Christian World Pulpit, vol. viii., p. 321. Acts 3:16.—Homiletic Quarterly, vol. iii., p. 461; vol. xix., p. 301.
The Danger and Results of Unbelief.
I. How St. Peter came to have a right to make allowance for the Jews. When the Apostle states that what the Jews did, they did through ignorance, he must be considered as conveying the idea that they were not acquainted with the actual character and dignity of Christ. They did not crucify Him as the Messiah, the Son of the Living God, but as one who pretended to be the Messiah, and who blasphemed in calling Himself the Son of God. But were the Jews, then, innocent in this their ignorance? We may not venture to say this; we may not venture to think that St. Peter implied this; for this would evidently impeach the whole course of Christ's ministry on earth, representing His miracles as defective credentials, inadequate to the establishing of the character which He claimed to Himself. The Jews, beyond all question, if ignorant, were to blame for their ignorance. They might have known, they ought to have known, that Jesus was the Christ, and ignorance can only be an excuse when we do not ourselves cause it, whether through wilfully neglecting means of obtaining information, or cherishing prejudices which bar out the truth. It was not in crucifying Christ, but in rejecting the final evidence afforded by the descent of the Holy Ghost, that they perpetrated the sin for which they were cast off.
II. Contrast the case of the modern unbeliever with that of the Jews and judge whether it be an exaggerated charge which would fix on the latter the far greater criminality. The Jew crucified Christ whilst His appearance was that of an ordinary man; we crucify Him afresh when He has assumed the glory which He had from the beginning with the Father. It was the Son of Man on earth who was crucified by the Jew; it is the Son of God in the heavens who is crucified by ourselves. Christ had not then given the most touching proof of His love and His compassion. He had not yet died for His enemies; neither was it understood, even by His disciples, and much less by His adversaries, that the death which He was willing to undergo was to serve as a propitiation for the sins of the whole world. The Jew, at the time, and in the act referred to in our text, had not the power of sinning such a sin as any one of us sins, when, through not believing in Christ, he crucifies Him afresh. It is Christ's having been once crucified in the flesh, which gives such immeasurable heinousness to His being crucified again in the spirit.
H. Melvill, Penny Pulpit, No. 1498.
References: Acts 3:17.—Contemporary Pulpit, vol. ix., p. 170. Acts 3:17, Acts 3:18.—Homiletic Quarterly, vol. iii., p. 462. Acts 3:19.—Spurgeon, Sermons, vol. xiv., No. 804; J. G. Rogers, Christian World Pulpit, vol. vii., p. 8; Homiletic Magazine, vol. xiii., p. 8; W. Hay Aitken, Around the Cross, pp. 33, 49. Acts 3:19.—J. H. Thorn, Laws of Life after the Mind of Christ, 2nd series, p. 105. Acts 3:19, Acts 3:20.—R. S. Turner, Christian World Pulpit, vol. i., p. 264; H. W. Beecher, Ibid., vol. ii., p. 172; Homiletic Magazine, vol. xix., p. 115.
Acts 3:19, Acts 3:21
The Restitution of All Things will be:
I. A clearing away of suffering. Earth shall be restored to its original beauty; its face shall be wiped from tears; its scarred and stained countenance shall be radiant again with a more than Eden loveliness: for it is one of those "all things" which must receive restitution when the heaven which has received Him shall send Jesus back.
II. We pass to a thought not less bright, and far more practical, when we say that man, his soul and body, his very being and life, is among these "all things" which are awaiting a restoration. Who that has seriously tried the struggle to be holy, the warfare under Christ's banner with indwelling, obstinate, inveterate sin, has not found himself vexed and irritated, if not reduced to despair, by perpetual failure; has not felt times without number that without a promise he would surrender, he would capitulate on the instant, and that the promise which keeps him fighting is not more that of "strength as his day," than that of victory in the end? If there be a restoration of all things at the Advent, and amongst these "all things" I am, then I will arise, if need be, from a thousand falls in one day, cast down, but not destroyed.
III. That restitution of all things which thus affects earth and the man, has an aspect, finally, towards God. It is one of the express revelations of the times of refreshing, that then the conscious presence, the spiritual Shechinah, the Divine companionship, will be restored. In the light of that sun all lesser luminaries will pale, if not vanish. That only can live there which can bear the light of God. Sin will be destroyed, and all that is of it; selfish affection, creature worship, idolatrous love. All that then survives will have fallen into its place by instinct; all other love will shine in the love of God; stronger, more intense than ever, yet entirely pure, entirely devout, absolutely sinless and selfless. In the prospect of that admission into the very presence of God, let us be willing to endure now the difficulty of the pursuit and the delay of the attainment. If we give up the search, we must abandon the hope, if we will only seek on, we shall surely find.
C. J. Vaughan, Temple Sermons, p. 66.
References: Acts 3:19-21.—Homiletic Quarterly, vol. iv., p. 35; Homilist, 1st series, vol. v., p. 260; T. L. Cuyler, Christian World Pulpit, vol. xiv., p. 352; J. Keble, Sermons for Lent and Passiontide, p. 318; C. J. Vaughan, Church of the First Days, vol. i., p. 111. Acts 3:20.—Contemporary Pulpit, vol. vii., p. 368. Acts 3:21.—S. Martin, Pulpit, vol. iii., No. 1625. Acts 3:22.—Clergyman's Magazine, vol. ii., p. 217. Acts 3:22-26.—Homiletic Quarterly, vol. iv., p. 36.
Here, in few words, is the plan proposed by our heavenly Father to make us happy, a plan well worthy to be considered.
I. God does not secure happiness to his people by making all of them rich. Instead of saying "Blessed are ye rich," he says, "Blessed are the poor."
II. Our heavenly Father does not propose to make us happy by bestowing on us the empty honours of the world.
III. God's plan for making His people happy does not consist in affording them a large share of worldly pleasure.
IV. There can be no salvation for us unless we are delivered from our sins. God only makes men happy by making them holy. The object of the coming of the Son of God in the flesh was that "He might save His people from their sins."
J. N. Norton, Old Paths, p. 159.
I. The boldness and loftiness of the claim which is here made for Jesus Christ.
II. The dawning vision of a kingdom of world-wide blessings.
III. The purely spiritual conception of what Christ's blessing is "To bless you in turning away every one of you from his iniquities."
A. Maclaren, A Year's Ministry, 1st series, p. 245.
References: Acts 3:26.—W. Hay Aitken, Around the Cross, p. 97; Homilist, 2nd series, vol. iv., p. 377; C. J. Vaughan, Church of the First Days, vol. i., p. 130. Acts 3:26.—Christian World Pulpit, vol. iv., p. 190; T. M. Herbert, Sketches of Sermons, p. 55.
The Lame Man Healed
We are reminded by this incident:
I. That there are some things more valuable than money. Peter with his gift of healing was of infinitely greater service to this lame man than if he had possessed the riches of Croesus. The moment wealth becomes an end to be sought simply for its own sake, it ceases to be a blessing.
II. That fidelity is the true kindness in the end. Mark how pointedly Peter here addresses the multitude. He charges home upon them, in unmistakable terms, the crucifixion of the Lord Jesus Christ. Faithfulness is to be tempered with justice in all things, but especially when we are dealing with the unconverted.
III. That the enjoyment of times of refreshing from God's presence is inseparably connected with our return to God.
W. M. Taylor, Peter the Apostle, p. 185.
References: Acts 3—J. Oswald Dykes, Preacher's Lantern, vol. iv., p. 385. Acts 4:1, Acts 4:2.—Parker, City Temple, vol. iii., p. 335. Acts 4:1-4.—Homiletic Quarterly, vol. iv., p. 37.
Nicoll, William R. "Commentary on Acts 3:4". "Sermon Bible Commentary". https://www.studylight.org/commentaries/sbc/acts-3.html.

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text:
Q: replaceText Google Apps Script pass through blank values to Google Doc Merge I'm doing something like a mail merge - taking values from google sheets (actually script array) and replacing placeholder variables in a google doc template. I have about 50 of these. Below are three of them. The format looks like this:
body.replaceText("{Property Address}", propertyAddress);
body.replaceText("{Lead Type}", leadType);
body.replaceText("{Deal Type}", dealType);

The problem I'm running into is sometimes a variable can be blank - e.g. say leadType is blank or "". There is an error occurring if it's blank. What I want to do is if leadType has a value, then replace {Lead Type} in the Google Doc with the value. Else if leadType = null or "" or undefined etc., then replace {Lead Type} in the Google Doc with "". I'd rather not have to do an if/then statement for each of these 50 lines. Is there an easy way to have either a value or a blank push through to the sheet (I want a blank to go through because I don't want the final sheet to be printed with {} variables)

A: How about using the OR operator?
body.replaceText("{Property Address}", propertyAddress || "");
body.replaceText("{Lead Type}", leadType || "");
body.replaceText("{Deal Type}", dealType || "");


Pulagam is an Indian traditional dish made with rice and toor dal. We make this on festival days to offer (Naivedyam) Goddess Lakshmi.
2&1/2 cups of water. Use more water if you want more soft rice.
Put a small sauce pan on medium heat. Add ghee to the pan. Fry the washed rice and dal in ghee for 2 minutes. Then take these into your cooker vessel and add water. Add a little turmeric powder and salt for your taste. Turn on the electric cooker and cook until it is done. Once it is done it will turn off automatically. Add 1 tsp of ghee to pulagam and mix it gently. You can serve this hot with any raita, chutney or pickle.
Pulagam is made in different ways in different regions. This is a basic recipe. You may also make it with moong dal instead of toor dal.
Whole grain brown rice will provide you with essential Vitamin B, Vitamin E, fiber and selenium. Toor dal is a good source of protein and fiber.


text:
LADY SCOT
Ripped Bodice
Love's Sweet Arrow
Also by Natasha Lester:
The Three Lives of Alix St. Pierre, January 2023
Hardcover / e-Book
The Riviera House, September 2021
Paperback / e-Book
The Paris Secret, September 2020
The Three Lives of Alix St. Pierre
Natasha Lester
Readers Selection
On Sale: January 10, 2023
Kindle: B09ZB6BVWV
Historical fiction at its best!
An unforgettable and "fascinating tale" of an orphan turned WWII spy turned fashion icon in Paris (Kate Quinn, New York Times bestselling author of The Diamond Eye).
Alix St. Pierre. An unforgettable name for an unforgettable woman. She grew up surrounded by Hollywood glamor, but, as an orphan, never truly felt part of that world. In 1943, with WWII raging and men headed overseas to fight, she lands a publicity job to recruit women into the workforce. Her skills—persuasion, daring, quick-witted under pressure—catch the attention of the U.S. government and she finds herself with an even bigger assignment: sent to Switzerland as a spy. Soon Alix is on the precipice of something big, very big. But how far can she trust her German informant…?
After an Allied victory that didn't come nearly soon enough, Alix moves to Paris, ready to immerse herself in a new position as director of publicity for the yet-to-be-launched House of Dior. In the glamorous halls of the French fashion house, she can nearly forget everything she lost and the dangerous secret she carries. But when a figure from the war reappears and threatens to destroy her future, Alix realizes that only she can right the wrongs of the past …and finally find justice.


text:
var HelloMessage = React.createClass({
    render() {
        return
Hello {this.props.name}
;
    }
});
ReactDOM.render(
    <HelloMessage name="React" />,
    document.getElementById('react')
);


text:
Arctic Monkeys Released Warp Speed Chic Film
Celebrity, Entertainment, Music, Watch
by Onn Editor
Earlier this year, the Arctic Monkeys released Tranquility Base Hotel & Casino, a concept album set in a casino piano bar on the moon. Now the band's longtime film partner Ben Chappell (who has directed several of their videos) was at the French mansion where they crafted the songs, and this week released a trippy short companion film to the album with a title Warp Speed Chic.
In a statement, Chappell explained that he joined the band in France just as the songs were coming together.
"I was still only hearing bits and pieces of the tunes as they were being recorded," he said. Then, almost a year later, he went to their show in July in Lyon, France, to the film the finished versions. " I put the live songs next to my shots of them recording and found it to be quite nice," he said. "With that as a starting point, I then used James Ford's instrumental mixes of the album and I put an assortment of the images we created for this album overtop, giving a glimpse into the last year and the full progression of Tranquility Base Hotel & Casino."
Director Ben Chappell shot the entire film on a 16mm camera while the band spent time in France crafting the album. "After spending a year documenting Tranquility Base Hotel & Casino I wanted to try and pull it all together in a short film," Chappell said.
The documentary shows the progression of the band recording songs like "The Ultracheese" in a crowded living room, to ultimately performing it onstage, and gives a behind-the-scenes view as the band films the music video for "Four Out Of Five."
The documentary also contains shots of Arctic Monkeys in concert and shooting videos and photos for the album.
Tags: Arctic Monkeys, Featured
Onn Editor
Mikaela Shiffrin's Unbelievable Killington Cup Journey! An Astonishing Win By Half A Second That Secured Shiffrin's Hat-Trick
Rolls-Royce Demonstrated The World's First Fully Autonomous Ferry With The Help Of Finferries!
Onn Editor December 6, 2018


text:
Nineteen News
News You Can Trust
July 11, 2019 World News
Queen to be protected from Brexit constitutional crisis, vows former British PM John Major
Posted by 19&travel
Former PM John Major said a no-deal Brexit would force the Queen into a British constitutional crisis.
Former British prime minister Sir John Major has vowed to save Queen Elizabeth II from a constitutional crisis, if Boris Johnson moves to suspend parliament in order to deliver a no-deal Brexit.
Sir John Major said a no-deal Brexit may trigger the suspension of Westminster
To do that, the British PM would have to seek permission from the Queen
Sir John said that would place her in a position "no serious politician" should consider
Mr Johnson, the current favourite to be Britain's next prime minister, has refused to rule out suspending, or proroguing, parliament to ensure the United Kingdom leaves the European Union on October 31, with or without a deal.
Sir John told the BBC, if he was to do so, the move would leave the Queen with no other choice but to provoke a constitutional crisis in one of the world's oldest and most stable democracies.
"If her first minister asks for that permission, it is almost inconceivable that the Queen will do anything other than grant it," Sir John said.
"She is then in the midst of a constitutional controversy that no serious politician should put the Queen in the middle of."
As a constitutional monarch, the Queen's executive authority is designed to sit above politics.
As such, she does not have the power to make or pass legislation, but all legislation must be approved by her.
The same process applies in Australia under the auspices of the Governor-General who represents the Queen.
Last parliamentary suspension during English Civil War
Sir John accused Mr Johnson of hypocrisy for backing Brexit to secure more power for Britain's parliament, only to propose to sideline MPs when it suited him.
Why the monarchy still matters
In an era when representative government is despised and democratic accountability has resulted in the creation of undemocratic and unaccountable elites, we should not be surprised if monarchy becomes ever more popular, writes Philip Blond.
He said parliament had not been suspended since King Charles I did so during the English Civil War. Charles was eventually executed, in 1649.
"The idea of proroguing parliament is utterly and totally unacceptable from any British parliamentarian or democrat," Sir John said.
"If that were to happen, there would be a queue of people who would seek judicial review. I for one would be prepared to go and seek judicial review."
The former prime minister, who is backing Mr Johnson's prime ministerial rival Jeremy Hunt, said he was speaking in a personal capacity.
Mr Johnson responded by saying it would be "very odd" to give the judiciary a say over Brexit.
"I think everybody is fed up with delay and I think the idea of now consecrating this decision to the judiciary is really very, very odd indeed," he said.
Johnson fails to rule out suspending parliament over Brexit
The question of suspending parliament was raised during a televised debate between Mr Johnson and Mr Hunt, the Foreign Secretary, on Tuesday evening.
While Mr Hunt categorically ruled it out, Mr Johnson said he would "not take anything off the table".
Brexit victory was never an option
The defeat of Theresa May's deal will go down as the biggest in British modern history, writes Samantha Hawley.
Votes in Parliament have indicated a majority of MPs are against a no-deal Brexit because ofconcerns it would cripple supply chains and damage trade.
The pound was trading near its lowest level for more than two years on Wednesday (local time), even with better-than-expected readings on the economy, as fears about a no-deal Brexit grew stronger.
On Tuesday, MPs narrowly approved a measure that could make it harder for the next prime minister to suspend parliament.
House Speaker John Bercow has said it was "blindingly obvious" that the next prime minister would not be able to sideline parliament, adding:
"Parliament will not be evacuated from the centre stage of the decision-making process on this important matter."
Sir John said there was a risk Britain would not be ready to leave the bloc in October, and Mr Johnson lacked leadership qualities.
He followed other party heavyweights in questioning whether the former London mayor was fit for the highest office.
"National leaders look first at the interests of the country — not first at the interests of themselves," he said.
Between 1990–1997, the Major administration was plagued by Conservative Party disputes over Europe, which saw Britain crash ignominiously out of the European Exchange Rate Mechanism, the predecessor to the single currency, in 1992.
Queen to be protected from Brexit constitutional crisisvows former British PM John Major
Australian mother makes urgent plea for Government help from inside Syrian al-Hawl detention camp
Islamic State recruit from Melbourne languishes in Kurdish prison as his father begs for his release
Hello my beautiful people, how are you going? Donate $5 to buy me a coffee so I have fuel I need to keep producing great content! Thank you so much Calvin
Meatball Soup
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HOW TO TAKE CARE OF YOUR SKINS
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5 Ways Gardening is Therapeutic in Fighting Dementia
10 EASY COOK RECIPES


text:
Q: How to scroll using an external USB trackpad that is not a magic-mouse? In some cramped situations I would prefer to not use the macBook pro internal trackpad but rather an external one as shown

The right/left clicks and the trackpad area do work fine for traversing the screen and clicking. But then how to scroll the current application window?  On the internal trackpad that would be done by two fingers on the trackpad.
The Settings | Trackpad only applies to the internal trackpad of the macBook pro.  There is also an "Configure bluetooth Trackpad" : but this is a usb.
Is this possible? I do not want to use a bluetooth because I have bad issues with the bluetooth freezing the macOS.
I have a 2020 macBook pro 16" and am on Catalina.

A: I suspect that the trackpad settings/bluetooth settings may not be all that useful for this.
Firstly if you do a system report (Apple icon ► About This Mac ► System Report...), you can check to see if the USB device is recognised. This seems to be the case since your saying it recognises left and right clicks.
My only suggestion is that this may require some third party software to properly recognise it. For example: SteerMouse. Similar to if you have ever installed a joy stick for a simulator, it usually requires the actual third party software to recognise it.

A: Looking at the picture of your trackpad it looks that scrolling is done by sliding the finger around the right border of the pad. Have you checked whether this works? Not sure if it supports multitouch, which would be required for the "apple gestures".
There's two Apps I would consider checking out.

*

*Better Touch Tool (paid) - which allows to customize touch gestures in great detail and even add your own. But AFAIK only works on Apple devices. Still worth a try.

*Karabiner Elements (free) - which allows to create very complex mappings, but it is complex to operate. You can easily create mappings where mouse movent is turned into scrolling while a key is held down.  There is a Guide for hackintoshes which may help.


If you are not at the home page, click on HOME button (1). Next click on CASUAL (2).
That changes screen. Next click ALL PUBLIC AND PRIVATE CLUBS (3).
There are some more clicks needed ;) Click on ALL PRIVATE CLUBS (4).
In the list of all those clubs locate BBO Intermediate Advanced Club, managed by bbo_iac. Click SHOW TABLES of that room (5).
That's because web BBO version never shows all tables in any room, you have to choose one of the horizontal tabs at the top.
If you choose OPEN TABLES, it will list tables with empty seats (6).
If you choose FULL TABLES, it will only list completely full tables (7).
And, if you click on FRIENDS (8), listed will be tables where at least one player (not kibber) is on your friend list.
You also can always start the new table (button just over chat area).
Click on the table you want to join.


text:
Q: How I create a View for more lists? It is my first question and I a noob in Sharepoint but I get a task to set a view for a list. I explain the idea. Our company has a teamsite where we can see who have task and who do this task.
Now I want to create a view that i can see my tasks only. The Problem is the list has under lists with task by a hyperlink to a other list.
Can I set a view with a filter about more list in sharepoint?

If I click on "Werk" in the first picture i jump to picture 2 and I want a view for all lists.
I need help :(


text:
<?php

class AmbajuTable extends Doctrine_Table
{
}


Are you planning on buying a condo, townhouse, or a detached house in Scarborough Ontario? Andrew Shevtsov is the real estate agent for you.
When buying with Andrew, all of your fees, closing costs and taxes will be broken down and explained to you. There are no surprises when working with Andrew.
Andrew Shevtsov knows every neighborhood in Scarborough like the back of his hand. Want to make sure you buy in the right area? Contact Andrew today with any questions.
Andrew is always one text, email or call away from answering all of your questions. Found a property you want to view? Contact Andrew to help you out.
When you decide to buy a property, your main concern is finding a home that meets all your needs and most importantly, your budget. Meeting those requirements can be a challenging task in the Scarborough market which is why working with the right agent is so important. Andrew Shevtsov is an expert local real estate agent that knows the Scarborough market like the back of his hand. Whether you are buying a condo, a house, or anything in between, Andrew will know precisely how to match you to your dream home. His expert negotiation skills will help you not only seal the deal but also get you the best price possible.
The first step to buying any property is getting pre-approved by your bank for a mortgage so that you know the exact budget you can afford. The next step is to create a realistic wishlist of all your desired property characteristics. Once you have those two things, you can present them to your real estate agent and start viewing properties that meet your requirements. When you find a home that you want to purchase, you make an offer and negotiate until it is accepted.
Assuming that you are already pre-approved to buy a home within a specific budget, you can own a property in as little time as a week. Of course, that would be considered the minimum amount of time. Most property searches can last months, and once you do finally find the right property and agree on a price, you will have to wait for the closing period to end before you take possession of the property. Closing periods take around three months, but they can be negotiated down to any timeframe.
Our office is located in Scarborough Ontario, and we specialize in serving the entire region. These neighborhoods include the following; Birch Cliff, Oakridge, Cliffside, Cliffcrest, Scarborough Village, Guildwood, West Hill, Port Union, West Rough, Rough Hill, Highland Creek, Woburn, Bendale, Scarborough Junction, Clairlea, Malvern, Hillside, Dorse Park, Wexford, Maryvale, L'Amoreaux, Agincourt, and Milliken.
How do you view a property that is for sale?
When you find a property that is for sale, and you wish to view it in person, your best option is to go with a local real estate agent that represents you. Just tell your agent the address of your desired properties, and they will be able to schedule the necessary appointments.
Get any of your real estate related questions answered by expert real estate agent Andrew Shevtsov by simply calling or texting him directly, or sending him an email using the contact form below.


text:
\section{Introduction}

Cosmic evolution is an accumulation of substantial changes including
dark energy and gravitational collapse, which has developed the
interest of researchers to debate on a variety of problems of
cosmology as well as gravitational physics. When the fuel in the
core of a star runs out and there is not enough pressure to
counteract the strong gravitational pull, the star collapses,
resulting in the formation of new compact objects. The rapid
expansion of the cosmos is confirmed through several astrophysical
observations such as supernova type Ia and cosmic microwave
background. Alternative gravitational theories support expanding
behavior of the universe. These theories are considered the most
favorable and remarkable techniques to discover hidden attributes of
the cosmos. These approaches are constructed by adding higher order
curvature invariants in the geometric part of the Einstein-Hilbert
action. The $f(\mathfrak{R})$ gravity is the natural extension of
general relativity (GR) which discusses some novel aspects of cosmos
\cite{3a}.

Many researchers have shown a keen interest in the idea of curvature
and matter coupling. These modified proposals explain the rotation
curves of galaxies and various cosmic eras. The conservation law
does not hold in these theories, assuring the existence of an extra
force on particles. These theories are assumed to be an excellent
candidate to understand the dark cosmos. The minimal
curvature-matter coupling was developed in \cite{4a} named as
$f(\mathfrak{R},T)$ gravity, while non-minimal coupled theory is
$f(\mathfrak{R}, {T},{\mathfrak{R}}_{\gamma\delta}
{T}^{\gamma\delta})$ gravity \cite{5a}. The presence of
singularities in GR is a major problem due to their prediction in
high energy regions, where GR is not viable due to quantum effects
and there is no specific approach in quantum theory. Accordingly, a
new modification of GR has been developed by adding the analytic
function ${T}_{\gamma\delta}{T}^{\gamma\delta}=\mathcal{T}^{2}$ in
the functional action of $f(\mathfrak{R})$ theory \cite{7a}, dubbed
as $f(\mathfrak{R},\mathcal{T}^{2})$ gravity. This modified theory
is also known as energy-momentum squared gravity (EMSG) which
includes squared components of the matter variables, that can be
used to examine several interesting astrophysical consequences. In
the early universe, this theory has a regular bounce, i.e., finite
maximum energy density as well as small-scale parameter \cite{8a}.
As a result, the big-bang singularity can be resolved using this
non-quantum prescription.

It is worth noting that this gravity overcomes the singularity of
spacetime without affecting the cosmological evolution. Board and
Barrow \cite{9a} investigated exact solutions with perfect matter
configuration in the presence of a specific EMSG model. Nari and
Roshan \cite{10a} studied dense objects that were stable as well as
physically realistic. Bahamonde et al. \cite{11a} explored different
EMSG coupling models (\textit{minimal}/\textit{non-minimal}) and
found that these models unveil mysteries of the cosmos. This theory
incorporates significant work and illustrates numerous cosmic
implications \cite{12a}.

A hypothetical highly compact star free from a singularity is
suggested as a promising substitute for the black hole, which may be
developed by scrutinizing the essential concept of Bose-Einstein
condensation, known as gravastar. Mazur and Motolla \cite{13a}
proposed this unique model which provides the corresponding solution
of the Einstein field equations, they fabricated this cold and dark
object as a gravitationally vacuum star or gravastar. Several
researchers are interested to study the structure of gravastar
because this model is expected to resolve two major issues
associated with black hole structure: the singularity formation and
the information paradox. The gravastar structure comprises of three
different domains, interior region, thin-shell, and exterior region.
The connection between state variables in the inner domain is of the
type $\varrho=-\mathcal{P}$, where $\varrho$ and $\mathcal{P}$
represent energy density and pressure, respectively. These matter
configuration yields non-attractive force which is responsible to
produce enough pressure to resist the collapsing phenomenon. This
interior zone is guarded by a thin-shell that holds stiff fluid
obeying the following relationship $\varrho=\mathcal{P}$. The inner
domain experiences inward-directed force exerted by the shell,
consequently hydrostatic equilibrium is maintained and a
singularity-free compact object is formed. The exterior domain is
completely vacuumed and satisfies the equation of state (EoS)
$\varrho=0=\mathcal{P}$.

There is no observational evidence in the support of gravastar, but
few indirect evidence justifies the existence of gravastar
structure. Sakai et al. \cite{d1} provided the criteria to detect
the gravastar by investigating their shadows. Kubo and Sakai
\cite{d2} used gravitational lensing to detect gravastar by
observing maximum luminosity effects. But, these effects are not
found in black holes with equal mass. Moreover, using
interferometric LIGO detectors, a cosmic event GW150914 observed a
ringdown signal \cite{d3}-\cite{d4}. These signals are released by
objects having no event horizon, thus, strongly suggesting the
presence of gravastar. Recently, an image taken by the first M87
Event Horizon Telescope has been examined and found likely to
gravastar \cite{d5}.

Visser and Wiltshire \cite{5aa} investigated the stability of
gravastars in GR corresponding to specific EoS and found dynamically
stable gravastar structures. Carter \cite{6aa} demonstrated
non-singular solutions of gravastars to examine their various
characteristics. Horvat and Iliji\'{c} \cite{7ddd} studied the
gravastar configuration by taking inner de Sitter and outer
Schwarzschild black hole. There is extensive literature on gravastar
solutions with different matter contents \cite{9aa}-\cite{14aa}.
Ghosh et al. \cite{cd} analyzed higher-dimensional gravastars and
results were compared with 4-dimensional analog model. The
astonishing findings of gravastars inspired the research community
to analyze their physical attributes in modified theories
\cite{29aa}-\cite{65aa}.

Electromagnetism plays a crucial role in the study of structure
evolution and stability of collapsing celestial objects. To maintain
the equilibrium state of a stellar object, a star needs an enormous
amount of charge to overcome the strength of the gravitational pull.
Lobo and Arellano \cite{cg1} developed gravastar solutions
incorporating nonlinear electrodynamics and investigated its
significant structural properties. Horvat et al. \cite{cg2}
discussed charged gravastar and computed surface redshift, the EoS
parameter as well as the speed of sound for the model under
consideration. Turimov et al. \cite{cg3} provided a brief study on
slowly rotating gravastars by taking into account extremely
magnetized perfect matter. Usmani et al. \cite{cg4} used conformal
motion to investigate gravastar by taking charged interior and
Reissner-Nordstrom metric in the exterior. It may be noted that
charged interior behaves as an electromagnetic mass model and plays
a key role in the stability of gravastar structure by generating
gravitational mass. Rahaman et al. \cite{cg5} explored gravastar
structure in 3-dimensional spacetime with the contribution of charge
and investigated several viable features for such compact
structures. Ghosh et al. \cite{cg6} established a charged gravastar
model in a higher-dimensional manifold and found that physical
attributes ensure the viability of the model. Sharif and Javed
\cite{7dddd} investigated the stability of gravastars in the
framework of quintessence as well as regular black hole geometries
and obtained a continuously increasing profile of physical
characteristics corresponding to thickness of thin-shell.

Sharif and Waseem \cite{cg8} analyzed charged gravastar structure
using conformal motion in realm of $f(\mathfrak{R}, T)$ gravity.
Yousaf et al.  \cite{cg7} studied charged gravastar in the same
gravity and discussed its stability. Bhatti et al. \cite{w2}
explored the stability of charged gravastar in the modified gravity.
Bhar and Rej \cite{w3} studied gravastar admitting conformal motion
to analyze the contribution of charge on the stability of the
considered model. Bhatti and his collaborators \cite{w4} also
investigated gravastars in the realm of ${f(\mathcal{G})}$ gravity,
where $\mathcal{G}$ defines Gauss-Bonnet invariant. They
investigated different attributes of gravastar with and without
electromagnetic field. Various attributes of gravastar solutions are
also analyzed through the gravitational decoupling technique
\cite{w5}. We have studied gravastar structure with Kuchowicz metric
in the context of EMSG gravity \cite{w6} . Recently, Sharif and
Saeed \cite{w7} studied charge-free gravastar accepting conformal
motion in the background of $f(\mathfrak{R},\mathcal{T}^{2})$
gravity and discussed the behavior of various physical attributes.
Motivated by numerous works presenting effects of charge, we are
interested to analyze the influence of charge on the gravastar model
in $f(\mathfrak{R},\mathcal{T}^2)$ gravity.

This paper explores the influence of charge on the geometry of
gravastar in the background of $f(\mathfrak{R},\mathcal{T}^{2})$
gravity. We analyze its structure for the specific EMSG model using
three different domains and also discuss physical attributes
graphically. The paper is organized in the following format. Section
\textbf{2} demonstrates the basic formalism of this theory under the
impact of charge. Section \textbf{3} investigates the structure of
charged gravastar with relevant EoS for each region and section
\textbf{4} reflects various physical features of charged gravastar
including EoS parameter, entropy, proper length as well as energy.
We summarize the results in the last section.

\section{Energy-momentum Squared Gravity}

The action of EMSG theory with the inclusion of charge is classified
as \cite{7a}
\begin{equation}\label{1}
\mathcal{I}=\frac{1}{2\kappa^2}\int
d^4xf\left(\mathfrak{R},{\mathcal{T}}^{2} \right){\sqrt{-g}}+\int
d^4x (\mathcal{L}_{m}+\mathcal{L}_{e})\sqrt{-g},
\end{equation}
where $g$ exhibits determinant of the metric tensor and
$\kappa^{2}=8\pi$ is the coupling constant. Moreover,
$\mathcal{L}_{e}=\frac{-1}{16\pi}\mathcal{F}^{\gamma\delta}\mathcal{F}_{\gamma\delta}$
where
$\mathcal{F}_{\gamma\delta}=\varphi_{\delta,\gamma}-\varphi_{\gamma,\delta}$
represents the electromagnetic field tensor and $\varphi_{\gamma}$
demonstrates the four potential. In comparison to GR, the action
includes additional degrees of freedom which enhance the chances to
get analytical solutions. Due to the existence of an extra force and
matter-dominated era, this theory offers a platform to gain
significant outcomes. The variation of the action with respect to
the metric tensor provides the equations of motion
\begin{equation}\label{2}
\mathfrak{R}_{\gamma\delta}f_{\mathfrak{R}}-\frac{1}{2}g_{\gamma\delta}f+g_{\gamma\delta}\Box
f_{\mathfrak{R}}-\nabla _{\gamma}\nabla_{\delta}f_{\mathfrak{R}}
=\kappa^{2}T_{\gamma\delta}-\Theta_{\gamma\delta}f_{\mathcal{T}^{2}}+\kappa^{2}\mathrm{E}_{\gamma\delta},
\end{equation}
where $f\equiv f(\mathfrak{R}, \mathcal{T}^{2})$, $\Box=
\nabla_{\gamma}\nabla^{\gamma}$, $f_{\mathfrak{R}}= \frac{\partial
f} {\partial \mathfrak{R}}$, $f_{\mathcal{T}^{2}}= \frac{\partial f}
{\partial\mathcal{T}^{2}}$,
\begin{eqnarray}\label{3}
\Theta_{\gamma\delta}
=-2\mathcal{L}_{m}\left(T_{\gamma\delta}-\frac{1}{2}g _{\gamma\delta
}T\right)-4\frac{\partial^{2}\mathcal{L}_{m}}{\partial
g^{\gamma\delta}
\partial g^{\theta\eta}}T^{\theta\eta}-TT_{\gamma
\delta}+2T_{\gamma}^{\theta}T_{\delta\theta},
\end{eqnarray}
and $\mathrm{E}_{\gamma\delta}$ is the electromagnetic
energy-momentum tensor given by
\begin{equation}\label{}
\mathrm{E}_{\gamma\delta}=
\frac{1}{4\pi}\left(\mathcal{F}^{\theta}_{\gamma}\mathcal{F}_{\delta\theta}-\frac{1}{4}g_{\gamma\delta}
\mathcal{F}^{\theta\eta}\mathcal{F}_{\theta\eta}\right).
\end{equation}

The covariant divergence of Eq.(\ref{2}) turns out to be
\begin{equation}\label{4}
\nabla^{\gamma}T_{\gamma\delta}=\frac{1}{2\kappa^{2}} \left[-f_{
\mathcal{T}^{2}}g_{\gamma\delta}{\nabla^{\gamma}}{\mathcal{T}^{2}}+2{\nabla^{\gamma}}(f_{
\mathcal{T}^{2}}{\Theta_{\gamma\delta}})+\nabla^{\gamma}\mathrm{E}_{\gamma\delta}\right].
\end{equation}
Due to the coupling of matter and geometry, the conservation law is
failed in this theory. The field equations of EMSG reduce to
$f(\mathfrak{R})$ and GR when  $f(\mathfrak{R},\mathcal{T}^{2})=
f(\mathfrak{R})$ and $f(\mathfrak{R},\mathcal{T}^{2})=\mathfrak{R}$,
respectively. In the investigation of different astrophysical and
cosmological scenarios, the distribution of matter is crucial. To
examine features of the charged gravastar, we use isotropic matter
distribution
\begin{equation}\label{5}
T_{\gamma\delta}=\left(\mathcal{P}+\varrho\right)\emph{U}_{\gamma}\emph{U}_{\delta}-g_{\gamma\delta}\mathcal{P},
\end{equation}
where $\emph{U}_{\gamma}$ depicts four-velocity. The matter
distribution has two possible choices of $\mathcal{L}_{m}$, either
we can take it as $-\varrho$, or the other choice is
$\mathcal{L}_{m}=\mathcal{P}$. For minimally coupled theories of
gravity, the distribution is unaffected by the choice of matter
lagrangian \cite{pp}. Here, we take $\mathcal{L}_{m}=\mathcal{P}$
and using Eqs.(\ref{3}) and (\ref{5}), it follows that
\begin{eqnarray}\nonumber
\Theta_{\gamma\delta} = -\left(3\mathcal{P}^2+\varrho^2+4
\mathcal{P}\varrho\right)\emph{U}_{\gamma}\emph{U} _{\delta}.
\end{eqnarray}
Manipulating Eq.(\ref{2}), we have
\begin{equation}\label{6}
\mathbb{G}_{\gamma\delta}=
\frac{1}{f_{\mathfrak{R}}}\left(T_{\gamma\delta}^{c}+\kappa^{2}T_{\gamma\delta}
+\kappa^{2}\mathrm{E}_{\gamma\delta}\right)=T_{\gamma\delta}^{eff},
\end{equation}
where $\mathbb{G}_{\gamma\delta}=\mathfrak{R}_{\gamma\delta}
-\frac{1}{2}\mathfrak{R}g_{\gamma\delta}$ is the Einstein tensor,
$T_{\gamma\delta}^{c}$ denotes the additional impact of EMSG and
$T_{\gamma\delta}^{eff}$ expresses the effective stress-energy given
as
\begin{equation}\label{7}
T_{\gamma\delta}^{eff}=\frac{1}{f_{\mathfrak{R}}}\bigg\{\kappa^{2}T_{\gamma\delta}-g_{\gamma\delta}\Box
f_{\mathfrak{R}}+\nabla_{\gamma}\nabla_{\delta}
f_{\mathfrak{R}}-\Theta_{\gamma\delta}f_{\mathcal{T}^{2}}+
\frac{1}{2}g_{\gamma\delta}\left(f-\mathfrak{R}f_{\mathfrak{R}}\right)\bigg\}.
\end{equation}

The field equations become highly non-linear when derivatives of
multivariate functions are involved and obtaining exact solutions is
more challenging. The solution to non-linear equations is obtained
by employing a particular $f(\mathfrak{R},\mathcal{T}^{2})$ linear
model given as \cite{7a}
\begin{equation}\label{8}
f(\mathfrak{R},\mathcal{T}^{2})=\mathfrak{R}+\zeta \mathcal{T}^{2},
\end{equation}
where $\mathcal{T}^{2}=3\mathcal{P}^{2}+\varrho^{2}$ and $\zeta$ is
an arbitrary model parameter. The inclusion of $\mathcal{T}^{2}$ in
this modified gravity produces more wider form of GR than the
$f(\mathfrak{R})$ and $f(\mathfrak{R},T)$ theories of gravity. The
functional form (\ref{8}) describes the $\Lambda$CDM model and has
extensively been used to solve many cosmological issues \cite{ab}.
It depicts three major epochs of the universe: radiation-dominated,
matter-dominated and de Sitter-dominated, while the obtained
solutions indicate rapid expansion. The corresponding field
equations are
\begin{equation}\label{9}
\mathbb{G}_{\gamma\delta}=8\pi T_{\gamma\delta}+\frac{1}{2}\zeta
g_{\gamma\delta} \mathcal{T}^{2}-\zeta f_{
\mathcal{T}^{2}}{\Theta_{\gamma\delta}}+8\pi\mathrm{E}_{\gamma\delta}.
\end{equation}
The results of GR are retrieved for $\zeta=0$.

To study the interior region of charged gravastar, we choose static
spherical spacetime as
\begin{equation}\label{10}
(ds)^{2}_{-}=e^{\chi(r)}dt^{2}-e^{\xi(r)}dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta
d\phi^{2}.
\end{equation}
The corresponding non-zero Einstein tensor components are given as
\begin{eqnarray}\label{11}
\mathbb{G}_{0}^{0}&=&\Pi(r)\left(e^{\xi}-1+{\xi'r}\right),
\\\label{12}
\mathbb{G}_{1}^{1}&=&\Pi(r)\left(e^{\xi}-{\chi'r}-1\right),
\\\label{13}
\mathbb{G}_{2}^{2} &=&\Pi(r) \left
(\frac{2r}{4}({-{\chi'}+\xi'})+({\chi'\xi'}-2{\chi''}-{\chi'^{2}})\frac{{r}^{2}}{4}\right),
\end{eqnarray}
where $\Pi(r)= e^{-\xi}r^{-2}$ and prime symbolizes radial
derivative. The modified equations are obtained by inserting
Eqs.(\ref{5}) and (\ref{11})-(\ref{13}) in (\ref{9}) as
\begin{eqnarray}\label{14}
&&{\xi'r}-1+e^{\xi}
=\frac{r^{2}}{e^{-\xi}}\big[8\pi\varrho+\frac{\zeta}{2}
(3{\varrho^{2}}+9{\mathcal{P}^{2}}+8\mathcal{P}\varrho)+\frac{{q}^{2}}{{r}^{4}}\big],
\\\label{15}
&&e^{\xi}-{\chi'r}-1=\frac{r^{2}}{e^{-\xi}}\big[-8\pi
\mathcal{P}+\frac{\zeta}{2}({\varrho^{2}}+3{\mathcal{P}^{2}})-\frac{{q}^{2}}{{r}^{4}}\big],
\\\nonumber
&&\frac{r}{2}({\xi'}-{\chi'})-(2{\chi''}+{\chi'^{2}}-{\chi'\xi'})
\frac{r^{2}}{4}=\frac{r^{2}}{e^{-\xi}}\big[\frac{\zeta}{2}({\varrho^{2}}+3{\mathcal{P}^{2}})-8\pi
\mathcal{P}+\frac{{q}^{2}}{{r}^{4}}\big],\\\label{16}
\end{eqnarray}
where $q$ indicates the charge of the interior sphere defined by
\begin{equation}\label{}
q(r)=4\pi\int_{0}^{r}r^{2}\sigma(r)e^{\chi/2}dr,\quad
E(r)=\frac{r^{-2}q(r)}{4\pi }.
\end{equation}
Here, $E(r)$ and $\sigma$ represent the electric field intensity and
surface charge density, respectively.

The non-conserved equation (\ref{4}) provides
\begin{equation}\label{17}
\frac{d\mathcal{P}}{dr}+\chi'(\frac{\varrho+\mathcal{P}}{2})+\aleph^\star-\frac{qq'}{4\pi
r^{4}}=0,
\end{equation}
where $\aleph^\star$ shows the significant contribution of EMSG as
follows
\begin{equation}\nonumber
\aleph^\star=\frac{\zeta}{2}\left(3\mathcal{P}^2+\varrho^2+4
\mathcal{P}\varrho\right)\chi'+\zeta(\varrho\varrho'+3\mathcal{P}\mathcal{P}').
\end{equation}
Using Eq.(\ref{14}), we obtain
\begin{equation}\label{18}
e^{-\xi}=\frac{r-2m}{r}-\frac{\zeta}{3}\Big(\frac{3{\varrho^{2}}
+9{\mathcal{P}^{2}}+8\mathcal{P}\varrho}{2}\Big)r^{2}-\frac{1}{r}
\int\frac{q^{2}}{{r}^{2}}dr,
\end{equation}
where $m=\int4\pi {\varrho}r^{2}dr$ is the gravitational mass and
$\mathcal{H}(r)=\frac{1}{r} \int\frac{q^{2}}{{r}^{2}}dr$ depicts
effect of charge. This provides a relationship between matter
variables and the radial metric coefficient of the considered
spacetime. The hydrostatic equilibrium equation (\ref{17}) reads as
follows
\begin{equation}\nonumber
\frac{d\mathcal{P}}{dr}=\frac{-\frac{\chi'}{2}(\varrho+\mathcal{P})-\zeta
\frac{\chi'}{2}\left(3\mathcal{P}^2+\varrho^2+4
\mathcal{P}\varrho\right)-\zeta\varrho\varrho'++\frac{qq'}{4\pi
r^{4}}}{3\mathcal{P}\zeta+1},
\end{equation}
where $\chi'$ is calculated from Eqs.(\ref{15}) and (\ref{18}) reads
as
\begin{equation}\nonumber
\chi'=\frac{r\big(8\pi
\mathcal{P}-\frac{\zeta}{2}(\varrho^{2}+3{\mathcal{P}^{2}}+\frac{q^{2}}{r^{4}})\big)
+{\frac{1}{r}\big(\frac{2m}{r}+\frac{\zeta}{2}(3{\varrho^{2}}+
9{\mathcal{P}^{2}}+8\mathcal{P}\varrho+\mathcal{H}(r))\big)\frac{r^{2}}{3}}}{1-\frac{2m}{r}
-\frac{\zeta}{2}(3{\varrho^{2}}+9{\mathcal{P}^{2}}+8\mathcal{P}\varrho)-\mathcal{H}(r)}.
\end{equation}
This reduces to the Tolman-Oppenheimer-Volkoff equation of GR for
$\zeta=0$.

\section{Structure of Charged Gravastars}

In this section, we investigate charged gravastar geometry by
considering three regions (inner, thin-shell and outer) satisfying
particular EoS. The shell of negligible thickness covers internal
domain of the gravastar lying in the range
$\mathbf{R}_{2}=\mathbf{R}+\epsilon>r>\mathbf{R}=\mathbf{R}_{1}$,
where negligible thickness of the shell is indicated by
$\mathbf{R}_{2}-\mathbf{R}_{1}=\epsilon$. The radii of the inner as
well as outer regions of charged gravastars are represented by
$\mathbf{R}_{1}$ and $\mathbf{R}_{2}$, respectively. The three
regions of charged gravastar geometry with specific EoS for each
region is given by
\begin{itemize}
\item Inner region $(\mathcal{J}_{1})$\quad $\Rightarrow$\quad $ \mathbf{R}_{1}\geq
r>0$ with $\varrho=-\mathcal{P}$,
\item Thin-shell $(\mathcal{J}_{2})$\quad $\Rightarrow$\quad $\mathbf{R}+\epsilon\geq
r\geq \mathbf{R}_{1}$ with $\mathcal{P}-\varrho=0$,
\item Outer region $(\mathcal{J}_{3})$\quad $\Rightarrow$\quad $r>\mathbf{R}+\epsilon$
with $\varrho=0=\mathcal{P}$.
\end{itemize}

\subsection{Inner Region}

The interior region of charged gravastar satisfies an EoS
$(\mathcal{W}=-1=\mathcal{P}/\varrho)$ that leads the relation
between matter variables as $\mathcal{P}=-\varrho$. The negative
pressure is responsible for outward directed repulsive force to
overcome the pull exerted by thin-shell on the core of spherical
charged gravastar. This particular choice of $\mathcal{W}$ is also
called dark energy EoS. Using this form of EoS in Eq.(\ref{17}), we
get $\varrho=\varrho_{c}$ (where $\varrho_{c}$ is constant) and
hence $\mathcal{P}=-\varrho_{c}$. This relation shows that pressure
and density remain same in the interior of charged gravastar.
Inserting this equation in Eq.(\ref{14}), we acquire the radial
metric function as
\begin{equation}\label{21}
e^{-\xi}=1-\mathcal{H}(r)-\frac{2{\varrho_{c}}{r^{2}}}{3}(4\pi+\zeta{\varrho_{c}})+\frac{C_{1}}{r},
\end{equation}
where $\mathcal{H}(r)$ provides the effect of charge and $C_{1}$
denotes the constant of integration. The singularity-free solution
is obtained at the center of gravastar by setting $C_{1}=0$. Thus we
have
\begin{equation}\label{22}
e^{-\xi}=1-\frac{2{\varrho_{c}}{r^{2}}}{3}(4\pi+\zeta{\varrho_{c}})-
\mathcal{H}(r).
\end{equation}
Using Eqs.(\ref{14}) and (\ref{15}), we can present the correlation
of unknown metric functions as
\begin{equation}\label{23}
e^{\chi}={C_{2}}e^{-\xi},
\end{equation}
where $C_{2}$ corresponds to integration constant. The mass of inner
domain is
\begin{equation}\label{24}
M=\frac{\mathbf{R}^{3}(4\pi\varrho_{c}+\zeta{\varrho_{c}^{2}})}{3}+\int\frac{q^{2}}{r^{2}}dr.
\end{equation}

\subsection{The Intermediate Shell Region}

The non-vacuumed shell of a charged gravastar is filled with an
ultra-relativistic fluid that obeys the EoS $\mathcal{P}-\varrho=0$.
In connection with cold baryonic universe, Zel'dovich introduced the
idea of stiff matter configuration \cite{58aa}. Several researchers
have employed this type of matter distribution and obtained
remarkable results \cite{59aa}-\cite{63aa}. For this region, finding
exact solution to the field equations is difficult. However, in the
limit specified for this domain, i.e., $0<e^{-\xi}\ll1$, such
solutions can be found by taking into account the EoS of this
region. This demonstrates that the smooth matching of inner and
outer domains yields thin-shell presenting the intermediate region
of charged gravastar. By implementing this constraint, the
corresponding field equations can be written as
\begin{eqnarray}\label{25}
\frac{de^{-\xi}}{dr}&=&\frac{2}{r}-\frac{2q^{2}}{r^{3}}-12{\varrho_{0}}^{2}\zeta
r,\\\label{26}
\big(\frac{3}{2r}+\frac{\chi'}{4}\big)\frac{de^{-\xi}}
{dr}&=&\frac{1}{{r}^{2}}-\frac{2q^{2}}{r^{4}}-12{\varrho_{0}}^{2}\zeta.
\end{eqnarray}
Integration of Eq.(\ref{25}) yields
\begin{equation}\label{27}
e^{-\xi}={C_{3}}-6{\varrho_{0}}^{2}\zeta
{r}^{2}+2\ln{r}-2\int\frac{q^{2}}{r^{3}}dr,
\end{equation}
where ${C_{3}}$ indicates another constant of integration. Solving
Eqs.(\ref{25}) and (\ref{26}) to find  $\chi'$ and substituting it
in Eq.(\ref{17}) corresponding to stiff matter configuration
\begin{equation}\label{28}
\mathcal{P}=\varrho=\frac{1}{e^{\chi}}\int \varpi
{e^{\chi}}dr+\mathcal{S}e^{-\chi},
\end{equation}
where $\varpi$ depicts the additional effect of charge and EMSG
gravity and $\mathcal{S}$ is another integration constant. This
relationship implies that fluid distribution at the outer layer of
thin-shell domain is denser than in the inner domain of shell. The
evolution of energy density is illustrated in Figure \textbf{1}. It
can be seen that the energy density is positive across the shell.
\begin{figure}\center
\epsfig{file=cdp.eps,width=0.5\linewidth} \caption{Behavior of
$\varrho$ $(km^{-2})$ versus thickness of thin-shell.}
\end{figure}

\subsection{Outer Region}

We assume that both pressure and density are zero in this region,
i.e., $\varrho=0=\mathcal{P}$. The outer region of spherical charged
gravastar is vacuum and is described by the Reissner-Nordstrom
metric reads as
\begin{equation}\label{29}
(ds)^{2}_{+}=\mathfrak{P}(r) dt^{2}-r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})-\frac{dr^{2}}{\mathfrak{P}(r)},
\end{equation}
where
$\mathfrak{P}(r)=\frac{r-2\mathcal{M}}{r}+\frac{\mathcal{Q}^{2}}{{r}^{2}}$
and $\mathcal{M}$ corresponds to the whole mass of gravastar. In
order to examine celestial bodies, inner and outer spacetimes must
be perfectly matched. The structure of charged gravastar is mainly
composed of three regions named as $\mathcal{J}_{1}$,
$\mathcal{J}_{2}$ and $\mathcal{J}_{3}$, where $\mathcal{J}_{2}$
acts as a connection between $\mathcal{J}_{1}$ and
$\mathcal{J}_{3}$. The smooth matching of the interior and exterior
geometries is ensured by Israel matching criterion \cite{64aa}. The
continuity of metric coefficients is maintained but it is found that
their derivatives may possess discontinuity at the hypersurface.
Considering Lanczos equations, the stress-energy tensor of matter
surface is computed as
\begin{equation}\label{30}
\mathcal{S}^{\lambda}_{\beta}=\left(\delta^{\lambda}_{\beta}
\mu^{\alpha}_{\alpha}-\mu^{\lambda}_{\beta}\right)\frac{1}{8\pi},\quad
\lambda,\beta=0,2,3,
\end{equation}
where $\lambda, \beta$ denote hypersurface coordinates and
$\mu_{\lambda\beta}=\mathbb{K}^{+}_{\lambda\beta}-\mathbb{K}^{-}_{\lambda\beta}$
implies extrinsic curvature discontinuity. The extrinsic curvature
corresponding to $\mathcal{J}_{1}$ and $\mathcal{J}_{3}$ is defined
as
\begin{equation}\label{31}
\mathbb{K}^{\pm}_{\lambda\beta}=-\left(\frac{\partial^{2}
x^{\tau}}{\partial\Phi^{\lambda}\partial\Phi^{\beta}}+\Gamma^{\tau}_{\mu\nu}
\frac{\partial x^{\mu}\partial x^{\nu}}
{\partial\Phi^{\lambda}\partial\Phi^{\beta}}\right)\Upsilon^{\pm}_{\tau},
\end{equation}
where $\Upsilon^{\pm}_{\tau}$ and $\Phi^{\lambda}$ represent the
unit normals and intrinsic coordinates of the hypersurface,
respectively.

The line element of charged static spherically symmetric geometry is
given as
\begin{equation}\label{32}
ds^{2}=\mathcal{N}(r)dt^{2}-\frac{dr^{2}}{\mathcal{N}(r)}-r^{2}d\theta^{2}
-r^{2}\sin^{2}\theta d\phi^{2}.
\end{equation}
The corresponding unit normals are of the form
\begin{equation}\label{33}
\Upsilon^{\pm}_{\tau}=\pm\left|g^{\mu\nu}\frac{\partial\mathcal{N}
}{\partial x^{\mu}}\frac{\partial\mathcal{N}}{\partial
x^{\nu}}\right|^{-\frac{1} {2}}\frac{\partial\mathcal{N}}{\partial
x^{\tau}},
\end{equation}
\textrm{where} $\Upsilon^{\tau}\Upsilon_{\tau}=1$. For perfect fluid
matter configuration, we obtain
$\mathbb{S}_{\lambda\beta}=$diag$(\Omega,$ $-\Psi,-\Psi)$. Hence,
the surface density turns out to be
\begin{equation}\label{34}
\Omega=-\frac{1}{4\pi\mathbf{R}}\Big[\sqrt{1-\frac{2\mathcal{M}}{\mathbf{R}}+
\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}}
-\sqrt{1-\frac{(4\pi+\zeta{\varrho_{c}}){2{\varrho_{c}}{\mathbf{R}^{2}}}}{3}+\mathcal{H}(r)}\Big].
\end{equation}
Using $\mathcal{J}_{1}$ and $\mathcal{J}_{3}$ as domains of charged
gravastar, we obtain respective expression of surface pressure as
\begin{equation}\label{35}
\Psi=\frac{1}{8\pi\mathbf{R}}\Big[\frac{{1-\frac{\mathcal{M}}{\mathbf{R}}}}
{\sqrt{1-\frac{2\mathcal{M}}{\mathbf{R}}+\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}}}-
\frac{1-\frac{(4\pi+\zeta{\varrho_{c}}){4{\varrho_{c}}{\mathbf{R}^{2}}}}
{3}+\mathcal{H}(r)+\mathcal{H'}(r)}{\sqrt{1-\frac{(8\pi+2\zeta{\varrho_{c}})
{{\varrho_{c}}{\mathbf{R}^{2}}}}{3}+\mathcal{H}(r)}}\Big].
\end{equation}
The mass inside gravastar is calculated through Eq.(\ref{34}) as
follows
\begin{equation}\label{36}
{m}_{\textit{shell}}=4\pi\mathbf{R}^{2}\Omega=\mathbf{R}
\Big[{\sqrt{1-\frac{(4\pi+\zeta{\varrho_{c}})
{2{\varrho_{c}}{\mathbf{R}^{2}}}}{3}+\mathcal{H}(r)}}
-\sqrt{1-\frac{2\mathcal{M}}{\mathbf{R}}+\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}}\Big].
\end{equation}
Finally, the total mass of charged gravastar is
\begin{eqnarray}\nonumber
\mathcal{M}&=&{m}_{\textit{shell}}{\sqrt{1-\frac{(4\pi+\zeta{\varrho_{c}})
{2{\varrho_{c}}{\mathbf{R}^{2}}+\mathcal{H}(r)}}{3}}}+\frac{(4\pi+\zeta{\varrho_{c}})
{2{\varrho_{c}}{\mathbf{R}^{3}+\mathcal{H}(r)}}}{3}\\\label{37}
&-&\frac{{m}_{\textit{shell}}^{2}}{2\mathbf{R}}
-\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}.
\end{eqnarray}

\section{Attributes of Charged Gravastars}

This section deals with the study of characteristics of charged
gravastar, i.e., the EoS parameter, entropy, proper length and
thin-shell energy through graphical analysis. We have analyzed these
features of thin-shell for negative as well positive values of the
model parameter for three different values of charge. This
discussion will show viability of the spherical charged gravastar
structure in the context of EMSG.

\subsection{The EoS Parameter}

The EoS for shell of charged gravastars at $r=\mathbf{R}$ is
\begin{equation}\label{ww}
\mathcal{W}=\frac{\Psi}{\Omega}.
\end{equation}
Replacing the values of $\Psi$ and $\Omega$ in Eq.(\ref{ww}), we
have
\begin{equation}\label{jj}
\mathcal{W}(\mathbf{R})=\frac{\frac{1}{8\pi\mathbf{R}}\Big[\frac{{1-\frac{\mathcal{M}}{\mathbf{R}}}}
{\sqrt{1-\frac{2\mathcal{M}}{\mathbf{R}}+\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}}}-
\frac{1-\frac{(4\pi+\zeta{\varrho_{c}}){4{\varrho_{c}}{\mathbf{R}^{2}}}}
{3}+\mathcal{H}(r)+\mathcal{H'}(r)}{\sqrt{1-\frac{(8\pi+2\zeta{\varrho_{c}})
{{\varrho_{c}}{\mathbf{R}^{2}}}}{3}+\mathcal{H}(r)}}\Big]}{-\frac{1}{4\pi\mathbf{R}}
\Big[\sqrt{1-\frac{2\mathcal{M}}{\mathbf{R}}+
\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}}
-\sqrt{1-\frac{(4\pi+\zeta{\varrho_{c}}){2{\varrho_{c}}{\mathbf{R}^{2}}}}{3}+\mathcal{H}(r)}\Big]}.
\end{equation}
We impose some additional constraints on $\mathcal{W}(\mathbf{R})$,
i.e.,
$\frac{2\mathcal{M}}{\mathbf{R}}+\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}<1$
or $2\mathcal{M}<\mathbf{R}$. The relationship between
$\mathcal{M}$, $\mathcal{Q}$ and $\mathbf{R}$ is obtained as
$\mathcal{Q}>\sqrt{2\mathcal{M}\mathbf{R}-\mathbf{R}^{2}}$ and
$2\mathcal{M}<\mathbf{R}$. Taking positive values of surface density
as well as pressure results in positive value of $\mathcal{W}$. We
attain $\mathcal{W}(\mathbf{R})\approx1$ for large values of
$\mathbf{R}$. The structure of gravastar is similar to compact
objects for appropriately bigger values of  $\mathbf{R}$. Moreover,
by inserting specific value of $\mathbf{R}$ in Eq.(\ref{35}) may
result $\Psi=0$, which presents dust shell.

\subsection{Entropy of Thin-Shell}
\begin{figure}\center
\epsfig{file=cE1.eps,width=.5\linewidth}\epsfig{file=cE2.eps,width=.49\linewidth}
\caption{Entropy of the shell with thickness (km) corresponding to
different values of charge.}
\end{figure}

Mazur and Mottola \cite{13a} investigated the inner region and
concluded that entropy density is zero which represents reliability
of condensate phase. In intermediate region, the entropy of charged
gravastar is calculated as
\begin{equation}\label{ea}
S=\int^{\mathbf{R}_{2}}_{\mathbf{R}_{1}}4\sqrt{e^{\xi(r)}}\pi
r^{2}\mathbb{V}(r)dr,
\end{equation}
where $\mathbb{V}(r)$ shows the entropy density given as
\begin{equation}\label{}
\mathbb{V}(r)=\frac{\curlyvee^{2}K^{2}_{\mathcal{B}}\mathbb{T}(r)}{4\pi
\hslash^{2}}=K_{\mathcal{B}}
\left({\frac{\mathcal{P}}{2\pi}}\right)^{\frac{1}{2}}\frac{\curlyvee}{\hslash},
\end{equation}
$\curlyvee$ is a dimensionless constant. The Planckian units are
taken into account $K_{\mathcal{B}}=\hslash=1$ so that Eq.(\ref{ea})
becomes
\begin{equation}\label{}
S=\curlyvee({8\pi})^{\frac{1}{2}}\int^{\mathbf{R}_{2}}_{\mathbf{R}_{1}}\frac
{({C_{3}}-6{\varrho_{0}}^{2}\zeta
{r}^{2}+2\ln{r}-2\int\frac{q^{2}}{r^{3}}dr)^{\frac{-1}{2}}}{(\frac{1}{e^{\chi}}\int
\varpi {e^{\chi}}dr+\mathcal{S}e^{-\chi})^{\frac{-1}{2}}}{r}^{2}dr.
\end{equation}
It is too difficult to obtain an analytical solution to this
equation, therefore, we solve it numerically. Figure \textbf{2}
depicts the disorderness of gravastar and shows that the entropy of
shell increases with the increment in thickness. The entropy is
proportional to the small thickness of thin-shell. It has maximum
value at the outer surface of thin-shell and zero thickness yields
the value of entropy to be zero. The disorderness increases with the
increasing amount of charge, thus allowing the formation of a more
stable structure. The model parameter affects the entropy such that
its positive value provides higher entropy as compared to its
negative value.

\subsection{Length of Charged Shell}
\begin{figure}\center
\epsfig{file=nl.eps,width=.5\linewidth}\epsfig{file=pl.eps,width=.5\linewidth}
\caption{Proper length (km) of the charged shell against thickness
(km).}
\end{figure}

The proper length of shell region is calculated from
$\mathbf{R}_{1}$ to $\mathbf{R}_{2}=\mathbf{R}+\epsilon$, where
$\epsilon\ll1$ indicates negligible change. The mathematical formula
of length of intermediate-shell is given by
\begin{equation}\label{}
\mathbb{L}=\int^{\mathbf{R}_{2}}_{\mathbf{R}_{1}}\sqrt{e^{\xi(r)}}dr
=\int^{\mathbf{R}_{2}}_{\mathbf{R}_{1}}\frac{dr}{\sqrt{{C_{3}}-6{\varrho_{0}}^{2}\zeta
{r}^{2}+2\ln{r}-2\int\frac{q^{2}}{r^{3}}dr}}.
\end{equation}
We obtain the numerical solution of this equation and its plot is
shown in Figure \textbf{3}. The left panel of Figure \textbf{3}
shows a continuous increment in the length of thin-shell by
increasing the value of charge with $\zeta=-0.025$. However,
$\zeta=0.020$ leads to the rapid increase in length of the shell for
similar values of charge, presented in the right panel of Figure
\textbf{3}.

\subsection{Thin-Shell Energy}

In the core of charged gravastar, the presence of dark energy is
confirmed due to negative pressure which is responsible for
repulsive force. The mathematical representation of the energy is
\begin{equation}\label{ey}
\varepsilon=\int^{\mathbf{R}_{2}}_{\mathbf{R}_{1}}4\pi\varrho
r^{2}dr.
\end{equation}
Plugging the value of surface density in Eq.(\ref{ey} yields
\begin{equation}\label{}
\varepsilon=\int^{\mathbf{R}_{2}}_{\mathbf{R}_{1}}4\pi
{r}^{2}\left(\frac{1}{e^{\chi}}\int \varpi
{e^{\chi}}dr+\mathcal{S}e^{-\chi}\right)dr.
\end{equation}
This increases by enhancing its thickness as shown in Figure
\textbf{4}. The energy plots corresponding to both values of $\zeta$
show linear profile with the increment of charge.
\begin{figure}\center
\epsfig{file=cen.eps,width=.5\linewidth}\epsfig{file=cep.eps,width=.5\linewidth}\caption{Plots
of energy (km) of charged thin-shell versus thickness of the shell
(km).}
\end{figure}

\section{Conclusions}

In this manuscript, we investigate the structure of charged
gravastar in the background of EMSG. We have considered charged
sphere in the interior with EoS $\varrho=-\mathcal{P}$ and
Reissner-Nordstrom black hole in the exterior region with vacuum
($\varrho=\mathcal{P}=0$). The interior region is surrounded by a
thin-shell with EoS $\varrho=\mathcal{P}$. In the intermediate
shell,  non-repulsive force counterbalances the pull exerted by the
charged sphere and assures the formation of singularity-free compact
object. It is worth mentioning that solutions corresponding to dark
energy EoS show connection of charged gravastars with dark stars.

It is worth here that solutions obtained through dark energy EoS
indicate that charged gravastars are linked with dark stars.

The main results are summarized as follows.
\begin{itemize}
\item The increasing density depicts that the shell's
external boundary is denser than the internal boundary (Figure
\textbf{1}).
\item The real value of the EoS parameter is obtained when
$\frac{2\mathcal{M}}{\mathbf{R}}+\frac{\mathcal{Q}^{2}}{\mathbf{R}^{2}}<1$.
This is a necessary condition for the stable and physically viable
gravastar structure.
\item The entropy of charged thin-shell is proportional to its thickness,
thus greater thickness means higher entropy (Figure \textbf{2}). The
increment of electromagnetic strength increases its entropy as well.
\item
The proper length of the charged thin-shell indicates increasing
behavior with the thickness of the shell (Figure \textbf{3}). The
gravastar length goes on increasing with a higher value of the
charge.
\item
The energy of the inner boundary is less than that of the outer
domain. The thickness of the shell is shown to be linearly related
to the energy of the shell (Figure \textbf{4}).
\end{itemize}

It is worthwhile to mention here that the length, energy and entropy
of a gravastar increase with respect to thickness of the shell as
compared to the uncharged case \cite{w7}. Our analysis follows a
consistent increasing trend of the physical properties in the
presence of charge, presented in GR as well as other modified
theories of gravity \cite{7dddd,cg8,w3,w4}. The matching of the
charged inner sphere and Reissner-Nordstrom metric produces
gravitational mass which behaves as an electromagnetic mass model.
The charge provides an outward-directed force, therefore, an
additional repulsive force helps the gravastar from collapsing into
a singularity. Thus, the presence of charge generates a more stable
structure as compared to the uncharged model \cite{w7}. Furthermore,
we have found that the physical attributes of gravastar have greater
values in EMSG as compared to GR and other modified theories of
gravity \cite{7dddd,cg8,w3,w4}. We conclude that
$f(\mathfrak{R},\mathcal{T}^{2})$ gravity successfully discusses
charged gravastar model.

redpajama_set_name: RedPajamaArXiv


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Thomas O'Loghlen, lawyer and deputy chairman Melbourne and Metropolitan Tramways Board, and Jane Mary Hutchinson.
Marital Status: (1) Edna Florence Davy; 2d.; (2) 1935 Joan Nicholls; 2d.
Called to Bar 1915; Australian Imperial Force 1915-1919, subaltern Army Service Corps, France, Belgium; KC 1939 Vic., 1942 New South Wales, 1955 Papua-New Guinea and Tasmania; counsel assisting Royal commission Banking Systems 1936; president Law Council of Aust. 1947-1948; Leader Vic. Bar 1946-1952; honorary life member Vic. Bar 1966; chairman British Memorial Fund; executive member Aust.-Asian Association; founding president Toorak RSL 1950; president Registered Clubs Association 1953-1971; president Yorick Club 1930-1931,1947-1949, life member 1952.